Sunday, July 24, 2011

Grow up or shut up

I’ve pointed out that the argument so many atheists like to attack when they purport to refute the cosmological argument -- namely “Everything has a cause; so the universe has a cause; so God exists” or variants thereof -- is a straw man, something no prominent advocate of the cosmological argument has ever put forward.  You won’t find it in Aristotle, you won’t find it in Aquinas, you won’t find it in Leibniz, and you won’t find it in the other main proponents of the argument.  Therefore, it is unfair to pretend that refuting this silly argument (e.g. by asking “So what caused God?”) is relevant to determining whether the cosmological argument has any force.    

I’ve also noted other respects in which the cosmological argument is widely misrepresented.  Now, in response to these points, it seems to me that what a grownup would say is something like this: “Fair enough.  I agree that atheists should stop attacking straw men.  They should avoid glib and ill-informed dismissals.  They should acquaint themselves with what writers like Aristotle, Aquinas, Leibniz, et al. actually said and focus their criticisms on that.”  But it would appear that Jason Rosenhouse and Jerry Coyne are not grownups.  Their preferred response is to channel Pee-wee Herman:  “I know you are, but what am I?” is, for them, all the reply that is needed to the charge that New Atheists routinely misrepresent the cosmological argument.  

In particular, Rosenhouse, who is curiously silent on this charge -- though it is, after all, the point at issue -- has decided to change the subject.  Like the teenager who, when caught red-handed with the evidence of his drug use, responds by criticizing Mom and Dad’s drinking habits, Rosenhouse  works himself into an adolescent dudgeon over how I’ve allegedly misrepresented Robin Le Poidevin.  And how exactly have I misrepresented him?  That is never made clear.  I said that Le Poidevin presents a variation of the straw man as if it were “the basic” cosmological argument.  And he does.  I said that Le Poidevin presents the “more sophisticated versions” he considers later on in his book as “modifications” of that straw man.  And he does.  I did not deny that Le Poidevin addresses these more sophisticated versions.  I explicitly noted that he does.  Nor did I say that Le Poidevin claimed that Aristotle, Aquinas, Leibniz, et al. actually defended the straw man argument themselves.  Indeed, I quoted Le Poidevin as acknowledging that “no-one has defended a cosmological argument of precisely this form.”  Rather, I said that readers who are unfamiliar with what Aristotle, Aquinas, Leibniz, et al. actually wrote are liable to come away from a discussion like Le Poidevin’s with the false impression that what the major defenders of the cosmological argument are up to is, essentially, merely an attempt at a patch-up job on a manifestly feeble argument.  And I explicitly said that I was not claiming that Le Poidevin was deliberately trying to give this false impression, but rather that he should know better. 

So, again, how exactly did I misrepresent him?  It turns out that Rosenhouse’s real complaint, to the extent he has any, reduces to another adolescent trope.  My problem, you see, is that I need to lighten up.  I’m “overreacting” to what was merely a “pedagogical” exercise on Le Poidevin’s part.  Beginning his treatment of the cosmological argument with the straw man was simply Le Poidevin’s gentle way of ”introducing” a complicated topic to undergraduates.  

Well, we all know why this dodge won’t work.  Suppose a creationist writer began his exposition of Darwinism by presenting the claim that “Monkeys gave birth to humans” as “the basic” claim of the theory, of which the “more sophisticated versions” of Darwinism he would consider later were variants.  Naturally, he would have little trouble showing that this claim (which no Darwinist has ever made) is false.  But suppose he defended this odd approach as merely a “pedagogical” technique for “introducing” Darwinism to his readers.  And suppose he also held that any biologist who finds this procedure outrageous is merely “overreacting.”  Rosenhouse and Co. would, quite rightly, be unimpressed.  And neither should we be impressed by Rosenhouse’s lame defense of Le Poidevin. 

To be sure, Rosenhouse thinks he has preempted such a comparison: 

The claim that a monkey gave birth to a human is not an oversimplified version of Darwinism that might serve as a helpful stepping stone into a complex topic.  It is just a completely made up idea tossed off specifically to make evolution look foolish.  

But of course, this is no answer at all, but merely reinforces my point.  For the straw man version of the cosmological argument we’ve been discussing is no less a completely made up idea, one tossed off to make the cosmological argument look foolish.  How do we know this?  Well, here’s some pretty good evidence: First, as I keep pointing out -- and, you will note, as Rosenhouse and his ilk never deny (because they can hardly deny it) -- Aristotle, Aquinas, Leibniz, and the other prominent defenders of the cosmological argument never gave the straw man argument.  Second, the only people who ever do pay the straw man argument much attention are atheist critics of the cosmological argument, and they typically present precisely it as a reason to dismiss the cosmological argument as foolish.

So, the cases are parallel and Le Poidevin’s procedure is no more defensible than that of our imaginary creationist.  No doubt Rosenhouse, as is his wont, will at this point just stamp his foot some more.  “The cases are not parallel!  Are not!  Are not not not not not!” 

But then, Rosenhouse isn’t one to notice when he’s merely making unsupported assertions or begging the question.  For example, in an earlier post, he had written: 

Feser seems rather taken with [the cosmological argument], but there are many strong refutations to be found in the literature.  Off the top of my head, I found Mackie's discussion in The Miracle of Theism and Robin Le Poidevin's discussion in Arguing for Atheism to be both cogent and accessible. 

I then pointed out that this merely begged the question against defenders of the cosmological argument, which (given the context) it quite obviously does.  But the obvious is never obvious enough for Rosenhouse, who in his latest post writes: 

My point was simply that I think the cosmological argument is not very good, and that I think Mackie and Le Poidevin provided cogent and accessible refutations of it.  How could I have been clearer?  I have no idea what question I was begging by expressing those particular opinions. 

Well, Prof. Rosenhouse, here’s a clue:  Whether the cosmological argument is “not very good” and whether writers like Le Poidevin and Mackie have actually “refuted” it are precisely what is at issue between yourself and defenders of the cosmological argument like me.  And merely to assume some proposition which is at issue instead of arguing for it -- as you did when, in response to my advocacy of the cosmological argument, you asserted matter-of-factly that the argument had been “refuted” by the likes of Mackie and Le Poidevin -- is a textbook instance of what logicians call “begging the question.”  But then, in between all those volumes on Aquinas and Leibniz you haven’t read, it seems there are a few logic textbooks you haven’t gotten to either.

Those who are interested in other curious examples of undefended assertion are directed to the rest of Rosenhouse’s post.  But beyond providing us with Exhibit 2,345 of the Higher Cluelessness that is the New Atheism, Rosenhouse’s remarks on this controversy are absolutely devoid of interest.  As I’ve said, in response to the points I made in my earlier post, an atheist who is also a grownup would at least be happy to acknowledge that atheists should not attack straw men and should deal instead with what the major defenders of the cosmological argument actually said.  Yet Rosenhouse can’t even bring himself to do that much before launching into his botched “Gotcha!” exercise.  And that pretty much says it all.

609 comments:

  1. E.H. Munro said...
    Would you like to borrow my glasses? That might help you see more clearly.

    It was so very kind of you to offer to help me see the philosophy of mathematics more clearly. I wish I could find your glasses had better focus than they seem to.

    I did not say that his view was rare or outrageous, I said that it was anti-rational. And it most undeniably is.

    You are of course welcome to your opinions, even when you offer so little evidence to support them.

    I can see that someone else dealt with a good portion of my objections to this, but I'd like to note one thing further, namely that your statement isn't really true. The non-euclidian geometries would be true even if we lived in a Newtonian universe.

    That was my statement. They would be formally true. Euclidean geometry is formally true even though we live in a non-Euclidean universe. However, my understanding of a necessary truth is that it is more than merely formally true.

    Now, if you mean to say that the only rational position is that models like Euclidean geometry are true in some more-than-formal fashion that elevates it to the status of necessary truth, you'll need to pull out a bigger gun than "I said" and "undeniably".

    ... this returns to my point to djindra that all that really happened with the acceptance of relativity is that physicists found new ways to use all that nifty math that they'd learned.)

    More properly, physicists learned from mathematicians some nifty math that turned out to be a good model.

    Mathematicians work with concepts that have no observable counterpart in "reality" on a regular basis, and always have.

    Hence, formally true.

    Euclidian geometry is perfectly valid in a two dimensional system, but when did Euclid ever live in such a system?

    Euclidean geometry also works for a flat three-dimensional space.

    Mathematicians have never really relied on sensory evidence.

    Some schools of mathematicians would disagree with you.

    Put another way, even in your three object universe two plus two would still equal four because if one of those objects were a mathematician he could work it out.

    Sure, but the models where "2 + 2 = 1" or "2 + 2 = 0" or even "2 + 2 = infinity" would be more valid. All such models are just as formally true as "2 + 2 = 4".

    Now, if the sentient object were djindra, it might remain undiscovered, but it would still be true.

    If no object sets up the formal system in shich a statement can be expressed, how can that statement be true? The statement has no inherent truth outside the system.

    ReplyDelete
  2. ACuriousMind,

    Then, for the sake of claification:

    But when you say "2+2=1", the "+" in that statement is not the same as in "2+2=4", though both are truths inside their respective axiomatic systems.

    In addition to the "+", the "2" and the "=" are also not the same between the two statements.

    ReplyDelete
  3. One Brow,


    In addition to the "+", the "2" and the "=" are also not the same between the two statements.


    I would agree that the "2" is not the same. But why is the "=" different? Isn't it expressing equivalence both times? (Not that this matters for the point in question, but I'm, well, curious)

    The statement has no inherent truth outside the system.
    Isn't it rather that the statement is not even possible without the system?

    Besides that, I pretty much think we're on the same side here. But you seem to have understood djindra differently than I have. He claims (as I understand him) we cannot do math with numbers that are too large to fit into our heads and that we could not conceive of a system where "2+2=4" were it not that the counting of entities in our world behaved like this, while you are only defending the idea that "2+2=1" would be more sensible a statement (regarding reality) than "2+2=4" in a modular world (which I totally agree with), right?

    The disagreement here is not about our own opinions, but about djindra's opinions on math (about which I wouldn't be as harsh as E.H.Munro, but neither do I think they are a possible view of math).

    ReplyDelete
  4. One Brow what is the difference between something being formally true verses Necessarily true?


    Because taken at face value djindra's belief is clearly irrational.

    2+2=5 cannot ever be true in any reality. 2+2=4 cannot fail to be true in any reality.

    Even if we lived in a Reality or alternate universe where whenever some intelligent being added a set of two objects with another set of two identical objects & some macro quantum process creates an additional object from the Cosmic Ather. In such a world 2+2=5 would still not be true nor could it be.

    It would merely be a world where 1+(2+2)=5. A world which has a property of +1 but that's it.

    Your thoughts?

    ReplyDelete
  5. @BenYachov:

    2+2=5 cannot ever be true in any reality.

    By our usual definition of addition within the field of real numbers, you are right. The thing is that the inhabitants of a reality in which an object mysteriously appears when we put two and two together would, in order to describe their reality, not express it in 2+2+1 = 5. For them, our addition bears no connection to their reality (just as many mathematical models bear no connection to our reality). They would use another vector space (loosely speaking), in which an addition operation delivers 5 if operating on 2 and 2 (and write it, of course, 2+2=5). Of course, they could study our addition, and agree that 2+2=4 is valid within our formal system. 2+2=4 is not necessarily a true statement about reality - it is only necesssarily true in the sense that it's always true (i.e validly decucted) within the field of real numbers with the operations defined as we're used to. This latter notion of "necessarily true" is what one could call formally true.

    I agree, however, that djindra seems to be irrational, because he also denies the formal truth of 2+2=4 without evidence, if I understand him correctly.

    ReplyDelete
  6. The problem here, One Brow, is that you aren't arguing djindra's position on mathematics, and unlike ACM you've been here long enough to know better. Djindra has, repeatedly, claimed that mathematics are purely empirical and that its truths are only truths insofar as they match up to sensory observation.

    But what two actual two dimensional entities was Euclid observing? What ideal right triangles was Pythagorus observing when he wrote his equation? I'll be honest, I can only make those happen on a computer screen, give me a technical pencil, compass, and paper and those right triangles of mine won't be ideal.

    While it's easy to say that Euclidian geometry breaks down on three dimensional surfaces like spheres, Euclid himself was working on three dimensional surfaces that only approximated planes. So when djindra tells us that he rejects platonic ideals in math he's essentially rejecting whole huge gobs of the discipline. That may not strike you as anti-rational, but you're going to be pretty much alone there.

    Also, your artificial distinction between necessary & formal truths (I think the terms you're looking for are necessary and contingent truths) has nothing to do with djindra's position, because he isn't saying that math is a necessary truth, he's saying that it's purely contingent and necessarily wrong if it doesn't match sensory input. And that is simply not true. And I take his position as seriously as I do that of young earth creationists, because that's about where it falls.

    ReplyDelete
  7. ACurious Mind,

    You say:

    "By our usual definition of addition within the field of real numbers, you are right. The thing is that the inhabitants of a reality in which an object mysteriously appears when we put two and two together would, in order to describe their reality, not express it in 2+2+1 = 5. For them, our addition bears no connection to their reality (just as many mathematical models bear no connection to our reality). They would use another vector space (loosely speaking), in which an addition operation delivers 5 if operating on 2 and 2 (and write it, of course, 2+2=5). Of course, they could study our addition, and agree that 2+2=4 is valid within our formal system. 2+2=4 is not necessarily a true statement about reality - it is only necesssarily true in the sense that it's always true (i.e validly decucted) within the field of real numbers with the operations defined as we're used to.

    Just to clarify the positions at stake here: do you believe mathematics is something we discover or is it something we invent?

    ReplyDelete
  8. Also of help (possibly) is p. 274 of Feser's TLS, n. 10.

    ReplyDelete
  9. @Josh:

    We invent mathematics. Sometimes we're inspired by reality, sometimes we aren't. But we never depend on it.
    If we go and discover what parts actually are applicable to reality, that's physics.

    Or meant you something else with invent and discover?

    ReplyDelete
  10. ACurious Mind,

    No, I think we understand on the same terms. Would you say, then, that mathematics depends for its truth on man's invention? Are the abstract numbers purely mental in nature?

    ReplyDelete
  11. Josh

    Would you say, then, that mathematics depends for its truth on man's invention? Are the abstract numbers purely mental in nature?

    As far as I can say, yes to both.
    I mean, it's not that these mathematical ideas are lying around until someone stumbles upon them.

    ReplyDelete
  12. The thing is that the inhabitants of a reality in which an object mysteriously appears when we put two and two together would, in order to describe their reality, not express it in 2+2+1 = 5. For them, our addition bears no connection to their reality (just as many mathematical models bear no connection to our reality). They would use another vector space (loosely speaking), in which an addition operation delivers 5 if operating on 2 and 2 (and write it, of course, 2+2=5).

    I think we're going to have to agree to disagree here, because even by djindra's Sola Empirica! standard I think that's incorrect. I tend to think that in this alternate universe that at least one observer would be smart enough and curious enough to count to five and wonder where the fifth object came from after he only put four into the bowl.

    ReplyDelete
  13. @E.H.Munro:

    Why would they wonder if it always happened this way? It would be perfectly natural to them. They would have evolved in a way that makes them feel that it's obviously right that way. But as I think that the hypothetical thoughts of a hypothetical being in a hypothetical and weird universe are not really the core issue here, I will not go on a crusade about that, especially since I cannot even properly imagine such a universe (being raised in a universe where 2+2=4 is real (most of the time, anyways))

    ReplyDelete
  14. ACuriousMind said...
    But why is the "=" different? Isn't it expressing equivalence both times?

    Equvilence under clock arithmetic is more general than under peano arithmetic. When partitions formed by the equivalence relations are different, I take that to mean the equivalence relations are different.

    Isn't it rather that the statement is not even possible without the system?

    There are systems where the statement could be true, false, or meaningless (among other possibilities).

    But you seem to have understood djindra differently than I have.

    I always try to interpet people as if they make sense, and sometimes this means actively looking for similar ideas. I have no intention of speaking for djindra, but I wanted to point out what his ideas reminded me of. It may be his English vocabulary is not up to expressing his thoughts.

    He claims (as I understand him) we cannot do math with numbers that are too large to fit into our heads and that we could not conceive of a system where "2+2=4" were it not that the counting of entities in our world behaved like this, while you are only defending the idea that "2+2=1" would be more sensible a statement (regarding reality) than "2+2=4" in a modular world (which I totally agree with), right?

    I took djindra's point to be that our formal ability to add basically incomprehensible numbers by rote did not render those numbers comprehensible, and that absent a real-world instantiation, we would never understand what a real-world instantiation of those numbers would mean. I don't think he is arguing with the ability to make formal manipulations. That interpretation seems unnecessarily harsh.

    ReplyDelete
  15. @Djindra:

    ""Exercise left for the readers to discover what surface the equation describes."

    IOW, it means whatever the reader wants it to mean. That "x=y+z" does not express an objective truth."

    Can you explain how you jump from my "it is an equation that describes a specific surface in three-dimensional space" (or words to such effect) to yours "it means whatever the reader wants it to mean"?

    ReplyDelete
  16. BenYachov said...
    One Brow what is the difference between something being formally true verses Necessarily true?

    Calculus: Classical logic.
    Axiom 1: Ben Yachov is an atheist.
    Axiom 2: All atheists are 3 feet tall.

    At this point, "Ben Yachov is 3 feet tall" is a formal truth. I do not see it as a necessary truth.

    Because taken at face value djindra's belief is clearly irrational.

    I tend to find terms like "clearly irrational" to be unpersuasive.

    2+2=5 cannot ever be true in any reality. 2+2=4 cannot fail to be true in any reality.

    ACurioousMind has already discussed a possible world where "2+2=5" is a model that reflects the underlying reality. I have mentioned one where "2+2=4" cold be considered meaningless, therefore not true.

    Even if we lived in a Reality or alternate universe ... In such a world 2+2=5 would still not be true nor could it be.

    Why not?

    It would merely be a world where 1+(2+2)=5. A world which has a property of +1 but that's it.

    Your thoughts?


    I disagree. We decide what a "2" is, what "+" does, etc. Therefore, in a reality where everytime you put 2 things with 2 other things a fifth appears, you set up mathematics in that reality to reflect it.

    ReplyDelete
  17. Josh said...
    Just to clarify the positions at stake here: do you believe mathematics is something we discover or is it something we invent?

    We create mathematics, but I think "invent" is wrong. Once we have created a mathematical system, can can discover new features of that suystem we have created that were not origianlly intended, but it is still our creation.

    ReplyDelete
  18. E.H. Munro said...
    TDjindra has, repeatedly, claimed that mathematics are purely empirical and that its truths are only truths insofar as they match up to sensory observation.

    This is not far from the "new empiricist" idea of mathematics.

    But what two actual two dimensional entities was Euclid observing?

    Under this interpretation, Euclid's construction is an approximation of reality, not a model.

    So when djindra tells us that he rejects platonic ideals in math he's essentially rejecting whole huge gobs of the discipline.

    Not at all. Mathematicians can be realists, empricists, formalists, and a few other things; none of this impacts the ability to do the math. It just changes the interpretation of the results.

    That may not strike you as anti-rational, but you're going to be pretty much alone there.

    Thank you for your analysis of the uniqueness of my ideas in the field of mathematical philosophy.

    Also, your artificial distinction between necessary & formal truths (I think the terms you're looking for are necessary and contingent truths)

    You think wrongly. I've already offered an example of a formal truth that was not necessary. The proerties of gravity could well be an example of a necessary truth that is not formal.

    ... he's saying that it's purely contingent and necessarily wrong if it doesn't match sensory input.

    Can math that does not match sensory input (assuming the senses are accurate) be true for the situation being secribed?

    ReplyDelete
  19. @One Brow:

    I think you are right about the distinction between formal and necessary truths, though I think the philosophers around here rather think of your formal truth as the necessary one, which may be a problem in understanding each other.

    Can math that does not match sensory input (assuming the senses are accurate) be true for the situation being described?

    You're introducing the condition "for the situation being described". If we're describing a real world situation, it's physics, not mathematics (though we use mathematics), and therefore concerned with a different truth (namely, the one that cares about corresponding to reality), not the formal one. I do not imply you are oblivious to that, but I suggest spelling it out when you are changing the meaning of "truth" in order to avoid misunderstandings. It may very well be that this kind of imprecision is also the case with djindra, I cannot judge that.

    Under this interpretation, Euclid's construction is an approximation of reality, not a model.

    What is a model if not an approximation? If you're saying that a model is an exact description of reality, without approximation, then I want to know whether you can present such a model.
    If you can, Nobel Prize awaits you (and all physicists will scream at you for not having told them earlier). ;)
    If not, of what use is the term "model"?

    And you're right, Djindra's insistence that we cannot add beyond the count of entities in the universe (or the number our brains can store) is exactly compatible with Mill's "new empiricism". But this does not make it less crazy.
    For example:
    Mathematicians regularly deal with infinites (hell, Cantor even found a way to talk about the size of infinites meaningfully) - where does that idea come from empirically? Infinties are exactly what haunts the nightmares of physicists because they keep showing up in equations about reality where they are really not supposed to (i.e anywhere). It took them twenty years to figure out how to get rid of them after discovering the basic equations of QED (winning the guys who did it a Nobel Prize). Infinites do certainly not come from empirical things. They are pretty much the definition of a mathematical concept that has no meaning in reality. Empiricism may have been valid for what Mill new of mathematics, but it certainly does not suffice for what mathematics is today (perhaps not even back then)

    On a side note, it is quite interesting where a discussion starting from the cosmological argument has taken us. Humans seem certainly not designed for thinking in straight lines, especially not on the internet. ;)

    ReplyDelete
  20. >I disagree. We decide what a "2" is,

    So you are a nominalist? Since I am clearly talking about the actual number 2 not the numeral symbol "2" that is used to represent this many stars found in between these brackets (* *).

    You do remember we are all realists here right?

    The number two is an objective thing. It has little to do with what symbol we choose to represent it or what symbol we choose to represent addition.

    Forgive me I realize you are trying to answer as honest as you can but you should just own up to the fact you hold a nominalist view and defend that.

    Unless you are such an extreme nominalist you can't own up to the objective definition of a nominalist which would explain why it is so frustrating to have a discussion with you.

    Maybe you will do a better job of it then djindra.

    ReplyDelete
  21. >Calculus: Classical logic.
    Axiom 1: Ben Yachov is an atheist.
    Axiom 2: All atheists are 3 feet tall.

    At this point, "Ben Yachov is 3 feet tall" is a formal truth. I do not see it as a necessary truth.

    Here I will further quibble with your terminology. Clearly it is a necessary truth given the axioms but not an actual truth since Axiom 2 is doubtful & axiom i is ambigous.

    (I am a Strong Atheist in regards to the existence of any and all Theistic Personalist concepts of Deity).

    I looked up Formal truth and it's defined as a Logical truth. Logical truth is given the axioms a necessary truth.

    So again we need some clearly defined terms here.

    >ACurioousMind has already discussed a possible world where "2+2=5" is a model that reflects the underlying reality. I have mentioned one where "2+2=4" cold be considered meaningless, therefore not true.

    Imagining such a world and conceiving of it are two different things.

    I can analogously conceive of infinity since I can deduce counting can never end and all numbers together are Infinity. Even thought I can't comprehend all numbers.

    But I cannot conceive of any possible world where 2+2=5. It has no meaning and no logic.

    ReplyDelete
  22. That is all for now. My kids are driving their mother up the wall & I have to go lay down the Law.

    ReplyDelete
  23. ACuriousMind,

    "The surface described by x=y+z is the plane which contains all points whose coordinates fulfill the equation."

    We could indeed find values for x,y, and z that fall in a plane but we could also find values that fall outside that plane. If x and y are considered Cartesian coordinates and z is the 3rd axis (the dimension rising above the plane), then consider these values:

    x=y+z

    1=1+0
    2=1+1
    3=1+2
    4=1+3

    This describes a horizontal line that slopes upward off the 2 dimensional paper at 45 degrees.

    Now consider:

    x=y+z

    1=0+1
    2=1+1
    3=2+1
    4=3+1

    That describes a diagonal line on the x/y plane that hovers 1 unit above the 2 dimensional plane.

    These two lines are not parallel and are not in the same plane.

    ReplyDelete
  24. BenYachov

    "But my point is there is no conceivable or possible reality where 2+2=4 is not true or that 2+2=5 is true."

    We conceive that when we watch a magician put a rabbit out of his hat.

    Nevertheless, your ability or inability to conceive does not affect the nature of an unknown universe. This argument concerning the "necessary" nature of all possible worlds is really an attempt to pass ourselves off as an all-knowing, universe-creating God.

    ReplyDelete
  25. E.H. Munro,

    "Someone (I believe it was either Ben or TheOFloinn) pointed out to him that even in a universe where if you placed two objects in a bowl and upon adding two more that a fifth spontaneously materialised that 2+2 would still equal four because the actual description of what was happening was 2+2+1=5 ..."

    Somewhere along the line you have forgotten what "equality" means.

    It is, btw, related to what we see as "equal" in the material world. That's where the concept comes from.

    ReplyDelete
  26. E.H. Munro,

    "I did not say that his view was rare or outrageous, I said that it was anti-rational. And it most undeniably is."

    It most certainly is deniable. It's irrational to believe mathematical truths condense out of nothing and fall to earth like manna.

    ReplyDelete
  27. "Why would they wonder if it always happened this way? It would be perfectly natural to them. They would have evolved in a way that makes them feel that it's obviously right that way. But as I think that the hypothetical thoughts of a hypothetical being in a hypothetical and weird universe are not really the core issue here, I will not go on a crusade about that, especially since I cannot even properly imagine such a universe"

    Again, I think we'll need to agree to disagree here, the thought experiment concerned our new empirical universe where a fifth object magically appeared every time someone tossed four into a container. I'm fairly confident that sooner or later someone would notice that inside the container was an object that they hadn't placed there, and thence begin pondering just where it came from.

    ReplyDelete
  28. This comment has been removed by the author.

    ReplyDelete
  29. "It's irrational to believe mathematical truths condense out of nothing and fall to earth like manna."

    I'm not even sure what the heck that's supposed to mean. If you're telling us that everything in math is observable through sensory perception you're simply wrong. Even the new empiricists admit that large sections of mathematics have no corresponding sensory input and try to weasel around it via a priori assumptions and circular arguments. (Mostly by claiming that all theoretical math originates in the brain and that even lacking a corresponding sensory input it must be a biochemical process therefore making it an empirical one. But this would only be true if you believed that physicalism was a necessary truth.)

    ReplyDelete
  30. I guess I don't follow what all the fuss is about. It seems to me that Moderate Realists aren't committed to a Platonic realm of number any more than any other universal. And I also don't see how one can't grasp that if all the minds that supposedly "create" mathematical truth were to go out of existence, that mathematical propositions would still be true. What is the problem? Nominalists, let me hear from ya!

    ReplyDelete
  31. OneBrow and djindra seem incapable of separating the math from the physics.

    Explicitly "equals" does not mean the same as "causes", such that when they say 2+2=5 they are describing a situation where two things and two things causes five things, not 2+2=5 (which is clearly irrational).

    Would 2+2=4 be useful in such a magical realm? Probably not. But it would still be true because we're not talking about the numerals "2" and operator symbols "+" and "=" but numbers and the proposition expressed by 2+2=4, 010+010=100, zwei und zwei ist vier, and so on.

    @ACuriousMind:

    Given what you've said about inventing/discovering math I'd be very interested in your answer to a question.

    In The Last Superstition, Dr. Feser offers a list of claims, embodying metaphysical assumptions, that science takes for granted. Among his list: there are objective laws of logic and mathematics that apply to the objective world outside our minds.

    Do you deny this claim? If not, I have a follow on question: how do you account for realism of propositions, numbers, mathematical objects, etc?

    ReplyDelete
  32. djindra wrote:
    We conceive that when we watch a magician put a rabbit out of his hat.

    Actually, we imagine that when watching magicians.

    If bright enough we could conceive the trick itself, the mechanics of which will fool the simple minded viewer into imagining the desired effect.

    ReplyDelete
  33. @ACuriousMind:

    "And you're right, Djindra's insistence that we cannot add beyond the count of entities in the universe (or the number our brains can store) is exactly compatible with Mill's "new empiricism". But this does not make it less crazy."

    It is worse than that. Mr. Djindra's position, as he stated it, has nothing to do with whatever philosophy of mathematics we hold onto. Mathematically speaking, one could pigeonhole Mr. Djindra in the ultrafinitist camp, which is an extreme form of constructivism where proofs are resource-bounded. One could view ultrafinitism as the mathematics of the "feasibly computable", that is, essentially a sub-discipline of theoretical computer science dealing with deterministic polynomially-bounded algorithms. There are *severe* foundational problems with taking this position; for what concerns me here, the most obvious one is that physicists commonly use non-constructive mathematical principles (think of all the functional analysis needed for quantum physics) and I am afraid they would not like being told that they cannot use such things. Physicists are even more oblivious than your average practicing mathematician of foundational mathematical problems. But even ultrafinitism is probably not tenable for Mr. Djindra; in order to make sense of it as the mathematics of the "feasibly computable" one has to detour into abstract mathematics that has no connection with reality, and thus, by Mr. Djindra, it is "meaningless".

    ReplyDelete
  34. E.H. Munro,

    "If you're telling us that everything in math is observable through sensory perception you're simply wrong."

    No, I'm not saying that. I have never said that. I am saying that mathematics utterly depends on its postulates. I keep on asking, where do they come from? Do they drop from the heavens like manna? Are they totally disassociated from the way we perceive nature? What makes us assume they are true? In fact, what makes us assume equality is meaningful at all? I think it's irrational to think these postulates and the rules we operate by are totally arbitrary. I think it's irrational to think math just happens, by accident, to describe nature. And I think it's irrational to assume math is all there is and nature follows from it. Yet these seem to be the positions of some.

    ReplyDelete
  35. ACuriousMind said...
    ..., but I suggest spelling it out when you are changing the meaning of "truth" in order to avoid misunderstandings.

    You're right, I'll try to be more careful.

    What is a model if not an approximation?

    In pratical terms, there may be little difference. In interpretation, the model is what happens in the ideal (so the closer you get to the ideal, the more accurate the model becomes), the approximation offers no such assurance.

    Mathematicians regularly deal with infinites (hell, Cantor even found a way to talk about the size of infinites meaningfully) - where does that idea come from empirically?

    I don't think Mill's empiricism means it's impossible to talk about infinities, only that it's impossible to do so with any any idea that what we say aboiut them reflects any sort of reality. For example, if we had empirical knowledge of infinities, we would know whether the Axiom of Choice was true or false. It would not be any more undecidable than the Axiom of Pairing. At least, that is the impression I get from what the position means.

    ReplyDelete
  36. BenYachov said...
    So you are a nominalist?

    Not as the term has been described in TLS (although I'm not sure anyone is a nominalist under that depiction). I could see myself being a conceptualist or a realist, I have not firmly made up my mind.

    Since I am clearly talking about the actual number 2 not the numeral symbol "2" that is used to represent this many stars found in between these brackets (* *).

    I can see there being a real form of "red" that instantiates on objects as a reasonable position. However, "2" is the count of the numbers of a set, and set are themselves arbitrary determined. So, I find it difficult to grant the same status to "2" that I can see as possible for "red".

    You do remember we are all realists here right?

    "All"? I think not.

    The number two is an objective thing.

    Under what meaning of objective?

    Forgive me I realize you are trying to answer as honest as you can but you should just own up to the fact you hold a nominalist view and defend that.

    It's so nice that you are here to tell me what my views are. Otherwise I'd have a completely different opinion of the views I hold. Lucky me.

    ReplyDelete
  37. BenYachov said...
    Here I will further quibble with your terminology. Clearly it is a necessary truth given the axioms but not an actual truth since Axiom 2 is doubtful & axiom i is ambigous.

    In all the reading I have done so far, I have never gotten the impression that a necessary truth was one that simply came from axioms. People discuss necessary truths as those which occur in all possible worlds, or are true by their construction, or similar things.

    For example, the Axiom offered, even if true, would have been classified as contingently true, in my experience. Therefore the conclusion from them is also contingent, not necessary.

    So again we need some clearly defined terms here.

    Agreed.

    Imagining such a world and conceiving of it are two different things.

    Go on.

    ReplyDelete
  38. @One Brow:

    Again, you have chosen the more charitable position in interpreting what we disagree about. It's fine, but until djindra tell us if he only denies that math without empirical evidence can reflect reality or if he denies that math without empirical evidence can be done, we're left with what we have. From what he has written above, I gather it is the latter, but the lack of clear positioning is a problem here.

    ReplyDelete
  39. @OneBrow:

    "For example, if we had empirical knowledge of infinities, we would know whether the Axiom of Choice was true or false. It would not be any more undecidable than the Axiom of Pairing. At least, that is the impression I get from what the position means."

    What is an undecidable axiom? What is "empirical knowledge" of finite sets? In what sense they justify the finitary versions of say, the axiom of extensionality, regularity or restricted comprehension?

    ReplyDelete
  40. @djindra:

    First of all, I took my CAS and plotted x=y+z. It showed a plane. Basic vector geometry tells me that 0=-x+y+z is a plane that contains the point (0;0;0) and has a normal vector of [-1,1,1]. x=y+z is equivalent to [x,y,z]*[-1,1,1] = 0, which is a plane. Period.

    These two lines are not parallel and are not in the same plane.

    Ok, let's do this carefully:


    1=1+0
    2=1+1
    3=1+2
    4=1+3


    This line is (using vectors):
    [x,y,z] = [1,1,0] + [1,0,1] * t,
    where t is a real number.


    1=0+1
    2=1+1
    3=2+1
    4=3+1


    This line is (using vectors):
    [x,y,z] = [1,0,1] + [1,1,0] * r,
    where r is a real number.

    Let's check where/if they intersect:
    [x,y,z] = [1,1,0] + [1,0,1] * t and
    [x,y,z] = [1,0,1] + [1,1,0] * r.

    A bit of math gives us:
    [1,1,0] + [1,0,1] * t = [1,0,1] + [1,1,0] * r

    Doing basic algebra we get:
    [0,1,-1] = [-1,0,-1] * t + [1,1,0] *r

    Solving the linear equation system this represents, we find:
    r=1 and t=1.

    So the lines intersect. Two intersecting lines always lie inside a plane. Check for yourself that this is the plane I gave above.
    Perhaps you should learn a bit math before you jump into mathematical philosophy.

    ReplyDelete
  41. djindra said...
    We could indeed find values for x,y, and z that fall in a plane but we could also find values that fall outside that plane.

    Not with Cartesian coordinate and Euclidean geometry. Of course, notihing about "x=y+z" implies either one.

    1=1+0
    2=1+1
    3=1+2
    4=1+3

    ...
    1=0+1
    2=1+1
    3=2+1
    4=3+1

    These two lines are not parallel and are not in the same plane.


    The lines intersect at 2=1+1, so they are in the same plane (any pair of intersecting lines are coplanar).

    ReplyDelete
  42. @grodrigues:

    What is an undecidable axiom?

    It is what it says - it's an axiom of which we cannot decide whether it is true, false or meaningless in reality.

    What is "empirical knowledge" of finite sets?
    Take some apples. You know got a finite set. See which characteristics of sets apply to this real set and you have gained empirical knowledge which set theory is applicable to reality. Simple.

    In what sense they justify the finitary versions of say, the axiom of extensionality?

    I think it's clear that the apples the real set contains uniquely define this set. Does the axiom really say anything more than that?

    ReplyDelete
  43. jack bodie said...
    OneBrow and djindra seem incapable of separating the math from the physics.

    jack bodie seems incapable of distinguishing an explanation of a positon from an endorsement/adoption of that position, even with specifc statements indicating that it is an explanation.

    Explicitly "equals" does not mean the same as "causes", such that when they say 2+2=5 they are describing a situation where two things and two things causes five things, not 2+2=5 (which is clearly irrational).

    Whether the grouping of objects together or not is the actual cause for the fifith object to appear is not relevant to the discussion.

    However, in a universe where such things happen, 2 + 2 = 5 is the only rational model.

    Would 2+2=4 be useful in such a magical realm? Probably not. But it would still be true ...

    True based on how we define "2", etc., absolutely. It's formally true.

    In The Last Superstition, Dr. Feser offers a list of claims, embodying metaphysical assumptions, that science takes for granted. Among his list: there are objective laws of logic and mathematics that apply to the objective world outside our minds.

    Do you deny this claim?


    Though you did not ask me, I deny it.

    If not, I have a follow on question: how do you account for realism of propositions, numbers, mathematical objects, etc?

    I feel no need to account for that which may not exist.

    ReplyDelete
  44. @jack bodie:

    There are objective laws of logic and mathematics that apply to the objective world outside our minds.

    This is difficult. What do you mean with objective?
    If you mean that the world works the way the laws predict without us having found out the laws, then I think the statement is true.
    If you mean that there really is math involved in what happens in nature and that there really are "laws" as we imagine it, I don't know. I cannot know. But I'd say there are not. Things happen - and we found a clever method to describe how. It works. This is all we can ever achieve ,I think. We can never say that the laws do "exist". Things behave according to them, but whether our laws are merely descriptive or really explanatory, we can never know.

    On further thought, I don't think this is an assumption science makes.
    Laws are not thought to be reality, they are (nowadays very accurate) approximations of reality. "Truth", confident truth, is unaccessible by the epistemology I understand t obe the scientific. Only falsification is possible, not verification, and therefore we only rule out the imprecise hypotheses that do not work.
    If there were two different frameworks that are equally simple, make the same (precise) predictions and leave nothing uncovered (i.e. are the desired "theory of everything") but fundamentally different assumptions about the underlying reality, we could not decide between the two. They are both as true as they could ever get - and "objective" reality has no meaning if we cannot decide between these two frameworks. Yet science would be at its end.

    I don't know whether I answered your question, but this is the best I can do.

    ReplyDelete
  45. grodrigues said...

    ACurouisMind answered three questions, I'll take the other two.

    In what sense they justify the finitary versions of say, the axiom of ...regularity

    If we compose a set of apples, and then look at an individual apple, the Componets that make up the individual apples are not members of the set of apples.

    or restricted comprehension?

    If we have a set of apples, we can discuss the (possibly empty) subset of yellow apples.

    ReplyDelete
  46. @ACuriousMind:

    ""What is an undecidable axiom?"

    It is what it says - it's an axiom of which we cannot decide whether it is true, false or meaningless in reality."

    I am not trying to be difficult, but genuinely trying to understand what you are saying here. What does it mean to say that an axiom is true or false in reality? Are you employing some sort of Popperian falsifiability standard? If yes, pray tell, how could we falsify say, the axiom of extensionality?

    The issue is made even more difficult, because the word axiom has two distinct meanings: a descriptive one, for example when we say "axioms of group theory" we are just singling out a particular family of objects we want to study, and a prescriptive one, when we say "axioms of first order Peano arithmetic" which, at least morally, attempts to describe a single "object". Of course, Goedel dashed these hopes, and thus in part, the first meaning is the more prevalent, nevertheless while it is easy to concoct "alternative" first order PA's by adding extra axioms, the exercise is of interest mainly to logicians.

    "Take some apples. You know got a finite set. See which characteristics of sets apply to this real set and you have gained empirical knowledge which set theory is applicable to reality. Simple."

    I am not going to press this matter further, but your dismissal as "simple" will simply not do, because there is a chasm between "take some apples" and "You know (sic) got a finite set" because the operation of grouping is a mental operation so you have to do a lot more philosophical work to use expressions like "real set" and justify them.

    ReplyDelete
  47. OneBrow, Aug 3 4.24PM:
    "Therefore, in a reality where everytime you put 2 things with 2 other things a fifth appears, you set up mathematics in that reality to reflect it."

    OneBrow, Aug 4 6.48AM:
    "Whether the grouping of objects together or not is the actual cause for the fifith object to appear is not relevant to the discussion."

    jack bodie:
    When you bring it up it's relevant, when I do it isn't? Is this your like jibberjabber about how we define '2' and we define '+' and we define '='? You define relevant?

    The relevance is 4 does not equal 5. If you claim that "in a reality where everytime you put 2 things with 2 other things a fifth appears" then the = in your 2+2=5 means causes, or grants, or reveals, or somesuch. Whatever you choose it is not the same as = meaning equals.

    If you don't redefine everything the way you want then you'll find you can't explain sense into the nonsense of 2+2=5.

    ReplyDelete
  48. @grodrigues:

    You are right about the distinction between the two kinds of axioms.
    Your descriptive ones could also be called definitions, right? I don't think they have any real meaning at all. That is, they are neither true nor false because they are simply labels which we attach to things arbitrarily.
    Your prescriptive ones are the ones in question, because they are truth claims which can or can not be true in reality, right?
    The axiom of extensionality, I think, is the former, because, in a way, it defines the criterion of uniqueness. The axiom of extensionality therefore seems to me to have no truth value at all assigned to it.
    The axiom of choice, however, is the latter - it states something about choosing from sets, and ultimately states that every set can be made well-ordered. E.g. it makes the weird statement that we can tell two socks apart if there is an infinite number of socks pairs. If infinite things existed in reality, we could go and see whether we can or cannot choose socks from these pairs. As there are no infinite sets in reality, the axiom is undecidable.

    Note that set theory is a field in which I am not fluent, and someone more knowledgeable may smash me into bits here.

    because there is a chasm between "take some apples" and "You know (sic) got a finite set"

    A set is a collection of things (in naive set theory). Take some apples. They are things. Talk about the collection of apples. You are now talking about the finite set of apples. I see no chasm here, but maybe I am missing something.

    ReplyDelete
  49. @ACuriousMind:

    Interesting reply, thank you for taking the time.

    It seems as though you started by marginally favouring the claim but, on further reflection, rejected it. I hope I understood you correctly.

    If it is not the case that science takes that claim for granted, does every scientific inquiry have to start by proving that logic and math apply to the particular case in hand each and every time?

    ReplyDelete
  50. The relevance is 4 does not equal 5. If you claim that "in a reality where everytime you put 2 things with 2 other things a fifth appears" then the = in your 2+2=5 means causes, or grants, or reveals, or somesuch. Whatever you choose it is not the same as = meaning equals.

    2+2=5 holds true with "=" meaning "equals" in an additive ring whose addition operation works like that. "=" never means "actually causes". In the strange alternate reality where objects appear when putting others together, 2+2=5 is an accurate description of what happens, meaning the result of performing the addition operation that is applicable to this universe on 2 and 2 yields 5. As pointed out above, this is not the addition operation we use normally, but I have given an example of vector spaces with different additions somewhere (way) above. Addition is not specified by yielding certain result, but by having certain properties (associativity, commutativity, neutrality to a null element...). The "+" in 2+2=4 is only one of an infinite possibilites to add. Of course, it is the one reflecting our reality, but that does not make it special somehow.

    Of course they are not equal. You seem to be puzzled by the fact that "we" say:
    2+2=4
    And "they" say:
    2+2=5
    Which would yield 4=5. This is not true, because the "+"s are not the same operations. They are only supposed to have some properties in common, not to be actually the same.

    ReplyDelete
  51. @jack bodie:

    If it is not the case that science takes that claim for granted, does every scientific inquiry have to start by proving that logic and math apply to the particular case in hand each and every time?

    In a way, yes. But isn't this already included in the normal process by drafting/using a (mathematical) theory for the case at hand and observing that it is not falsified in that case? If logic and math weren't applicable to some case, the attempt to use them would utterly fail every single time. Such a case is not yet known to me, but it is not impossible that it occurs (though it would drive scientists mad even more than reality already does).

    ReplyDelete
  52. @ACuriousMind:

    "Your prescriptive ones are the ones in question, because they are truth claims which can or can not be true in reality, right? The axiom of extensionality, I think, is the former, because, in a way, it defines the criterion of uniqueness. The axiom of extensionality therefore seems to me to have no truth value at all assigned to it."

    Then forget about the axiom of extensionality and consider the axiom of pairing or the axiom of foundation or whatever other set-theoretical axiom you care to name. Restrict to finite sets if you want to. You still have not clarified what it means to say that it is true or false.

    And once we leave the comfort zone of finitary, naive set theory, what then? Is the axiom of infinity true? You have just stated that we have no empirical knowledge of infinite sets. And yet, huge swaths of *applied mathematics* would go down the drain without them. Now what? And what about alternative foundational schemes for mathematics, are they true? Is the free topos true?

    "A set is a collection of things (in naive set theory). Take some apples. They are things. Talk about the collection of apples. You are now talking about the finite set of apples. I see no chasm here, but maybe I am missing something."

    Apples are things, but the set of apples is not a thing in the same sense that apples are things. That is the chasm you have to bridge.

    ReplyDelete
  53. @One Brow

    >Not as the term has been described in TLS (although I'm not sure anyone is a nominalist under that depiction). I could see myself being a conceptualist or a realist, I have not firmly made up my mind.

    Thank you for your candor.

    But you also seem to shift back and forth between them which is why discussion and debate with you on most topics has been frustrating for me and for others. Might I suggest you at least when you argue make a definite choice if only for the purpose of argument.

    Just some friendly advice. Thanks again.

    @djindra
    >We conceive that when we watch a magician put a rabbit out of his hat.

    You are making Hume's mistake of conflating imaginaton with the power to conceve. Did you even read TLS or did you just skim it?

    @jack bodie

    You summed up rather neatly why 2+2=5 can never be true since 4=5 cannot be true.

    Long live Moderate Realiam!

    Now I have to read more of this thread it's facinating.

    ReplyDelete
  54. @One Brow

    Redefining "=" to be synonymous with "causes" is no better than equating "the actual color property of an object" with the "light absorption properties" of an object. Especially when discussing expansive properties.

    I agree with Jack you do redefine things in mid argument. It makes rational discussion with you frustrating if not impossible.

    ACouriousMind is trying to avoid it. Take lessons from him.

    Again just some friendly advice.

    ReplyDelete
  55. @ACuriousMind:
    Apologies that you're having to walk me through this.

    In your weird vector example you used the notation "+" to make it clear that the addition operation was not the same as our +.

    To simply write 2+2=5 doesn't draw that same distinction.

    Are you saying that the distinction makes no difference?

    ReplyDelete
  56. @grodrigues:

    As I said, set theory isn't something I am very familiar with, so I am a bit puzzled that the existence of certain sets is an axiom (which both the axiom of pairing and the axiom of infinity are about, if I understand correctly).
    Concerning the axiom of pairing:
    Sets can contain sets. I got two sets A and B. So I can define a set that contains A and B. For me, it follows from the definition of sets and is therefore a deduced statement, not an axiom.
    Concerning the axiom of infinity:
    The are infinite natural numbers. I can define a set to contain the natural numbers. So there is a set that is infinite. Again, I don't see the need to treat this as an axiom, it too seems deduced.

    the set of apples is not a thing in the same sense that apples are things.
    True, but treating it as one is getting us predictions, which in turn turn out to be useful and accurate (or not).
    And this objection takes us back to another problem - the apples really aren't things either. They are made of trillions of particles, the set of which we call "apple". This kind of objection is recursive, until we hit ur-elements (elements of sets that are not themselves sets). One could now reasonably say that the ur-elements of reality are the fundamental particles (as far as we know), but then how do we justify talking about apples when they are sets of particles and you claim that sets do not exist in the same sense their elements exist. (I do not deny that apples exist, mind you)
    So before we discuss the existence of sets, we first have to get an agreement about what it means for something to exist.

    PS: The longer this discussion is going on, the more lucid it becomes to me what I really know and what I don't know. This is fun. Though potentially not coming to and end anywhere soon...

    ReplyDelete
  57. @jack bodie:

    I understood intuitively that when talking about 2+2=5 that + there is not the same operation as the + in 2+2=4, and I think I never have implied that it is.
    Though if that is the source of confusion, I apologize for not having made that sufficiently clear.
    The point is that both times the + is applicable as a representant of "putting things together" in the respective worlds, but not in the other. Yet, inhabitants of either world could study both arithmetics and examine their properties without being hindered by the fact that one of them does not correspond to their reality. Remember that this whole discussion about/with djindra originally is about the question whether math is based on reality or not (at least, that's what I am talking about. If that's not the point in question, anybody please let me know). And many vector spaces are based on nothing real, as I see it.

    And now I'm off to get a beer. I'll return tomorrow. (Don't worry, it's evening where I live.) ;)

    ReplyDelete
  58. @ACuriousMind

    >The longer this discussion is going on, the more lucid it becomes to me what I really know and what I don't know.

    My advice is keep going since the above statement shows you are slowly but surely acquiring the mind of a philosopher.

    Gotta love the beginning of wisdom.

    "I don't Know"

    ReplyDelete
  59. This comment has been removed by the author.

    ReplyDelete
  60. @ACuriousMind:

    "As I said, set theory isn't something I am very familiar with, so I am a bit puzzled that the existence of certain sets is an axiom (which both the axiom of pairing and the axiom of infinity are about, if I understand correctly). Concerning the axiom of pairing: Sets can contain sets. I got two sets A and B. So I can define a set that contains A and B. For me, it follows from the definition of sets and is therefore a deduced statement, not an axiom. Concerning the axiom of infinity: The are infinite natural numbers. I can define a set to contain the natural numbers. So there is a set that is infinite. Again, I don't see the need to treat this as an axiom, it too seems deduced."

    So if you are not familiar with it, maybe you should abstain from making definite pronouncements? Basing one's philosophy on ignorance is always a dangerous thing. I hasten to add that ignorance has never prevented me from opening my mouth on any subject whatsoever.

    It seems that by your own account all the ZFC axioms (except for the ones you do not know, of course) are nothing but definitions but earlier you concurred that ZFC axioms were prescriptive and that making truth judgments about them is valid. So which is it?

    The ZFC axioms can roughly be divided in three categories (I am making up terminology as I go along):

    1. Descriptive ones, like the axiom of extensionality that state that sets behave in a certain way or the axiom of foundation that rules out certain pathologies.

    2. Existential ones, that state that certain sets exists like the pairing axiom or the axiom of infinity.

    3. Everything else, which could be called technical axioms, like the axiom of replacement.

    And to take on one of your examples, no, you cannot go from the statement "for every natural number there is one greater than it" (this is a formulation of the infinity of the natural numbers) to the *existence of an infinite set* of natural numbers; no logical principle allows you to do that. I repeat my question: in what sense are any of these axioms true or false in reality?

    And to repeat another of my questions: if the question of the truthfulness of ZFC axioms is valid, then the same must be true of other foundational schemes. Is the free topos true or false in reality? Lambek advocates the free topos as an adequate foundation for mathematics, able to satisfy the logicist, the moderate intuitionist, the moderate Platonist and the moderate formalist.

    I also would like to add that standard mathematical justifications for the ZFC axioms do not make appeal to "reality" but to such things as the intuitive conception of the Von-Neumann hierarchy, reflection principles, etc. Shame that mathematicians forgot to consult all you hardened empiricists for justifying their foundational schemes.

    "So before we discuss the existence of sets, we first have to get an agreement about what it means for something to exist."

    You are making a mess of it all and not responding to my objection. Since I said I would not press this matter further, I stop here.

    ReplyDelete
  61. Hi ACuriousMind,

    You wrote: "In a way, yes. But isn't this already included in the normal process by drafting/using a (mathematical) theory for the case at hand and observing that it is not falsified in that case?"

    But now I'm really confused.

    Math and logic cannot be proven using physics. That they apply is a metaphysical claim and all the drafting/theorising/hypothesizing in the world won't get you out of the circle of applying unproven tools (eg, correct reasoning in the form of logic) to prove that those self same tools do, indeed, apply.

    Dr. Feser must be right: science presupposes logic and math, just as (I think) the logic of rational discourse starts from self-evident first principles of thought that it cannot itself prove.

    And if you do agree that science presupposes these laws then I guess some account must be made of them as it would be strange, I think you'd agree, to say they aren't real even as you rely on them to reach that conclusion?

    ReplyDelete
  62. ACuriousMind wrote:
    "Though if that is the source of confusion, I apologize for not having made that sufficiently clear."

    No I don't think the fault is yours; more likely I was being a little obtuse. And you're also right that this was started by the question of whether math is based on physical reality.

    I still think, however, that OneBrow and djindra are playing fast and loose with meanings.

    Even if we look at it from their reality perspective, in our world, each of the four things we bring together has a cause, or explanation if you rather, outside of the operation of 'putting them together'. This is true before the operation, and this is true after the operation.

    In the alternate reality one of the five things has no explanation outside of the operation of 'putting together' the other four. The operation itself explains, or causes, the fifth one. And causes doesn't mean equals.

    ReplyDelete
  63. ACuriousMind,

    Oops, I missed the intersect. It's been a long, long time since I took vector geometry but I'm thinking the equation to describe a plane is more complex than x=y+z. Nevertheless, this mapping of x=y+z into 3d space helps demonstrate what I've been saying all long. We try to make sense of the abstract equation by using it to represent physical dimensions of the world we live in. We could easily, and arbitrarily, decide z is on the same plane as x and y. This puts the points on a different plane. But there's no reason these variables have to represent dimensions. They could represent money: Sales revenue(x) = Gross profit(y) + cost of goods sold(z). My point is that the equation itself has no inherent meaning. Its "truth" is, at most, formal but meaningless. We must impose meanings on the variables before the equation itself acquires meaning. This implies the equation is a skeleton for meaning -- a tool for expressing something else. If we were strictly interested in the beautiful syntax we could plug in any pair of numbers, solve for the third, then reduce the equation down to a meaningless 0=0.

    But let's say we want "x=y+z" to describe a plane. (The following is my primary motivation in discussing this topic though it may not interest you) I see no reason why we should give the equation more ontological significance than the word "plane."

    ReplyDelete
  64. jack bodie,

    "In The Last Superstition, Dr. Feser offers a list of claims, embodying metaphysical assumptions, that science takes for granted. Among his list: there are objective laws of logic and mathematics that apply to the objective world outside our minds."

    There are objective laws of nature that apply to the objective world outside our minds. For the most part those laws work logically and with mathematical precision. And when we use math to describe the world we can verify it empirically. That positive feedback gives us the confidence and motivation to continue using math in this manner. So it is not "taken for granted." Its usage ultimately depends on verification.

    Suppose we ask a mathematician to describe falling bodies prior to Galileo. The mathematician might come up with an equation that gives more weight to heavier bodies. Then Galileo performs his experiment proving the math did not work. Which do we reject, the mathematical formula or the physical evidence? Was the mathematician's beautiful formula ever true?

    ReplyDelete
  65. @jack bodie:

    I made a mistake above. You are right, science presupposes that logic (and parts of mathematics) can be applied to the world. It starts from the logicians observation that induction never provides certain truth, but that all of our thoughts about how the world is stem from inducing from our limited set of data, and therefore dismisses the idea that anything about reality can be proven - hypotheses can only be disproven, and those that are not are believed to be true until contrary evidence is presented.

    And if you do agree that science presupposes these laws then I guess some account must be made of them as it would be strange, I think you'd agree, to say they aren't real even as you rely on them to reach that conclusion?

    I struggle with the "real" in this question. They are real in the sense that all real things seem subject to them. But in how far does that make them themselves real? I do not think that the laws of physics or of logic exist. They are human creation, and I do not believe that something in nature really is made of - or depends on - differential equations or wave functions. The behaviour of things is described by these laws, but this does not make the laws themselves reality, in my view.

    ReplyDelete
  66. @djindra:

    OK, your hypothetical mathematician fails aerodynamics.

    Suppose we ask Galileo to perform his experiment without using any math and without, say, the law of noncontradiction.

    How does Galileo's investigation even proceed?

    ReplyDelete
  67. @djindra:

    Suppose we ask a mathematician to describe falling bodies prior to Galileo.
    If we do so, we ask him to step out of the mathematical realm of formal truths and make him a physicist. His formula then is a scientific hypothesis, making a truth claim about reality, not a purely mathematical thing.

    We must impose meanings on the variables before the equation itself acquires meaning.

    The point is, simply saying "x=y+z" is not math. Without a definition of the symbols used, it is as meaningful as "kgjwrkfg", you're right there. Saying "'x=y+z' describes a surface within the threedimensional Euclidean space" is math. Now the symbols are defined, and we can study their properties.
    You said:
    We try to make sense of the abstract equation by using it to represent physical dimensions of the world we live in.
    Well, if you insist, I can also say that "'x=y+z' describes a surface within the ten-dimensional (named x1 to x10) vector space over the real numbers, with x=2*x4, y=4*x8 and z=x4+x8-3*x2". This is also mathematically meaningful, but has nothing to do with the dimensions of the world we live in. It has no connection to reality at all.

    ReplyDelete
  68. I feel the need to heckle here a little.

    Watching a rational Atheist and rational Theist interact with an irrational Atheist is most entertaining.

    ReplyDelete
  69. BenYachov said...
    But you also seem to shift back and forth between them which is why discussion and debate with you on most topics has been frustrating for me and for others.

    To people who insist on firmly definied categories, I'm sure it does seem like shifting.

    Might I suggest you at least when you argue make a definite choice if only for the purpose of argument.

    No, I don't think I will. I see no reason to try to fit into categories I don't accept for the sake of conforming to other people's beliefs. I think that will inhibit discussion, not promote it.

    Redefining "=" to be synonymous with "causes" is no better than equating "the actual color property of an object" with the "light absorption properties" of an object. Especially when discussing expansive properties.

    Which is why I corrected jack_bodie about his use of "cause". Nice to see you agree with my correction.

    ReplyDelete
  70. jack bodie said...
    OneBrow, Aug 3 4.24PM:
    "Therefore, in a reality where everytime you put 2 things with 2 other things a fifth appears, you set up mathematics in that reality to reflect it."

    OneBrow, Aug 4 6.48AM:
    "Whether the grouping of objects together or not is the actual cause for the fifith object to appear is not relevant to the discussion."

    jack bodie:
    When you bring it up it's relevant, when I do it isn't?


    I did not bring up cause, and there is no trace of it in the first post from which you quoted. You brought up cause, and I corrected you. Saying "A" happens, then "B" happens, so "A" causes "B" is a logical fallacy.

    ReplyDelete
  71. ACuriousMind said...

    Minor quibbles, not at all on point.

    E.g. it makes the weird statement that we can tell two socks apart if there is an infinite number of socks pairs.

    Actually, it means no matter how many pairs you have, you can always take one sock from each pair.

    The axiom of choice is fundamentally equivalent to saying an infinite crossproduct of non-empty sets always has an element.

    If infinite things existed in reality, we could go and see whether we can or cannot choose socks from these pairs. As there are no infinite sets in reality, the axiom is undecidable.

    Technically, undecidabiltiy refers to it's relationship to ZF set theory (or some similar theory). Some axioms in ZF are similarly unverifiable from an empirical standpoint, but are not considered undecidable.

    ReplyDelete
  72. @One Brow:

    If you mean the technical idea of undecidability (or independence), then whey did you let my answer to grodrigues pass?

    I said:

    What is an undecidable axiom?

    It is what it says - it's an axiom of which we cannot decide whether it is true, false or meaningless in reality.


    I thought we were talking about math's connection to reality here, and you said that if we had empirical knowledge of infinite sets, then the axiom of choice would not be undecidable anymore. How does empirical knowledge change the fact that the axiom of choice is undecidable in ZF?

    ReplyDelete
  73. @OneBrow:
    I've looked at the situation you described, and I've looked at it.

    I see an event "A" (2 things put together with 2 other things) and a second event "B" (a fifth object appears) which occurs as a consequence of the first (everytime you put 2 thing with two other things) ...

    From your description it is clear:
    A precedes B in all cases.
    The is 100% correlation between A and B.
    A never produces not-B.

    It is not clear, but implied that:
    A and B are contiguous in space-time.
    A is necessary for B.

    Tell me how I should define causation because so far it's walking and quacking like a duck.

    The fact is, you want to imagine a world where 2 material things put with 2 other material things gives you (sorry, can't think of another phrase) 5 material things. This is not the same as ACuriousMind's addition-like operations on numbers.

    I explained in an earlier comment how, to my mind, the fifth thing had no cause but the operation itself. Perhaps you could give me a counter-example?

    ReplyDelete
  74. A post just disappeared on me. Sorry it this turns out repetitious.

    grodrigues said...
    I think it would be good to remember that discussions of what a strictly empirical view would entrail is not endorsement of that view. As far as I can tell, ACuriousMind is a formalist, like myself.

    Then forget about the axiom of extensionality and consider the axiom of pairing or the axiom of foundation or whatever other set-theoretical axiom you care to name.

    Well, the axioms of union and powersets would be prescriptive rather than descriptive. However, I think it is obvious how an empiricist would verify such axioms, at least on a case-by-case basis.

    Is the axiom of infinity true?

    To a strict empiricst, no.

    You have just stated that we have no empirical knowledge of infinite sets. And yet, huge swaths of *applied mathematics* would go down the drain without them.

    I agree with your analysis here. This is why it will remain an unpopular position; the mathematics is not useful enough.

    ReplyDelete
  75. grodrigues said...
    Lambek advocates the free topos as an adequate foundation for mathematics, able to satisfy the logicist, the moderate intuitionist, the moderate Platonist and the moderate formalist.

    I'm interested. Is there an online site you would recommend?

    ReplyDelete
  76. ACuriousMind said...
    I thought we were talking about math's connection to reality here, and you said that if we had empirical knowledge of infinite sets, then the axiom of choice would not be undecidable anymore. How does empirical knowledge change the fact that the axiom of choice is undecidable in ZF?

    If we had empirical knowledge of the truth of the Axiom of Choice, it would have been a foundational part of ZF to begin with, rather than being left outside it. If we had empirical knowledge of the falsity of the Axiom of Choice, it would have been a foundational part of ZF to begin with, rather than being left outside it. What I meant was that it would never have been considered undecidable because it had already been decided, much like the Axiom of Infinity has been decided.

    ReplyDelete
  77. jack bodie said...
    Tell me how I should define causation because so far it's walking and quacking like a duck.

    Usually, on this blog, causation refers to things that come from the natures of the causes and effects. Inks color paper because of the nature of paper and inks.

    The notion of mere correspondance as causation is one that Feser railed against in TLS, from my recollection.

    I explained in an earlier comment how, to my mind, the fifth thing had no cause but the operation itself. Perhaps you could give me a counter-example?

    In the universe where whenever 2 apples are put next to 2 apples, God creates a fifth apple, every time. The cause is not the 2 paris of 2 apples.

    ReplyDelete
  78. @One Brow
    >To people who insist on firmly definied categories, I'm sure it does seem like shifting.

    I don't insist on belief in those catagories. But I insist on rational and orderly consistant discussion with common termonology.

    >No, I don't think I will. I see no reason to try to fit into categories I don't accept for the sake of conforming to other people's beliefs.

    No you are being asked to be consistant. Why is that so hard?

    >I think that will inhibit discussion, not promote it.

    With whom? So far everyone seems to have a problem figuring out what you are really saying. So it's all of us but not you?

    >Which is why I corrected jack_bodie about his use of "cause". Nice to see you agree with my correction.

    Which post have you done this I don't recall? He accused you of conflating "causes" with "equals" & as far as I know I agree with him. Are you saying you don't conflate "causes" with equals or you do but somehow it legitimate?

    What's the deal here?

    ReplyDelete
  79. @OneBrow:

    "I think it would be good to remember that discussions of what a strictly empirical view would entrail is not endorsement of that view. As far as I can tell, ACuriousMind is a formalist, like myself."

    If you do not mind my curiosity, why are you defending a position which you do not endorse?

    "Well, the axioms of union and powersets would be prescriptive rather than descriptive. However, I think it is obvious how an empiricist would verify such axioms, at least on a case-by-case basis."

    Obvious? Maybe I am being thick here, but you have clarified exactly nothing.

    First, as I remarked to ACuriousMind, a set of apples is not a thing in the same sense that apples are. In order to verify any set-theoretical axiom you have to map the mathematical concept of set into reality, in other words, the verification itself depends on an interpretative act. And if it depends on an interpretative act, how can it be meaningful to say that they are true or false in reality?

    Second, what does "on a case-by-case basis" mean? Performing experiments? Then, assuming you surpass the first hurdle, pray tell me, how can you even perform an experiment if you do not have the *concepts* of counting or set? What does it mean to verify the induction axiom of first order PA on a case by case basis if you do not even have the concept of counting?

    From nothing, nothing comes. ACuriousMind already conceded that logic and mathematics precedes the empirical disciplines in a response to Jack Brodie. Are you seriously arguing that we can observe the outside world and perform experiments without any sort of theoretical scaffolding and conceptualization?

    "I agree with your analysis here. This is why it will remain an unpopular position; the mathematics is not useful enough."

    There has been very little analysis on my posts, mainly the stating of a few, paltry facts. I have even avoided stating my philosophical position (I favor moderate realism, but in a Thomist blog that is hardly surprising). And isn't there an extraneous "no" in that last sentence? If not, then you have me confused.

    ReplyDelete
  80. Why do I get the feeling this is the whole "Property of having a Color" vs "Light absorption Properties" via Color is an expansive property debate all over again?

    One Brow in the scenario I originally described (i.e alternate reality where adding 2 objects to another 2 object causes a 5th object to appear etc) it is clear 2+2 causes 5 but clearly doesn't equal 5.

    Thus there is no conceivable reality where 2+2=5.

    You may believe otherwise motivated by radical skepticism and equivocation. But you can't believe so rationally.

    ReplyDelete
  81. @OneBrow:

    ""Lambek advocates the free topos as an adequate foundation for mathematics, able to satisfy the logicist, the moderate intuitionist, the moderate Platonist and the moderate formalist."

    I'm interested. Is there an online site you would recommend?"

    Interested in what? In Lambek's position? In the free topos?

    Note: I should have added the "moderate" qualifier to the logicist also, because of the axiom of infinity (in its topos-theoretic form of the existence of a natural numbers object).

    I just mentioned the free topos because it was the first example that came to my mind. There are many foundational schemes out there; not to mention the fact that set theory itself is a mathematical discipline in active development: Woodin's programme of finding the Ultimate L axioms, large cardinal axioms, the fact that some set-theorist experts think that there is no uniquely defined concept of set but a whole multiverse of them, etc., etc. and etc.

    ReplyDelete
  82. >In the universe where whenever 2 apples are put next to 2 apples, God creates a fifth apple, every time. The cause is not the 2 paris of 2 apples.

    It's not what kind of cause? It's not the formal cause? The efficient cause? Material Cause? Final Cause?

    What kind of cause is it not?

    ReplyDelete
  83. Of course in my original example I didn't invoke "god". I appealed to a natural feature of that reality.

    "Macro Quantum event where 5th object arises out of the Cosmic Ather."

    ReplyDelete
  84. @OneBrow:
    I'm not sure if you do or don't endorse the position of 2+2=5 in Unrealverse but you're certainly a zealous advocate!

    In anycase when you wrote "where whenever" this makes it a consequence of your operation of putting two material things with two other material things. It may not be the proximate cause but A still causes B.

    If your alter-god is just randomly creating apples and it sometimes coincides with people putting 4 apples in a bucket, then you're right: cause is irrelevant but it's also the case that 2+2=4 so it doesn't help you (or, rather, the position you're trying to defend).

    ReplyDelete
  85. @BenYachov:
    "Macro Quantum event where 5th object arises out of the Cosmic Ather."

    Ha ha! I see, yes of course: the quantum makes it not only plausible but cool.

    ReplyDelete
  86. @One Brow:

    If we had empirical knowledge of the truth of the Axiom of Choice, it would have been a foundational part of ZF to begin with, rather than being left outside it. If we had empirical knowledge of the falsity of the Axiom of Choice, it would have been a foundational part of ZF to begin with, rather than being left outside it.

    Are you saying the criterion for statements for being included as a foundational part in ZF is whether or not we have empiricial knowledge of their truthfulness? Contains ZF only axioms for which we have such evidence (ignoring the fact that we would first have to see how we can define a set in reality, as grodriguez showed me above)

    ReplyDelete
  87. @BenYachov:

    One Brow in the scenario I originally described (i.e alternate reality where adding 2 objects to another 2 object causes a 5th object to appear etc) it is clear 2+2 causes 5 but clearly doesn't equal 5.

    And here we go again:
    2+2 causes nothing. 2+2 has no real meaning until you declare that the operation of addition works like the mechanism of putting things together. 2+2 is not something you find in reality until you claim that the mathematical symbols represent real entities, which is not required to claim that there are spaces in which 2+2=5 is a formal truth. Declaring that this addition now represents the mechanism of putting things together in our alternate reality, we find it reflects this reality, as we never find a situation where something other than 5 things show up. So in this reality, 2+2=5 is true, but using another operation of addition. This 2+2 is not equal to our 2+2, but it equals 5 in their system (I don't know why we should now start the debate about causes again in an alternate reality...). Vice versa, our (still formally true) 2+2=4 is no true statement about their reality, if we again assume the addition represents putting things together. 2+2, with our addition, has no representant in the alternate world.

    ReplyDelete
  88. @ACouriousMind

    >2+2 causes nothing.

    Naturally which is why it can never equal 5 in any concevible reality. In the example of the alternate universe I gave it does cause 5 but it still equals 4 and does not equal 5.

    Even empirically in that Universe one can intellectively exclude counting the 5th object that appears and conclude 2+2=4.

    Equating "cause" and "equal" is not legitimate here. It is one of the errors of nominalism that you can change the nature of a thing by merely changing it's name or term. That is not true see TLS.

    You can't say in the alternate universe "their" math has 2+2=5. Since for all we know they might have a formula 2+2 C 5 to account for the auto +1 nature of their universe. C meaning "causes".

    I can program a computer to automatically +1 to any 2+2 calculation but there is no logical way it can produce a true 5 from a 2+2 by itself without that additional programing. Even reprogramming the computer to make the numeral symbol "5" now represent this many objects in brackets[****] is still 2+2=4.

    Renaming and replacing the numeral symbol "4" with "5" doesn't change that reality, that essence that nature. 2+2=4 is true by nature, necessity and actually.

    If we assume addition is putting things together then we would conclude we put together 4 objects and the universe merely added 1 more.

    So I maintain logically 2+2=5 cannot be true in any concevable reality or universe. 2=2=4 cannot fail to be true.

    Anybody is free to believe what they like but as far as I am concerned if they deny this truth they might as well say the Earth is 5000 years old and was made in 144 hours and be done with it.

    ReplyDelete
  89. edit: 2+2=4 cannot fail to be true.

    Sorry the plus symbol and the equal symbol are on the same key and sometime the shift key doesn't work.

    ReplyDelete
  90. @ACuriousMind:

    At my own peril, I am going to butt in this discussion and add the following points:

    1. Are you just not shifting the terms and arbitrarily renaming things? If that is the case, then you have not advanced your case a single iota.

    2. If you are not shifting the terms, then 2 + 2 = 5 is a contradiction, because 2 + 2 is "just" a short-hand for counting a set of 4 and since 4 is not equal to 5, 2 + 2 can never equal 5.

    3. If 2 + 2 were equal to 5 it would mean that whenever we had 4 objects put together (which is a mental operation, not a physical one) a fifth would pop out of nothing. This does not cinch the fact that 2 + 2 equals 4 and besides, it is a metaphysical impossibility. Or to put it in other words, if you want to argue that it is a metaphysical possibility, you have work to do.

    4. Are you saying that there can be arithmetics where 2 + 2 = 5? Sure, one can redefine +, for example as m + n = m ++ n ++ 1 where ++ is the "conventional" + and in this case, sure 2 + 2 = 5 but we have just arbitrarily redefined things and 2 ++ 2 still equals 4. Redefining does not change the nature of the thing.

    5. Unless of course, you are a nominalist. But you still have a lot of work to do, especially in a Thomist blog inhabited by moderated realists.

    6. Notice also that very, very little, and I stress the very little, suffices to prove that 2 + 2 equals 4 so, even if we redefine arithmetic such that 2 + 2 equals 5, we would end up with a useless concept with nothing resembling arithmetic, so why call it arithmetic? In other words, I am back to 1. We have just renamed things as if renaming changes the nature or the essence of the thing itself.

    ReplyDelete
  91. @BenYachov and grodriguez:

    I agree with most of what you said, but it seems we are talking about different things. You are absolutely right that counting two things and counting two others always yields 4, no matter what may happen in reality if you actually put them together. So, 2+2=4 is always true as a representant of counting things. But wasn't the whole example of the world where a fifth object appeared meant to show that statements in mathematics do not necessarily bear resemblances to reality (i.e. 2+2=5 (with a redefined +, I admit) bears no connection our our reality), but are nevertheless mathematical truths? At least, that is what I perceived. We are "shifting the terms" because the goal was never to establish that the truth of 2+2=4 within the field of real numbers depends on the reality we live in - but neither does the truth of 2"+"2=5 within whatever space it is true in. Mathematical truth does not depend on reality. The reality changes only which parts of mathematics we apply to it and which we don't.
    I was not trying to change the nature of things by redefining them. I was trying to show that there are many different operations of addition which exist in mathematics, but of which the vast majority has nothing to do with our reality, and some of which may be used by inhabitants of other world to model their reality. I wasn't equating these operations, and I certainly did not want to discuss causes in this context.
    Probably the whole thing with the alternate reality wasn't really the simplest way to do this...

    ReplyDelete
  92. >But wasn't the whole example of the world where a fifth object appeared meant to show that statements in mathematics do not necessarily bear resemblances to reality......

    Rather the truths of mathematics don't rely absolutely on empiricism. I don't equate "reality" with what solely can be verified empirically.
    Plus it is self evident not all truths need to be known empirically.

    ReplyDelete
  93. BenYachov,

    "You are making Hume's mistake of conflating imagination with the power to conceive. Did you even read TLS or did you just skim it?"

    Do you think I take Feser's word for it? Let's forget for a moment that Merriam-Webster lists "imagine" as one meaning of "conceive." Let's see if Feser is consistent on this issue.

    In TLS Feser asserts "imagining something and conceiving it in the intellect simply aren’t the same thing." He claims that imagining something is "to form a certain mental image" but to conceive something is to form "a coherent intellectual idea of it." Well, that's vague to say the least. Feser is sure this ill-defined difference between imagining and conceiving was not understood by Hume.

    But this is Feser in his "Edwards on infinite causal series" post:

    "Suppose a 'time gate' of the sort described in Robert Heinlein’s story 'By His Bootstraps' were possible. Suppose further that here in 2010 you take a stick and put it halfway through the time gate, while the other half comes out in 3010 and pushes a stone. The motion of the stone and the motion of the hand are not simultaneous -- they are separated by 1000 years -- but we still have a causal series ordered per se insofar as the former motion depends essentially on the latter motion."

    So Feser expects us to believe we cannot form a coherent intellectual idea of a ball suddenly forming on a table -- and doing so without any known cause -- yet we can form a coherent intellectual idea of a stick that we push through a time porthole and "cause" a per se effect.

    In "Philosophy of Mind" Feser is quite fond of the conceivability argument. There we are expected to have a coherent intellectual idea of a man who stares at himself in the mirror only to discover he has no eyes with which to see. This scenario is supposedly non-contradictory. Then we are expected to conceive of hacking off the top of our skull to find we have no brain. Still we can "intellectually grasp" the idea that we continue thinking.

    After that, it's very hard to take Feser's position as genuine on this issue. In fact, it's hard to separate it from intellectual dishonesty.

    Plus, I wonder why Feser failed to footnote Hume on the bowling ball scenario. From what I've been reading of Hume lately (I do check up on these sources) I think Feser has created a fake Hume -- a convenient whipping boy who is more imagined than real.

    ReplyDelete
  94. @djindra
    >Do you think I take Feser's word for it? Let's forget for a moment that Merriam-Webster lists "imagine" as one meaning of "conceive." Let's see if Feser is consistent on this issue.

    Where as someone with intelligence would have consulted the Stanford Encyclopedia of Philosophy on Imagination. Since the subject is philosophy after all.

    QUOTE"To imagine something is to form a particular sort of mental representation of that thing.......It is also sometimes distinguished from mental states such as conceiving and supposing, on the grounds that imagining S requires some sort of quasi-sensory or positive representation of S, whereas the contrasting states do not.

    You are worse than useless djindra.

    Perhaps you could uses Merriam-Webster to explain Quantum Physics to ACouriousMind?

    I'm sure he would find your looney blather as convincing as the rest of us do.

    Now let the grown ups talk son.

    ReplyDelete
  95. ACuriousMind,

    "I can also say that 'x=y+z' describes a surface within the ten-dimensional (named x1 to x10) vector space over the real numbers, with x=2*x4, y=4*x8 and z=x4+x8-3*x2". This is also mathematically meaningful, but has nothing to do with the dimensions of the world we live in. It has no connection to reality at all."

    But I don't claim math cannot be nonsensical. Doing proper English we can still create fairy-tales. Any language can express nonsense, fantasy or obscifate. That does not imply the language is disconnected from reality. It especially does not imply the language finds "truth" that way.

    ReplyDelete
  96. ACuriousMind,

    On an undecidable axiom:

    "It is what it says - it's an axiom of which we cannot decide whether it is true, false or meaningless in reality."

    The fact that there is such a special category implies that decision via reality is desirable and that decision by reality is a factor at other times.

    ReplyDelete
  97. jack bodie,

    "Tell me how I should define causation because so far it's walking and quacking like a duck."

    Suppose I toss two apples in a bucket. Then I toss in two more. I expect to find five apples in the bucket when I get around to counting. But when I do count, I find there are only four apples. Should I ask what caused one apple to disappear?

    Suppose I expect to count 4 apples and I do count 4. Which pair of apples caused 4 to appear? The first pair? The second? Is it proper to talk of any apple causing anything in this scenario? Is "4" an effect, a property, or something else?

    But let's say 2+2 apples cause 4 apples. This must be the Final Cause of 2 apples. I suppose both pair of apples have that Final Cause of 2+2 cause 4. Just to show how smart Final Cause is, we throw in another apple and to our amazement the pairs of apples have lost that Final Cause of 2+2 cause 4. The 4 apples have gained a new Final Cause! Now 4+1 causes 5! We might ask where this Final Cause comes from. Just when did the apples assume that new Final Cause? But we must be resigned to the fact that we cannot know such things. Final Cause just is. Final Cause is not caused at all. Accept and behave yourself. If tomorrow, in some distant universe, Final Cause of 2 apples plus 2 apples causes 5, well, that's just Final Cause doing its thing.

    ReplyDelete
  98. BenYachov,

    "Where as someone with intelligence would have consulted the Stanford Encyclopedia of Philosophy on Imagination. Since the subject is philosophy after all."

    And you provide yet another hypocritical stance.

    First, I used Feser's own definition in criticizing him, not Websters. But more importantly, Feser uses the "common sense" understanding of concepts when it suits him. There is no reason I need to accept your definition -- especially when that definition is vague to the point of uselessness -- as Feser shows himself.

    ReplyDelete
  99. BenYachov,

    "Thus there is no conceivable reality where 2+2=5."

    I see you believe your mind possesses the ability to control what may or may not exist. Do you bend spoons too?

    ReplyDelete
  100. djindra writes:

    First, I used Feser's own definition in criticizing him, not Websters.
    August 6, 2011 5:20 PM

    and

    Do you think I take Feser's word for it? Let's forget for a moment that Merriam-Webster lists "imagine" as one meaning of "conceive."
    August 6, 2011 5:20 PM

    It's obvious to those of us with some higher education and even some amateur knowledge of philosophy that Feser "definition" isn't unique to him.

    Indeed if djindra really read TLS instead of skimming it he would know Feser's critique of Hume isn't even his, it's Anscombe's.

    See pages 105-106 and footnotes 23&24 of Chapter 3.

    djindra has dropped all pretense at this point that he is anything but a proof-texting troll without a coherent thought in his head.

    ReplyDelete
  101. That is pages 105-106 etc from THE LAST SUPERSTITION.

    Wow djindra your reading comprehension skills suck as badly as my spelling and grammar.

    Which would make them really bad.

    ReplyDelete
  102. Let's imagine a different universe. In this place there exists only one type of thing, a monad. Monads exist independently. They don't like each other and try to avoid other monads. So if one comes too close they both try to fly away from each other at high speed. Nevertheless, the escape mechanism is not foolproof. So when they fly away to avoid each other they occasionally bump into another by mistake. When this happens, the two colliding monads spontaneously generate a third monad. So as long as the monads remain apart, normal math applies. If we see one monad in the area and another drifts into view, we see two monads and 1+1=2. But if circumstances are just right and the two collide, we have 3 monads. So under pressure, 1+1=2 until it equals 3. Looking at it from the POV of God, the universe went from a collection of x monads to x+1. But it was not achieved in the normal mathematical way. It was not x+1=(x+1). It was x + an accident = (x+1). This universe needs a very different math than ours. It would need a combination of our math with a collision addition. Nevertheless, it still has a certain weird logic and conceivability to it.

    ReplyDelete
  103. BenYachov,

    "Indeed if djindra really read TLS instead of skimming it he would know Feser's critique of Hume isn't even his, it's Anscombe's."

    Do you seriously expect me to believe it's not Feser's view as well? Every time Feser mentions Hume I can almost see his gag reflex.

    ReplyDelete
  104. BenYachov,

    I'm speling challunged as well. I don't fult yu fo that.

    ReplyDelete
  105. >Do you seriously expect me to believe it's not Feser's view as well?


    Who do you think he learned it from?

    Wow you are thick son.

    ReplyDelete
  106. @djindra
    >So Feser expects us to believe we cannot form a coherent intellectual idea of a ball suddenly forming on a table -- and doing so without any known cause --

    I reply: Naturally, go look up the footnotes and read Anscombe's two part philosophical paper for yourself.

    Of course you would need to know how to look up journals in a college library and frankly djindra I'm not sure you know how to tie your shoes.

    >yet we can form a coherent intellectual idea of a stick that we push through a time porthole and "cause" a per se effect.

    I reply: I don't see how pushing a stick is a hard thing to conceive?

    Besides if you really read the post "Edwards on infinite causal series" & not skim it you would see Fesder is imagining a "time gate" as a device for his argument on infinite series requiring a first mover in the top down sense.

    Feser never says you can't use imagination to create ad hoc conventions to do thought experiments in philosophy. He merely tells us imagination and intellectual conception are not the same.

    Much like I imagine a universe where nature causes a 5th object appears and is added to four objects that have just been added together. It's to illustrate you really can't conceive of 2+2=5.
    Your still conceiving of 1+(2+2)=5.

    You don't get that? It's really not that hard. You attempts to play gotcha are just sad at this point.

    ReplyDelete
  107. @djindra:

    But I don't claim math cannot be nonsensical.

    Then what was your point when talking about the numbers larger than we can hold in our head?

    On undecidability:

    The fact that there is such a special category implies that decision via reality is desirable and that decision by reality is a factor at other times.

    Well, I must admit that I jumped on this without first looking up the mathematical meaning of this -if one does, one finds that "undecidable" is equivalent to "independent" and means that a certain statement can neither be proven true nor wrong inside a certain axiomatic system. This notion has nothing to do with reality, which is why One Brow consequently confused me with his "empirical knowledge", which would change nothing about the undecidability of any statement.

    But let's say 2+2 apples cause 4 apples.

    Would please everyone stop to claim that mathematical expressions cause anything? 2+2 causes nothing. Tossing the apples together may cause something, but the addition bears no necessary connection to the process of putting things together. Addition is abstractly defined as a mathematical operation that has certain properties (commutativity, associativity, neutrality to the null element,...). It may be used to represent putting things together because that process also has these properties - but addition as such is nothing real. It's a symbol, and cannot cause anything.
    I really don't get how "2+2 causes..." could ever be a meaningful statement. It is "Putting things together" that causes something, not "2+2".

    This universe needs a very different math than ours.

    Are you finally conceding with your monad universe that we have mathematics that are unfounded in reality (such as additions in vector space which may accurately model this weird universe)?

    I really have no idea anymore what your position actually is.

    ReplyDelete
  108. @BenYachov:

    I am a bit puzzled why "imagining" vs. "conceiving" suddenly is such a big deal here, and I am not sure whether the distinction is really meaningful:

    You say on imagining:
    on the grounds that imagining S requires some sort of quasi-sensory or positive representation of S, whereas the contrasting states do not.

    And on Feser's infinite series you write:
    you would see Feser is imagining a "time gate"

    How has he gained quasi-sensory representation of a time gate?
    Or: If we can get a representation of such a weird thing - of which kind then are the things we cannot imagine, but only conceive of?

    It's to illustrate you really can't conceive of 2+2=5.

    If the + there is our natural addition, then not. But I don't think anybody wanted that we conceive of 2+2=5 using that addition. The point is that, in order to reflect the alternate reality, we would use a space in which addition works like 2"+"2=5. And we can conceive of such a space although nothing in our reality is like that. (Particle physics might resemble this a bit, but uses a far more complicated math, and really isn't like this at all)
    Mathematics (e.g. of vector spaces) has no foundation in reality - it is not empirical.
    Hasn't this been demonstrated now?
    What is the disagreement still about?

    ReplyDelete
  109. @ACuriousMind:

    ""Mathematics (e.g. of vector spaces) has no foundation in reality - it is not empirical."

    Hasn't this been demonstrated now? What is the disagreement still about?"

    Who said the quoted phrase? I cannot find it anywhere, but I presume you attribute it to Mr. Djindra. If I am right, according to Mr. Djindra, and quoting from a post above,

    "But I don't claim math cannot be nonsensical. Doing proper English we can still create fairy-tales. Any language can express nonsense, fantasy or obscifate. That does not imply the language is disconnected from reality. It especially does not imply the language finds "truth" that way."

    So I suppose this means that it is "nonsensical", "fairy-tales", "fantasy" or "obscifate" (sic). It also has no "truth" content -- the "" in this particular case are from Mr. Djindra, so only he knows what exactly they mean.

    Pity you cannot do quantum mechanics without linear spaces. And it is no good saying that because of *this*, linear spaces are based upon "reality" or are "empirical", because first that would be inverting the historical path of discovery. Second, you cannot point to reality and say "here is a linear space". Third, the quantum state vector is *not* measurable according to the Copenhagen interpretation. So not only is a linear space an *artifact* of the theory, but the *elements* of the linear space themselves are also an *artifact* and not directly measurable by the Copenhagen Interpretation -- think epicycles in the old Ptolemaic theory. Oh well, so there goes the mathematics that underlies quantum mechanics to the shelf of "nonsensical". What this says of quantum mechanics? Well, it is expressed in a "nonsensical" language so... Numerous other examples could be given.

    Note: for the record, I dislike the Copenhagen Interpretation intensely, but this is a whole different debate.

    ReplyDelete
  110. @ACuriousMind:

    "If the + there is our natural addition, then not. But I don't think anybody wanted that we conceive of 2+2=5 using that addition. The point is that, in order to reflect the alternate reality, we would use a space in which addition works like 2"+"2=5."

    You have made this point repeatedly, but I think you should be a little more careful. In order to prove that 2 + 2 = 4 we only need the existence of a unit element (to be able to define 1, 2, 3, 4, etc.) and associativity of +, so in order to have a "space" (your word, not mine) in which 2 + 2 = 5 the operation would have to be non-associative. Associativity is the tiniest of constraints on a binary operation, it just means that parenthesis do not matter. So let us not exaggerate the claims that we could have addition in which 2 + 2 = 5.

    None: I hasten to add that there are important non-associative binary operations, e.g. the Lie bracket.

    ReplyDelete
  111. @Djindra:

    "Let's imagine a different universe."

    I do not know what you think your example purports to prove, but it proves absolutely nothing, in particular it does not prove that "It would need a combination of our math with a collision addition. Nevertheless, it still has a certain weird logic and conceivability to it." And if you think it does, just note that what you describe already happens in our universe. Take a first order Feynman diagram where a positron and an electron collide to give birth to a photon.

    One thing seems certain to me though: you giving that example proves Ben Yachov right that you do not know the difference between imagine and conceive.

    ReplyDelete
  112. >I am a bit puzzled why "imagining" vs. "conceiving" suddenly is such a big deal here, and I am not sure whether the distinction is really meaningful:


    All the more reason to read The Last Superstition.

    It is absolutely meaningful and the major difference between classical philosophy vs modern. Aquinas vs Hume.

    I might say more later but the house is a mess and my brain is fried from to much FALLOUT:NEW VEGAS.

    When FALLOUT 4 comes out one day I am getting the PC over the XBOX if only so I can Mod the crap out of it.

    Cheers.

    ReplyDelete
  113. grodrigues,

    "Take a first order Feynman diagram where a positron and an electron collide to give birth to a photon."

    Apples and oranges. H and 2 Os collide to form water too. But neither have anything to do with my example.

    ReplyDelete
  114. BenYachov,

    "Feser is imagining a 'time gate' as a device for his argument on infinite series requiring a first mover in the top down sense."

    So he's 'imagining' a 'time gate' to give substance and permission to 'conceive' of his silly example. But even good fiction needs a certain level of consistency.

    "Feser never says you can't use imagination to create ad hoc conventions to do thought experiments in philosophy."

    He's just irritated when Hume supposedly does it, but not when he and his friends do it. It's A-okay then.

    read Anscombe's two part philosophical paper for yourself.

    Is Anscombe's paper about hypocrisy? Unless it is then it's not pertinent.

    "Much like I imagine a universe where nature causes a 5th object appears and is added to four objects that have just been added together. It's to illustrate you really can't conceive of 2+2=5."

    In that vein, you can't conceive of a Pure Actuality called God. You can't conceive of how that god moves anything without being material itself. You can't conceive of a form called "soul." You can't conceive of absolute nothingness. You can't conceive of infinity. You can't conceive of a universe other than our own which operates according to different rules than our own. You can't conceive of a lot of things you persist in imagining. I'm on to your game.

    ReplyDelete
  115. BenYachov,

    Me: "Do you seriously expect me to believe it's not Feser's view as well?"

    You: "Who do you think he learned it from?"

    Now that's dull, even for you. First, Feser begins this section with "Now Hume famously attacks this principle..." at the top of page 105. Once he rants for two paragraphs about the vague distinction between imagining and conceiving, then he brings up Anscombe with, "For another thing..." So it's unclear where he gets his idea.

    Nevertheless, you point out one serious deficiency in Feser. Virtually everything he writes is directly attributed to someone else. He often does this intentionally. I'm not saying there is anything deceptive about it. What I am saying is that he is not an original thinker. Therefore he's boring. The only thoughts that are distinctly his own are those of a political nature. And even that would be okay if he was creative about it. But he isn't. He's just irritated. And that's all that comes through. The one plus is that he covers a lot of bases and loosely points a lot of directions. For those interested, it's a starting place from which to look more closely into the matters.

    ReplyDelete
  116. @Djindra:

    ""Take a first order Feynman diagram where a positron and an electron collide to give birth to a photon."

    Apples and oranges. H and 2 Os collide to form water too. But neither have anything to do with my example."

    Photons, electrons and positrons, unlike water molecules, are not composite objects but simple ones, so yes, my example applies.

    ReplyDelete
  117. ACuriousMind,

    "Then what was your point when talking about the numbers larger than we can hold in our head?"

    My point was not aimed directly at you. As I've mentioned before, my focus is about the supposed ontological being of math truths and numbers. I take it as a given that even most Platonists would be hesitant about giving ontological status to flights of fancy. That would force them into giving ontological status to unicorns, Greek gods and Freddy Kruger.

    Btw, my point was not simply about numbers in our head or the number of objects we can count; it was about how numbers are limited by the medium itself. Any substance has this problem. Sooner or later there is no more substance and nowhere else to record. To reach that next number seems to me to be a legitimate paradox. And I don't care what number base we use or how many objects we think might exist, it's still the same paradox.

    "Would please everyone stop to claim that mathematical expressions cause anything? 2+2 causes nothing."

    I agree but it gave me a good way of getting in a jab at "Final Cause."

    ReplyDelete
  118. ACuriousMind,

    "Are you finally conceding with your monad universe that we have mathematics that are unfounded in reality (such as additions in vector space which may accurately model this weird universe)?"

    No. I still think a math depends on reality to make it sensible to us. I still can't swallow this notion that math is a "true" universe of its own. Maybe it can cut off its own legs and strike off in any direction as free as a bird. I don't dispute that it (and any language) can try to do that. I dispute that such findings necessarily mean or necessarily are something. They might be beautiful fictions, but then again, they might wrap around and eventually reinforce basic notions -- notions (I say) are based on our experience in the real world in the first place. My focus is on meaning and truth. Faithfully following syntax and algorithms cannot guarantee that, IMO. So I disagree with what you wrote earlier:

    "How can your syntactically correct program not have a semantic? The computer does something when it executes your program, doesn't it? That's the meaning. It hasn't to be useful, but it has a semantic - otherwise the computer would not know what to do."

    I dispute that syntax necessarily leads to semantics. The computer (as of today) understands nothing. It goes through an algorithm as blind as a bat. I'm saying a mathematician who merely constructs beautiful algorithms could easily be doing the same thing. The algorithms and equations work, but are they true in a meaningful way and how can we be sure?

    ReplyDelete
  119. >Virtually everything he writes is directly attributed to someone else. He often does this intentionally.

    So that is your story now? First it's Feser giving his evil ideas that are somehow wrong (though you have yet to explain why outside of citing Webster) now he is giving everyone else's ideas.

    As entertaining is your hysterical meltdown is do make up your mind.

    ReplyDelete
  120. @djindra
    >In that vein, you can't conceive of a Pure Actuality called God.

    I can't conceive of the number "Infinity" unequivocally as it is in it's essence but I can intellectually infer counting without end and I have a concept of "every single number there is or could be in math which is infinite".

    So what is the problem?

    >You can't conceive of how that god moves anything without being material itself. You can't conceive of a form called "soul." You can't conceive of absolute nothingness. You can't conceive of infinity.......You can't conceive of a lot of things you persist in imagining. I'm on to your game.

    I reply: But I can know these things analogously & coherently. I can infer their existence intellectually by what in fact I can know exists via my senses and what they tell me about how change happens.

    I can't coherently know that any possible reality can exist where 2+2=5 is true or 2+2=4 fails to be true. Since that doesn't describe anything coherent. Counting forever is coherent. Logically deducing that there is in mathematics an infinite number of numbers is coherent if incomprehensible. But 2+2=5 is not coherent thus it doesn't exist.

    It's not hard genius.

    BTW imagination still isn't the same as comprehension.

    Live with it troll boy.

    ReplyDelete
  121. BenYachov,

    "So that is your story now? First it's Feser giving his evil ideas that are somehow wrong (though you have yet to explain why outside of citing Webster) now he is giving everyone else's ideas."

    Is this you imagining or conceiving? because either way you're not comprehending.

    ReplyDelete
  122. BenYachov said...
    >No, I don't think I will. I see no reason to try to fit into categories I don't accept for the sake of conforming to other people's beliefs.

    No you are being asked to be consistant. Why is that so hard?


    I can be consistent without adopting a particular categorization. I acknowledge this may seem inconsistent to those who only think in specific categorizations.

    With whom? So far everyone seems to have a problem figuring out what you are really saying. So it's all of us but not you?

    I disagree that everyone has a problem. grodrigues and ACuriousMind have asked quesitons that indicate substantial understanding of at least much of what I am saying.

    >Which is why I corrected jack bodie about his use of "cause". Nice to see you agree with my correction.

    Which post have you done this I don't recall?


    August 4, 2011 6:48 AM

    He accused you of conflating "causes" with "equals" & as far as I know I agree with him. Are you saying you don't conflate "causes" with equals

    I am saying that any discussion of cause is irrelevant tot he discussion of the model of mathematics that would be created.

    Why do I get the feeling this is the whole "Property of having a Color" vs "Light absorption Properties" via Color is an expansive property debate all over again?

    Because that was teh one time you were right, and so now you need every argument to be that one?

    One Brow in the scenario I originally described (i.e alternate reality where adding 2 objects to another 2 object causes a 5th object to appear etc) it is clear 2+2 causes 5 but clearly doesn't equal 5.

    Describe what that is clear.

    Thus there is no conceivable reality where 2+2=5.

    In the universe described previously, that is how mathematics would work.

    It's not what kind of cause? It's not the formal cause? The efficient cause? Material Cause? Final Cause?

    If you want to make a serious case for a cause of some sort (a tangent that is still irrelevant), it up to you to make it. In a universe where whenever 2 apples wind up next to two other apples, a fifth suddenly is created by God, in what fashion are the original aples a cause of the fifth? Material, formal, efficient, final? I'm not holding my breath for your answer.

    Of course in my original example I didn't invoke "god". I appealed to a natural feature of that reality.

    "Macro Quantum event where 5th object arises out of the Cosmic Ather."


    Again, the answer is yours. In what way would the original apples be the material/formal/efficient/final cause of a quantum event? Not that this is relevant.

    >2+2 causes nothing.

    Naturally which is why it can never equal 5 in any concevible reality. In the example of the alternate universe I gave it does cause 5 but it still equals 4 and does not equal 5.


    So, you agree it causes nothing ("Naturally"), and then say it causes 5. Is this the type of consistency I am supposed to exhibit?

    Even empirically in that Universe one can intellectively exclude counting the 5th object that appears and conclude 2+2=4.

    All that means is that the mathematics would disassociate counting and adding, to that degree.

    ReplyDelete
  123. Equating "cause" and "equal" is not legitimate here. It is one of the errors of nominalism that you can change the nature of a thing by merely changing it's name or term. That is not true see TLS.

    Addition, and mathematics generally, is not a thing. Addition, and mathematics generally, is a process that we invent to describe the universe in which we live. When physicists showed that spacetime has negative curvature, that did not change the Euclidean model of space, because the Euclidean model was never a thing to begin with, whose nature could be discovered. We just changed to a more useful model of space on occasions where we needed more accuracy. Geometry classes all over the world still teach teh Euclidean model, because it is easy and sufficiently accurate for daily life for most professions.

    I can program a computer to automatically +1 to any 2+2 calculation but there is no logical way it can produce a true 5 from a 2+2 by itself without that additional programing.

    Programming was required for the computer to do 2+2=4 in the first place. It's programmed wither way.

    You are worse than useless djindra.

    Your own quote from the SEP demonstrated that the requirements for conceiving something were less regorous than for imagining it, not more rigorous. You won the argument that there was a difference, and in doing so defeated your own position that the requirements for conceiving required more rigor than that for imagination. Bravo.

    I reply: I don't see how pushing a stick is a hard thing to conceive?

    I think it's that whole "through a time portal" thing that you would have trouble conceiving (in manner which claim to be necessary).

    I can't coherently know that any possible reality can exist where 2+2=5 is true or 2+2=4 fails to be true. Since that doesn't describe anything coherent. Counting forever is coherent. Logically deducing that there is in mathematics an infinite number of numbers is coherent if incomprehensible. But 2+2=5 is not coherent thus it doesn't exist.

    The limitations on your ability to see coherence are not prescriptive on those who can see more.

    ReplyDelete
  124. grodrigues said...
    If you do not mind my curiosity, why are you defending a position which you do not endorse?

    I absolutely do not mind your asking any question at all, and there are very few I will refuse to answer as directly as I can manage. However, since I do not feel I am defending a particular position of mathematical empiricism, but only discussing it, I find this one difficult to answer directly. I can say that I am discussing it because carefully examing ideas you do not agree with is important to my learning process.

    First, as I remarked to ACuriousMind, a set of apples is not a thing in the same sense that apples are. In order to verify any set-theoretical axiom you have to map the mathematical concept of set into reality, in other words, the verification itself depends on an interpretative act. And if it depends on an interpretative act, how can it be meaningful to say that they are true or false in reality?

    It seems to me that if you take the strict empiricist view we are discussing, you can pull out 5 distinct apples, and then arrange them to form all 32 separate groupings, thus emprically verifying the powerset axiom. Now, in creating the model, adopting the power set axiom is of course an intuitive leap based on empirical data, but that's true of any empirical branch of knowledge.

    Second, what does "on a case-by-case basis" mean? Performing experiments? Then, assuming you surpass the first hurdle, pray tell me, how can you even perform an experiment if you do not have the *concepts* of counting or set? What does it mean to verify the induction axiom of first order PA on a case by case basis if you do not even have the concept of counting?

    Well, we do choose our axioms to reflect the mathematics we already want to perform, and I don't see how this is different for the empircist. They already have a naive notion of counting, subsets, etc. that they wish to create a model for.

    From nothing, nothing comes. ACuriousMind already conceded that logic and mathematics precedes the empirical disciplines in a response to Jack Brodie. Are you seriously arguing that we can observe the outside world and perform experiments without any sort of theoretical scaffolding and conceptualization?

    Back when I had babies, they were quite able to observe the outside world and perform experiments without knowledge fo mathematics or logic, and for that matter so can my beagles. mathematics and logic give us a structure that alloes us to perform these experiements in a more rigorous, controlled structure, but it not necessary for the existence of experiemntation or observation.

    "I agree with your analysis here. This is why it will remain an unpopular position; the mathematics is not useful enough."

    There has been very little analysis on my posts, mainly the stating of a few, paltry facts.


    The selection of the appropriate facts can, and often does, itself convey the analysis of a position.

    ReplyDelete
  125. And isn't there an extraneous "no" in that last sentence? If not, then you have me confused.

    Sometimes i drop a "no", but I don't see where I did in that sentence. Where would you expect one?

    Interested in what? In Lambek's position?

    Yes. I'm curious why he thinks it would unite so many disparate opinions.

    At my own peril, I am going to butt in this discussion and add the following points:

    1. Are you just not shifting the terms and arbitrarily renaming things? If that is the case, then you have not advanced your case a single iota.


    Acually, the shifted terms and arbitrary renaming is the case.

    We have just renamed things as if renaming changes the nature or the essence of the thing itself.

    Mahematics has no nature or essence to be changed.

    You have made this point repeatedly, but I think you should be a little more careful. In order to prove that 2 + 2 = 4 we only need the existence of a unit element (to be able to define 1, 2, 3, 4, etc.) and associativity of +, so in order to have a "space" (your word, not mine) in which 2 + 2 = 5 the operation would have to be non-associative.

    Agreed.

    Associativity is the tiniest of constraints on a binary operation, it just means that parenthesis do not matter. So let us not exaggerate the claims that we could have addition in which 2 + 2 = 5.

    None: I hasten to add that there are important non-associative binary operations, e.g. the Lie bracket.


    Exactly.

    Photons, electrons and positrons, unlike water molecules, are not composite objects but simple ones, so yes, my example applies.

    Actually, in djindra's example, the original two monads were not detroyed, as they are in the Feynman diagram.

    ReplyDelete
  126. djindra said...
    The fact that there is such a special category implies that decision via reality is desirable and that decision by reality is a factor at other times.

    The ultimate criteria for inclusion include usefulness as well as reality.

    ReplyDelete
  127. jack bodie said...
    In anycase when you wrote "where whenever" this makes it a consequence of your operation of putting two material things with two other material things. It may not be the proximate cause but A still causes B.

    Perhaps you can answer Ben yachov's quesiton about what sort of cause it is?

    If your alter-god is just randomly creating apples and it sometimes coincides with people putting 4 apples in a bucket, then you're right: cause is irrelevant but it's also the case that 2+2=4 so it doesn't help you (or, rather, the position you're trying to defend).

    Please explain why a being of that universe would use a mathematics where 2 + 2 = 4.

    ReplyDelete
  128. ACuriousMind said...
    Are you saying the criterion for statements for being included as a foundational part in ZF is whether or not we have empiricial knowledge of their truthfulness?

    I would not say it is "the criterion", but it would certainly be "a criterion". For example, the powerset axiom was included without question because it was not derviable from the other axioms and also empircally verfiable. The axiom of infinity was extremely useful, and so included. In the grand scheme of things, the Axiom of Choice is not particularly useful nor empirically verfiable, and so not included.

    Do you think that is the Axiom of Chjoice were empircally verifiable, it would have been left out of ZF?

    Probably the whole thing with the alternate reality wasn't really the simplest way to do this...

    I would have been surprised if you had gotten a different response, regardless of approach.

    ReplyDelete
  129. @OneBrow:

    ACuriousMind is right in that the start of this discussion was the odd idea (that you may or may not endorse) that math is empirical, or in some way jerry-rigged from what is materially real.

    Now, as soon as you bring apples or oranges or anything material into a conversation that should have been about arithmetic in order to prove your (or maybe not your) point, you're no longer just talking about the mathematical operation. "Imagine," you're saying, "a place where whenever you put two things together with two other things a fifth thing appears."

    Do you deny that is what you're saying? (How can you deny it when you outright said it?)

    Why then is it wrong to point out that you haven't described 2+2=5 at all, when you haven't? You're the one who's got it backwards by assuming math follows empirical observation. And by choosing the scenario you did, you made cause relevant.

    How? Well put aside your demands for a full accounting in terms of Aristotle's four causes for one moment; it is, in any case, just a transparent attempt to sidestep the obvious: pick up any manual on scholastic philosophy and you'll find they're happy to define a cause as whatever a thing is positively dependent upon either for its reality or for its coming into existence.

    Now we haven't fiddled with your scenario when pointing out that the 4 original apples have causes outside of the act of bringing them together; that "I toss two apples into a bucket" (as you or djindra said) is a cause for those (and then the next two) apples being in the bucket; and that the fifth apple appears only as a consequence of the act of bringing together the original 4. And all you can do in an attempt to deny this is demand we embellish your nonsense description in order for it to make logical sense? Please. You should crawl before demanding someone takes you ice-skating, and as this apparently isn't a position you endorse anyway you should be free to try understanding our objections, instead of reflexive gainsaying.

    As far as other discussion goes, you and djindra are alone in denying that the physical sciences presuppose math and logic. That does not surprise me.

    To be frank, given ACuriousMind's remarks about the (non)reality of laws of math and logic, I'm more interested in what kind of existence he or she thinks Hawking means in, "Because there is a law like gravity, the universe can and will create itself"?

    ReplyDelete
  130. jack bodie said...
    ACuriousMind is right in that the start of this discussion was the odd idea (that you may or may not endorse) that math is empirical, or in some way jerry-rigged from what is materially real.

    To be clear, again, I'm a formalist. To me, mathematics no more empirical than chess. It's a big game where we all agree to play by certain rules, and can change the rules at need by agreement. Over the years, the rules of the game have been very well honed, so that matheamtics is now highly useful.

    Now, as soon as you bring apples or oranges or anything material into a conversation that should have been about arithmetic in order to prove your (or maybe not your) point, you're no longer just talking about the mathematical operation.

    There is no independent entity that can be the object of "the mathematical operation", so I was never talking about "the mathematical operation" to begin with, just one of many different possible mathematical operations with similar goals.

    Why then is it wrong to point out that you haven't described 2+2=5 at all, when you haven't?

    Actually, we are describe a scenario where the mathematics would be created in such a way that 2+2=5.

    You're the one who's got it backwards by assuming math follows empirical observation.

    We create mathematics, in part, to be empircally useful.

    And by choosing the scenario you did, you made cause relevant.

    In what way? You'll need to offer more detail. You use the idea of dependency below, but never describe a dependency.

    How? Well put aside your demands for a full accounting in terms of Aristotle's four causes for one moment;

    I find it amusing how Ben Yachov assign the position of jack brodie to me to ask me to defend it, and in return jack bodie assigns to me the question Ben Yachov wants answered in defense of his idea. It's almost as if they can't conceive of disagreeing with each other on this point, so the points of disagreement must belong to me.

    ReplyDelete
  131. it is, in any case, just a transparent attempt to sidestep the obvious: pick up any manual on scholastic philosophy and you'll find they're happy to define a cause as whatever a thing is positively dependent upon either for its reality or for its coming into existence.

    In a universe where God chooses to create a fifth apple whenever two apples are placed next to two apples, in what way is God dependent upon the placing of the apples? In a universe where a quantum fluctuation happens to create an apple at such a time, in what way is the quantum fluctuation dependent on the placing of the apples? Again, I see no cause, nor the relevancy of cause.

    Now we haven't fiddled with your scenario when pointing out that the 4 original apples have causes outside of the act of bringing them together;

    Is that relevant is some way?

    And all you can do in an attempt to deny this is demand we embellish your nonsense description in order for it to make logical sense? Please.

    I don't think anyone is asking to make logical sense of the mysteriously apearing apples.

    You should crawl before demanding someone takes you ice-skating, and as this apparently isn't a position you endorse anyway you should be free to try understanding our objections, instead of reflexive gainsaying.

    You mean, unlike the way you have completely missed the point about mathematics, in order to focus on the cause of the apples?

    As far as other discussion goes, you and djindra are alone in denying that the physical sciences presuppose math and logic. That does not surprise me.

    You are far from alone in your refusal to directly address the notion that experimentation and observation require no mathematical or logical basis for babies, dogs, etc. That does not surprise me. Instead, you scoff without addressing an idea. That also does not surprise me.

    To be frank, given ACuriousMind's remarks about the (non)reality of laws of math and logic, I'm more interested in what kind of existence he or she thinks Hawking means in, "Because there is a law like gravity, the universe can and will create itself"?

    I won't speak for ACuriousMind. My personal understanding is that because it is possible for spacetime to behave in this fashion, it is possible for spacetime to create itself.

    ReplyDelete
  132. @OneBrow:

    "If you want to make a serious case for a cause of some sort (a tangent that is still irrelevant), it up to you to make it. In a universe where whenever 2 apples wind up next to two other apples, a fifth suddenly is created by God, in what fashion are the original aples a cause of the fifth? Material, formal, efficient, final? I'm not holding my breath for your answer."

    Not a direct answer to your question, but in such a universe 2 + 2 would still equal 4. Your scenario only inserts a Demiurge that mischievously pops apples just to confuse the poor humans. *He* would know the truth.

    "It seems to me that if you take the strict empiricist view we are discussing, you can pull out 5 distinct apples, and then arrange them to form all 32 separate groupings, thus emprically verifying the powerset axiom. Now, in creating the model, adopting the power set axiom is of course an intuitive leap based on empirical data, but that's true of any empirical branch of knowledge."

    Then you just conceded my point that set-theoretical axioms are not empirically verifiable because you cannot point to reality and say here is a set without having a model of what a set is in reality. But if a set in reality is interpreted via a model, there is no meaning to the phrase "the set-theoretical axioms are true of reality". You got things backward, mathematical concepts prop the theories not the other way around. And your last point is obviously not true, because the empirical sciences are first and foremost based on sensory data (even if mediated by complicated instruments) which is obviously not the case for mathematical knowledge.

    "Well, we do choose our axioms to reflect the mathematics we already want to perform, and I don't see how this is different for the empircist. They already have a naive notion of counting, subsets, etc. that they wish to create a model for."

    This is completely circular and does not answer my questions.

    "Back when I had babies, they were quite able to observe the outside world and perform experiments without knowledge fo mathematics or logic, and for that matter so can my beagles. mathematics and logic give us a structure that alloes us to perform these experiements in a more rigorous, controlled structure, but it not necessary for the existence of experiemntation or observation."

    But since Babies or Beagles cannot articulate this knowledge and communicate it to subject it to external validation and count it as empirical knowledge this is irrelevant for your case.

    ReplyDelete
  133. @OneBrow (continued):

    "Interested in what? In Lambek's position?

    Yes. I'm curious why he thinks it would unite so many disparate opinions."

    The position of Lambek is quoted in a paper by Landry and Marquis,

    http://umontreal.academia.edu/JeanPierreMarquis/Papers/188871/Categories_In_Context_Historical_Foundational_and_Philosophical

    Jump to page 14.

    "Mahematics has no nature or essence to be changed."

    That is itself a philosophical position and a contentious one at that, so you should not pass it up as a self-evident truth without further justification. And no, I am not a Platonist.

    "Actually, in djindra's example, the original two monads were not detroyed, as they are in the Feynman diagram."

    This is irrelevant to my point. There is absolutely no reason Mr. Djindra has given us to suspect that the intelligent inhabitants of such a universe would have to develop a "collision addition", especially if the normal state of affairs is the same as in our own universe. They could just follow our route and have some sort of explanatory theory still based on perfect ordinary mathematics for the extraordinary events. In other words, imagining is all fine and dandy, but substantiating your fairy-tale fantasies (to use a favorite expression of Mr. Djindra) is another thing altogether.

    "The axiom of infinity was extremely useful, and so included. In the grand scheme of things, the Axiom of Choice is not particularly useful nor empirically verfiable, and so not included."

    Huh? Classical analysis can hardly get off the ground without countable dependent choice, and full choice (or something slightly weaker) is needed for many results fundamental in various areas of mathematics: Hahn-Banach, Tychonoff, Krein-Milman, Prime Ideal theorem, etc. Heck, even constructivists (e.g. Bishop) accept some form of countable choice.

    And the not-empirically verifiable is a useless insertion because the mathematical justifications of ZF(C) do not make appeals to the empirically verifiable reality.

    ReplyDelete
  134. OneBrow (and not BenYachov) wrote:
    If you want to make a serious case for a cause of some sort (a tangent that is still irrelevant), it up to you to make it. In a universe where whenever 2 apples wind up next to two other apples, a fifth suddenly is created by God, in what fashion are the original aples a cause of the fifth? Material, formal, efficient, final? I'm not holding my breath for your answer.

    See my post above; I explained how the putting together of the four apples is the cause of the fifth apple in the sense of the fifth apple being positively dependent upon that act for its coming into existence.

    In a universe where God chooses to create a fifth apple whenever two apples are placed next to two apples, in what way is God dependent upon the placing of the apples?

    Yes, now your god 'chooses to' create the fifth apple. Again, a transparent attempt to weaken the causal link that I suspect you can see but now must deny for all you're worth. Certainly even you would admit that this wasn't your first scenario, or your second ("where whenever 2 apples are put next to 2 apples, God creates a fifth apple, every time"). Anyway when he chooses to create no extra apple, or to create a dozen extra, come back to me - you'll have a new problem. Until then you can answer your own question by finding out what whenever actually means, and by asking why you call it the fifth apple and not independent whimsy apple number #whatever. (Hint: in your account before your god makes any choice about a fifth apple, he knows there are four together.)

    As to the Hawking quote perhaps I should've been clearer (though I expect ACuriousMind would have understood) - unlike you, ACuriousMind admits that the physical sciences presuppose objective laws of math and logic that apply to the objective world outside of our minds, BUT he denies that they are in any sense 'real'. Given that Hawking explicitly chose the law of gravity for his quote he must presuppose either (1) matter exists, (2) the law of gravity exists, or both. My question isn't about the physicist's mistaken understanding of nothing but specifically what ACuriousMind believes 'exists' means in (2).

    ReplyDelete
  135. grodrigues,

    "There is absolutely no reason Mr. Djindra has given us to suspect that the intelligent inhabitants of such a universe would have to develop a "collision addition", especially if the normal state of affairs is the same as in our own universe."

    In my example intelligent inhabitants would be impractical since I stipulated that the monads repel each other. They could never form composites. I suppose I could grant they have monad souls. And if they have souls maybe they could have minds that contemplate monad existence and why the number of their enemies keeps growing.

    "They could just follow our route and have some sort of explanatory theory still based on perfect ordinary mathematics for the extraordinary events."

    I suppose they could but why would they when it's so much easier to recognize the collision component, with mathematical precision and certainty, always adds one to their numbers?

    ReplyDelete
  136. jack bodie,

    "you and djindra are alone in denying that the physical sciences presuppose math and logic."

    Give us time, we'll soon be a movement. And even though I sense OneBrow does not fully embrace the orthodoxy, I welcome his sect.

    (Btw, my position is that our nature and our interaction with the universe predisposes us to think logically and mathematically.)

    ReplyDelete
  137. @Djindra:

    ""They could just follow our route and have some sort of explanatory theory still based on perfect ordinary mathematics for the extraordinary events."

    I suppose they could but why would they when it's so much easier to recognize the collision component, with mathematical precision and certainty, always adds one to their numbers?"

    So your answer is a conjecture about the thinking processes of intelligent inhabitants, that are "impractical" anyway, in a totally implausible scenario? And why is not that a fairy-tale fantasy? But in the end this is all irrelevant, because "to recognize the collision component" and "always adds one to their numbers" they would have to have in the first place a standard arithmetic to always add "one to their numbers".

    ReplyDelete
  138. jack bodie said...
    OneBrow (and not BenYachov) wrote:
    If you want to make a serious case for a cause of some sort (a tangent that is still irrelevant), it up to you to make it. .... Material, formal, efficient, final


    I congratulate you on demonstrating you have the ability to determine who typed something.

    See my post above; I explained how the putting together of the four apples is the cause of the fifth apple in the sense of the fifth apple being positively dependent upon that act for its coming into existence.

    A depdency that is 1) irrelevant to the point being made, and 2) does not exist, regardless.

    Yes, now your god 'chooses to' create the fifth apple. Again, a transparent attempt to weaken the causal link that I suspect you can see but now must deny for all you're worth.

    Actyually, I don't care a fig about the causal argument at all, I just enjoy seeing you tie yourself in knots. I was unaware you were thinking of a God who was so controlled by human actions that They were forced to create apples on human whims.

    Certainly even you would admit that this wasn't your first scenario, or your second ("where whenever 2 apples are put next to 2 apples, God creates a fifth apple, every time").

    Frankly, I would never bring the notion of a God who was forced to create apples from the actions of humans into a board of Thomists.

    Anyway when he chooses to create no extra apple, or to create a dozen extra, come back to me - you'll have a new problem.

    Really? What problem?

    (Hint: in your account before your god makes any choice about a fifth apple, he knows there are four together.)

    In what manner is that a cause?

    As to the Hawking quote perhaps I should've been clearer (though I expect ACuriousMind would have understood) - unlike you, ACuriousMind admits that the physical sciences presuppose objective laws of math and logic that apply to the objective world outside of our minds, BUT he denies that they are in any sense 'real'.

    Unlike you, Im humble enough to know that my interpretation of ACuriousMind's words may not reflect his actual thoughts. I have not understood his words as you seem to. However, I'm not inclined to try to explain him.

    Given that Hawking explicitly chose the law of gravity for his quote he must presuppose either (1) matter exists, (2) the law of gravity exists, or both.

    Neither is a viable option.

    ReplyDelete
  139. OneBrow:

    Seriously you accuse me of mistaking your demand for BenYachov's. I point out where you made the demand by cutting and pasting part of your post and you respond with snark? A simple "oh I forgot I wrote that" would be the response of most adults.

    Once again the dependency exists in every scenario you outlined. It's relevant because you failed to show 2+2=5.

    As to the other stuff, drodrigues has explained how you simply injected a Demiurge. You say you would never do X even as your scenario shows you doing X - do I believe you or my lying eyes?

    As to how knowledge of four apples is a cause... Would coming to know there is a bomb under your chair cause you to make a choice? Is there something controversial about that idea?

    ReplyDelete
  140. >Once again the dependency exists in every scenario you outlined.

    One Brow does this all the time. He redefines what he writes in mid discussion.

    Which is why it's impossible to dialog with him.


    It's also why nobody takes his "critique" of TLS seriously.

    He claims he answered Feser on top down essential causality & challenges me to read it. I read it and pointed out to him putting little motors on the box cars is no different than having them pulled by a Locomotive it just concedes the first cause argument.
    He pettifogs the issue by changing the subject to the physics of how an unpowered box car transfers momentum to the Caboose it's towing while ignoring the question of where the initial actuality comes from.

    Round and Round even if you are civil it is impossible to have a rational consistent discussion with One Brow.

    He is like djindra sans the paranoid left wing conspiracy nuttyness and bitter hate. But he is just as confusing.

    ReplyDelete
  141. grodrigues,

    I appreciate you taking your time to discuss this topic.

    grodrigues said...
    Not a direct answer to your question, but in such a universe 2 + 2 would still equal 4. Your scenario only inserts a Demiurge that mischievously pops apples just to confuse the poor humans. *He* would know the truth.

    The truth of apple creation ex nihilo, sure. The Demiurge can also use any mathematics he dreams up.

    Then you just conceded my point that set-theoretical axioms are not empirically verifiable because you cannot point to reality and say here is a set without having a model of what a set is in reality.

    We have an empircal experience of gathering things into a collection. The model would be a formal representation of it. This would not be any different than having an experience of things falling to the ground, and then making a model called the Theory of Gravitation.

    But if a set in reality is interpreted via a model, there is no meaning to the phrase "the set-theoretical axioms are true of reality".

    I actually described looking at powersets by physical manipulation of a collection.

    You got things backward, mathematical concepts prop the theories not the other way around.

    I agree. I am not sure why you think that was reversed in my example.

    And your last point is obviously not true, because the empirical sciences are first and foremost based on sensory data (even if mediated by complicated instruments) which is obviously not the case for mathematical knowledge.

    In my example, the subsets were physically constructed.

    This is completely circular and does not answer my questions.

    Which question in particular?

    But since Babies or Beagles cannot articulate this knowledge ... as empirical knowledge this is irrelevant for your case.

    I'm curious what you think my point is.

    The position of Lambek is quoted in a paper by Landry and Marquis,

    Thank you.

    That is itself a philosophical position and a contentious one at that, so you should not pass it up as a self-evident truth without further justification. And no, I am not a Platonist.

    You may have missed this, but we are on a board where contentious philosophical positions get passed off as self-evident truths regularly :). However, I will endeavor to be more careful in my discussions with you, and I'm sure you'll do the same.

    They could just follow our route and have some sort of explanatory theory still based on perfect ordinary mathematics for the extraordinary events.

    Except, there are no extraordinry events, just normal events. I agree that an alternate path would be to have a mathematics like our and a physics that tries to account for monad creation. If fact, both would probably exist.

    substantiating your fairy-tale fantasies

    I acknowledge we wil never likely to be able to confirm what real residents of an imaginary universe might create with respect to mathematics.

    "The axiom of infinity was extremely useful, ..."

    Huh? Classical analysis can hardly get off the ground without countable dependent choice,


    I apologize for being unclear. By "useful", I meant that the various applications in the sciences required structures dependent on the axiom of infinity (such as calculus). I am not aware there were any such application requiring dependent choice a hundred years ago.

    Heck, even constructivists (e.g. Bishop) accept some form of countable choice.

    Cool.

    And the not-empirically verifiable is a useless insertion because the mathematical justifications of ZF(C) do not make appeals to the empirically verifiable reality.

    Fine. However, if the Axiom of Choice were empirically verifiable, do you think their would ever have been a question of it's inclusion in set theory axioms?

    ReplyDelete
  142. jack bodie said...
    Seriously you accuse me of mistaking your demand for BenYachov's.

    Seriously, that would be impossible, becasue I have no demand regarding the notion of causation in this argument, because I think causation is irrelevant to this argument. I have enjoyed watching you and Ben Yachov contradict each other and pin your differences to me, but I don't demand that you do so.

    I point out where you made the demand by cutting and pasting part of your post and you respond with snark?

    I supposed I could have responded more directly and seriously, but I was trying to be light-hearted. After all, it should have been obvious tht I was merely quoting Ben yachov's demand back at Ben Yachov, not formulationg one of my own, by the fact that I used an indirect quote of the list of causes he presented and that I quoted in the very same post you pasted from. Of course, a serious discussion would have included comments on your ability to read, interpret context, probably partisanship, etc. I generally prefer not to go that route, because I always try to think the best of people, even when they make it difficult to do so.

    A simple "oh I forgot I wrote that" would be the response of most adults.

    I have no idea whether most adults would have noticed that I was throwing Ben Yachov's words back at Ben Yachov or not. I only know that the context of that was clear, and yet you missed it. I did not forget what I wronte. I do know the difference between suggesting a standard and suggesting that a person meet their own standard. I have no evidence you are aware of that difference.

    Once again the dependency exists in every scenario you outlined. It's relevant because you failed to show 2+2=5.

    Once again, you confuse correlation with dependency, and in a manner that goes back to the mechanical notion of causation that Dr. Feser railed against in TLS. If you claim a dependency and wish to convince others of it, demonstrate the dependency, not teh correlation.

    As to the other stuff, drodrigues has explained how you simply injected a Demiurge.

    Again, not really relevant to the mathematics the being of such a dimension would construct.

    You say you would never do X even as your scenario shows you doing X - do I believe you or my lying eyes?

    Actually, what I'm saying, and have been saying, is that I don't care about X becasue X is not relevant, wheile nonetheless challenging you to prove your own assertions about X.

    As to how knowledge of four apples is a cause... Would coming to know there is a bomb under your chair cause you to make a choice? Is there something controversial about that idea?

    It may or may not, there would be any other significant factors (like my curent desire to keep living). Of course, there are teh issue of potential effects of a bomb. Are you saying there are similar potential effects regarding the bringing together of four apples that would cause a reation in some putative God/quantum flucuation?

    ReplyDelete
  143. BenYachov said...
    One Brow does this all the time. He redefines what he writes in mid discussion.

    What term have I re-defined in this conversation? Can you back your accusation up?

    Which is why it's impossible to dialog with him.

    Many people dialog with me, some even productively. Maybe it's you. Who was the last person you had a productive dialog with even though you fundamentally disagreed?

    It's also why nobody takes his "critique" of TLS seriously.

    Whatever you wish to tell yourself.

    He claims he answered Feser on top down essential causality & challenges me to read it. I read it and pointed out to him putting little motors on the box cars is no different than having them pulled by a Locomotive it just concedes the first cause argument.

    The First Cause is the metaphorical locamotive. When the metaphorical locamotive is removed, the First Cause disappears from the actual argument.

    He pettifogs the issue by changing the subject to the physics of how an unpowered box car transfers momentum to the Caboose it's towing while ignoring the question of where the initial actuality comes from.

    First, there is no caboose. Second, there is no initial acutality. I answer the question by noting it rests on a false assumption. Since there is no initial actuality, there is no need to answer where it came from.

    Round and Round even if you are civil it is impossible to have a rational consistent discussion with One Brow.

    Again, maybe its you. Many people have rational, consistent discussions with me.

    But he is just as confusing.

    I have no reason to think your experience with me is anything else.

    ReplyDelete
  144. >What term have I re-defined in this conversation? Can you back your accusation up?

    He also likes to pretend he is not doing it.

    For example:

    >where whenever 2 apples are put next to 2 apples, God creates a fifth apple, every time".

    vs

    >In a universe where God chooses to create a fifth apple whenever two apples are placed next to two apples, in what way is God dependent upon the placing of the apples?


    Nice slight of hand.

    Now his story is "I think causation is irrelevant to this argument."

    >Again, maybe its you. Many people have rational, consistent discussions with me.

    Are they sober or just humoring you?

    One Brow pick a topic. State your view clearly & be consistent for once in your life.

    Please?

    ReplyDelete
  145. Of course I never brought up God in my original example that off topic evolving tangent was introduced by you One Brow.

    I was arguing with djindra against his nominalist claim that 2+2=4 wasn't a necessary truth.

    Plus his view that 2+2=4 was based solely on empirical truth.

    At this point you can't seem to make up your mind as to wither you are defending hsi view or criticizing it.

    I think as per usual you are here just to bust chops.

    You are just like djindra sans the paranoid political wackyness.

    ReplyDelete
  146. >First, there is no caboose.

    Yeh the example given by Feser is can an infinite number of un-powered box cars pull a caboose? That is the question it asks.

    >Second, there is no initial acutality. I answer the question by noting it rests on a false assumption.

    No you changed the argument because you have no rational answer to it.

    >Since there is no initial actuality, there is no need to answer where it came from.

    I can for the life of me see how that conclusion can be reached giving the example of little motors on the box cars since they would also metaphorically represent the first cause and the initial actuality.

    You make no sense at all.

    ReplyDelete
  147. edit:that is I can't for the life of me etc

    ReplyDelete
  148. OneBrow:

    You have serially implied that my understanding of causation is mistaken; somehow out of place in a blog followed by Thomists; irrelevant to the scenario you gave (as well as the two later modifications you made).

    You have critiqued The Last Superstition on your own blog so I presume you've read it.

    You also claim to "always try to interpet people as if they make sense"

    I'd love to take you at your word but all of these things cannot be true.

    Dr. Feser himself, on page 274 of TLS in endnote 10, points out that people who suggest that if the physical world were set up in such a way that whenever we put two objects together with two other objects, a fifth magically appeared among them, have not described a case where 2+2=5. I quote: "For by their own account what they've described is not 2 and 2 equaling 5, but rather the act of placing 2 objects together with 2 other objects (which makes 4 objects total) suddenly and magically causing a new fifth object to appear ("X causes Y" doesn't mean "X equals Y.")"

    Now you can claim a superior understanding of what Thomists can, ought to, or usually, bring up on this blog with respect to caustion but I think you'll struggle to make your case that something I learned from Dr Feser doesn't fit.

    You can claim you knew about this endnote (after all you read TLS to critique it), but then your comments to me suggesting that I'm mistaking correlation and causation, and in a way explicitly rejected by Dr. Feser, would appear culpably misleading.

    And you can claim that you always try to interpret people as if they make sense, but your deliberately obtuse denial of dependency/causation in this answer to the position you've been defending would suggest that you only make that effort for your preferred stances (even as you warn us not to believe you endorse them).

    I'm quite happy to engage civilly with people who disagree with me; unless they do so dishonestly.

    ReplyDelete
  149. BenYachov said...
    >What term have I re-defined in this conversation? Can you back your accusation up?

    He also likes to pretend he is not doing it.


    No pretense needed.

    For example:

    >where whenever 2 apples are put next to 2 apples, God creates a fifth apple, every time".

    vs

    >In a universe where God chooses to create a fifth apple whenever two apples are placed next to two apples, in what way is God dependent upon the placing of the apples?


    Nice slight of hand.


    Did you personally, honestly think that in the first paragraph, I meant that God did not make a choice in creating the extra apple, but that God's hand was forced by human activity?

    Now his story is "I think causation is irrelevant to this argument."

    Now, and from the moment jack bodie first brought the subject up. I can quote post and timestamp, if necessary.

    Are they sober or just humoring you?

    How like you to cast out insults rather than discuss.

    One Brow pick a topic. State your view clearly & be consistent for once in your life.

    Please?


    For me, the topic is currently whether mathematics would look different in a universe where behavior was different, such as one where when two apples were placed next to two apples, there would be five apples. I'll be glad to stick to that topic, if you think you can do so.

    Of course I never brought up God in my original example that off topic evolving tangent was introduced by you One Brow.

    True enough.

    I was arguing with djindra against his nominalist claim that 2+2=4 wasn't a necessary truth.

    It's not a nominalist claim, or at least not exlusively a nominalist claim.

    Plus his view that 2+2=4 was based solely on empirical truth.

    I agree that confuses an emprical basis for developing a model with an empirical basis fof selecting a model.

    At this point you can't seem to make up your mind as to wither you are defending hsi view or criticizing it.

    Why does it have to be one or the other? I choose neither, and don't care to take sides. That leaves me free to discuss implications and justifications as I see fit.

    I think as per usual you are here just to bust chops.

    I take your opinion with all the respect you have earned.

    >First, there is no caboose.

    Yeh the example given by Feser is can an infinite number of un-powered box cars pull a caboose? That is the question it asks.


    There is no real-life item analogous to the caboose in Feser's metaphor. Therefore, asking what can pull the caboose is like asking what the temperature of dragonfire from the moon is.

    >Second, there is no initial acutality. I answer the question by noting it rests on a false assumption.

    No you changed the argument because you have no rational answer to it.


    "the question rests on a false assumption" is a rational answer. If you disagree, prove that the question does not rest on a false assumption.

    I can't for the life of me see how that conclusion can be reached giving the example of little motors on the box cars since they would also metaphorically represent the first cause and the initial actuality.

    There is no first cause and no initial actuality to be represented in the boxcars, just continuing causation and continuing actuality.

    ReplyDelete
  150. jack bodie said...
    I'd love to take you at your word but all of these things cannot be true.

    Well, they can not all be true when you adopt certain other axioms, but perhaps those other axioms are false?

    Dr. Feser himself, on page 274 of TLS in endnote 10, ... is not 2 and 2 equaling 5, but rather the act of placing 2 objects together with 2 other objects (which makes 4 objects total) suddenly and magically causing a new fifth object to appear ("X causes Y" doesn't mean "X equals Y.")"

    Then Dr. Feser, himself, is using the model of causation that Dr. Feser, himself, rails against when he speaks of Hume's mechanistic philosophy. I have no trouble accepting that Dr. feser, himself, applies such notions inconsistently.

    Of course, perhaps I'm wrong about Dr. Feser's criticism of Hume's notion of causation. Perhaps Dr. Feser really thinks that "Actions of type B always follow actions of type A, so A causes B" is a valid model of causation. Can you point out an explicit endorsement of that model of causation from TLS, or any other work of Dr. Feser?

    Now you can claim a superior understanding of what Thomists can, ought to, or usually, bring up on this blog with respect to caustion ...

    I can, but I won't, because I'm well aware I'm a novice.

    but I think you'll struggle to make your case that something I learned from Dr Feser doesn't fit.

    Can you make a case it does fit? Do you dispute my characterization of Dr. Feser's opinions on models of causation that are strictly correlative?

    You can claim you knew about this endnote (after all you read TLS to critique it), but then your comments to me suggesting that I'm mistaking correlation and causation, and in a way explicitly rejected by Dr. Feser, would appear culpably misleading.

    I'll gladly admit I'd forgotten all about the endnote. However, the existence of the endnote does not change teh contents of Dr. Feser's criticisms of Hume's model of causation.

    ... but your deliberately obtuse denial of dependency/causation in this answer to the position you've been defending ...

    You hacve still made no case that the mathematics created by natives of such a world would would reflect anything other than 2 + 2 = 5.

    ReplyDelete
  151. @OneBrow:

    "Not a direct answer to your question, but in such a universe 2 + 2 would still equal 4. Your scenario only inserts a Demiurge that mischievously pops apples just to confuse the poor humans. *He* would know the truth.

    The truth of apple creation ex nihilo, sure. The Demiurge can also use any mathematics he dreams up."

    I sincerely have no idea how you think that answers the objection. 2 + 2 = 4 in that universe, notwithstanding the puzzlement of their inhabitants. The Demiurge is just playing tricks (and because of that, we should name it OneBrow).

    "We have an empircal experience of gathering things into a collection. The model would be a formal representation of it. This would not be any different than having an experience of things falling to the ground, and then making a model called the Theory of Gravitation."

    You are just running around in a circle and I envision no way of breaking it, so why don't we put this issue to rest? And besides, you actually do not support the position you are defending so it is not like it really matters. I mean, it is really bizarre a self-avowed nominalist / formalist defending what you are defending, but hey, whatever rocks your boat.

    "By "useful", I meant that the various applications in the sciences required structures dependent on the axiom of infinity (such as calculus). I am not aware there were any such application requiring dependent choice a hundred years ago."

    Let me see... a hundred years ago puts us in the beginning of the twentieth century. I suppose it depends on what one means by "application", but my contention remains much the same, for in order for classical analysis to get off the ground, let alone to be applied, you already need countable dependent choice. Numerous examples can be given, but here goes one of the simplest: the equivalence of the Heine and the Cauchy definition of limit needs countable choice.

    "And the not-empirically verifiable is a useless insertion because the mathematical justifications of ZF(C) do not make appeals to the empirically verifiable reality.

    Fine. However, if the Axiom of Choice were empirically verifiable, do you think their would ever have been a question of it's inclusion in set theory axioms?"

    I do not know how to answer this question, not the least, because as I have repeated many times, I do know what it means for an axiom to be empirically verifiable (and your clarifications have clarified nothing for me). And I once again stress that Choice is accepted by the majority of practicing mathematicians, who if nothing else are eminently practical people and just want to go on with their business of understanding their subject of expertise. About their knowledge of foundations I insert a quote from Timothy Chow over at MO (this came about because of Voevodsky's programme and a possible inconsistency on PA):

    "For most mathematicians, "ZFC" is just an arbitrary trigraph that is cited when the need arises to specify a particular foundation for mathematics. I daresay many people who toss the trigraph around couldn't even state all the axioms of ZFC precisely. If we scale back to some other system that goes by some other trigraph, it won't take much retraining to learn the new trigraph. For most researchers, that will be the only impact on their day-to-day work."

    ReplyDelete
  152. grodrigues,

    Thank for your your comments.

    I find it interesting that in a discussion with a moderate realist, I find a position being taken that notions like set have no emprical basis. Is not any given collection of books an instantiation of the form "set" to you, in the say a pair of books would be an instantiation of "2"? Are you saying instantiations of this sort are non-empircal? Can you create a set of books, or not? Can you create a subset of that set, or not?

    grodrigues said...
    I sincerely have no idea how you think that answers the objection. 2 + 2 = 4 in that universe,

    However, why would people use that model, in preference to a model where 2 + 2 = 5?

    The Demiurge is just playing tricks (and because of that, we should name it OneBrow).

    Flattery will get you nowhere.

    You are just running around in a circle and I envision no way of breaking it, so why don't we put this issue to rest?

    Why? You don't think there is some real phenomenon that can be called a collection?

    And besides, you actually do not support the position you are defending so it is not like it really matters.

    I agree that it doesn't really matter, but I am learning a lot from the discussion, so I find value in continuing. If you do not, OK.

    I mean, it is really bizarre a self-avowed nominalist / formalist defending what you are defending, but hey, whatever rocks your boat.

    Formalist, yes. Nominialist, not as far as I can tell. I see a lot more relity in universal like "red" than in models like "2".

    Numerous examples can be given, but here goes one of the simplest: the equivalence of the Heine and the Cauchy definition of limit needs countable choice.

    I'm fairly sure you don't need that particular equivalence to construct a useful calculus. In a typical introductory class, we don't even discuss the Heine definition.

    Please understand, I do not disagree that countable choice or dependent choice are useful from the viewpoint of mathematicians. I'm just not sure their was any practical value (that is, value to applications) 100 years ago. Their value came from a theoretical/foundational aspect.

    About their knowledge of foundations I insert a quote from Timothy Chow

    That was a good quote, and largely conformed to my notions.

    ReplyDelete
  153. >Did you personally, honestly think that in the first paragraph, I meant that God did not make a choice in creating the extra apple, but that God's hand was forced by human activity?

    What does this even have to do with the discussion regarding 2+2=4 being a necessary truth 2+2=5 being impossible in any conceivable reality?

    At this point it is impossible to determine what the heck you are even talking about.

    Your delusion of being understood not withstanding.

    Beside I asked you a direct question as to what type of causality was lacking in your "god" scenario and you refused to answer. You shifted the burden of proof.

    If you are saying there is no causality of any kind in regards to the 2+2 then simply say so.

    Stop being a dick about it.

    Clearly God as he is Classically Known would be the ultimate cause of all things including the 2+2. The 2+2 can be the formal cause of God creating 5 in so much as God freely wills 5 if 2+2 happens. Of course Classic Theology teaches God must follow his own will by necessity. Naturally nothing can compel God to create 5 but he can freely will to create a 5th object if two are added to two.

    But in such a reality 2+2 would still not equal 5. God can't make 2+2=5. Can't God do anything? Of course but 2+2=5 doesn't describe anything. It describes nothing. Adding a whole new meaning to the phrase "There is nothing God cannot do.".

    Get a clue One Brow.

    ReplyDelete
  154. @OneBrow:

    "I find it interesting that in a discussion with a moderate realist, I find a position being taken that notions like set have no emprical basis."

    If by "empirical basis" you mean that mathematical concepts like "set" are borne out of our experience with the outside world, then sure they do, I never disputed that. What I dispute is the crank talk of Mr. Djindra that concepts that were not directly borne out of real-world experience are "meaningless" and other, equally loony ideas, of empirical verifiability, that are just the expression of an irrational scientism.

    "I sincerely have no idea how you think that answers the objection. 2 + 2 = 4 in that universe,

    However, why would people use that model, in preference to a model where 2 + 2 = 5?"

    Questions about the thought processes of would-be inhabitants of implausible universes are impossible to answer. I will say though, that just imagining that we could have an arithmetic where 2 + 2 = 5 does not make it an actual possibility. I have seen no argument to that effect. On the contrary, for various reasons that I have adumbrated here and there, notions like numbers and sets are "primitive" that even the imagining of 2 + 2 = 5 probably requires a prior conception of ordinary arithmetic. And an arithmetic where 2 + 2 = 5 violates even the most basic rules of arithmetic so from my POV it is useless mathematics, equally useless to build models of reality -- I have seen no argument to the contrary other than the power of free imagination at work.

    "You are just running around in a circle and I envision no way of breaking it, so why don't we put this issue to rest?

    Why? You don't think there is some real phenomenon that can be called a collection?"

    Collections or sets, in the mathematical sense, are beings of reason. But they are not beings in the same sense that a collection of apples is. You cannot point anywhere to reality and say here is a set; you can only point to particular instantiations of the concept. A collection of apples has extension, weight, etc. while a collection has neither. You can probe a collection of apples via the senses and perform experiments on it; on collections, the only organ that can probe it is reason itself. The ontology and phenomenology of beings of reason also depends on what you allow yourself to call reality. If you are a die-hard materialist, saying that "there is some real phenomenon" to a collection is something of a stretch, to put it mildly.

    ReplyDelete
  155. @OneBrow (continued):

    "I mean, it is really bizarre a self-avowed nominalist / formalist defending what you are defending, but hey, whatever rocks your boat.

    Formalist, yes. Nominialist, not as far as I can tell. I see a lot more relity in universal like "red" than in models like "2"."

    There is a clear lineage between nominalists and formalists. I once read someone asserting (I forgot the source) that most mathematicians are Platonists but when pressed to defend their position, they fall back to formalism. This accords to my own tiny biased observational sample and anecdotal evidence. I would add, somewhat tongue-in-cheek, that formalists are nominalists that have forgotten their philosophical source.

    "I'm fairly sure you don't need that particular equivalence to construct a useful calculus. In a typical introductory class, we don't even discuss the Heine definition."

    A typical introductory calculus class does not aim to teach the mathematical intricacies of real analysis. That they can afford such a luxury is because the basis are well covered and any blanks can be filled by just pointing to the appropriate textbooks.

    As for applications, my argument is that insofar as applied mathematics, e.g. calculus, needs to make mathematical manipulations, and these manipulations are justified only in a rigorous conception of the real line, applications will use choice. To give you a specific example I would need to know the status of applied mathematics (itself a fuzzy concept) at the beginning of the 20th century, which I don't. And a rigorous conception of the real line simply cannot avoid using some form of choice. Although I am no historian and cannot certify it with 100% certainty, I daresay that the founding fathers of rigorous calculus in the 19th century (Cauchy, Bolzano, Riemann, Weierstrass, etc.) used countable choice naively, as a self-evident principle. It is thrust upon you as soon as you deal with sequential extraction principles, (sequential) compactness or completeness. The awareness that the implicit needed to be made explicit only came later.

    ReplyDelete
  156. BenYachov said...
    What does this even have to do with the discussion regarding 2+2=4 being a necessary truth 2+2=5 being impossible in any conceivable reality?

    Nothing. Not a thing. It's completely irrelevant. I'm perfectly will to drop the subject.

    At this point it is impossible to determine what the heck you are even talking about.

    Your delusion of being understood not withstanding.


    Well, I never claimed that you understood me.

    Beside I asked you a direct question as to what type of causality was lacking in your "god" scenario and you refused to answer. You shifted the burden of proof.

    1) I don't think causality is relevant to the argument at all.
    2) I'm not claiming any sort of causality exists or does not exist.
    3) I expect those who claim causality (whoever irrlevant that claim may be) would justify their claim before asking me to counter arguemnts for a claim I don't think is relevant and don't take a position on, regardless.

    If you are saying there is no causality of any kind in regards to the 2+2 then simply say so..

    I'm not saying it exists. jack bodie says it exists.

    Clearly God as he is Classically Known would be the ultimate cause of all things including the 2+2. The 2+2 can be the formal cause of God creating 5 in so much as God freely wills 5 if 2+2 happens. Of course Classic Theology teaches God must follow his own will by necessity. Naturally nothing can compel God to create 5 but he can freely will to create a 5th object if two are added to two..

    Thank you for making an interesting contribution to this discussion.

    I'm confused, though. The descriptions of foraml cause that I have seen refer to things like what makes a ball a ball (sphericity, size, etc.). I can see, in this line of reasoning, how this can be inertpreted as a form of 2 being instantiatyed by a pair of apples, and a 5 being instantiated by 5 apples, but I don't see how that extends to 2 + 2 being a formal cause for a freely willing God. Could you expand on that, please?

    But in such a reality 2+2 would still not equal 5. God can't make 2+2=5. .

    The rules of addition, and what addition means, are set by inttelligent being, and can be changed by intelligent beings. Any putative God can redefine addition if They so desire.

    ReplyDelete
  157. grodrigues said...
    If by "empirical basis" you mean that mathematical concepts like "set" are borne out of our experience with the outside world, then sure they do,

    Good. We can work from there.

    What I dispute is the crank talk of Mr. Djindra ...

    I'm still not sure what djindra means when he says things like that. I don't think he is as extreme as you seem to think he is, but more experience with djindra may persuade me your evaluaiton is correct.

    Questions about the thought processes of would-be inhabitants of implausible universes are impossible to answer.

    OK.

    And an arithmetic where 2 + 2 = 5 violates even the most basic rules of arithmetic so from my POV it is useless mathematics, equally useless to build models of reality

    Well, not all mathematics exists to build models of reality. But even if it did, we don't know what shape reality will take. There were probably mathematicians of the prior millenium who felt that Lobachevskian geomety was useless to build models of reality, then it turned out to be a better model of reality than Euclidean geometry. I don't think we'll ever say that 2+2=5 is a better model for our reality, but why should we limit our free imagination on that basis?

    You cannot point anywhere to reality and say here is a set; you can only point to particular instantiations of the concept.

    This is what I expected. Then, given such an instantiation of a set of 5 distinct apples, you can instantiate the 32 distinct subsets of those apples, correct? Why do you feel that does not qualify as an empircal verification of the powerset axiom?

    If you are a die-hard materialist, saying that "there is some real phenomenon" to a collection is something of a stretch, to put it mildly.

    I'm not sure what you mean by materialist, as opposed to naturalist or physicalist. I'm currently agnostic on the postion that saying "there is a collection of books on the table" means more than just "there are books on the table", although my natural inclination is that the first sentence does say something the second does not.

    There is a clear lineage between nominalists and formalists.

    I'm a bastard, I guess.

    I would add, somewhat tongue-in-cheek, that formalists are nominalists that have forgotten their philosophical source.

    Somewhat more seriously, any pure nominalism would include formalism.

    A typical introductory calculus class does not aim to teach the mathematical intricacies of real analysis. That they can afford such a luxury is because the basis are well covered and any blanks can be filled by just pointing to the appropriate textbooks.

    I agree. Are you saying that without the equivalence of Cauchy and Heine limits, the calculus would not be useful?

    I do agree that countable (and probably dependent) choice is essential to getting a good understanding of the real line, and I see the full Axiom of Choice as reflective of reality. What I don't see is that rejecting countable choice would render the mathematics unusable. You might have not have limits for any arbitrary Cauchy sequence, but you can still find limits for the ones you need to find limits for.

    ReplyDelete
  158. @One Brow

    The second cause according to Aristotle is the formal cause. This cause pertains to the essence or "pattern" of something. It can mean it's physical appearance but not always. It doesn't always have to refer to physical objects.

    Thus the "pattern"/Form here is in terms of operation. God wills that He create a fifth object whenever two objects are added to two more objects. Thus the adding of 2+2 is the formal cause of God creating a fifth object as He has willed.

    ReplyDelete
  159. Ben Yachov,

    I have no objection to that. So in such a universe where 2 + 2 is a formal cause of a fifth object appearing, why would the residents use a mathematics where 2 + 2 = 4, instead of 2 + 2 = 5?

    ReplyDelete
  160. @OneBrow:

    "I'm still not sure what djindra means when he says things like that. I don't think he is as extreme as you seem to think he is, but more experience with djindra may persuade me your evaluaiton is correct."

    You are probably right on the second statement and about the first, I think not even Mr. Djindra knows exactly what he means. For example, in his latest response to Jack Bodie he says "my position is that our nature and our interaction with the universe predisposes us to think logically and mathematically." Now there's a vacuous platitude for you. Shrug shoulders.

    "Well, not all mathematics exists to build models of reality."

    You will find no disagreement here.

    "I don't think we'll ever say that 2+2=5 is a better model for our reality, but why should we limit our free imagination on that basis?"

    I am not sure I understand your question, but see my next paragraph as it probably will clarify my position.

    "Then, given such an instantiation of a set of 5 distinct apples, you can instantiate the 32 distinct subsets of those apples, correct? Why do you feel that does not qualify as an empircal verification of the powerset axiom?"

    I don't think the question is meaningful. Sets are beings of reason; it is a category mistake to think that the power set axiom can be falsified. Let me put things this way: suppose you gave me a falsifiability criterion for the power set axiom and also an experiment that indeed falsified it. What then? Well, my answer would be simply that you got things backwards and your model is botched and you should use other mathematical objects than sets, because it pertains to the very nature of sets that the power set axiom is true. That is why this talk about verifying mathematical axioms is either strictly circular or is nothing but an operation in renaming things. Mathematics, while certainly informed in many ways by real-word considerations, precedes the empirical sciences, just like the axiom that the universe is ordered and, at least in part, understandable by our minds, is a necessary pre-condition for doing empirical science in the first place.

    "I'm not sure what you mean by materialist, as opposed to naturalist or physicalist."

    Sorry about my sloppyness, English is not my primary language. Replace materialism with naturalism or physicalism.

    ReplyDelete
  161. @OneBrow (continued):

    "Are you saying that without the equivalence of Cauchy and Heine limits, the calculus would not be useful?

    I do agree that countable (and probably dependent) choice is essential to getting a good understanding of the real line, and I see the full Axiom of Choice as reflective of reality. What I don't see is that rejecting countable choice would render the mathematics unusable. You might have not have limits for any arbitrary Cauchy sequence, but you can still find limits for the ones you need to find limits for."

    For the first question, some things would certainly break down. What could be salvaged is still useful? I would have to go back to the books to answer that question. But remember that the equivalence of the two definitions of limit was just an example of the need for choice; many more could be given.

    More interesting is your last sentence in the second paragraph, as I believe it goes right to the heart of the question: your hope is naive and actually we do need completeness (that is, every Cauchy sequence has a limit). Why? Because the solution to many *practical* problems amounts to construct a sequence of approximations to the would-be solution of the problem and then prove that the sequence is Cauchy. By completeness, you have a solution without actually exhibiting it. Depending on how the sequence was constructed, you may even end up with an explicit algorithm to construct said solution. This is a recurrent pattern, for example, in numerical analysis (and is there anything more practical than that?). An even more obvious example: how do you know that the square root of 2 exists in the real line? At bottom, this is just an application of completeness.

    I should add that completeness, like all concepts that guarantee that certain objects exist, is genuinely deep and subtle. Compactness, even more. It is also why you will never see them mentioned in introductory calculus courses; students are unable to appreciate what is involved.

    ReplyDelete
  162. The rules of addition, and what addition means, are set by inttelligent being, and can be changed by intelligent beings. Any putative God can redefine addition if They so desire.

    Not a very deific act... as anyone can redefine addition in any manner that they wish. The problem is getting everyone else to agree with that definition. 2 + 2 = 4 (and not 5) is true only because it follows from applying the rules of addition (which are commonly agreed upon) to the set of Z (as per its commonly accepted definition).

    Changing that is akin to saying that one has changed an orange into a scone by saying that scone means "spherical fruit with a rough skin that comes from the C sinensis tree.

    ReplyDelete
  163. grodrigues,

    "I think not even Mr. Djindra knows exactly what he means. For example, in his latest response to Jack Bodie he says 'my position is that our nature and our interaction with the universe predisposes us to think logically and mathematically.' Now there's a vacuous platitude for you."

    Your feigned ignorance is amusing. I've asked repeatedly, Where do mathematical postulates, axioms and/or rules come from? The alternatives to my platitude are these platitudes: a) They're true because mathematicians agree they're true (convention); or b) They're handed down from the gods (revelation). Those alternatives are just not good enough. So speak of loony-ness when you come up with a straight answer to my question.

    ReplyDelete
  164. Stone Top,

    I largely agree with what you posted. Reading into your tone (always dagerous in text interchange), you may have meant that changing the conventions was inappropriate, and I don't see a reason to consider them so. But that difference would not affect our agreement on what you did post.

    ReplyDelete
  165. grodrigues said...
    I don't think the question is meaningful. Sets are beings of reason; it is a category mistake to think that the power set axiom can be falsified.

    I agree. Perhaps verificaton is the wrong word to use when there can be no falsification. Maybe it is something closer to empircal inspiration. Axioms like the powerset axiom would be empirically inspired, while those like the axiom of infinity would not be empirically inspired.

    Mathematics ... is a necessary pre-condition for doing empirical science in the first place.

    I agree that mathematics does not depend on empirical studies, but empirical studies are not required to use mathematics (I have read a few that do not seem to use any). I largely agree with what you wrote, I thought it went just a little too far down that path.

    Sorry about my sloppyness, English is not my primary language. Replace materialism with naturalism or physicalism.

    It's not your sloppiness. Were I to rate your English, it would be excellent.

    There are three basic categories (with many variations, no doubt). People who think only physical objects and the physical effects they generate are real, people who think physical objects the relationships and patterns that they form are real, and an overall category that includes both. I have seen any of those three called physicalist, naturalist, or materialist.

    But remember that the equivalence of the two definitions of limit was just an example of the need for choice; many more could be given.

    Of course.

    By completeness, you have a solution without actually exhibiting it. Depending on how the sequence was constructed, you may even end up with an explicit algorithm to construct said solution.

    However, such an algorithm rarely arrives at an exact solution in a finite number of steps. Nor, for pratical purposes, does it need to. You take that precess to as many decimal places as you need.

    I agree that notions like completeness and compactness are important for understanding these things. However, perhaps I see the notion of "practical application" differently.

    ReplyDelete
  166. djindra said...
    I've asked repeatedly, Where do mathematical postulates, axioms and/or rules come from? The alternatives to my platitude are these platitudes: a) They're true because mathematicians agree they're true (convention); or b) They're handed down from the gods (revelation). Those alternatives are just not good enough.

    I disagree. Convention is more than good enough. Convention is inspired by empirical notions and by necessity, but convention is all that is possible, and suffices.

    ReplyDelete
  167. Here’s my two cents.

    Mathematics and logic are rules that are abstracted from our experience of the empirical world, and become incorrigible due to the centrality that they play in our conceptual frameworks. They are due to a combination of real patterns in nature and our agreed-upon conventions.

    Mathematicians are able to violate their own rules by being creative and imaginative. Making two parallel lines meet on a curved surface is one example. Imaginary numbers is another one. These examples show that what was once considered impossible and contradictory became not only possible, but essential to understanding the natural world through relativity and quantum mechanics. So, the rules of mathematics are not utterly rigid and incorrigible after all.

    ReplyDelete
  168. And one more thing.

    There are two senses to the idea of “truth in mathematics”.

    One is truth-relative-to-mathematics (i.e. mathematical truth1), and the other is truth-relative-to-the-world (i.e. mathematical truth2).

    The former is in the same category as truths relative to a fictional universe, such as Harry Potter, in which given the initial assumptions, the narrative fits in a coherent fashion, but does not necessarily correspond to reality. This would involve mathematical theorems that have not yet been shown to be relevant to explaining the natural world.

    The latter is the more important category, because it involves mathematical truths1 that have also been shown to be operative in the empirical world, and thus are mathematical truths2.

    ReplyDelete
  169. @OneBrow:

    "I don't think the question is meaningful. Sets are beings of reason; it is a category mistake to think that the power set axiom can be falsified.

    I agree. Perhaps verificaton is the wrong word to use when there can be no falsification. Maybe it is something closer to empircal inspiration. Axioms like the powerset axiom would be empirically inspired, while those like the axiom of infinity would not be empirically inspired."

    Since I have already agreed that much mathematics is inspired (to use your own words) by real-world considerations I guess we can close this matter as settled. Although I am not sure where exactly to put the boundary of what counts as "empirically inspired".

    "By completeness, you have a solution without actually exhibiting it. Depending on how the sequence was constructed, you may even end up with an explicit algorithm to construct said solution.

    However, such an algorithm rarely arrives at an exact solution in a finite number of steps. Nor, for pratical purposes, does it need to. You take that precess to as many decimal places as you need."

    Ack, sorry, that last sentence came out chopped. What I wanted to write was: "Depending on how the sequence was constructed, you may even end up with an explicit algorithm to construct *an approximation to any desired accuracy to* said solution." This is all that is needed in practice. But to pull off this strategy, completeness is essential since it is what guarantees that the sequence of approximations *does* converge to the solution. The proof of Cauchyness amounts to an estimation of the error term, so if the sequence was constructed in a reasonably explicit manner, you can usually turn the proof into an explicit algorithm that on a finite number of steps constructs an approximation to the solution with the desired accuracy. Of course the algorithm may be still be completely ineffective, but this is a whole different problem.

    ReplyDelete
  170. @Djindra:

    "Your feigned ignorance is amusing. I've asked repeatedly, Where do mathematical postulates, axioms and/or rules come from? The alternatives to my platitude are these platitudes: a) They're true because mathematicians agree they're true (convention); or b) They're handed down from the gods (revelation). Those alternatives are just not good enough."

    b) is silly and you know it so why bring it up? I mean, do you know of any religious tradition that included as part of its divine revelation any tidbit of mathematical knowledge? And no, the Pythagorean school does not count as "religious tradition". a) is true in part, if "convention" is properly understood. Contrary to what your question implies, those are not the only two options. That you think they are, only betrays the fact that you are speaking out of ignorance, and by ignorance I mean not just ignorance of mathematics itself, but also of its nature and history.

    To add to what I said, the justification of mathematical axioms can be roughly divided in three layers:

    1. An appeal to intuitive conceptions and per force, to a shared consensus.

    note: though there is no universal consensus on the foundations of mathematics (since those are the axioms most relevant for this discussion), mathematicians of widely different persuasions can still communicate and understand each other's work. This is important to realize, so as not to fall in the silly relativism talk.

    2. Specifically mathematical justifications. This is the process where mathematics turns on itself and works itself out, so to speak. So in the case of ZF(C) you get appeals to intended models such as the Von-Neumann hierarchy, reflection principles and all the vast body of work of modern set theory at unlocking new set-theoretical axioms to decide outstanding problems such as the continuum hypothesis. Scott has a paper of 1974, "Axiomatizing Set Theory", with a very clear discussion of how to motivate and justify the set-theoretical axioms.

    3. Appeals to your favored philosophical school.

    ReplyDelete
  171. >why would the residents use a mathematics where 2 + 2 = 4, instead of 2 + 2 = 5?

    Now you are back to conflating "cause" with "equals". I see no reason why they would confuse the two since they would intellectually be able to distinguish between the four objects they brought together vs the gratuitous one your version of "god" created after adding 2+2.

    ReplyDelete
  172. Reading into your tone (always dagerous in text interchange), you may have meant that changing the conventions was inappropriate, and I don't see a reason to consider them so.

    My point was that the debate over 2+2=4 or 2+2=5 is rather meaningless without considering the larger rule set involved. 2+2=4 is "true" in the mathematical sense because the rules for the set of Z say that it is true... the only way to change that is to change the underlying rules.

    ReplyDelete
  173. grodrigues,

    We can dismiss "Appeals to your favored philosophical school" as a non-answer. "Specifically mathematical justifications" fall under the category of circular reasoning and again this is not serious. (We could easily ask where ZFC gets notions of sets and equality in the first place.) Appeals to intuition might fall into the category of revelation (you misunderstand if you think I meant "from the gods" as more than metaphor), or deep empirical truths we've learned since birth (or evolved).

    Also, why do you think we have ZFC in the first place? What was the motivation for looking for a better foundation?

    ReplyDelete
  174. @Djindra:

    "We can dismiss "Appeals to your favored philosophical school" as a non-answer."

    Speak for yourself. If you want to dismiss philosophy as irrelevant to this particular question, then go ahead, but please, do not drag the rest of us with you.

    ""Specifically mathematical justifications" fall under the category of circular reasoning and again this is not serious."

    Tell that to the mathematical community.

    "Appeals to intuition might fall into the category of revelation (you misunderstand if you think I meant "from the gods" as more than metaphor), or deep empirical truths we've learned since birth (or evolved)."

    Here you are not disputing that we do have intuitions as I stated, you are just listing the only two possible sources (in your view) for them. By the way, this question is also eminently philosophical.

    "Also, why do you think we have ZFC in the first place? What was the motivation for looking for a better foundation?"

    If you do not know the answer, get yourself a book on the history of mathematics. If you do know the answer, then what exactly do you want to know from me?

    ReplyDelete
  175. grodrigues,

    Thank you for the discussion. I learned a coupe of new ways to look at things..

    grodrigues said...
    ... This is all that is needed in practice. But to pull off this strategy, completeness is essential since it is what guarantees that the sequence of approximations *does* converge to the solution.

    I agree with you in principle, because I am also a mathematician by training. However, the pratical people I know wouldn't care about the guarantee, just the ability to get a sufficient approximation. Hence, at least this particular result would not be needed, for them.

    ReplyDelete
  176. BenYachov said...
    >why would the residents use a mathematics where 2 + 2 = 4, instead of 2 + 2 = 5?

    Now you are back to conflating "cause" with "equals".

    Not at all. Cause is irrelevant. Why would they not use a mathematics that describe their reality?

    I see no reason why they would confuse the two since they would intellectually be able to distinguish between the four objects they brought together vs the gratuitous one your version of "god" created after adding 2+2.

    Which does not answer the question of why they would not use a mathematics thatdescribes their reality, one where 2 + 2 = 5?

    ReplyDelete
  177. StoneTop,

    Point acknowledged. I agree.

    dguller,

    A good contribution, and pleasant to read.

    ReplyDelete
  178. One Brow

    You are a hopeless sophist.

    Hopeless.

    A universe with the property of 1+(2+2)=5 is not and is never and can never be a universe of 2+2=5.

    That's the point.

    No such reality may exist at all.

    You are hopeless and you have nothing to teach me or anybody.

    ReplyDelete
  179. The "rules" are very simple. 2+2=4 always. If "god" or the natural laws of some fictional alternate universe add another object that is 1+(2+2)=5. But there can be no universe or reality where 2+2=5.

    Too deny this is to deny logic. No and's if's or but's.

    Nominalism is clearly an inferior philosophy that leads to irrationality.

    It's not hard people.

    ReplyDelete
  180. BenYachov said...
    A universe with the property of 1+(2+2)=5 is not and is never and can never be a universe of 2+2=5.

    Mathematics is not a property of universes to begin with. However, if it were a property of a universe, it could change when the universe changed.

    You are hopeless and you have nothing to teach me or anybody.

    A discussion with you sans some pointless comment on my worth, abilities, or knowledge would seem incomplete.

    The "rules" are very simple. 2+2=4 always.

    We make the rules for our convenience, we can change the rules.

    But there can be no universe or reality where 2+2=5.

    That depends on the rules we use to define "+".

    Too deny this is to deny logic. No and's if's or but's.

    I am aware this is your opinion, but I find it lacking authority or persuasiveness.

    Nominalism is clearly an inferior philosophy that leads to irrationality.

    Thank you for bringing up a completely irrelevant point.

    ReplyDelete
  181. OneBrow:

    You're just begging the question and asserting a whole crock beliefs without arguing for them.

    If I've understood the current evolution of the world you've asked us to imagine we are now at a Universe where 2 things put with 2 other things gives 4 things with an extra thing created to make a total of 5 things - your question, then, is why would the inhabitants of this imagined world not 'create' a mathematics where 2+2=5.

    Well, several reasons: no doubt they'd have Sesame Street for the juveniles of this Alter-world and everyone would be able to look at your five apples and sing, "One of these things is not like the others." This would encourage them to take account of that fact in any systems they actually did create. Yes, I am just asserting this but you seem ok with the move.

    More seriously when you ask "why they would not use a mathematics thatdescribes their reality, one where 2 + 2 = 5?" you've stolen a few bases in asserting that it does reflect their reality. In fact you've contradicted your own description of their reality which is that 2+2 (things) = 4 (things) plus one other thing. You seem to be insisting that we further imagine them to be adept in math but prevented from practising philosophy or natural science, or even satisfying basic curiosity about the world around them. Well, it's your imaginary world so I concede, yes - they could be ignorant and wrong, thereby choosing to believe that 2+2 could equal 5.

    But they could also be as smart as, say, grodrigues and discover that their reality is, in fact, better reflected by taking account of the Demiurge that chooses to create a fifth apple every time they put 2 apples with 2 others. How exactly does their level of intelligence make the case that 2+2=5?

    ReplyDelete
  182. At this point Jack, One Brow is no better than djindra sans the weird left-wing conspiracy theories and paranoia.

    ReplyDelete
  183. Look, all other nonsense aside I think there's a distinction to be made between a useful heuristic for counting (as per your alt-reality inhabitants) and discovering the logical truths of mathematics.

    @BenYachov: yup, and all for a position I'm not even sure he endorses! Can you imagine if we were talking about something he had deeply held beliefs about?

    ReplyDelete
  184. @OneBrow:

    "But to pull off this strategy, completeness is essential since it is what guarantees that the sequence of approximations *does* converge to the solution.

    I agree with you in principle, because I am also a mathematician by training. However, the pratical people I know wouldn't care about the guarantee, just the ability to get a sufficient approximation. Hence, at least this particular result would not be needed, for them."

    Yes, but "sufficient approximation" to what? If it does not approximate the solution, it is not very "practical", is it? And how do you know it approximates the solution? This is where completeness enters as it establishes two things: it proves indeed that there *is* a solution and it gives you a sequence approximating the solution. Depending on the details of the construction, an algorithm can be extracted.

    ReplyDelete
  185. @dguller:

    "The former is in the same category as truths relative to a fictional universe, such as Harry Potter, in which given the initial assumptions, the narrative fits in a coherent fashion, but does not necessarily correspond to reality. This would involve mathematical theorems that have not yet been shown to be relevant to explaining the natural world."

    There are several problems with this view. First, a Platonist would disagree with you as he ascribes a definite reality to the mathematical objects. So are you arguing that Platonism is false? Or even more extremely, as your simile with fiction seems to imply, are you arguing that Formalism is true? Now, I am not defending any position (although for the record, I am theist, Christian and, surprise surprise, favor an AT moderate realism), I am just pointing out that there is an implied philosophical position in this paragraph and an argument needs to be made.

    If you restrict "reality" to "natural world" as you do in the second sentence, then your notion of truth is either useless or suffers from other problems -- see my next paragraph.

    "The latter is the more important category, because it involves mathematical truths1 that have also been shown to be operative in the empirical world, and thus are mathematical truths2."

    First there is a value judgment in the first sentence that not everyone agrees with, and is a fine example of what the great literary critic Northrop Frye called the Archimedes fallacy. But more importantly, what do you mean by mathematical truths operative in the empirical world? That they are used in our mathematical models of the natural world? If yes, then there are problems to overcome. For example, Newtonian classical mechanics posits that space is a 3d Euclidean space. Does this mean that Euclidean geometry is true? Well, we know that Newtonian theory is wrong so does that mean that "operative truths" are historically contingent? That is a very bizarre notion of truth.

    One more example. It is well known that classical mechanics can be reformulated as roughly, a subdiscipline of symplectic geometry -- see for example the gem that is V. Arnold's "Mathematical methods of classical mechanics". Does that mean that symplectic geometry is "operatively true"? And if no one had made the reformulation, would that mean that symplectic geometry would not be "operatively true"? That is a bizarre notion of truth that depends on the vagaries of mathematical history and progress.

    ReplyDelete
  186. jack bodie said...
    Well, several reasons: no doubt they'd have Sesame Street for the juveniles of this Alter-world and everyone would be able to look at your five apples and sing, "One of these things is not like the others."

    Why do yuou think this would lead to them creating a mathematics that did not reflect their reality?

    This would encourage them to take account of that fact in any systems they actually did create.

    In what way?

    Yes, I am just asserting this but you seem ok with the move.

    We're all just speculating.

    More seriously when you ask "why they would not use a mathematics thatdescribes their reality, one where 2 + 2 = 5?" you've stolen a few bases in asserting that it does reflect their reality. In fact you've contradicted your own description of their reality which is that 2+2 (things) = 4 (things) plus one other thing.

    So, 4 things plus 1 other thing is not 5 things? That seems to be splitting a very fine hair.

    You seem to be insisting that we further imagine them to be adept in math but prevented from practising philosophy or natural science,

    I don't recall on insisting any such thing.

    or even satisfying basic curiosity about the world around them. Well, it's your imaginary world so I concede, yes - they could be ignorant and wrong, thereby choosing to believe that 2+2 could equal 5.

    Why is that wrong?

    But they could also be as smart as, say, grodrigues and discover that their reality is, in fact, better reflected by taking account of the Demiurge that chooses to create a fifth apple every time they put 2 apples with 2 others.

    Why does that change their method of addition?

    How exactly does their level of intelligence make the case that 2+2=5?

    As far as I know, it doesn't.

    Out of all those paragraphs, the only time you addressed the main point is to say it was "wrong". You offered no reason for it to be wrong, no justification, except that it was so wrong only stupid people would believe it. Do you have a justification for that position?

    Look, all other nonsense aside I think there's a distinction to be made between a useful heuristic for counting (as per your alt-reality inhabitants) and discovering the logical truths of mathematics.

    Yes, that's part of the distinction. A number that exists for counting (and possibly subtraction) in this alternate reality does not exist for at least one sort of addition.

    yup, and all for a position I'm not even sure he endorses!

    I'm definately a formalist. I firmly believe that math (and logic, for that matter) are human constructions designed to help us look at our world.

    ReplyDelete
  187. grodrigues said...
    Yes, but "sufficient approximation" to what?

    To the number of inches on the side of your garden, for example. It will typicaly not matter is your garden is 2 square feet, 1.99999999998 square feet, or 2.00000000001 square feet, so once you know the length of the a side to within a certain precision, it does not matter, on a pratical level, where there exists a true square root of two or not.

    If it does not approximate the solution, it is not very "practical", is it?

    Again, this is the way I naturally think. However, for pratical applicaitons, it seems to be unnecessary.

    ReplyDelete
  188. Grodrigues:

    >> First, a Platonist would disagree with you as he ascribes a definite reality to the mathematical objects. So are you arguing that Platonism is false?

    Yup.

    >> Or even more extremely, as your simile with fiction seems to imply, are you arguing that Formalism is true?

    Not necessarily. All that I am saying is that if you start with a series of assumptions and add rules of inferences from those assumptions, then you will develop a set of conclusions. The conclusions will be true within the context of that system, of course, but that does not mean that they have a real existence independent of the system itself. That is why I brought up fiction, which would meet this criterion, but clearly does not exist in any meaningful sense.

    >> If you restrict "reality" to "natural world" as you do in the second sentence, then your notion of truth is either useless or suffers from other problems -- see my next paragraph.

    I think that if a proposition makes no impact upon the empirical world in any way, then it is hard to say that it refers to any genuinely real state of affairs.

    >> But more importantly, what do you mean by mathematical truths operative in the empirical world? That they are used in our mathematical models of the natural world? If yes, then there are problems to overcome. For example, Newtonian classical mechanics posits that space is a 3d Euclidean space. Does this mean that Euclidean geometry is true? Well, we know that Newtonian theory is wrong so does that mean that "operative truths" are historically contingent? That is a very bizarre notion of truth.

    Perhaps I should clarify.

    What I mean is that if a mathematical theorem can be shown to reliably model empirical reality, then we can probably say that that theorem is true in the sense of corresponding to a state of affairs in reality. If there is a mathematical theorem that can be shown to be falsified by the facts, then the theorem can be called false, even if it is true within mathematics.

    With regards to your example of Newtonian mechanics and Euclidean space, I would say that they are both good approximations in various macroscopic contexts, and since they can both be seen to be special cases of more general theories, then their truth can be preserved, but only in the context of specific scenarios. Newtonian mechanics is not wrong, but only not universal. It works perfectly well within its domain of macroscopic objects moving in Euclidean space. Similarly, DNA is the medium of genetic information being transmitted within living organisms on earth. However, it does not follow that DNA is how genetic information is transmitted in all living organisms in the universe. Does that mean that DNA is false? Of course not, but only not a universal phenomenon.

    I hope that helps.

    ReplyDelete
  189. OneBrow:

    Out of all those paragraphs, the only time you addressed the main point is to say it was "wrong". You offered no reason for it to be wrong, no justification, except that it was so wrong only stupid people would believe it. Do you have a justification for that position?

    You'll have to help me out: what is the main point? Every paragraph but the one that restates your position (and a single sentence where I opine on the root of your error) offers you an answer to your question "why would they not use a mathematics that describes their reality, one where 2+2=5?"

    Each of the following three paragraphs points out that your imaginary world is not, despite the convoluted scenario you've engineered, one where 2+2=5; and so the justification for my alternatives, or saying your inhabitants are wrong, is implied in your question - ie, the inhabitants are trying to describe their reality. I assume you meant describe their reality correctly.

    Is there some paragraph-to-content quotient that I've violated?

    ReplyDelete
  190. jack bodie said...
    You'll have to help me out: what is the main point?

    Mathematics is a tool created by humans to be used to help explain the world, with no inherent truths of its own.

    Every paragraph but the one that restates your position (and a single sentence where I opine on the root of your error) offers you an answer to your question "why would they not use a mathematics that describes their reality, one where 2+2=5?"

    Funny, all I found were objections to using the notion of addition we have constructed in our reality as being applicable to the inhabitants of the other reality, and some ideas on the notion of counting (which is a distinct notion from addition). Not once did you seem to discuss the actual reasons they would have mathematics that resembled ours.

    Each of the following three paragraphs points out that your imaginary world is not, despite the convoluted scenario you've engineered, one where 2+2=5;

    You seem to be talking about the formal truth of statement after the notion of addition has been constructed, not the construction of addition itself. That would be non-responsive to my argument. "+" has the meaning we attach to it, no more and no less.

    and so the justification for my alternatives, or saying your inhabitants are wrong, is implied in your question - ie, the inhabitants are trying to describe their reality. I assume you meant describe their reality correctly.

    Yes, correctly. Why is their description, using the "+" they have created for addition in their reality, of 2+2=5, wrong?

    Is there some paragraph-to-content quotient that I've violated?

    None of which I'm aware. I'll email the paragraph-to-content police and ask them.

    ReplyDelete
  191. @dguller:

    "What I mean is that if a mathematical theorem can be shown to reliably model empirical reality, then we can probably say that that theorem is true in the sense of corresponding to a state of affairs in reality. If there is a mathematical theorem that can be shown to be falsified by the facts, then the theorem can be called false, even if it is true within mathematics."

    The talk about falsifying "mathematical theorems" is circular or meaningless. I have already hashed this out with OneBrow so I am not going to repeat the arguments.

    "I think that if a proposition makes no impact upon the empirical world in any way, then it is hard to say that it refers to any genuinely real state of affairs."

    Since you define "genuinely real" as having an impact upon the empirical world, you are just restating your belief that Platonism is false. Actually, you are stating more. Are you saying that there is no reality apart from what is described by the empirical sciences?

    "With regards to your example of Newtonian mechanics and Euclidean space, I would say that they are both good approximations in various macroscopic contexts, and since they can both be seen to be special cases of more general theories, then their truth can be preserved, but only in the context of specific scenarios. Newtonian mechanics is not wrong, but only not universal. It works perfectly well within its domain of macroscopic objects moving in Euclidean space."

    The view that Newtonian mechanics is not universally correct is problematic as for example, it contradicts the basic principle of relativity that signals cannot travel faster than light.

    So if I understand you, because Euclidean geometry is used in a physical theory that despite contradicting some fundamental principles, is a good approximation to some slice of reality, it can be certified as "operatively true"? If yes, then I do not see how the notion of "operatively true" is anything more than useless, because almost every branch of mathematics has a connection with some more or less practical activity. Hardy could extol the virtues of Number Theory as the purest of pure mathematics, with none of the impure and grubby stains of the real world. With the advent of cryptography not even Number Theory is safe from being muddied by reality.

    And of course, this still means that being "operatively true" is contingent upon the history and progress of science. Maybe you should abstain from using such big words as "True" and "False"?

    "Similarly, DNA is the medium of genetic information being transmitted within living organisms on earth. However, it does not follow that DNA is how genetic information is transmitted in all living organisms in the universe. Does that mean that DNA is false? Of course not, but only not a universal phenomenon."

    What does it mean to say that DNA, which is a concrete object of our physical reality, is true or false?

    ReplyDelete
  192. Grodrigues:

    >> The talk about falsifying "mathematical theorems" is circular or meaningless. I have already hashed this out with OneBrow so I am not going to repeat the arguments.

    I only meant that it is possible to show that mathematical assumptions can be false, even those that were thought to be indubitable, such as that parallel lines cannot meet, or that there is no square root to -1. The rejection of both of these assumptions was necessary to develop forms of mathematics that have been essential to both relativity and quantum mechanics. This is important, because there may be assumptions that we currently hold to be indubitable and impossible to doubt, but which can be doubted and rejected.

    >> Since you define "genuinely real" as having an impact upon the empirical world, you are just restating your belief that Platonism is false.

    Even Platonism required the Forms to make an impact upon the empirical world. That was the point of the analogy of the cave, I believe. In other words, the beings outside the cave were the sources of the shadows upon the walls within the cave, but the fact was that there were supposed to be the shadows in the first place.

    >> Actually, you are stating more. Are you saying that there is no reality apart from what is described by the empirical sciences?

    I am saying that even if there was a reality beyond the empirical world, unless it made some kind of impact upon the empirical world, then we couldn’t know it at all.

    >> The view that Newtonian mechanics is not universally correct is problematic as for example, it contradicts the basic principle of relativity that signals cannot travel faster than light.

    Newtonian mechanics only breaks down at very tiny masses and at very high speeds, especially those close to the speed of light. In most other contexts, it works just fine, and even the equations of relativity become Newtonian equations at the right sizes and speeds. Since it works for those sizes and speeds, it can be said to be true for those sizes and speeds. It does not follow that it is true for all sizes and speeds. That is what relativity is, for the most part.

    >> So if I understand you, because Euclidean geometry is used in a physical theory that despite contradicting some fundamental principles, is a good approximation to some slice of reality, it can be certified as "operatively true"? If yes, then I do not see how the notion of "operatively true" is anything more than useless, because almost every branch of mathematics has a connection with some more or less practical activity.

    That is great. For those branches of mathematics that have a measurable connection to empirical reality and practical activity, I would say that they have more right to be considered true than those branches of mathematics that have no such connection. However, my point remains that it is this connection that is the source of mathematical truth, and that without it, mathematics is just spinning its wheels in the air without traction.

    >> And of course, this still means that being "operatively true" is contingent upon the history and progress of science. Maybe you should abstain from using such big words as "True" and "False"?

    It’s a good thing that I never used True or False, but only their more humble cousins, true and false.

    >> What does it mean to say that DNA, which is a concrete object of our physical reality, is true or false?

    Thanks for pointing out some imprecision on my part. I meant to say the theory of DNA as the mechanism of genetic heritability in biological organisms. This theory is clearly true on our planet, given its evolutionary history, but it does not follow that it is necessarily true on all planets with living entities upon them. My point was that this truth, if it is true, does not mean that we can say that the theory of DNA being the mechanism of genetic heritability is false.

    Hope this helps.

    ReplyDelete
  193. First, a Platonist would disagree with you as he ascribes a definite reality to the mathematical objects. So are you arguing that Platonism is false?

    A quite reasonable position to take... as it falls to the Platonist to provide evidence for her beliefs.

    ReplyDelete
  194. @dguller:

    There is a lot of sloppy talk in your post, but I will concentrate on the following paragraph:

    "That is great. For those branches of mathematics that have a measurable connection to empirical reality and practical activity, I would say that they have more right to be considered true than those branches of mathematics that have no such connection. However, my point remains that it is this connection that is the source of mathematical truth, and that without it, mathematics is just spinning its wheels in the air without traction."

    You insist on using adjectives like "true" and "false" (no capitalization this time) in an equivocal manner. That some branch of mathematics has some practical application does not make it more true in any reasonable sense of the word. In particular, your statement "without it, mathematics is just spinning its wheels in the air without traction" is one of those sloven illiteracies that only someone who does not know or understand mathematics can produce.

    All the supposedly objective talk about a practical "connection that is the source of mathematical truth" runs counter to all the facts of mathematical experience and is nothing but a subjective value judgment, a thin veneer over an empiricist prejudice. It commits the fallacy of putting one's favorite study into a causal relationship with whatever interests him less, all the while giving one the illusion of explaining the subject. Mathematics is an independent and autonomous discipline, with its own conceptual framework and its own standards of value. It is not the handmaiden of whatever discipline you think more important.

    I emphasize that this is not my opinion, it is a *fact*. You disagree with the current state of affairs? Feel free to complain; I am sure the mathematical community will be thrilled to hear your opinions.

    ReplyDelete
  195. grodrigues,

    I emphasize that this is not my opinion, it is a *fact*.

    LOL! Wouldn't it be nice if emphasis is all it took to make opinion a fact.

    ReplyDelete
  196. djindra said...
    LOL! Wouldn't it be nice if emphasis is all it took to make opinion a fact.

    I have tried to understand adn learn more about your empirical point of view, and tried to give you the benefit of the doubt. I can even see where some versions of a strictly empirical notion could be acceptable/useful.

    However, I agree that "Mathematics is an independent and autonomous discipline, with its own conceptual framework and its own standards of value. It is not the handmaiden of whatever discipline you think more important." is a fact. You can offer personal judgements about the values of one result or another, but that won't change the autonomy of the pratice of mathematics.

    ReplyDelete
  197. grodrigues,

    "That some branch of mathematics has some practical application does not make it more true in any reasonable sense of the word."

    It not only makes it true but is critical in confirming it can be true.

    "'without it, mathematics is just spinning its wheels in the air without traction' is one of those sloven illiteracies that only someone who does not know or understand mathematics can produce."

    Your objection is a sloven ignorance that only someone who does not understand truth, meaning and/or presuppositions can produce.

    "All the supposedly objective talk about a practical 'connection that is the source of mathematical truth' runs counter to all the facts of mathematical experience and is nothing but a subjective value judgment, a thin veneer over an empiricist prejudice.

    Yet you are the subjectivist. You believe mathematicians can dream up internally consistent, subjective procedures without any basis in anything else whatsoever yet subsequent results are guaranteed to be true. That's kind of what astrologers and psychoanalysts do. I think mathematics has a more solid foundation than that.

    ReplyDelete
  198. grodrigues,

    Me:"We can dismiss 'Appeals to your favored philosophical school' as a non-answer."

    You: "If you want to dismiss philosophy as irrelevant to this particular question, then go ahead, but please, do not drag the rest of us with you."

    If it's just a matter of appealing to a favored philosophical school, I choose empiricism in this matter. Clearly you don't accept that. So your "answer" says nothing.

    Here you are not disputing that we do have intuitions as I stated, you are just listing the only two possible sources (in your view) for them. By the way, this question is also eminently philosophical.

    Of course it is. You misunderstand if you think I don't acknowledge that.

    "If you do not know the answer, get yourself a book on the history of mathematics. If you do know the answer, then what exactly do you want to know from me?"

    You are evading the fact that ZFC was created out of the debate over mathematical foundations that arose in the early 20th century. If math had the solid foundation you suggest, we would have never seen that crisis.

    ReplyDelete