## Friday, June 28, 2019

### Frege on what mathematics isn’t

Mathematics is an iceberg on which the Titanic of modern empiricism founders.  It is good now and then to remind ourselves why, and Gottlob Frege’s famous critique of John Stuart Mill in The Foundations of Arithmetic is a useful starting point.  Whether Frege is entirely fair to Mill is a matter of debate.  Still, the fallacies he attributes to Mill are often committed by others.  For example, occasionally a student will suggest that the proposition that 2 + 2 = 4 is really just a generalization from our experience of finding four things present after we put one pair next to another – and that if somehow a fifth thing regularly appeared whenever we did so, then 2 and 2 would make 5.

A comparable thesis from Mill that is criticized by Frege is the claim that the proposition that 1 + 2 = 3 is made true by our experience of finding that a group of objects that looks like OOO can be separated into two groups that look like O and OO.  Frege jokes that in that case it is a good thing that all the objects in the world aren’t nailed down, otherwise we wouldn’t be able to separate them and thus it wouldn’t be true that 1 + 2 = 3.  In fact, of course, 1 + 2 = 3 would still be true even in this scenario, in which case Mill’s account is wrong.

Someone might accuse Frege of begging the question here, but he is not.  Consider again my imagined student’s suggestion that if whenever we put two pairs of things together we regularly found that this left us with five things, we would judge that 2 + 2 = 5.  A little reflection shows that this is not necessarily the case.  For there are at least two ways we might describe such a scenario.  We might, as the student proposes, characterize it as a world in which 2 + 2 = 5.  But we might instead characterize it as a world in which 2 + 2 = 4 but where there is a strange causal law operating that ensures that bringing two pairs of things together to make a collection of four will immediately generate a fifth thing.

Now, which of these is the correct description of the student’s scenario?  Experience itself cannot tell us, because any set of experiences is consistent with either description.  I would say, and Frege would say, that we can know a priori that the first description is wrong, because the proposition that 2 + 2 = 5 is just nonsense.  But put that aside for present purposes.  What matters is the fact that we can make sense of the difference between these two alternative descriptions of the scenario despite the fact that experience cannot tell in favor of one rather than the other.  And that entails that there is more to the content of the proposition that 2 + 2 = 4 than a mere description of what we happen to experience.

That, I submit, is Frege’s point.  The proposition that 2 + 2 = 4 says more than merely that putting two pairs of things together will regularly give you four things, because we can describe a scenario in which it is still true that 2 + 2 = 4 even when putting two pairs of things together will regularly give you five things.  That scenario would not by itself entail that 2 + 2 = 5, as opposed to merely entailing the operation of a weird causal law.  And by the same token (and to return to Frege’s own example), the proposition that 1 + 2 = 3 says more than merely that a collection that looks like OOO can be separated into parts that look like O and OO, because we can describe a scenario in which no such separation is physically possible and yet it is still the case that 1 + 2 = 3.  Mill’s account fails to capture even the meaning of the proposition that 1 + 2 = 3, let alone the grounds for judging it true.

A second and related problem with Mill’s view, notes Frege, is that it cannot account for examples that don’t involve collections of physical objects which we might know via sensory perception and separate into smaller parts.  He gives the example of there being three methods of solving a certain equation.  Methods of solving an equation are not physical objects that we might literally perceive to be lumped together as a collection, or which we might physically separate into parts (one part consisting of two of the methods, with the other part being the remaining method sitting off by itself).

A third problem is that Mill’s account is a non-starter when applied to facts concerning large numbers like 777,864.  Obviously, it is absurd to suppose that our grasp of the proposition that 770,001 + 7,863 = 777,864 is grounded in experience of finding that whenever groups of 770,001 things are lumped together with groups of 7,863 things, we find that the resulting collection has 777,864 things in it.

Mill says several other things which Frege shows cannot be right.  For example, there is the claim that number is merely a property that a bundle of things has, alongside its color, shape, or the like.  Hence a pile of ten red pens can be said to be ten, just as it can be said to be red.  As Frege would ask, why say that the pile has the property of being ten, as opposed to being twenty (on the grounds that if we distinguish the pens from their pen caps, we get twenty things)?  Or why not say that it has the property of being in the billions (on the grounds that we get such a number when we distinguish the particles out of which the pens are composed)?

Furthermore, Frege points out, 1,000 grains of wheat remain 1,000 grains even after they are sown far and wide and no longer form a bundle.  Nor would Mill’s account explain what the number 0 is, since it obviously can’t be a property that a pile of pens or a bundle of grains of wheat has, the way Mill thinks being ten and being a thousand are properties of such collections.  And then there is the fact that we can apply number to things that are not physical objects that might be lumped together into a bundle (for example, the number of ways to prove a theorem, the number of concepts one entertained Wednesday morning, or the number of events occurring right now).

Mill also says that even 1 = 1 can be false, insofar as one one-pound weight does not always weigh exactly the same as another.  This is like the fallacy committed by the student who thinks that 2 + 2 = 4 is merely a generalization of the observation that when we put pairs of things together we typically find that the result is a collection of four things.  As Frege says, it simply gets wrong what a claim like 1 = 1 is asserting.  It isn’t an empirical claim to the effect that, as a matter of contingent fact, any item that we happen to characterize as weighing one pound will always be exactly equal in weight to any other item we happen to characterize as weighing one pound.  One reason why this is a mistake is, of course, that typically we are only approximating when we characterize something as weighing a pound.  But the deeper point is that, even if we were not speaking merely approximately, the claim would not be a mere description of the empirical facts.  If it turned out that no two things were ever exactly of the same weight, that would not entail that it is false that 1 = 1.  It would entail only that this arithmetical truth does not strictly describe anything in the empirical world.

As Frege says, Mill’s error is to suppose that arithmetical claims are inductive generalizations from particular cases, and to confuse what are in fact applications of arithmetical truths with empirical evidence for those truths.  When we stick one pair of apples next to another to yield four apples, we are not assembling one further bit of empirical evidence in a way that gives additional inductive support for a contingent general claim to the effect that 2 + 2 = 4.  Rather, we are applying a necessary truth, and one that is already known a priori, to a specific case.  And the same thing is true of our application of the proposition that 1 = 1 to the comparison of two weights and the like.

Again, it would be a mistake to accuse Frege of begging the question against Mill.  He isn’t stomping his foot and refusing to listen to empirical evidence against a contingent generalization to the effect that 1 = 1.  Indeed, it would beg the question against Frege to characterize the situation that way, because his point is precisely that the proposition that 1 = 1 is not a contingent empirical generalization in the first place.  His point is that when Mill characterizes arithmetical statements that way, he is changing the subject.  He is no longer talking about the proposition usually expressed by “1 = 1,” but rather about some empiricist-friendly ersatz.  Mill is really just ignoring the arithmetical truth that 1 = 1 and talking instead about a very different sort of claim while using the same symbols.

The actual situation, for Frege, is that it is only because we already have an independent grasp of the meaning and truth of arithmetical propositions that we know how to apply them to concrete empirical cases.  Frege gives the example of pouring 2 unit volumes of liquid into 5 unit volumes of liquid.  We judge that this will yield 7 unit volumes of liquid only given the absence of some chemical reaction or other causal factor that might alter the volume.  We don’t “work up” from the specific empirical cases to the general arithmetical proposition but rather “work down” from the arithmetical proposition to a description of what is really going on in the specific empirical cases.

1. If in fact no two objects in the physical world were ever the same weight, we couldn't know that unless it were known to be true that 1 = 1 (having selected one of the objects in the physical world as our unit of weight measurement).

2. I wonder why the names of modern philosophers are "household names" (at least in the household of lay intellectuals), while one has to dig deep to find out that these philosophers have been given adequate response--and by adequate I don't mean "overwhelmingly convincing" but just that the response are up to the same level of rigor. It seems like there could be an "alternative" history which is a line of thinkers who are famous for having "debunked" modern thought.

1. I think this is a special case of a general truth, viz. that original ideas are generally better known than any specific criticisms of them.

This is deducible, because understanding a criticism requires understanding the idea criticized, while the reverse is not true. Furthermore, a single idea may have many different criticisms, all of which compete for the popularity of "debunking" it.

2. Confirmation bias? We are eager to accept the modern ideas. Adherence to the classical ideas has baggage; it has implications and makes demands. Better to live in a meaningless universe where we can do whatever we want and be the king (or so it goes).

3. Okay, but I'm still left wondering how, on the Aristotelian account, we arrive at knowledge that 2 + 2 = 4? On the Platonic account, of course, no problem: "2" and "4" and "+" are immaterial forms that our minds can grasp, and once we do, 2 + 2 can't be anything but 4.

1. Vince,

“2”, “4”, “+”, and “=“ are immanent forms that we abstract from the objects in question. I don’t think Dr. Feser is denying the Scholadtic thesis that “Whatever is in the mind is first in the senses”.

However, once you abstract the form from the matter and behold it in your intellect, you can treat the form like a Platonic pure form (with the caveat that it is dependent on an intellect for its existence, thus leading to Augustine’s Proof for God). Then once you understand the universal form, you can apply it...universally. Therefore you can move from the particular to the universal.

2. I recall there being a study on this, but yes you grasp quantity in a way akin to the way you grasp triangularity or to the way you grasp living thing. I is from the things themselves but not in an empiricist manner. It is not just the collection and arrangement of images in the imagination confused with conceptualization. Recall empiricism tends to be tacitly nominalist or else weakly conceptualist. Rather in the abstracta is now had as what it is in the way that it comes to be actualized in the mind and in this form it can be used in mathematics. You really do receive in the mind from the things '1' '2' 'π' etc. You are not simply receiving images of salt shakers, pens, and the like and arranging them in some strange way to do mathematics.

4. Dr. Feser here's some trivia you might enjoy.

In the sitcom The Big Bang theory, Sheldon actually quotes Frege as having proven psychologism wrong.

Many other times, BBT seems to mirror the ever present scientism we see in pop culture, but as in rewatching It, it does bring up some philosophy. It's mentioned reductio ad absurdum and Buridians ass.

If I'm not mistaken where it's set isn't too far from you either. I've always thought how cool it'd be if Feser hit Sheldon with structural realism to dent his enthusiasm in string theory explain the fundamental questions.

Anyway, Sheldon would approve of this blog post on Frege!

5. You should send this to my mathematician friends. They still have this firm belief that somehow empirical induction "just gotta" have a place in math or it isn't "a real science."

I guess Mill is drowning like Leonardo di Caprio. Oh and spoiler alert.

1. Maths is even better than science, insofar as its knowledge is more certain. Your friends should not worry about that.

6. Quote:" Obviously, it is absurd to suppose that our grasp of the proposition that 770,001 + 7,863 = 777,864 is grounded in experience of finding that whenever groups of 770,001 things are lumped together with groups of 7,863 things, we find that the resulting collection has 777,864 things in it."

Couldn't an empiricist respond that our grasp of the proposition is rooted in our imagination because it is our imagination (and our other material cognition) which calculates the numbers, and knows them as represented with physical numerical symbols?

Quote:" But we might instead characterize it as a world in which 2 + 2 = 4 but where there is a strange causal law operating that ensures that bringing two pairs of things together to make a collection of four will immediately generate a fifth thing. "

If such a causal law existed, and acted in an immaterial way - meaning that the fifth object is generated by direct actualisation and not by a material cause such as in chemical reactions - would such a causal law be an example of non-angelic immaterial causation? Or does A-T require that all non-material direct actualisation of potentials be something only purely non-material agents can do?

7. To supplement Dr. Feser’s 2 + 2 = 5 example:

Whenever we combine two hydrogen atoms with one oxygen atom, we do have, from an empirical point of view, a case of “2 + 1 = 1”. Of course no one interprets that mathematically. Rather, people regard it as a change in substantial form.

1. This is a +' operator operating on different logic. You could define a +' operator in programming languages with operator overloading to behave EXACTLY as you specified it by casuistric(sp?) description. It wouldn't be +, but it would be a mathematical operator with its own internally consistent rules.

8. Although it's years since I read Frege's The Foundations of Arithmetic, I still remember the thrill of seeing how a philosophical position, namely, mathematical empiricism, can be almost definitively refuted, despite the eventual problems with logicism as a philosophy of mathematics. Moreover, the depth and clarity of Frege's writing was a model for other philosophers. Still, I've always been disturbed by the implication that there are no natural units, that is, we count what is there only under the application of a concept. For instance, one wood, a hundred trees, a zillion cells, and so on. Does this then mean that the notion of, say, a person, as a natural single particular, is made conditional and relative by the a priori, non-empirical nature of arithmetic and its necessary application under concepts?

1. SleepyJackdaw (C. Boyd)June 28, 2019 at 5:40 PM

I think here we'd have to distinguish between natural units and natural unities. Number doesn't explain the unity of a tree, its substantial form does. But the number of substantial forms (or subsumed forms virtually, or forms of other substances) are counted by applying units mathematically. That number is agnostic towards forms doesn't imply that forms don't exist, but that mathematics is not a science or a system or anything of that sort whose business is to determine the natures of those unities it does end up counting.

2. Thanks, that makes sense. It seems to me that the concentration on language in analytic philosophy can easily slide into a kind of linguistic idealism, so much so that one ends up questioning whether it even makes sense to ask if something exists independently of the concepts, therefore the words, by which it is designated. This is how I felt after reading Wittgenstein's later philosophy, though I was no doubt misunderstanding much of it.

9. > Mathematics is an iceberg on which the Titanic of modern empiricism founders.

Are you implying that some older form of empiricism addressed this better? If so, which one?

10. "there are at least two ways we might describe such a scenario. We might, as the student proposes, characterize it as a world in which 2 + 2 = 5. But we might instead characterize it as a world in which 2 + 2 = 4 but where there is a strange causal law operating that ensures that bringing two pairs of things together to make a collection of four will immediately generate a fifth thing."

This works both ways. We might characterize this world as behaving like 2+2=4 because there is no other way it could behave, or we could characterize ourselves as believing 2+2=4 because there is no other way we could believe. We might think we know a priori that 2+2=5 is just nonsense, but our parents and teachers were careful to make our growing minds think it was nonsense.

There's no way to know we reach any conclusion a priori because it's impossible to separate ourselves from our lifetime of experience.

Algebra I teaches that 2+2=4 reduces to 0=0. IOW, it's a tautology. What weighty philosophical questions can be answered in tautologies?

"Furthermore, Frege points out, 1,000 grains of wheat remain 1,000 grains even after they are sown far and wide and no longer form a bundle."

I would point out that 1000 grains was arbitrary to begin with. There are billions and billions of grains of wheat. The 1000 was just a number of grains of interest at this time. Numbering is a description of things of interest. It's convenient for us to talk about 1000 grains of wheat as if they were 1000 identical things, which they are not. There's not even 2 of anything as far as we know. But we humans like to categorize and describe likenesses and relationships. Math is a very useful language for doing that. But it's no more than that.

It's not that 1=1 does not strictly describe anything in the empirical world, it's that it can do no more than describe our interests. Our interests are not necessary truths.

1. "Algebra I teaches that 2+2=4 reduces to 0=0. IOW, it's a tautology."

This is false, or in other words, if you want to count "2 + 2 = 4" as a tautology, it is a tautology in a different sense that "0 = 0" is a tautology. IOW, you are equivocating.

Stick to what you know and mathematics ain't it.

2. >This works both ways. We might characterize this world as behaving like 2+2=4 because there is no other way it could behave, or we could characterize ourselves as believing 2+2=4 because there is no other way we could believe. We might think we know a priori that 2+2=5 is just nonsense, but our parents and teachers were careful to make our growing minds think it was nonsense.

What fantasy planet do you live on where toddlers are inculcated mathematical identities? Were you taught from the Principia Mathematica before you moved on to Counting with the Cookie Monster? Children no more need to be taught that two greater than two is four than they do to suckle. If you tried to teach a child that 2+2=5, all you would end up teaching them is that the natural number denoted by 5 is the one referred to in English as "four". Psychology has seen a massive pushback against empirical models in recent decades precisely because it comports poorly with (funnily enough) empirical observations of learning.

>There's no way to know we reach any conclusion a priori because it's impossible to separate ourselves from our lifetime of experience.

What experience of Euler's possessed him to write e^ix = cos(x) + i * sin(x)? Even accepting that our experiences are prior to logic, it doesn't follow that everything must be strictly experiential in the same way that if we hold logic to be prior to experiences that the discomfort of burning your hand on a stove is a logical proposition.

>Algebra I teaches that 2+2=4 reduces to 0=0.

2+2=4 reduces to 0=0 in the same way that a dodecahedron reduces to a line. IOW, not at all.

>It's a tautology. What weighty philosophical questions can be answered in tautologies?

It's only a tautology if we already accept that 2+2 is necessarily identical to 4, which I thought was half the point in contention.

>I would point out that 1000 grains was arbitrary to begin with.

Try reading the paragraph above it.

>But we humans like to categorize and describe likenesses and relationships. Math is a very useful language for doing that. But it's no more than that.

It's just a language that every human on the planet (and, some scientists believe, many animals too) has comprehended for as long as they can remember with astounding lucidity and is communicable without any ambiguity.

>It's not that 1=1 does not strictly describe anything in the empirical world, it's that it can do no more than describe our interests. Our interests are not necessary truths.

So when is it convenient to represent the number one as not identical to itself? Numbers are useful precisely because they are completely univocal. If two were sometimes equal to three and rarely greater than forty-six then an elementary mathematics textbook would be completely inscrutable.

3. Quote:"It's just a language that every human on the planet (and, some scientists believe, many animals too) has comprehended"

While animals may have some mathematical capacity (such as counting, recognising relationships etc.) and may also generalise things and form easily knowable groups, their knowledge doesn't extend to knowing mathematics as mathematics - that is, as eternal immaterial truths and values.

This is also why Don Jindra has gotten off his rails there, too. The mistake he's made is reducing all mathematical knowledge to merely experiential induction and thereby forgetting that, even though it starts with contingent experience, it ends up being a window into the realm of the immaterial and essential.

To not see this, or to lack this knowledge, is to essentially end up with an animalistic form of mathematics that doesn't get beyond one's own experience.

4. grodrigues,

Please explain how your "tautology in a different sense" is different enough that it actually helps your case.

Gjon,

"Children no more need to be taught that two greater than two is four than they do to suckle."

Then why do flashcards exist?

I'm glad you mentioned suckling. Suckling is instinctual behavior. Even if we call it "a priori" knowledge, we surely cannot call instinct a necessary truth. Our species evolved to suckle at the breast of this world. We can't speak of things outside of the world of experience as if we had a detached, unbiased perspective.

"Psychology has seen a massive pushback against empirical models in recent decades precisely because it comports poorly with (funnily enough) empirical observations of learning."

Junk psychology, no doubt. There's a lot of that in the field. It reminds me of Chomsky's anti-empirical assertions about language learning. As I already mentioned, the issue goes deeper than the experience of an individual. I question that a species which evolved to make sense of this world has the competence to distinguish between something that's merely the behavior of brains versus a "necessary truth."

"What experience of Euler's possessed him to write e^ix = cos(x) + i * sin(x)?"

Complicating simple equations does not make the issue more complicated. Why did humans invent sines and cosines? They were interested in solving real-world problems, that's why. Otherwise it's just syntax manipulation -- Searle's Chinese Room for robots.

"It's only a tautology if we already accept that 2+2 is necessarily identical to 4."

It's simpler than that. The language of math allows 0=0 to be expressed as 2+2=4. But it's still algebraically identical to 0=0. The language allows us to complicate 0=0 to 0=4-4 or e^ix = cos(x) + i * sin(x), but it's still always a tautology.

I agree math is a precise language, a precious invention. But it's still a language.

"Numbers are useful precisely because they are completely univocal."

That's misleading. When I speak of "1" you might ask me which "1" I'm speaking of. Numbers are useful because they can represent anything. They are inherently ambiguous. They represent whatever we currently want them to represent.

JoeD,

"The mistake he's made is reducing all mathematical knowledge to merely experiential induction and thereby forgetting that, even though it starts with contingent experience, it ends up being a window into the realm of the immaterial and essential."

First, I said nothing about induction. My argument is that math is descriptive.

Second, if something "starts with contingent experience," at what point does this transform into "the immaterial and essential?" I see no point at which we can know that this transformation has occurred or could possibly occur.

5. @Don,

*I see no point at which we can know that this transformation has occurred or could possibly occur.*

Your own blindness to this is not an obstacle to reality. Anyone who understands any mathematical proposition such as 2 + 2 = 4, also understands it as a necessary truth that cannot fail to be, and also an immaterial truth that doesn't depend in any way on anything material.

This is a rather obvious observation. The eternal and immaterial truth of 2 + 2 = 4 is derived from practical and concrete examples in the real world, which are instantiations of this eternal truth. To deny that 2 + 2 = 4 is necessary or immaterial is absurd.

And the way in which our experience starts with material examples and ends in immaterial truth is basically the action of our immaterial intellect, which abstracts the form from particular instances in order to glimpse the eternal lying both beneath and above the material.

6. "Why did humans invent sines and cosines? They were interested in solving real-world problems, that's why. Otherwise it's just syntax manipulation -- Searle's Chinese Room for robots."

Translation: all mathematics was invented to solve practical problems, except the vast swaths that were not. Those are just "syntax manipulation". So declareth Don Jindra. By fiat.

"It's simpler than that. The language of math allows 0=0 to be expressed as 2+2=4. But it's still algebraically identical to 0=0. The language allows us to complicate 0=0 to 0=4-4 or e^ix = cos(x) + i * sin(x), but it's still always a tautology."

This is false. 0 = 0 is an instance of the tautology (or logical axiom, depends on how you carve things up) Ax x = x, or in words, that everything is self-identical. None of the other equations are tautologies in this sense neither can they proved from 0 = 0 or reduced in any way whatsoever to 0 = 0. This is a mathematical fact with a simple demonstration, not the ignorant opinion of some random ignorant internet know-nothing.

This answers your question to me. As a friendly suggestion: you really should stay silent about that you demonstrably know nothing of.

7. One Googolplex is a number that is symbolized by a one followed by one googol zeros and a googol is symbolized by a 1 followed by 100 zeros.

It is an absurdly huge number.

If I believed the views attributed to Mil then I have no real reason to believe Googolplex + 1 is greater than Googolplex till I counted it all out compared them.

Which is freakin insane.

8. grodrigues,

2x+2x=4x can be reduced to 0=0. It's simply false to say it cannot be. You're making yourself look foolish. If one is a tautology, the other is too. e=mc**2 can be reduced to 0=0 too. So it's a tautology. I'm not saying it expresses no meaning. It's not always obvious what identity is, that one thing is equivalent to another. We shouldn't make more out of equations than expressions of identity. That's all I'm saying.

9. "You're making yourself look foolish."

I know exactly what I am saying since mathematics is my profession, is what I took a phd on, etc. I explained why you are incorrect, but since you are an arrogant buffoon everything I said went over your head. I repeat: what I said can actually be proved and it is not even particularly difficult.

"We shouldn't make more out of equations than expressions of identity. That's all I'm saying."

No, this just betrays your ignorance. Identities can be highly non-trivial and not reducible to the triviality that everything, not just 0, is self-identical.

And e = mc^2, being a contingent law of our universe, is definitely not reducible to 0 = 0.

And I am over and done here. You continue to be the same know-nothing troll; it is hopeless to teach you even the most basic things.

10. grodrigues,

"I know exactly what I am saying since mathematics is my profession"

This issue is about meaning. I care that your profession is math in discussions about the meaning of math as much as you care my profession is computer software in discussions about the meaning of software.

"Identities can be highly non-trivial and not reducible to the triviality that everything, not just 0, is self-identical."

11. Don Jindra

Would it be fair summarise your position thus:

Without the symbol the object is a human perception and reflects all the phases under which human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations.

Should this be the case... I can find nothing wrong about it...

12. "This issue is about meaning."

The issue, as I framed my complaint, is exactly what I said is about, not what you imagine me saying. You simply do not understand mathematics qua mathematics and its essential nature (no it is not about syntax or symbol pushing), you make factually wrong mistakes that are easily provably so, and then you have the gall to shift the question and complain of misrepresentation.

And then again, this is history repeating itself, so my complaining is misplaced as well.

13. However Don Jindra

I do take issue concerning algebra and equations.

For example, from the standpoint of pure algebra:

0x^2 + bx + c =0 is not a quadratic but a linear equation.

0x^2 + 0x + c = 0 (c/=0) is not an equation but an incongruity.

0x^2 + 0x + 0 = 0 is not an equation but a tautology.

14. "0x^2 + 0x + c = 0 (c/=0) is not an equation but an incongruity.

0x^2 + 0x + 0 = 0 is not an equation but a tautology."

I suppose this is a matter of how we carve things up, but there is absolutely no reason to say the above two examples are not equations: the first just happens to have the empty set as solution set and the second the solution set is the universal set (since the equation makes sense in any ring, it would be the whole ring under consideration). This is in fact the best way to look at -- because then equations reduce to what is known as an equalizer, and whole new games, fruitful ones, can be played.

15. Don Jindra

Pure addition is just as much an idealization as E = mc^2

is false.

16. grodriques

No.

All it is saying is that the method derives from the proposition:

If a polynomial relation of degree n in x is satisfied by more than n distinct values of x, then the relation is an identity, i.e. is satisfied for any value of x.

17. grodrigues

Perhaps, this will help...

Simply, consider the the general quadratic function in one variable (like my example) given by the formula:

Q(x)=ax^2 + bx + c

Assume that the indeterminate coefficients a, b, c, range over all values, positive, negative, or zero.

Then the aggregate(Q) would include not only bona fide quadratic functions but also the functions in my previous comment.

18. grodrigues,

"You simply do not understand mathematics qua mathematics and its essential nature (no it is not about syntax or symbol pushing),"

Mathematic's "essential nature?" You beg the question and you don't even know you're doing it. You make it too easy to dismiss you.

19. Without dismissals you would have no arguments djindra.

20. Philip Rand,

"Without the symbol the object is a human perception and reflects all the phases under which human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations."

In think I go further than that. Without objects, there is no purpose for symbols. Symbols cannot be considered in isolation. There is no separate meaning to them. Therefore outside a material world (or better, outside human perceptions) there is no "necessary truth" to any symbol or any combination.

21. @Philip Rand:

"No."

It is so funny watching people trying to teach me mathematics, a mathematician, without even paying the courtesy of actually reading what I wrote.

"Perhaps, this will help..."

I am sorry but it is you that needs help. What you have written down is a family of functions that contains quadratic, linear and constant functions. What point do you imagine you are making? Because it does not invalidate anything I have said. And by the way, I started by saying "I suppose this is a matter of how we carve things up", so if calling it an equation offends thy tender sensibilities, by all means don't, the rest of the mathematical community will survive fine don't worry.

@Don Jindra:

"Mathematic's "essential nature?" You beg the question and you don't even know you're doing it. You make it too easy to dismiss you."

What a complete dumbass you are. First, I notice that you really are dishonest and have left by the wayside my actual complaints and shifted the discussion.

Second, do you deny that Mathematics is a real field of actual knowledge, like History, Physics or Literary Criticism, with its own specific proper object (no implied view on its status) and its own specific methods? That is what I mean by "essential nature". Now you may actually deny it, but since it is mathematicians that decide the future of mathematics, not random internet jackasses, your dumbass opinion has about as much intellectual weight as the sound a cow fart makes. Because your opinions are just that, not real, actual knowledge: nothing but a reflection of your own ignorance and prejudice.

22. grodrigues

Sorry grodrigues but, your problem cannot be evaded by an "appeal" to the equaliser procedure or any other formal devices, i.e. Harriet procedure, etc.

This is because, the difficulty you are faced with is inherent in the definition of the number zero (this is what Don Jindra exploited).

What I state is trivial for a mathematician to prove, i.e. simply consider the quadratic:

P(x)=(b-c)(x-a)^2 + (c-a)(x-b)^2 + (a-b)(x-c)^2

simple direct substitution gives:

P(a)=P(b)=P(c)=(a-b)(b-c)(c-a)

Therefore, the relation is satisfied by more than two values of x.

I am a theoretical physicist, expert in condensed matter physics, quantum ab initio EOS, and information-physics.

So, don't feel ashamed!

23. grodrigues: "I suppose this is a matter of how we carve things up"

Philip Rand: No, it is a matter of precision and accuracy.

24. Don Jindra

You state:Symbols cannot be considered in isolation.

That sounds like a necessary truth to me...

25. Philip Rand,

How does it sound necessary to you? I sure don't see it that way.

26. Don Jindra

Here are four propositions... indicate which are true & false.

1/ All symbols can be considered in isolation.
2/ No symbols can be considered in isolation.
3/ Some symbols can be considered in isolation.
4/ Some symbols cannot be considered in isolation.

27. grodrigues,

You don't like my POV, fine. But you in particular seem to have a need to make it personal.

I have not left your complaints by the wayside. You have not addressed mine. Frege understood the issue to be, What do numbers mean? I'm trying to stay on topic.

Your tactic was to start with a quibble about tautology. Then you threw out "e^ix = cos(x) + i * sin(x)" as if this sheds light on the meaning of "2". It doesn't. I pointed that out.

You said something to the effect that more complex equations can't be proven from 0=0. But that has nothing to do with what I'm saying. I'm saying equations have equal left and right sides. So that's "a phrase or expression in which the same thing is said twice in different words" -- words in this case being mathematical symbols, but symbols nonetheless.

This is a docudrama loosely based on how our argument went:

ME: Sam said his autobiography is his description of his own life. He doesn't realize that sentence is a tautology.

YOU: That's not *really* a tautology, you moron, because you can't get from there to "It's deja vu all over again."

I know what "essential nature" means around here, this being an essentialist site. But if you meant something less than that in this case, then it's probably no more relevant to this discussion than the "essential nature" of tic-tac-toe.

As to your credentials and the weight you think they deserve, mathematicians are not the only people on the planet who regularly manipulate symbols. I've been manipulating symbols for 40 years professionally. I think I have a good understanding of what I'm doing. My expertise on this topic doesn't play second fiddle to yours.

28. @Phil Rand:

"No, it is a matter of precision and accuracy."

Giggle. Sure, it is.

Your second iteration is absolutely irrelevant to what I said. Just go back and actually read what I wrote. I can spoon feed it to you if you need me to.

"I am a theoretical physicist, expert in condensed matter physics, quantum ab initio EOS, and information-physics."

This explains a lot. I was talking mathematics, don't feel ashamed if you cannot go much beyond calculus.

@Don Jindra:

This is a dishonest characterization of the dialectical situation. For if you really though it was mere quibbling and not addressing your point, then why you didn't you say so earlier? Instead of for example, coming out with: "2x+2x=4x can be reduced to 0=0. It's simply false to say it cannot be. You're making yourself look foolish.". So yeah, you took me up to task and dug in your heels, quibbling or no quibbling, about *mathematical facts*. Dismissing it as you do with "You said something to the effect that more complex equations can't be proven from 0=0. But that has nothing to do with what I'm saying." is once again, shameless dishonesty.

"My expertise on this topic doesn't play second fiddle to yours."

When the topic is mathematics, there is a difference between a mathematician and a coding monkey. Sorry, them's the breaks. What mathematicians do *cannot* be reduced to symbol manipulation, but is about *understanding* of their proper object, quite independently of the ontological status of mathematical objects. The fact that you keep repeating this mantra, shows the ignorance and shallowness of your so-called understanding. And this is not an ad-hominem since to discuss the central issue you want to discuss, the ontological status of mathematical objects (it is once again misleading and an indication of your ignorance that you talk in terms of "meaning"), it at least behooves a passing familiarity with what mathematicians *actually do* as opposed to whatever you imagine, based on whatever it was you learned 40 years ago. Pointing such failures is not an ad hominem because I never made the obvious fallacious move, but it goes towards showing that you do not possess any real knowledge or really, any evidence-based arguments.

29. @grodrigues

I think Phil Rand is a nut case. He tries to bug Dr. Dennis Bonnette over at the Strange Notions blog and pretends he knows how to answer Thomistic Philosophical arguments but his answers are just bizarre.

He is smoking something.

djindra is just being djindra.

30. grodrigues

Interesting...I gave you a precise and accurate answer concerning the definition of 0=0

Concerning 0=0, you tabled four positions:
1/ 0 = 0 is a tautology
2/ 0 = 0 is a logical axiom
3/ 0 = 0 reduces to an equaliser
4/ 0 = 0 is the universal set

Your final position is that "0=0" is equivalent to all four positions on account one can carve up 0=0 any way one chooses.

What is extremely interesting is that your position concerning 0=0 is directly in violation of the First Principle of Thomistic Metaphysics, i.e. the Principle of Non-contradiction.

I find it odd that I am in a dialogue with:
1/ An Atheist who is unaware he believes in necessary truths.
2/ A Thomist who is unaware that he does not believe in necessary truths.

Weird....

31. grodrigues

Concerning 0=0, you tabled four positions:
1/ 0 = 0 is a tautology
2/ 0 = 0 is a logical axiom
3/ 0 = 0 reduces to an equaliser
4/ 0 = 0 is the universal set

What I suggest you do is to plot each function on a graph.

Then compare the differences between all four conceptions of 0=0.

32. @Phil Rand:

"What I suggest you do is to plot each function on a graph."

Oh dear, with all the nutcases in all the blogs of the internet world I happened to bump the dimwit that is even more obnoxious and insufferable than me.

"Then compare the differences between all four conceptions of 0=0."

What you wrote is not even an accurate characterization of what I said. And in the cases where it is (or it could be patched to be) there is no inconsistency. 0 = 0 is quite obviously an *instance* of the logical axiom that everything is self-identical. If on the other hand, we interpret the symbol 0 as the constant function x |-> 0 then 0 = 0 is perfectly good equation with solution set (or equalizer) the universal set (say the real numbers if you take them for domain). A tautology is a formula in propositional calculus that is always true, which is obviously the case with 0 = 0 (depending on how you do things you may have to extend propositional calculus, say with the constant symbol 0, but this is a minor detail). There is no contradiction anywhere, except maybe in that empty skull of yours.

Your ignorant stupidity about these things is so obvious and blatant, that I suspect that "I am a theoretical physicist, expert in condensed matter physics, quantum ab initio EOS, and information-physics." is just bullshit. Despite my jab a theoretical physicist needs quite a lot more beyond calculus: GR needs differential geometry, QM needs functional analysis, representation theory, etc. That or the state of education in theoretical physics is a complete sham.

As the Son of Ya'Kov suggested, stop doing drugs, they are turning what little is left of your brain into mush.

33. grodrigues

A tautology is valid/true in ANY interpretation.

A) "2 + 2 = 4" is a tautology in a different sense that "0 = 0" is a tautology. (FALSE)

Let; a=b=1
ARITHMATIC: 1 + 1 = 2
GEOMETRY: 1 + 1 = sqrt(2)

B) equations are not tautologies (TRUE)
I demonstrated this.
C) e = mc^2 is not reducible to 0 = 0 (TRUE)
I demonstrated this.
D) e = mc^2 cannot be proved from 0 = 0 (TRUE)
I demonstrated this.

34. @Philip Rand:

I do not think it is worth cluttering the combox with further replies to the incoherent screaming coming from a padded cell.

Peace be with you.

35. grodrigues,

"Dismissing it as you do with 'You said something to the effect that more complex equations can't be proven from 0=0. But that has nothing to do with what I'm saying.' is once again, shameless dishonesty."

You're not listening. I suppose you give yourself permission to call me dishonest when you don't listen. This post was yet another evasion.

"When the topic is mathematics, there is a difference between a mathematician and a coding monkey."

Are you trying to be ignorant? It sure seems like you are. If you have an understanding of math "quite independently of the ontological status of mathematical objects" then I simply ask that you demonstrate that you do. But instead you're obsessed with making this personal.

36. @Don Jindra:

"You're not listening."

You're not talking. I laid out my reasons. Any response? No, so yeah, dishonesty.

"If you have an understanding of math "quite independently of the ontological status of mathematical objects" then I simply ask that you demonstrate that you do."

It is simply an empirical fact that mathematicians do mathematics and communicate with each other about the subject meaningfully, and yet disagree violently on what exactly is the status of mathematical objects. It is my impression that Platonism is over-represented (and many are atheists to boot) but you can find the whole spectrum of positions. This is not specific to mathematics. For an example, physicists do quantum mechanics all day long and yet they disagree violently what the theory is actually telling us about reality in itself.

You really are one of the most arrogant doofus I have ever met online, completely ignorant about the subject he pontificates about and completely unaware of his own ignorance.

"Are you trying to be ignorant? It sure seems like you are."

As opposed to you actually being one?

Enough bantering with an idiot.

37. Don Jindra

You are correct despite godrigues stating (twice):

None of the other equations are tautologies in this sense neither can they proved from 0 = 0 or reduced in any way whatsoever to 0 = 0. This is a mathematical fact with a simple demonstration

He has yet to provide any demonstration.

Remember, a man with experience is never at the mercy of a man with an argument.

38. grodrigues,

I have never said nor implied that mathematicians don't "do mathematics and communicate with each other about the subject meaningfully." If mathematicians disagree violently on the status of mathematical objects, why do you call me an "arrogant doofus" when I take sides? Seriously. Why?

You seem to think I have no respect for mathematics. But I have a great deal of respect for it and those who are good at it. I wish I remembered all the math I once knew.

I've been reading Frege's THE FOUNDATION OF ARITHMETIC.

Under "Is Number a property of external things?", #22, page 28, there is some text that's very similar to what I wrote some days ago:

"Baumann rejects the view that numbers are concepts extracted from external things: 'The reason being that external things do not present us with strict units; they present us with isolated groups or sensible points, but we are at liberty to treat each one of these itself again as a many.' And it is quite true that, while I am not in a position, simply by thinking of it differently, to alter colour or hardness of a thing in the slightest, I am able to think of the Iliad either as one poem, or as 24 Books, or as some large Number of verses. Is it not in totally different senses that we speak of a tree as having 1000 leaves and again as green leaves?"

Frege elaborates on this distinction between color and number and uses a card deck as an example. He ends the section with this:

"The Number 1, on the other hand, or 100 or any other Number, cannot be said to belong to the pile of playing cards in its own right, but at most to belong to it in view of the way in which we have chosen to regard it; and even then not in such a way that we can simply assign the Number to it as a predicate. What we choose to call a complete pack is obviously an arbitrary decision, in which the pile of playing cards has no say. But it is when we examine the pile in light of this decision, that we discover perhaps that we can call it two complete packs. Anyone who did not know what we call a complete pack would probably discover in the pile any other Number you like before hitting on two."

Yet when I say basically the same thing, I'm an "arrogant doofus."

Nevertheless, I think the "arrogant doofus" named Frege is correct about this. But I think it ultimately undermines his platonic conclusion. I'm reading on because I'm interested in seeing how he tries to resolve it.

39. Don Jindra

Yes. What Frege is concerned with are expressions of collection, aggregate.

However, what Frege is trying to explain in your quote is the distinction between the concrete preceding the abstract.

What he is trying to convey is that counting consolidates the concrete and thus the hetrogeneous notion of plurality into the homogeneous abstract number concept which makes mathematics possible.

Frege is not Platonic.

Symbols cannot be considered in isolation , i.e. The concrete and the abstract cannot be considered in isolation, undermines your own conclusion.

40. Don Jindra

Frege is considering this concrete and abstract relation from a Phenomenological standpoint.

Like Aquinas he would most likely agree with:
nihil in intellectu nisi prius sensu.

Which in many respects reflects your own position.

41. Philip Rand,

From Stanford's Encyclopedia of Philosophy:

"Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental. Platonism in this sense is a contemporary view. It is obviously related to the views of Plato in important ways, but it is not entirely clear that Plato endorsed this view, as it is defined here. In order to remain neutral on this question, the term ‘platonism’ is spelled with a lower-case ‘p’. The most important figure in the development of modern platonism is Gottlob Frege."

I'll have to read the whole book to see what Frege is up. My supposition is that he's trying to sell the idea that our abstract notion of "green" is indeed preceded by concrete "greenness" present in objects. But, more importantly, this is not the case with numbers. Numbers have no presence in concrete objects. One way or another, he'll probably make the argument that we "discover" numbers, and that we can discover only what's already there on some plane of existence. This is where I'll part ways with him.

"Symbols cannot be considered in isolation , i.e. The concrete and the abstract cannot be considered in isolation, undermines your own conclusion."

If you explain why you think so I'll explain why you're wrong.

By the way, "nothing in the intellect unless first in sense" does not fully describe my thinking. I believe in a human nature. Our sensing of the environment is regulated by that. IOW, we have intellectual predispositions.

42. Don Jindra

The statement:
Symbols cannot be considered in isolation.
is your own statement, not mine... you explain to me why it is wrong...

Stanford isn't accurate.

Frege's system concerns how thoughts can refer to particulars, i.e. he uses this concept of sense which is an abstract object; Frege was never particularly clear, but it is similar to sense-data infused with physical object interpretation.

It is a problem dealt with in the analytical theory of reference, i.e. the territory of Mill, Russell, Strawson, Kripke, Evans, etc.

Frege believed that you could solve the problem of non-existence only by treating them all as pure definite descriptions. These thoughts are referential in a way that exceeds their descriptive exactness, so that if they fail of reference there is no real thought at all.

You have not responded to:

Here are four propositions... indicate which are true OR false.

1/ All symbols can be considered in isolation.
2/ No symbols can be considered in isolation.
3/ Some symbols can be considered in isolation.
4/ Some symbols cannot be considered in isolation.

43. Don Jindra

Symbols cannot be considered in isolation.
Your proposition is conflating sign/symbol, i.e. abstract/concrete

44. Philip Rand,

"Symbols cannot be considered in isolation. Your proposition is conflating sign/symbol, i.e. abstract/concrete. This defeats your thesis."

I can't decipher what you intend to say. You seem to be suggesting something about signifier/signified. What do you think this has to do with me?

No symbols can be considered in isolation = true. Symbols, aka, abstract concepts, have no meaning apart from this material world. There's nothing self-defeating in this.

Stanford isn't accurate but you are? Unless you can show that Frege did not believe in abstract objects, and was speaking metaphorically or esoterically, I have to take his words at face value. He was a platonist.

45. Don Jindra

Given these four propositions, you have given:

1/ All symbols can be considered in isolation.(FALSE)
2/ No symbols can be considered in isolation.(TRUE)
3/ Some symbols can be considered in isolation.(FALSE)
4/ Some symbols cannot be considered in isolation.(FALSE)

Symbols cannot be considered in isolation.

LEMMA:Symbols, aka, abstract concepts, have no meaning apart from this material world.

Symobols = abstract concepts

Therefore, we have the proposition:
Abstract concepts cannot be considered in isolation.

Is this what your proposition means?

11. //Frege gives the example of pouring 2 unit volumes of liquid into 5 unit volumes of liquid. We judge that this will yield 7 unit volumes of liquid only given the absence of some chemical reaction or other causal factor that might alter the volume. We don’t “work up” from the specific empirical cases to the general arithmetical proposition but rather “work down” from the arithmetical proposition to a description of what is really going on in the specific empirical cases. /

I don't know about anyone else, but I did work up from counting on fingers, which is exactly what this example is. To find out what 2+2 is, I used my digits. I'm also pretty sure that with numbers like pi, humans did use empirical evidence to make more and more precise estimates of the number.

1. You did not work up per se, that:m is not a mere increasing refinement of the finger method. Nor, and this is very obvious in euclidean geometry as opposed to the more strictly empirical Babylonian and Egyptian geometries which preceded it, was the arrival of Greek mathematics proper simply a more refined form of drawing shapes in the sand. In both of these cases the operation ceases to be understood as the seen operation and instead becomes a sign or placeholder for concepts which are grasped abstractly with aid from the signs.

2. I did work up from that. I learned that adding 2 fingers to 2 other fingers made 4 fingers. I can't speak to others experience of learning arithmetic. Perhaps they learn in the abstract, I learned ny physical example.

I'm pretty sure that the study of pi began with something akin to putting a string around a circumfrance and measuring it against the diameter. Later study may have generalised out from that, but it started with string and sand.

3. Unknown, it's called the genetic fallacy, that is, a confusion between the history of how you came to know something and that something's meaning. That you learned to count on your fingers, toes or tentacles is irrelevant when it comes to the meaning of arithmetical terms and proofs. Frege argues that all arithmetic can be derived from logical laws, but certainly not that that was how we came to count. I felt the effects of gravity long before I knew anything about Newton's laws. Also, pi is certainly not worked out empirically in mathematics: it comes from a proof concerning all possible circles, as proved by the Greeks.

I learned arithmetic from counting physical objects. My knowledge of mathematics comes from that. My understanding of mathematics is from that. When I take 2 fingers and add 2 more fingers the result is a number I understand as 4 fingers. My concept of "fourness" directly relates to the physical, and is abstracted from that. So my knowledge of arithmetic does come from specific empirical examples, contra what Frege and Feser says.

5. What you are describing is working up from designating physical objects as units capable of being added, not working up from counting physical objects. Fingers are not separate physical objects but parts of your hand; the mere physical object tells you nothing about whether you should have taken fingers as 2 or simply as the 1 hand.

6. Unknown, I'm no expert (far from it), but I would say that you have to distinguish between how you, as well as I, came to know basic arithmetic and what it ultimately means. For instance, I think you would agree, at a minimum, that 1 + 1 is not just equal to 2, but is necessarily so; that the symbol '1' refers to the same number, namely 1, on every occasion of its use; and that numbers themselves, whatever they are, are not physical objects, such as fingers or apples. Given these truths, there is no way that you could derive the requisite notion of necessity, or the singular reference to the same number with the same symbol, or the non-physicality of numbers, merely from counting fingers, ergo counting fingers does not explain arithmetic. Still, the philosophy of mathematics has been one of the great and most mysterious areas of thought since Plato.

7. When I say you did not work up from the finger method I did not mean that you did not innd time start with the finger method rather I mean that the mathematics you have is not of itself operation upon the fingers in some very complicated form. This is trivial given that even with basic arithmetic 1 2 and the like are nothing like actual fingers. Rather, fingers are just signs for numbers which assist you in operating and holding them in memory.

8. This comment has been removed by the author.

9. Unknown,

"I'm pretty sure that the study of pi began with something akin to putting a string around a circumference and measuring it against the diameter."

I'm pretty sure you're correct. The ancients used a rough estimation of pi. That estimation was good enough for their purposes. Those purposes probably had nothing to do with "necessary truths." It's pretty absurd to estimate a 'necessary truth."

(Even today we don't use pi, we use an approximation to a certain number of digits.)

Brandon,

"Fingers are not separate physical objects but parts of your hand; the mere physical object tells you nothing about whether you should have taken fingers as 2 or simply as the 1 hand."

Exactly. I've been saying the same thing. But you draw the wrong conclusion from this. Concepts of objects such as fingers, noses, apples and trees are a subjective human perspective. Our grouping of these "objects" into sets is an extension of that perspective. That way of looking at reality is taught to us from birth. We're taught we have ten fingers and ten toes and not twenty fingertoes. We're taught a nose is not a finger even though it's a protrusion. We're taught that humans classify things in specific ways and if you want to talk, you need to learn how to refer to things like we want you to. We are taught that even though there are billions of fingers in the world, I have only ten of them. And those ten are conceptually the same even though they are not actually the same. The whole concept of "ten" is derived from a specific way of perceiving the world and talking about the world. It's not actually the world. No separate physical object of a set is actually the same. There are no sets in reality. Which set would reality "call" a set? Sets are a human conceit, a way of filtering too much input in actual reality from the reality that strikes our fancy here and now. Therefore math, which is just sets formalized, is a human conceit. It takes more than human conceit to make a "necessary truth."

This conceit becomes natural to how we think about everything. It's necessary for our survival but it's not necessary in a cosmic sense.

10. In other words, since what is true of mathematics would apply to logic, as well, you are epistemologically a traditionalist, in the nineteenth-century sense in which the term is applied to Bonald and Lamennais, taking intellectual life to be based on faith in the traditions of our ancestors.

12. JoeD,

"This is a rather obvious observation."

The gracious Mr. Feser has often pointed out that those who deny final cause can't help but speak in the language of final cause. I notice a similar thing with this issue. Those who claim math is detached from experience can't help but reference experience when making their case. You speak of my blindness, so I'll speak of yours. You're blind to the fact that observation is the only thing that allows you to "see" anything at all, either actually or metaphorically. Without experience, aka, observation, you would have no thought. This is so thoroughly part of us, it's easy to be blind to it. Yet you want me to be blind to it.

Yes, every thought is "derived from practical and concrete examples in the real world." Now take that to its obvious and "necessary" conclusion.

1. @Don,

"Those who claim math is detached from experience"

Nobody claims math is detached from experience.

Rather, it starts with experience, but ends in things that are beyond it.

"Without experience, aka, observation, you would have no thought."

And 2 + 2 is still 4 even when I'm asleep. And the law of non-contradiction still applies as well.

Unless, of course, you're one of those die-hard materialists who literally deny anything beyond matter.

"Yes, every thought is "derived from practical and concrete examples in the real world." Now take that to its obvious and "necessary" conclusion. "

Yes.

Namely, our thoughts at least partially transcend concrete experience.

Unless, of course, you're bold enough to think transcendent truths cease having meaning if every human being fell asleep.

2. Surely it's obvious that though experience allows us to say that something exists, it is not thereby true that everything that exists is experiencable. Kant of course made this point.

3. You don't know how stupid you sound.

Mathematics is not grounded in experience. People knew about prime numbers (= numbers with primality) before primality itself was ever observed in nature.

Saying that mathematics is grounded ("ground"--now you're making me use your down-to-earth flat language!) in experience because we need to use it to learn mathematics is like saying that mathematics is grounded in teachers because we need to use teachers to learn mathematics.

4. ErotemeObelus,

That some people here think I sound stupid is of no concern to me. I'm responding to ideas of questionable merit, so it comes with the territory.

"People knew about prime numbers (= numbers with primality) before primality itself was ever observed in nature."

I see no significance in that. Humans are inventive. We like naming things. We like finding patterns. I don't claim math is an unimaginative discipline. But prime numbers were found in practical experience: Some collections of objects can't be separated into piles other than 1. We don't need nature to show us that. Our desire to share leads naturally to dividing piles of stuff, and therefore prime numbers.

"Saying that mathematics is grounded in experience because we need to use it to learn mathematics is like saying that mathematics is grounded in teachers because we need to use teachers to learn mathematics."

Bad analogy. We are not in the classroom 24/7. But we are in the world of experience 24/7. I'm saying that there is no evidence anyone can detach his reasoning from this world of experience. Nobody has credible claims of an objective, detached view of it. The lofty mathematician is on no better plane than the hands-on plumber -- not in church, not in poetry, not in equations.

Andreas Smith,

"Surely it's obvious that though experience allows us to say that something exists, it is not thereby true that everything that exists is experiencable."

No, it is not obvious. This seems to beg the question because ultimately this is the issue: Is there more to reality than things that can be experienced directly or indirectly through instruments? Frege is arguing for a form of Platonism. I've always thought all forms of Platonism were bunk.

JoeD,

Yes, I'm one of those die-hard materialists who deny anything beyond matter.

I'm not sure what you mean by "transcendent truths." If you mean "laws of nature" then I agree those are outside human thought. They'll exist once we're gone. But a) I don't put math and logic in that category; and b) laws of nature are not necessary in the truly transcendent sense you probably mean.

5. Don Jindra, let me put it another way: the fact that we can experience only what is in itself experiencable says absolutely nothing about whether there are things which are not experiencable. There is nothing in your experience that tells you that experience alone is real or the route to reality and truth. Indeed, any argument about the veracity of experience would have to use principles that were not themselves verifiable by experience. That strikes me as pretty obvious (as it does almost every significant philosopher since Kant).

6. @Don Jindra,

Ah, so that's the problem you're struggling with.

Math and logic are obviously about truths which would be true even if nothing material existed. That is why I called them transcendent truths, since they are necessary, immaterial and immutable.

To deny that 2 + 2 would be 4 (or would have any meaning) even if nothing material existed is just...a textbook example of a fundamental error.

7. I don't claim math is an unimaginative discipline. But prime numbers were found in practical experience: Some collections of objects can't be separated into piles other than 1.

No they were not ignoramus.

Prime numbers were first described by Euclid (who never appealed empirical measurements but believed that things needed to be demonstrated formally). Euclid conceived of prime numbers in order to define the abstract idea of unique factorization.

8. Andreas Smith,

"the fact that we can experience only what is in itself experiencable says absolutely nothing about whether there are things which are not experiencable. "

Possibly true. But that doesn't tell us how to learn about things that are beyond experience. It doesn't tell us anything else is really there. When Frege points to numbers as one of those things, I say there's a more reasonable explanation that doesn't require more than experience. This number issue is just a different version of the supposed "problem of universals."

JoeD,

"Math and logic are obviously about truths which would be true even if nothing material existed."

My position is that it's mere hubris to talk about truths that might stick around even when nothing material exists. It's absurd to even talk about meaning in non-existence.

ErotemeObelus,

"No they were not ignoramus."

When people turn to ad hominem, I know I've won the argument. If you're going to attack someone, try mixing in some creativity.

Euclid's works contain the oldest surviving study of primes. He certainly did not discover primes.

9. @Don,

On the contrary, it would be hubris to think that truths don't outstrip experience, or that our experience and the things of experience exhausts all possible modes of being.

Your claim that meaning in non-existence is absurd is one of the worst of materialistic illusions, to say the least. I was pointing out that even if nothing material existed, these truths would still be true, which is quite obvious. These eternal truths themselves aren't objects of existence. They are not substances or possible "things" like the material things we know and experience. They aren't the type of thing that could or could not exist.

If you fail to see that eternal truths like those of mat and logic transcend contingent material existence, then....I don't know how to help you. Sometimes a man's reasoning is just so wrongheaded as to be virtually impossible to change practically.

10. When people turn to ad hominem, I know I've won the argument.

Pointing out your ignorance by a label is not an ad hominem. An ad hominem would be saying that because you're ignorant you must be wrong.

11. JoeD,

"On the contrary, it would be hubris to think that truths don't outstrip experience,"

I agree. But I don't argue truths outstrip experience. I argue that if we can ever know the whole of true, experience is the only reliable way to get there.

"or that our experience and the things of experience exhausts all possible modes of being."

I argue that whether or not there are several modes of being is irrelevant. There is only one mode of being we can ever know, and that mode is that which exposes itself to us through experience. All else is speculation.

"Your claim that meaning in non-existence is absurd is one of the worst of materialistic illusions, "

Please explain where the meaning in non-existence is to be found. It would require a strange definition of meaning. I doubt you could justify that definition.

I'm pointing out that if there is nothing, there can be no meaning and therefore there can be no truth. Truth from absolutely nothing would be tantamount to saying a null set has at least one object, that is, 0>=1.

Sure, you could say that I'm merely stuck in a materialistic POV. But this is the problem. If you want to make the case materialism is wrong, you've got to show how materialism fails. This math argument falls short because a materialist does not have to accept some of your assumptions, assumptions you have to have to make your case. To assert there are immaterial "eternal truths" merely begs the question. I would only grant that if there are eternal truths, they are eternal laws of nature, that is, physical laws. So the materialist and the dualist reach an impasse. You can accuse me of wrongheadedness but you can't make that stick unless you can prove your assumptions without begging the question.

ErotemeObelus,

"Pointing out your ignorance by a label is not an ad hominem."

If one merely claims someone is ignorant of some fact, that's not an ad hominem. You went beyond that.

12. @Don,

1)Well it depends on what you want to count as "experience". In some sense, everything is contained within our experience - and that also includes awareness of immaterial truth. We know there are immaterial truths, and in some sense they have to be "in our experience" in some way in order for us to know them.

But if you want to define "experience" as basically only being material and concrete, then of course you're going to be blind to immaterial truths since you've defined all our knowledge in such a way as to exclude it.

2) You seem to be expressing yourself too vaguely. I'm not claiming non-existence has "meaning", since non-existence isn't even the object of this discussion (immaterial truths are). Rather, even if nothing existed, certain necessary truths would still be / apply. And such things aren't "non-existence".

I'm pointing out that if there is nothing, there can be no meaning and therefore there can be no truth... This math argument falls short because a materialist does not have to accept some of your assumptions, assumptions you have to have to make your case. To assert there are immaterial "eternal truths" merely begs the question.

In other words, if nothing existed, 2 + 2 would not be 4, the laws of logic would have no meaning whatsoever, and no truth is actually necessary (because necessity implies eternity implies immateriality) . Your assumption is essentially to deny that there is such a thing as an necessary-immaterial-eternal truth, which means you deny that that mathematical truths have any meaning beyond the material world, or apply necessarily.

This is...wrong, and everyone who knows anything about math or logic sees that. If you can't see that, then that's your problem. There would be no point in further discussing this.

13. JoeD,

"We know there are immaterial truths"

Actually, we don't.

"But if you want to define 'experience' as basically only being material and concrete, then of course you're going to be blind to immaterial truths since you've defined all our knowledge in such a way as to exclude it."

So do we include mystical experience? Where do leaps of imagination turn into "immaterial truths?" Any worthwhile epistemology is gong to try to exclude phony evidence. Exclusion alone is not an argument against it.

"Your assumption is essentially to deny that there is such a thing as an necessary-immaterial-eternal truth, which means you deny that that mathematical truths have any meaning beyond the material world, or apply necessarily."

Absolutely. This is what I've been saying for years here. There are no truths, mathematical or otherwise, beyond some connection to the material world. This is what I meant with my reference to Searle's Chinese Room. Math disconnected from the material world is mere symbol manipulation. It has no meaning. Therefore it has no truth. E=mc**2 means nothing unless the variables stand for something -- in this case energy, mass and the speed of light. Similarly for 2+2=4. Two of what? Without the implicit reference to two of "something" rather than two of "nothing," these are meaningless pixels.

"This is...wrong, and everyone who knows anything about math or logic sees that."

This is so typical. You're just begging the question. Then you blame me when I refuse to make the same fallacy.

14. If one merely claims someone is ignorant of some fact, that's not an ad hominem. You went beyond that.

No I didn't. You could cross out the word "ignoramus" and it wouldn't affect my argument one bit. Therefore it wasn't an ad hominem

15. ErotemeObelus,

"No they were not ignoramus" is not an "argument." You're pandering to yourself.

13. The Axiom of Induction is one of those things that can't be rooted in experience.

1. Drew

Yes it can.

You are stranded on a desert island in the midst of the Pacific Ocean with the opportunity to float off a bottle with a note requesting rescue: it may not be knowable how probable it is that this action will be successful, but based on experience it may be known that it will be successful if anything is, and hence the strategy is rational.

14. Did Frege ever say what experience made his theory of truth itself true?

1. Machinephilosophy

Yes.

Frege used the term sense to ground his theories, i.e. the matter of experience.

15. Michael Huemer shows that there is no empirical knowledge without a priori assumptions.

16. This idea of Huemer is also in Einstein and Hegel [Logic in his encyclopedia i think around 162] and Einstein in his introduction to Russel. He was invited to make some comments and he said that you need the imput of pure reason and also empirical facts to have any kind of knowledge.

1. "Einstein in his introduction to Russel. He was invited to make some comments and he said that you need the imput of pure reason and also empirical facts to have any kind of knowledge."

There would be no General Relativity without pure mathematics; Einstein was not a mathematician so had to ask the help of his friend Marcel Grossmann to teach him differential geometry -- this is by no means an isolated instance, just probably the most notable one.

17. Mill didn't understand induction in mathematics or science. Unfortunately, we've still got his version of science, which ignores the goal of induction: definitions and postulates for deductive inference.

18. Regarding the number 0, it's interesting that it didn't exist in the classical world and was only imported from India (via the Muslims, I believe) in the medieval ages.

I've always thought that empirical fact was the easiest way to explain why the empiricist account is wrong.

Mill's account sounds much more like a description of the Roman system of numbers than the decimal system.

He is also remarkably crude because it's well-known - since Berkeley's critique of Newton in 'The Analyst' - that it is no easy feat to define with reference to experience, say, an infinitesimal.

And if Mill was really the empiricist that he claimed to be then had he not read his Berkeley, given that the latter was the founder of modern empiricism?

(I reckon Mill was in philosophy what Marx found him to be in economics: a gleaming mediocrity and the ideologue of an age).