tag:blogger.com,1999:blog-8954608646904080796.post2051478891193654114..comments2020-03-31T19:29:09.789-07:00Comments on Edward Feser: Grow up or shut upEdward Feserhttp://www.blogger.com/profile/13643921537838616224noreply@blogger.comBlogger609125tag:blogger.com,1999:blog-8954608646904080796.post-23326208326073592922012-07-20T02:15:01.381-07:002012-07-20T02:15:01.381-07:00doesn't cosmological argument say that "e...doesn't cosmological argument say that "everything that HAS A BEGINNING needs a cause"?<br />Or put it in different terms, only contingent beings need cause.<br />Since God does not have a beginning, and is not a contingent being, He does not have, nor need, a cause.Anonymoushttps://www.blogger.com/profile/05084506359778238012noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-7054858627554688572011-08-18T18:02:29.425-07:002011-08-18T18:02:29.425-07:00@dguller:
"There is only one of sense of tru...@dguller:<br /><br />"<i>There is only one of sense of truth when it comes to mathematics. What you call true2 is either meaningless or completely circular. That Euclidean Geometry is true of the world, just like that, without any extra qualifications, is a meaningless statement.</i><br /><br />Euclidean geometry states that the internal angles of a triangle will add up to 180 degrees. This is a truth of mathematics, or what I would call an example of true1. Say I was to draw a triangle with a ruler, and then add up the internal angles, and they added up to 180 degrees. I contend that I can say that the Euclidean geometric truth is equally true in the empirical world, and thus is true2."<br /><br />Since you cannot point to any triangles in the real world, to do what you propose you have to interpret what a mathematical triangle is in the real world, in other words, you have to have a model or a physical theory to mediate between the mathematical and the real world. The reason you are blinded to this simple fact is that the examples you have in mind are deceptively simple and the interpretative act seems so obvious that pointing out is just pedantry. But things are more complicated when you start talking about Hilbert spaces and self-adjoint operators (quantum mechanics), manifolds and Riemannian metrics (general relativity), connections on vector bundles (Yang-Mills gauge theories), spin networks (loop quantum gravity), etc.<br /><br />To take up your example; suppose you draw a triangle, measure the internal angles and do *not* get 180 degrees -- for example, if you drawed the triangle in the surface of the earth. You will *not* say that Euclidean geometry is "false"; you will simply realize that your model is botched and that using Euclidean geometry is wrong. Or to put it in other words, there is no falsifiability criterion for mathematical statements, which is obvious since they are not statements about the physical universe. That is why your talk is either meaningless, or if you add qualifications like you did in your example, then you are talking about truth of the model or physical theory, not truth of the Euclidean geometry, at least not in any reasonable sense of the word.<br /><br />"<i>As far as your last statement, I doubt very much that that is the sense of truth that most people think of, but that hardly matters.</i><br /><br />I know that when someone tells me that it is raining, what I think of is that the proposition, “it is raining”, is derived from assumptions according to the rules of logic and reason. But anyway, this “hardly matters”, unless you want to remain dry."<br /><br />My bad here. "Hardly matters" should be "hardly matters for the point I want to make", or "I am just nitpicking".grodrigueshttps://www.blogger.com/profile/12366931909873380710noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-84795460942555830042011-08-18T11:50:24.389-07:002011-08-18T11:50:24.389-07:00grodrigues:
This is a stupid (and yet, revealing...grodrigues:<br /><br /><i> This is a stupid (and yet, revealing of your confusion) analogy for the simple reason that *no* literary critic (or anyone else for that matter) has ever contended that the statements produced by the literary criticism are empirical truths or even more generally, truths about the "real world". Methinks, you are fighting mere figments of your imagination. </i><br /><br />First, that is irrelevant to the argument. Just because someone claims to be doing X does not mean that they are, in fact, doing X. A homeopath claims to be utilizing the law of opposites to heal illness, but that is not what they are doing at all, because it is all placebo.<br /><br />Second, literary critics’ work is operative on the assumption that they are describing a fictional world that does not exist in our world, but can still be objectively described and debated about. In a sense, this is similar to mathematicians, who study a mathematical world that also does not exist in our world, because its subjects are perfect and immutable, whereas our world is imperfect and mutable. I’m afraid that there are parallels between the two fields. And the bottom line is that I have no problem with either of them studying their respective worlds, but I reserve special respect for the conclusions of either that happen to also be operative in our empirical world, because we can use them to implement changes to better our lives.dgullernoreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-53627316092236764802011-08-18T11:50:04.203-07:002011-08-18T11:50:04.203-07:00Grodrigues:
You insist on the mistake of using t...Grodrigues:<br /><br /><i> You insist on the mistake of using the word "true" equivocally. </i><br /><br />How is that possible when I have specified unequivocally what the two senses of “true” are that I am using? <br /><br /><i> There is only one of sense of truth when it comes to mathematics. What you call true2 is either meaningless or completely circular. That Euclidean Geometry is true of the world, just like that, without any extra qualifications, is a meaningless statement. </i><br /><br />This may help.<br /><br />Euclidean geometry states that the internal angles of a triangle will add up to 180 degrees. This is a truth of mathematics, or what I would call an example of true1. Say I was to draw a triangle with a ruler, and then add up the internal angles, and they added up to 180 degrees. I contend that I can say that the Euclidean geometric truth is equally true in the empirical world, and thus is true2. <br /><br />It is like deducing a hypothesis from some assumptions, and then testing it to see if the world works in the way that the hypothesis predicts. The hypothesis, having been deduced from assumptions, is true1, but we are interested in seeing if it is equally true2, because we want to know if we can use that information in our lives in the material world. <br /><br />This is all that I mean, and I honestly cannot see how this is meaningless. Either mathematical truths, which are all true1, can also be confirmed to be operative in the empirical world, and thus be true2, or they cannot, and then they are just true1.<br /> <br /><i> As far as your last statement, I doubt very much that that is the sense of truth that most people think of, but that hardly matters. </i><br /><br />I know that when someone tells me that it is raining, what I think of is that the proposition, “it is raining”, is derived from assumptions according to the rules of logic and reason. But anyway, this “hardly matters”, unless you want to remain dry.<br /><br /><i> And for the record, what you call truth1 or mathematical truth, is also an independent reality in the sense that mathematical truth is not subjective or dependent on the vagaries of historical or cultural contingency. </i><br /><br />I can agree with that. So maybe I have to add a true3 to my zoo of truths, i.e. “not subject to variation according to historical and cultural changes”. In that sense, what is true1 and true2 would also be true3. <br /><br /><i> In fact, mathematical truth, contrary to empirical truth, is necessary truth and as such eternally true: if the premises are true (including the premises that are usually left implicit, like the basic axiomatic system in which we are working), the conclusions are true. </i><br /><br />And Harry Potter will have glasses and lightning bolt on his forehead for all eternity, given the assumptions of Rowling’s imagination. That is why I would put mathematical truths and literary truths in the same category of true1.dgullernoreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-45038045421264699362011-08-18T08:11:21.531-07:002011-08-18T08:11:21.531-07:00@dguller:
"As I mentioned above, there are a...@dguller:<br /><br />"As I mentioned above, there are a number of senses of “true” when it comes to mathematics. There is true1, which is true relative to the assumptions of a particular system. There is true2, which is true relative to real state of affairs in the world."<br /><br />You insist on the mistake of using the word "true" equivocally. There is only one of sense of truth when it comes to mathematics. What you call true2 is either meaningless or completely circular. That Euclidean Geometry is true of the world, just like that, without any extra qualifications, is a meaningless statement. What is true or false, as far as it concerns the outside physical, mind-independent world is that some model or physical theory that happens to use Euclidean geometry is true or false, in the sense that it is falsified or not by the observations -- what we can call empirical truth. That some model or physical theory is true or false has nothing to do with the mathematical truth of Euclidean geometry or whatever mathematical theory it happens to use; it is a category mistake to confound the two. <br /><br />"My argument is that mathematics is always true1, but not always true2. "<br /><br />This is not an argument, it is a statement, and as I said, a meaningless one.<br /><br />"I am more interested in truth2 than truth1, because truth2 actually refers to independent reality, and is what most people mean when they say that something is “true”."<br /><br />So you are interested in mathematics only as it concerns what we usually call the natural world? That is a fairly common attitude among non-mathematicians. As far as your last statement, I doubt very much that that is the sense of truth that most people think of, but that hardly matters. And for the record, what you call truth1 or mathematical truth, is also an independent reality in the sense that mathematical truth is not subjective or dependent on the vagaries of historical or cultural contingency. In fact, mathematical truth, contrary to empirical truth, is necessary truth and as such eternally true: if the premises are true (including the premises that are usually left implicit, like the basic axiomatic system in which we are working), the conclusions are true.<br /><br />"However, one can also say that literary criticism is equally “an independent and autonomous discipline, with its own conceptual framework and its own standards of value”."<br /><br />Indeed it is.<br /><br />"Does it follow that the conclusions of literary criticism are true2? Of course not." <br /><br />This is a stupid (and yet, revealing of your confusion) analogy for the simple reason that *no* literary critic (or anyone else for that matter) has ever contended that the statements produced by the literary criticism are empirical truths or even more generally, truths about the "real world". Methinks, you are fighting mere figments of your imagination.<br /><br />"If your focus of interest is truth1, then be my guest and knock yourself out. If your main interest is in coherence rather than correspondence, then I will not denigrate your interests or your efforts. All I can say is that I am equally interested, if not more so, in truth2 and correspondence to reality than in truth1 and logical coherence."<br /><br />Do not turn the tables; I never expressed a value judgment about what is more interesting or important so telling me to "knock myself out" is completely misplaced. You on the other hand, have made it quite clear what is more interesting and important to you. As I said, opinions like that are perfectly normal, even healthy and defensible, at least up to a certain point.grodrigueshttps://www.blogger.com/profile/12366931909873380710noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-16243490762375607352011-08-17T09:15:04.971-07:002011-08-17T09:15:04.971-07:00Grodrigues:
>> You insist on using adjectiv...Grodrigues:<br /><br />>> 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.<br /><br />As I mentioned above, there are a number of senses of “true” when it comes to mathematics. There is true1, which is true relative to the assumptions of a particular system. There is true2, which is true relative to real state of affairs in the world. That Harry Potter wears glasses is true1, but not true2, for example. Sometimes true1 is coextensive with true2, such as mathematical theorems that are applicable to empirical reality. In other words, all cases of true2 must be true1, but not all cases of true1 must be true2. <br /><br />My argument is that mathematics is always true1, but not always true2. I am more interested in truth2 than truth1, because truth2 actually refers to independent reality, and is what most people mean when they say that something is “true”. <br /><br />>> 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.<br /><br />I agree with everything that you just wrote. However, one can also say that literary criticism is equally “an independent and autonomous discipline, with its own conceptual framework and its own standards of value”. Does it follow that the conclusions of literary criticism are true2? Of course not. (See: Potter, Harry.) If your focus of interest is truth1, then be my guest and knock yourself out. If your main interest is in coherence rather than correspondence, then I will not denigrate your interests or your efforts. All I can say is that I am equally interested, if not more so, in truth2 and correspondence to reality than in truth1 and logical coherence.dgullernoreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-33714759483302881112011-08-17T05:47:42.454-07:002011-08-17T05:47:42.454-07:00djindra said...
I've tried to make it clear th...djindra said...<br /><i>I've tried to make it clear that my interest in this issue is about how math-truth relates to nominalism. </i><br /><br />That's simple. Mathematics is a formal discipline, like logic, philosophy, or law. Mathematical truths are determined by consistentcy/provability within the preferred axiom system using teh preferred truth calculus, much like the truths of philosophy, logic, or law.<br /><br /><i>If math is supposedly 100% autonomous then there is no standard by which we can test its truth claims. </i><br /><br />You are correct. Formal systems can be relatively reliable or unreliable models of reality, but they are not tested in ther sense scientific theories are tested, in part becasue the nature of formal systems is to make no empircal claims.<br /><br /><i>It might as well be astrology. </i><br /><br />Last I checked, astrology made empircal claims. So no, not like astrology.<br /><br /><i>So if grodrigues is correct, then mathematics poses no problem for nominalism. If I'm correct, it poses no more of a problem for nominalism than words like "apple" do. Either way, math itself is irrelevant in the nominalism debate.</i><br /><br />Since you can adopt a nominalistic or and realist apraoch to mathematics, and still do mathematics, I agree mathematics is irrelevant to the "nominalism debate". On the other hand, since nominalism and realism themselves are merely formal constructs, how could it be otherwise?One Browhttps://www.blogger.com/profile/11938816242512563561noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-50146682298990722282011-08-17T05:08:08.009-07:002011-08-17T05:08:08.009-07:00@djindra:
"If it's just a matter of appe...@djindra:<br /><br />"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."<br /><br />I have purposely strayed away from the philosophical foundations of mathematics. I explicitly stated that there are other sources to justify the axioms and, what is more important, I simply do not need to make appeals to philosophy to show that mathematics is an autonomous discipline, which is another, different question, from the justification of its axioms. The fixation with nominalism is yours, not mine -- I never set out to refute it, or to defend any philosophy of mathematics for that matter. Nominalism is simply irrelevant to the question of the autonomy of mathematics.<br /><br />As far as the core of the dispute, according to you, what I have said about the autonomy of mathematics is not a fact. Since we are talking about matters of fact, it should be pretty easy to establish that you are right and that I am wrong. You only have to do two things:<br /><br />1. Show that you understand what is meant by autonomous discipline.<br /><br />2. Show that mathematics is not an autonomous discipline.<br /><br />Responses such as an appeal to examples of applied mathematics do not work. I never disputed that there is such a thing as applied mathematics (this is misnomer, but for my purposes here let this pass) and it matters not one whit to the autonomy of mathematics. Or in other words, it only matters *if* you can show that *all* mathematics, as practiced by mathematicians today, is nothing but applied mathematics. Either way, put up or shut up.grodrigueshttps://www.blogger.com/profile/12366931909873380710noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-82172366392169042932011-08-16T19:41:53.493-07:002011-08-16T19:41:53.493-07:00One Brow,
"You can offer personal judgments ...One Brow,<br /><br /><i>"You can offer personal judgments about the values of one result or another, but that won't change the autonomy of the practice of mathematics."</i><br /><br />I've tried to make it clear that my interest in this issue is about how math-truth relates to nominalism. If math is supposedly 100% autonomous then there is no standard by which we can test its truth claims. It might as well be astrology. So if grodrigues is correct, then mathematics poses no problem for nominalism. If I'm correct, it poses no more of a problem for nominalism than words like "apple" do. Either way, math itself is irrelevant in the nominalism debate.Don Jindrahttps://www.blogger.com/profile/05550378223563435764noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-51765180894456062452011-08-16T19:32:20.795-07:002011-08-16T19:32:20.795-07:00grodrigues,
Me:"We can dismiss 'Appeals ...grodrigues,<br /><br />Me:<i>"We can dismiss 'Appeals to your favored philosophical school' as a non-answer."</i><br /><br />You: <i>"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."</i><br /><br />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.<br /><br /><i>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.</i><br /><br />Of course it is. You misunderstand if you think I don't acknowledge that.<br /><br /><i>"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?"</i><br /><br />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.Don Jindrahttps://www.blogger.com/profile/05550378223563435764noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-31589455290798438942011-08-16T19:13:01.804-07:002011-08-16T19:13:01.804-07:00grodrigues,
"That some branch of mathematics...grodrigues,<br /><br /><i>"That some branch of mathematics has some practical application does not make it more true in any reasonable sense of the word."</i><br /><br />It not only makes it true but is critical in confirming it can be true.<br /><br /><i>"'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."</i><br /><br />Your objection is a sloven ignorance that only someone who does not understand truth, meaning and/or presuppositions can produce.<br /><br /><i>"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. </i><br /><br />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.Don Jindrahttps://www.blogger.com/profile/05550378223563435764noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-59130282574121592312011-08-16T13:27:55.029-07:002011-08-16T13:27:55.029-07:00djindra said...
LOL! Wouldn't it be nice if em...djindra said...<br /><i>LOL! Wouldn't it be nice if emphasis is all it took to make opinion a fact.</i><br /><br />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.<br /><br />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.One Browhttps://www.blogger.com/profile/11938816242512563561noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-11118955185439506352011-08-16T06:35:38.621-07:002011-08-16T06:35:38.621-07:00grodrigues,
I emphasize that this is not my opini...grodrigues,<br /><br /><i>I emphasize that this is not my opinion, it is a *fact*.</i><br /><br />LOL! Wouldn't it be nice if emphasis is all it took to make opinion a fact.Don Jindrahttps://www.blogger.com/profile/05550378223563435764noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-66774210475473745652011-08-15T05:05:02.917-07:002011-08-15T05:05:02.917-07:00@dguller:
There is a lot of sloppy talk in your p...@dguller:<br /><br />There is a lot of sloppy talk in your post, but I will concentrate on the following paragraph:<br /><br />"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."<br /><br />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.<br /><br />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.<br /><br />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.grodrigueshttps://www.blogger.com/profile/12366931909873380710noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-38372218670611666872011-08-13T09:12:19.487-07:002011-08-13T09:12:19.487-07:00First, a Platonist would disagree with you as he a...<i>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?</i><br /><br />A quite reasonable position to take... as it falls to the Platonist to provide evidence for her beliefs.StoneTopnoreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-5498895879937500062011-08-12T18:25:28.482-07:002011-08-12T18:25:28.482-07:00Grodrigues:
>> The talk about falsifying &q...Grodrigues:<br /><br />>> 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.<br /><br />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.<br /><br />>> Since you define "genuinely real" as having an impact upon the empirical world, you are just restating your belief that Platonism is false. <br /><br />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.<br /><br />>> Actually, you are stating more. Are you saying that there is no reality apart from what is described by the empirical sciences?<br /><br />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. <br /><br />>> 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.<br /><br />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.<br /><br />>> 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. <br /><br />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.<br /><br />>> 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"?<br /><br />It’s a good thing that I never used True or False, but only their more humble cousins, true and false.<br /><br />>> What does it mean to say that DNA, which is a concrete object of our physical reality, is true or false?<br /><br />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.<br /><br />Hope this helps.dgullernoreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-10511568751473933162011-08-12T15:31:11.464-07:002011-08-12T15:31:11.464-07:00@dguller:
"What I mean is that if a mathemat...@dguller:<br /><br />"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."<br /><br />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.<br /><br />"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."<br /><br />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?<br /><br />"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."<br /><br />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.<br /><br />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.<br /><br />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"?<br /><br />"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."<br /><br />What does it mean to say that DNA, which is a concrete object of our physical reality, is true or false?grodrigueshttps://www.blogger.com/profile/12366931909873380710noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-43490083880256670302011-08-12T13:51:37.762-07:002011-08-12T13:51:37.762-07:00jack bodie said...
You'll have to help me out:...jack bodie said...<br /><i>You'll have to help me out: what is the main point? </i><br /><br />Mathematics is a tool created by humans to be used to help explain the world, with no inherent truths of its own.<br /><br /><i>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?"</i><br /><br />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.<br /><br /><i>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; </i><br /><br />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.<br /><br /><i>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.</i><br /><br />Yes, correctly. Why is their description, using the "+" they have created for addition in their reality, of 2+2=5, wrong?<br /><br /><i>Is there some paragraph-to-content quotient that I've violated?</i><br /><br />None of which I'm aware. I'll email the paragraph-to-content police and ask them.One Browhttps://www.blogger.com/profile/11938816242512563561noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-25567067697600359192011-08-12T12:49:46.309-07:002011-08-12T12:49:46.309-07:00OneBrow:
Out of all those paragraphs, the only ti...OneBrow:<br /><br /><i>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?</i><br /><br />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?"<br /><br />Each of the following three paragraphs points out that your imaginary world is <i>not</i>, 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.<br /><br />Is there some paragraph-to-content quotient that I've violated?jack bodienoreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-12884171525295732502011-08-12T12:00:28.893-07:002011-08-12T12:00:28.893-07:00Grodrigues:
>> First, a Platonist would dis...Grodrigues:<br /><br />>> 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? <br /><br />Yup.<br /><br />>> Or even more extremely, as your simile with fiction seems to imply, are you arguing that Formalism is true? <br /><br />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.<br /><br />>> 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.<br /><br />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. <br /><br />>> 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.<br /><br />Perhaps I should clarify. <br /><br />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.<br /><br />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.<br /><br />I hope that helps.dgullernoreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-64752366349775336382011-08-12T11:54:15.090-07:002011-08-12T11:54:15.090-07:00grodrigues said...
Yes, but "sufficient appro...grodrigues said...<br /><i>Yes, but "sufficient approximation" to what? </i><br /><br />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.<br /><br /><i>If it does not approximate the solution, it is not very "practical", is it? </i><br /><br />Again, this is the way I naturally think. However, for pratical applicaitons, it seems to be unnecessary.One Browhttps://www.blogger.com/profile/11938816242512563561noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-82722188464535972162011-08-12T11:48:22.614-07:002011-08-12T11:48:22.614-07:00jack bodie said...
Well, several reasons: no doubt...jack bodie said...<br /><i>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." </i><br /><br />Why do yuou think this would lead to them creating a mathematics that did not reflect their reality?<br /><br /><i>This would encourage them to take account of that fact in any systems they actually did create. </i><br /><br />In what way?<br /><br /><i>Yes, I am just asserting this but you seem ok with the move.</i><br /><br />We're all just speculating.<br /><br /><i>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. </i><br /><br />So, 4 things plus 1 other thing is not 5 things? That seems to be splitting a very fine hair.<br /><br /><i>You seem to be insisting that we further imagine them to be adept in math but prevented from practising philosophy or natural science, </i><br /><br />I don't recall on insisting any such thing.<br /><br /><i>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.</i><br /><br />Why is that wrong?<br /><br /><i>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. </i><br /><br />Why does that change their method of addition?<br /><br /><i>How exactly does their level of intelligence make the case that 2+2=5?</i><br /><br />As far as I know, it doesn't.<br /><br />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?<br /><br /><i>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.</i><br /><br />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.<br /><br /><i>yup, and all for a position I'm not even sure he endorses! </i><br /><br />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.One Browhttps://www.blogger.com/profile/11938816242512563561noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-24849886742195202302011-08-12T11:34:36.186-07:002011-08-12T11:34:36.186-07:00@dguller:
"The former is in the same categor...@dguller:<br /><br />"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."<br /><br />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.<br /><br />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.<br /><br />"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."<br /><br />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.<br /><br />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.grodrigueshttps://www.blogger.com/profile/12366931909873380710noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-81782830396368402352011-08-12T11:29:27.865-07:002011-08-12T11:29:27.865-07:00@OneBrow:
"But to pull off this strategy, co...@OneBrow:<br /><br />"<i>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><br /><br />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."<br /><br />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.grodrigueshttps://www.blogger.com/profile/12366931909873380710noreply@blogger.comtag:blogger.com,1999:blog-8954608646904080796.post-76709187428196787012011-08-12T11:23:23.194-07:002011-08-12T11:23:23.194-07:00Look, all other nonsense aside I think there's...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.<br /><br />@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?jack bodienoreply@blogger.com