tag:blogger.com,1999:blog-3772414331480635861.post734759721192854919..comments2024-07-01T20:08:17.142-07:00Comments on bylogos: Philosophers in Athensjohn bylhttp://www.blogger.com/profile/05766117392831032432noreply@blogger.comBlogger16125tag:blogger.com,1999:blog-3772414331480635861.post-37149144989989615172011-06-11T09:25:13.082-07:002011-06-11T09:25:13.082-07:00How can set theory (or other branches of math) be ...How can set theory (or other branches of math) be *useful* if they are not *true*?<br /><br />"<i>That itself is an inductive argument, and so it more or less begs the question.</i>"<br />The question-begging was self-conscious, and part of my point. What is the "usefulness" of questioning induction? Can you somehow prepare for a hypothetical day when the laws of nature no longer apply?<br /><br />I don't think your attempt to place an assumption of regularities behind/above/beyond God works. God is not bound to keep things regular, but is free to violate induction whenever he wants. It's called "miracles".<br /><br />Anyways, I'm glad you want to investigate the justification of math, rather than just math itself. You can't sustain atheism if you care about questions like that. (Unless, I guess, you go constructivist like Bishop)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-28000676558407697762011-06-11T03:50:06.519-07:002011-06-11T03:50:06.519-07:00confessionalouthouse,
I wouldn't even concede...confessionalouthouse,<br /><br />I wouldn't even concede that much. As an atheist mathematician-in-training, here's my position:<br /><br />First of all, I don't think the alleged problems have been clearly articulated. At least I still don't understand what they are supposed to be. For example, John Byl says, "The basic problem is: how do we know that our math is true?" But I would ask, what does it even MEAN for math to be "true"? Sure, individual mathematical statements have truth values, but I don't know how to make sense of assigning a truth value to something like set theory itself, or the entire field of mathematics, etc. I can say---and have said---that set theory is USEFUL. What more could we ask from it?<br /><br />Equally puzzling to me is how God is supposed to help solve whatever problems we might have. In his paper for instance, John Byl remarks that God "upholds all truths" (p83). Again, though, I'm not sure what that is supposed to mean. Maybe he's just saying that God causes the world to be a certain way. But how does that help us determine what is true and what is false? Or maybe he wants to say something like, on nontheism we have no reason to trust our truth-seeking faculties, whereas theism gives us a reason. So in that case his argument would be a sort of re-statement of Plantinga's evolutionary argument against naturalism (EAAN). Of course I think the EAAN has lots of problems, many of which have been pointed out by philosophers with credentials well beyond mine. ; )<br /><br />As for induction, I don't think it's appropriate to say that it has always worked in the past, and so we should continue to use it in the future. That itself is an inductive argument, and so it more or less begs the question. Instead I would simply point out that induction is part of what we mean by terms like "justification" and "warrant" and such. In other words, if you ask, "why do you believe X?" and I respond, "because I make an inductive inference to X," then that is a satisfactory response (provided my inference meets with an agreeable inductive standard). But more importantly, God doesn't help us with induction, either. Presuppositionalists sometimes want to say that God makes it the case that the future will be like the past, or something along those lines. But "makes" how? Using his causal powers, presumably. But causal powers only make sense if we already have an inductive standard in place. To say that God "causes" X to occur is to say that X is part of some regularity whereby God wills things to happen and they accordingly do. But then we still have to assume in advance that there are such things as regularities, which is just another way to express the inductive principle. Either way we have to rely on the assumption of regularities---i.e., on induction. Invoking the existence of God doesn't help us escape that.<br /><br />Anyway, I hope that's clear.<br /><br />--BenBen Wallishttps://www.blogger.com/profile/00131358613835119782noreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-16789699185581720482011-06-10T19:05:00.818-07:002011-06-10T19:05:00.818-07:00If I were an atheist, I would concede, maybe you&#...If I were an atheist, I would concede, maybe you're right, but so what? The inductive principle may not give us any certain guarantee that the "laws" we have observed in the past will continue to hold, but it's what we have, and it's worked so far, so why shouldn't we assume they will continue? What else should we do, just give up? My plan is to continue doing math and science, to continue depending on induction. You're not arguing that I shouldn't rely on induction, and I'm not worried about it, so what's the problem? You can have fun justifying math, if that interests you, I'll keep myself busy doing math (including using the Axiom of Choice and transfinite numbers).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-41903180862372454252011-06-10T16:17:25.214-07:002011-06-10T16:17:25.214-07:00In answer to your questions please read my article...In answer to your questions please read my article <br /><a href="http://www.csc.twu.ca/byl/matter_math_god.pdf" rel="nofollow">Matter, Math and God</a>, particularly the second part. <br /><br />The basic problem is: how do we know that our math is true? Math may be very useful. Yet, if it possibly contains inconsistencies,then mathematical models of, say, nuclear reactors, may turn out to be disastrously faulty.<br /><br />Brouwer and Bishop try to build up math using self-evident axioms and finite methods. This yields a consistent math but at the price of leaving out much classical math.<br /><br />The justification of classical math requires such things as the justification of the Axiom of Choice and the existence of transfinite numbers (higher orders of infinity).John Bylnoreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-72709839843467919962011-06-09T19:04:41.504-07:002011-06-09T19:04:41.504-07:00John Byl,
Thanks for the response! However, I mu...John Byl,<br /><br />Thanks for the response! However, I must still disagree with your assessment.<br /><br />Regarding Brouwer and Bishop: their arguments seem somewhat obscure to me, so it's difficult to tell what precisely they meant. However, as best I can make out, they objected that when math becomes too far idealized it loses its usefulness. So maybe God can make use of the idealized math if he is available to do so, but if we want to be able to use the math here on earth then we'll have to restrict ourselves to finite methods. Now, personally I don't buy into their claims. Few mathematicians do. It seems to me that non-constructivist math is enormously useful. But even if they were correct---if idealized math wasn't useful to us after all---then I don't see how the existence of God would change that. What usefulness does mathematics have with God that it hasn't got without?<br /><br />Also, you expressed some concern about the "validity" of math. But validity in what sense? We seem both to agree that math is incredibly useful, so it's valid in that sense. Are you perhaps doubting the validity of the arguments of mathematicians? But then how does God help us construct valid arguments?<br /><br />All in all I'm just really puzzled by how you think God helps us do/use/trust/etc. math. Hopefully you have time to say a few things about all this.<br /><br />Thanks again,<br /><br />--BenBen Wallishttps://www.blogger.com/profile/00131358613835119782noreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-19718137248964754602011-06-09T16:36:49.456-07:002011-06-09T16:36:49.456-07:00Interesting. Ya got me there. I don't know abo...Interesting. Ya got me there. I don't know about ZFC and I'm not sure what two-valued logic is (an anti-Go"del assumption that all propositions are decidable?)<br /><br />But still, I don't see how infinite sets are implausible from a naturalist view. I know loads of naturalists who work with {The Set of Integers} or {The Set of Real Numbers} all the time, and they don't find them implausible.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-85930002790050032632011-06-09T14:56:11.211-07:002011-06-09T14:56:11.211-07:00Actually, I was thinking of Go"del's 2nd ...Actually, I was thinking of Go"del's 2nd theorem: the consistency of a system S large enough to contain full arithmetic cannot be proven by methods within S. <br /><br />The problem then is to justify the soundness of full arithmetic and more advanced math, such as calculus. <br /><br />These can be derived from Zermelo-Fraenkel-Choice (ZFC) set theory but the ZFC axioms posit infinite sets and two-valued logic, which are plausible from a theistic perspective but not from a naturalist view.John Bylnoreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-13723373709576096712011-06-09T09:38:13.930-07:002011-06-09T09:38:13.930-07:00Trying again, if you are not immediately convinced...Trying again, if you are not immediately convinced of my self-evident brilliance, it is because my comment was better the first time...<br /><br />I am not convinced by Bishop's argument (or affirmation -- I don't know his argument) that mathematical operations can only be "carried out" (<i>done</i>) by an infinite being. Finite beings were doing just fine for millenia before Bishop came along, and continue doing just fine.<br /><br />As for Go"del, I don't see the point. <br />I don't think Go"del was asking epistemological questions about validity of proofs. He simply proved that there exist (must exist in systems with sufficiently complex axioms to provide elementary arithmetic) well-formed propositions which cannot be proven true or false, because they are neither true nor false, but a third category "undecidable"; self-contradictory whether you try to apply true or false to them (like "this statement is false"). Note, this is different from "We can't know whether any proof of proposition P is valid"; Go"del constructed a proposition P for which he was able to prove "P is not true and P is not false" (two implications of which are that P cannot be validly proven true, and P cannot be validly proven false).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-66338852622542743172011-06-09T09:25:24.480-07:002011-06-09T09:25:24.480-07:00#$%*@$)(%@$&()%@$ Stupid blogger/google trying...#$%*@$)(%@$&()%@$ Stupid blogger/google trying to make me sign up for a blogger account and lost my comment!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-34090775222840677352011-06-09T07:09:51.307-07:002011-06-09T07:09:51.307-07:00Mathematicians do not need to be philosophers in o...Mathematicians do not need to be philosophers in order to <em>do</em> math. The difficulty, however, is how to account for the <em>validity </em>of their math. Recall Godel's theorems. How do we know that a particular theorem is true? what counts as a valid proof?<br /><br /><a href="http://www.csc.twu.ca/byl/matter_math_god.pdf" rel="nofollow">I argue</a> that naturalism cannot account for classical mathematics. Bishop agrees. He writes:<br /><em>"Classical mathematics concerns itself with operations that can be carried out by God.I am doing my best to develop a secure philosophical foundation for current mathematical practice. The most solid foundation available at present seems to me to involve the consideration of a being with non-finite powers—call him God or whatever you will—in addition to the powers possessed by finite beings."</em><br /><br />Since Bishop doesn't believe in God, he is forced to find another basis for math. But his constructivism severely truncates math.John Bylnoreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-4374216465458126522011-06-07T10:52:54.056-07:002011-06-07T10:52:54.056-07:00Yes; are these mathematical systems he describes g...Yes; are these mathematical systems he describes going to take over and crumble mathematics because of failure to build from a Christian worldview? Or are they the result of a few non-mainstream sophists, who even the godless can clearly see are spinning their semantic wheels to no effect? I vote B.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-86796047048631756972011-06-07T07:30:19.471-07:002011-06-07T07:30:19.471-07:00confessionalouthouse,
Thanks for the link. It re...confessionalouthouse,<br /><br />Thanks for the link. It really disappoints me that a trained mathematician like Poythress could stray so far into that kind of nonsense. I guess it just goes to show that skilled mathematicians need not be good *philosophers* of mathematics.<br /><br />--BenBen Wallishttps://www.blogger.com/profile/00131358613835119782noreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-51123967902899679142011-06-06T19:52:54.434-07:002011-06-06T19:52:54.434-07:00Here is an article from Vern Poythress that talks ...<a href="http://frame-poythress.org/poythress_articles/1976Biblical.htm" rel="nofollow">Here is an article</a> from Vern Poythress that talks about that sort of stuff.<br /><br />But I agree with you; math "depends" on God in the trivial sense that if there were no God there would be no math (and no creation...). But belief in God is not necessary to do good and correct math. (Neither is it sufficient.)<br /><br />Also, I don't see that classical math is in any kind of crisis that it needs to be saved by constructivism or any other such.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-81698733440075184682011-06-05T07:33:14.295-07:002011-06-05T07:33:14.295-07:00In what way does classical mathematics depend on t...In what way does classical mathematics depend on the existence of a creator-deity? I hear this from time to time from presuppositionalists, but I don't know of any sense in which the statement can be defended.<br /><br />Even the constructivists seem to agree that invoking the existence of God won't save classical math. Bishop pointed out, for instance, that God can do all the math he wants, but it won't help us finite-minded human beings do OUR math. In other words, to the extent that we want to do math ourselves, we must rely on constructivist methods. Now, I don't agree with this view at all---I haven't found any good arguments for constructivism, or against classical math---but even assuming that the constructivists are correct, Christianity still won't help us reclaim any additional kinds of math. If there really is a problem with a classical approach, it doesn't seem that invoking God or Christianity helps us solve it.<br /><br />--Ben WallisBen Wallishttps://www.blogger.com/profile/00131358613835119782noreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-1629405443758680942010-01-15T09:43:07.976-08:002010-01-15T09:43:07.976-08:00Hi Ruberad
Welcome to my blog. You raise an inter...Hi Ruberad<br /><br />Welcome to my blog. You raise an interesting question. There are constructionist proofs for most classical theorems. These proofs can be much longer than classical ones, but once established those theorems can be utilized.<br /><br />Here is one possible constructive proof:<br /><br />Define x < y iff there is positive integer n such that y-x > 1/n.<br /><br />Define x neq y iff x 0).<br />Define x irrational iff for all rational a/b, x neq a/b. (Note that <br />this is stronger than not(x=y).)<br /><br />Consider rational a/b. We can prove (by usual means) that a^2 - 2b^2 <br />neq 0. (This is integer neq, of course.)<br />Now,<br /><br />abs( a/b - sqrt(2) ) = abs( (a/b + sqrt(2)) )^-1 b^-2 abs (a^2 - <br />2b^2) > ((abs(a)+2)b^2)^-1<br /><br />So we get a/b neq sqrt(2) for all rational a/b, so sqrt(2) is <br />irrational.<br /><br />http://cs.nyu.edu/pipermail/fom/2005-October/009237.htmlJohn Bylnoreply@blogger.comtag:blogger.com,1999:blog-3772414331480635861.post-82578034714636069412010-01-15T07:54:11.919-08:002010-01-15T07:54:11.919-08:00That must have been an interesting experience!
Ab...That must have been an interesting experience!<br /><br />About the "constructionist" mathematics; I have read a little bit about this before; doesn't their finiteness also disallow them from using the law of the excluded middle (i.e. not(not(A)) doesn't necessarily imply A)?<br /><br />If this is the case, then it is not just "advanced theorems" that are lost, but even very simple ones. I've been wondering for years whether it is possible to prove that sqrt(2) is irrational -- WITHOUT a proof by contradiction (i.e. law of excluded middle). The only proofs I can find or think of are all of the form:<br /><br />Suppose there exist integers p and q such that (p/q)*(p/q)=2<br /><br />...<br /><br />Contradiction! Therefore sqrt(2) is irrational.<br /><br />I mean, if there are only "finite, human constructions", I don't even see how we could make a statement like "sqrt(2) is irrational", but only "none of the integers I've ever heard of have a ratio of sqrt(2)"Anonymousnoreply@blogger.com