tag:blogger.com,1999:blog-4987609114415205593.post6964120924897480010..comments2021-06-19T18:51:04.369+01:00Comments on M-Phi: Ultra-finitism, types and tokensJeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-4987609114415205593.post-60078048543423395142013-12-25T01:28:36.240+00:002013-12-25T01:28:36.240+00:00I think three-realm finitism is a version of the i...I think three-realm finitism is a version of the idea <br />first realm is where most useful finite numbers are <br />second realm is where gaps of describability start to appear<br />third realm the only aspect of the numbers is their potential describability<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-24171479792907615152011-10-10T21:45:24.086+01:002011-10-10T21:45:24.086+01:00a, "Well, and Nelson explicitly rejects (U), ...a, "Well, and Nelson explicitly rejects (U), as I pointed out"<br /><br />But I think Nelson means "physical token" when he says "numeral". If so, he thinks, "each number has an actual physical token". <br /><br />"Science postulates a transfinite number of sets? Please..."<br /><br />Would you agree that if $\phi : M \rightarrow \mathbb{R}$ is a scalar field on a spacetime model $(M, g_{ab}, T_{ab})$, then the cardinality of $\phi$ is transfinite?<br /><br />"Anyway, it's pretty clear that this is the end of the road. Bye."<br /><br />Ah, that's just your strict finitism talking. There is no end of the road, so do return!Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-61436961137773444932011-10-10T17:43:07.135+01:002011-10-10T17:43:07.135+01:00"UF consists in the certain philosophical cla..."UF consists in the certain philosophical claims:<br /><br />(U) Each number has a physical token."<br /><br />Well, and Nelson explicitly rejects (U), as I pointed out. So which ultrafinitists are you talking about?<br /><br />"for science postulates a countable supply of numbers, a transfinite supply of sets, functions, sequences, along with proofs, derivations, languages, propositions, models, vector spaces, manifolds, Lie groups, fibre bundles, etc, etc."<br /><br />Perhaps. (Science postulates a transfinite number of sets? Please...) But as you said you have a Quinean perspective, you are willing to jettison all of that should mathematics suitable for science be developed which doesn't need all of that baggage. And as I said the ultrafinitist is willing to make the prediction that this is possible. <br /><br />"So, that experience is finite has no logical relevance to ontology."<br /><br />Sure it does. Think about it.<br /><br />"...unless one insists on an idealist reduction of the world to subjective experiences." <br /><br />That, IMHO, is a strawman.<br /><br />Anyway, it's pretty clear that this is the end of the road. Bye.anoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-5064190719353872452011-10-10T16:13:25.991+01:002011-10-10T16:13:25.991+01:00"So I think an ultrafinitist position can be ..."So I think an ultrafinitist position can be made compatible with yours."<br /><br />If we are to analyse a philosophical view, we must write it down as precisely as possible, isolate its basic assumptions and subject it to criticism. <br />UF consists in the certain philosophical claims:<br /><br />(U) Each number has a physical token.<br />(Fin-Tok) There is a largest, "finite", physical token.<br /><br />The crucial claim is (U), which is false, as it contradicts science: for science postulates a countable supply of numbers, a transfinite supply of sets, functions, sequences, along with proofs, derivations, languages, propositions, models, vector spaces, manifolds, Lie groups, fibre bundles, etc, etc. <br /><br />"After all, the world as we experience it is finite."<br /><br />I take science as the epistemically best account of what the world is like: science states (indeed, presupposes) that there are numbers, functions, sets, propositions, vector spaces, fibre bundles, manifolds, etc.. etc. <br />This is not finite. <br /><br />The role of experience is epistemic, not ontological. So, that experience is finite has no logical relevance to ontology, unless one insists on an idealist reduction of the world to subjective experiences:<br /><br />(Idealism) Each thing is subjectively experienced.<br />(cf., Berkeley's "esse est percipi")<br /><br />But such philosophical opinions have little rational basis and contradict science, for science itself tells me that there are physical objects which do not affect my sense organs.<br /><br />All in all, we simply have a philosophical opinion which contradicts science and has no greater rational basis than the belief that the world was created 14 seconds ago.Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-1319624812801922712011-10-10T10:06:57.399+01:002011-10-10T10:06:57.399+01:00So I think an ultrafinitist position can be made c...So I think an ultrafinitist position can be made compatible with yours. The ultrafinitist would, I think, be willing to predict that the non-core can be replaceable with core when scientific justifications are needed. After all, the world as we experience it is finite.anoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-33270122612547791942011-10-09T16:27:35.203+01:002011-10-09T16:27:35.203+01:00(As I just yesterday remembered, Shaughan Lavine&#...(As I just yesterday remembered, Shaughan Lavine's book on set theory has some sort of strictly finite set theory, based on Mycielski.)<br /><br />a, "Rather, there is a core of arithmetic which is analytically true and so does not depend on Quinean justifications."<br /><br />This is something I can drop on my epistemological picture, and replace "core" with a matter of epistemic degree, rather than a sharp semantic boundary. Some things will be more core than others, of course, yes.<br /><br />There are two classic examples which some (e.g., St Andrews neo-logicism) have been argued to be analytic:<br /><br />Hume's Principle: $\#(X) = \#(Y) \leftrightarrow X \sim Y$<br /><br />Extensionality: $\forall x \forall y(\forall z(z \in x \leftrightarrow z \in y) \rightarrow x = y)$<br /><br />Once pointed out, and given their role and use, which has been extensively explored, it seems reasonable to say that these are inside the Quinian core - but the core is epistemic, not semantic.<br /><br />"Also, you basically committed yourself to the view that if someone can come up with simpler axiomatizations that also work, then mathematics will need to change."<br /><br />Indeed. But such arguments are about "what cannot be logically ruled out with certainty". *That* standard applies only to the philosophy seminar room! Rather, rational thinking works with degrees of uncertainty, presumably constrained by a probabilistic credence function of some sort. So, we rationally assign negligible probability to the prediction that mathematicians will (for whatever reasons - internal consistency, coherence, sensory experience, physics, etc.) eventually reject $PA$.<br /><br />Perhaps a strictly finite set theory might in fact be the optimal approach for someone interested in strict finitism.Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-80386369597469452492011-10-08T06:36:43.270+01:002011-10-08T06:36:43.270+01:00"For me, however, the reasons are Quinian: we..."For me, however, the reasons are Quinian: we postulate things for scientific reasons to account for our flux of experience. "<br /><br />I think you want to be careful here, because if you mean that *all* of arithmetic is postulated, then (x)(y)(z)((x+y)=z=>(y+x)=z) may turn out to be false if our experiences change. This does not seem right to me. Rather, there is a core of arithmetic which is analytically true and so does not depend on Quinean justifications. Otherwise, I have a certain sympathy with your Quinean view that there is a non-core part which is postulated and is accepted because it has worked. If you accept this view as correct, though, then there are some natural questions: which part of mathematics is analytic and which postulates can only be justified because they work? Also, you basically committed yourself to the view that if someone can come up with simpler axiomatizations that also work, then mathematics will need to change.anoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-62625470469982882902011-10-07T22:12:30.127+01:002011-10-07T22:12:30.127+01:00a, thanks for the comment; your hint that I need t...a, thanks for the comment; your hint that I need to get some exact arguments from ultrafinitists written down is right! <br />Ultra-finitism just isn't there in the usual Phil Maths literature (though there is lots on nominalism), so I'd want to get a bit clearer what the view is. But I'm fairly sure that the epistemologically bottom level that they're happy with consists in tokens and token computations, rather than numbers; so, I think (3) is roughly the right interpretation - they're nominalists, rather than strict finitists.<br /><br />"(What is your argument?)"<br /><br />I take the point entirely about the burden of proof. (Though there are Russellian arguments that over-egged scepticism is as unreasonable as over-egged confidence.)<br />People will have difference grounds for accepting arithmetic (and more) as true; e.g., direct intuition, logicism, structuralism, modal constructiblity. For me, however, the reasons are Quinian: we postulate things for scientific reasons to account for our flux of experience. So, epistemologically, I see no difference of kind between my hands, the cells they are composed of, neutrinos and $\mathbb{R}^4$. (And, the homeric gods and characters from Dickens, but they do me no use in science.) I do have some sympathy with GĂ¶del's view that the human mind can cognize abstract mathematical reality, but I'll stick with the Quinian epistemology for the moment!<br /><br />Here's another argument, this time directed against some sort of ultra-finitist position. I written down a (token of a) formula $B$ with 61 symbols. There is a mathematical proof that it is inconsistent, but also a proofone that the shortest derivation of its inconsistency will never be tokened. The mathematical proof isn't ultra-finitistically acceptable. Is it inconsistent? If there is no token witness that $B$ is inconsistent, is a merely existential guarantee, based on mathematical considerations, good enough?<br /><br />Yes, I noticed what Nelson calls the finitary credo (which he goes on to argue against): I think he meant, "for any x, there is a numeral d such that Val(d) = x". This is a theorem of PA, of course, but it depends on a bit of induction (my guess is $I \Delta_0$).<br /><br />The usual justification of induction is: $\mathbb{N}$ is the closure of $\{0\}$ under successor. I guess that Nelson counts that as circular. <br /><br />But there's an epistemological stand-off here. <br />He rejects hypothetico-deductivism, and wants to build up from very meagre beginnings. I accept hypothetico-deductivism. I open a physics textbook and I accept the mathematical theory given as true; and I accept the mixed mathematical laws as approximately true. <br />I say, "How do you justify the predictions of quantum electrodynamics?". The reply is, "Our best theories are only useful instruments, and their utility doesn't imply their truth: also, like all past theories, they will be overthrown". And on it goes ...<br /><br />In the end, I think this becomes a debate about scientific instrumentalism vs. scientific realism (and connected to the theory of how mathematics is applied).Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-91260244856439603932011-10-07T21:07:14.767+01:002011-10-07T21:07:14.767+01:00Great - thanks for the pointer.
(In the post, I&#...Great - thanks for the pointer. <br />(In the post, I'm musing about how to figure out what mathematicians who are ultra-finitists (Nelson, Zeilberger and some others) mean when they sketch their philosophical views. Actually, I think (3) is closest to what they mean: the view is really nominalism. But I should probably write out some detailed quotes though.)<br /><br />I quickly read the Landini paper, but it's very dense in Russell-Frege notation! It describes a Russellian type-theoretic logicism, with an axiom of infinity, and then one can see what happens without that axiom (or with modifications). We think of our finitely many tokens as the individuals, and numbers are second-level properties rather than objects. But then we lose some arithmetic.Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-59013939636372342092011-10-07T20:10:15.865+01:002011-10-07T20:10:15.865+01:00Nelson in Elements (p. 11) gives the label "f...Nelson in Elements (p. 11) gives the label "finitary credo" to (U), which is spelled out as "for every number x there exists a numeral d such that x is d." However, he explicitly *rejects* the credo and, indeed, uses this rejection as the basis for his rejection of induction. (His argument is a bit tenuous because he seems to assume that the only possible justification for induction is the finitary credo. Therefore, by rejecting the credo, he has provided a satisfactory justification for a rejection of induction.)<br /><br />Maybe there are ultra-finitists who reason as you have said. My guess is there are as many reasons as there are ultra-finitists, so a discussion about what ultra-finitists think, as if there is some uniformity, is likely not to go very far.<br /><br />For instance I hold a certain dubiousness towards large numbers, but I certainly don't reason as you propose. On the contrary, I simply see a lack of good reasons to accept their existence. In short I think the burden of proof is clearly on the other side. If you think there are very large numbers, then it is you, by positing the existence of something, who must come up with a good reason why we should believe in them. (What is your argument?)anoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-31017058619312607612011-10-07T04:16:32.333+01:002011-10-07T04:16:32.333+01:00In "The Number of Numbers" (http://tinyu...In "The Number of Numbers" (http://tinyurl.com/3wrkfsq) Greg Landini discusses finitism in the context of the an interpretation of Russellian logicism that rejects the thesis that numbers are objects. Accordingly, the token/type point that you raise here may not affect the view Landini describes.jrshipleyhttps://www.blogger.com/profile/05991272871497674850noreply@blogger.com