Talk:Finitism

Odd
It's odd to me that the article doesn't mention Hilbert or the axiomatization of mathematics... or is this only Geometical Finitism?

reference:   --Graatz 18:01, 5 October 2005 (UTC)

Suggestion for expansion
The article on ultrafinitism is in bad shape, and should be either cleaned up or merged with this one. I say this because ultrafinitism is such a fringe philosophy. As such its inclusion here as opposed to being in a separate article would give some cohesion to the collection of articles on finitism.

History
Where did the concept originate? Who were the original contributers?

The idea is certainly a part of formalism. It seems much clarification is required for this to be a really useful article.

Countably infinite number of steps
I removed a claim about constructivists allowing a countably infinite number of steps in construction. I'm not sure what that was really trying to say, but it sure doesn't sound right. It almost sounds like it's saying the difference between finitism and constructivism is that the latter allows countable supertasks, and that is simply wrong. --Unzerlegbarkeit (talk) 13:25, 13 June 2008 (UTC)

Hilbert's finitism

 * An important thing to understand is that Hilbert's finitism is constrained solely on the length of mathematical proofs. Hilbert did not demand finitism of models, but instead he embraced the very source of transfinitism: "No one shall expel us from the Paradise that Cantor has created for us". It is not hard to understand that an infinitely long proof is impossible: a proof that never ends, is not a proof. Finitists deny the infinity of models too. According to Löwenheim-Skolem theorem LwS, all talk about innumerable infinite models can be substituted by the talk about numerably infinite models. Therefore, LwS is at least somewhat finitist in nature. []

I don't think so. Hilbert's finitism was full-blown finitism. His point was supposed to be that transfinite methods are safe as they could be justified as a conservative extension of finitary methods. Also, I don't know that LwS can be considered finitist. On the face of it, it certainly isn't. --Unzerlegbarkeit (talk) 15:20, 27 June 2008 (UTC)

The "God created the integers" quote may be unsourceable
This is Kronecker's most famous quote, so I was surprised when a scholar in the area told me that he has searched for its source and can't find it. If anyone can come up with the primary source -- something written by Kronecker himself or (at minimum) a direct witness who heard him say it -- I'd be grateful and will pass it on.

Yogi Berra once said that some of the best things he said he never said, and I fear Kronecker's line about God and the integers may be an instance of the same thing. --Jeffreykegler (talk) 16:38, 18 July 2008 (UTC)


 * People seem to say it was first printed in H. M. Weber's memorial article of him. For instance here for a quote and translation of the appropriate passag which sounds like it would meet your lesser requirement of a direct witness. --Unzerlegbarkeit (talk) 16:50, 18 July 2008 (UTC)


 * Thanks! That's an excellent source.  I hope someone with access to one of the cited texts will add it as a footnote.  If not, we can make do with an indirect citation via the FOM mailing list entry.--Jeffreykegler (talk) 18:33, 18 July 2008 (UTC)

Classical Finitism?
I would like to see more reputable sources for the words classical and strict finitism. As far as I know a finitist does not accept existence of (the set) of natural numbers as an actual totality, i.e. each natural number exists, but the set of natural numbers do not. 128.100.5.136 (talk) 22:24, 7 May 2010 (UTC)

Also Kronecker was what is called a semi-intuitionist, not a finitist. Just putting one line from him does not make him a finitist. 128.100.5.136 (talk) 22:27, 7 May 2010 (UTC)

Again, Skolem's work on PRA is after Hilbert-Bernays. They did not adopt it, it is William Tait's claim that PRA is a upper bound on what Hilbert considered finitism. Also it is not a semi-formal system. About Hilbert's views refer to his and Bernay's two volume book and his papers specially "on the infinite". A short explanation is available in first chapters of Kleene's book "Metamathematics". 128.100.5.136 (talk) 22:31, 7 May 2010 (UTC)

The question of potential infinite is not one from finistism, it is from semi-intuitionism and intuitionism. 128.100.5.136 (talk) 22:38, 7 May 2010 (UTC)

The are not enough reputable verifiable references for what is stated in this article. Either add good references to what is claimed here or remove them from the article. 128.100.5.136 (talk) 22:38, 7 May 2010 (UTC)

I don't think the Kronecker quote helps here since it's normally assumed that there are an infinite number of natural numbers. If God just made the naturals, he still made an infinite number of things. I suggest the quote be removed. —Preceding unsigned comment added by 92.23.100.38 (talk) 00:23, 12 August 2010 (UTC)

Most famous proponent
The source provided only claims Kronecker stated "God created the natural numbers, all else is the work of man." It doesn't claim he was the "most famous proponent of finitism." —Ruud 22:33, 22 January 2011 (UTC)
 * OK, dropping procedural matters for the moment, what is the heart of your concern? Do you think he might not have been a finitist, or that there might have been a more famous one?  In the latter case, who? --Trovatore (talk) 22:40, 22 January 2011 (UTC)
 * I happened to come across this article and that particular statement simply seemed a little "out of place", as if some editor made the claim based just on a single quote, while he may just have been minor proponent or not even that. The problem is that I currently have no way to judge whether it is true of false. —Ruud 00:48, 23 January 2011 (UTC)
 * Oh, I see. Now that you mention it that's a good point.  Does anyone have anything addressing whether Kronecker associated himself with (or intended to establish) finitism as a philosophical school per se, rather than just having certain views that seem to align with it? --Trovatore (talk) 01:06, 23 January 2011 (UTC)


 * There is a detailed article by Schappacher on Kronecker, for anyone who wants to pursue this. Tkuvho (talk) 13:41, 23 January 2011 (UTC)

Link to infinitism
I am removing the link to infinitsm in the see also section, since this article concerns it self with philosophy of mathematics and the other article with epistemology. There is little overlap. —Preceding unsigned comment added by 213.200.193.129 (talk) 17:08, 23 March 2011 (UTC)

Wittgenstein
I am a bit surprised to read about Wittgenstein being a finitist. This sentence would at least require some explanation, since I actually came to this article through "Foundations of mathematics", where pt 4. no. 61 : "Finitism and behaviourism are similar attitudes. Both say : Yet, there is only... Both deny existence of something ; both do this in order to escape some confusion." (My bad translation) — Preceding unsigned comment added by 46.249.224.172 (talk) 18:29, 17 May 2015 (UTC)

Names
The names of some finitists should be given. Galileo, Gauss and Poincare could be mentioned. — Preceding unsigned comment added by 86.172.85.252 (talk) 12:28, 29 August 2018 (UTC)

Finitism falsified by QFT
Finitism is wrong because of 1 + 2 + 3 + 4 + ⋯ =-1/12 = Zeta(-1) is true. Quantum mechanics only works in our reality with this relation, aka Renormalization. Finity and Infinity are correspondending, like wave and particle… Moreover every finite number is an infinite number in p-adic-system... --2.247.253.179 (talk) 03:20, 1 December 2018 (UTC)
 * This talk page is not the place to argue whether or not finitism is correct. This is the place to discuss improvements to the article.  If you have reliable sources that make the argument above, we can talk about whether it would improve the article to mention that, and if so, how. --Trovatore (talk) 20:23, 1 December 2018 (UTC)

Hilbert's Program
> Hilbert's goal of proving the consistency and completeness of set theory or even arithmetic through finitistic means turned out to be an impossible task due to Kurt Gödel's incompleteness theorems

Is it possible via infinitistic means?!

Constructivism
How Finitism is distinct from Constructivism isn't clear from article I would have thought Constructivism is a sub branch of "Finitism" because it seems to make more specific assertions.