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.

Examples welcome for better tractability!

I think the article would benefit from including some "practical" examples: e.g. what is the approximate size of the largest prime number which could be calculated and stored in our existing, ~14 billion year old and ~10**53 kg mass Universe, if taking in to account Bremermann's limit at maximum efficiency? Considering the great fascination with prime numbers even among the general public, it would make the article more approachable.

Of course some primes have compression tricks up their sleeves, like Mersennes, e.g. 2**136,279,841−1. However even that gives only temporary remedy, as "136,279,841" could simply be replaced with the largest number computable and storable in our universe that also gives a Mersenne prime.

Therefore, per finitism doctrine, there IS a largest prime number, because nothing infinite can fit in a finite Universe and any "theoretical construct" also takes place within the brains and computers of mathematicians, which are several magnitude smaller physical realities than our Universe, plus there is no proof other (possibly infinitely old and massive) universes exist. 94.21.160.96 (talk) 22:07, 16 November 2024 (UTC)

That is more relevant to the article Ultrafinitism; almost all of this article is concerned with other uses of the word "finitism", for which those concerns don't apply. However, I'm not convinced that what you suggest is feasible; it's one thing to say that there must be a limit to the size of numbers which can be expressed, and a very different thing to caculate, or even estimate, what that limit is. JBW (talk) 14:28, 17 November 2024 (UTC)

Infinite sets?

per the last two edits your defining finitism as not believing in infinite SETS. So I'm supposed to believe these people believe the number of whole numbers is not infinite, therefore presumably they believe their's only a finite number of numbers from 0,1,2,..., ? I don't think so. The editor is obviously thinking in terms of set theory. They do believe in infinite objects, the natural numbers are neverending,thus have the CHARACTERISTIC of infinite, but provably no SIZE.

    Consider an infinite island of various hotels, so one with one room numbered 1, the next 2 rooms first numbered 1 the next in position numbered 2, the next hotel rooms numbered in order of position 1,2,3 etc for all number of rooms.  Now for all the hotels the size of the hotel is the same as the room number of the last room except for 0, and empty lot. Now consider another hotel there having never-ending number of rooms so numbered, no last room so no size. Now consider a hotel with infinite number of rooms so numbered but having a last room.  The room number of that room is infinity thus the size of the hotel is a number infinity.

Finitists don't believe in that number or the set of natural numbers having a size or there being an infinite list of infinite numbers larger than the finites such as ∞a, ∞a+1, etc The editor presumably being a set theorist assumes any infinite object has a size, and also confused by ultrafinitists. Now the pertinent set theory articles should be edited, obviously the ordinal arithmetic nonsense is from the set theorists confusing never-ending-infinity with ending infinity, so 1 + omega = omega that is adding 1 element to the beginning of a well ordered never-ending list, presumably -1, but omega + 1 is not equal but larger than omega, adding 1 element to the end of a never-ending list, presumably omega the set well ordered of all natural numbers. That's obviously not possible since no end to add it to. And the set can't be ending since then the proof that 1 + omega = omega would not work, the tenant in the last room of the infinite hotel would be without a room. And the reason the set theorists start the natural numbers with 0 and formulate the set of natural numbers in the axiom of infinity is to force a set that provably has no size to have a size that's a number. That should be explained in Wikipedia. Each number is the set of previous numbers so for example 3 has 3 members, and the set of natural numbers has the same format so presumably is also a number, though axiomatically not in the set of natural numbers. Since each number is the number the size of the set that is that number likewise presumably for the set of natural numbers. So by the definition of number in the axiom of infinity the set of natural numbers is a number, and the size of that number is the size of that set, forcing omega and aleph0 to be the set of natural numbers, a number, and the size of the set of natural numbers thus causing it to have a size though as above provably having no size. Victor Kosko (talk) 18:53, 18 September 2025 (UTC)

@Victor Kosko: It is difficult to follow much of what you have written, but I will try to answer your first point. No, finitists don't "believe their's [sic] only a finite number of numbers from 0,1,2,..." Of course they believe there are infinitely many positive integers; what they don't believe is that it is meaningful to regard all those integers as constituting a single object called "an infinite set". JBW (talk) 11:05, 21 September 2025 (UTC)

Your not understanding the side issues in my post because you refuse to believe them.

Per your statement above that's the same as what I claimed, they believe in the set of natural numbers and that it's infinite, but not as an object since all objects have a size set theorists would insist it has a size. Believing in each and every member of the set is the same as believing in the set.

Second The phrase 'potential infinity' allows them to refer to the set as from 0 to infinity without there being an infinity, it's not a type of infinity. Wikipedia is wrong about that also and thus needs to be fixed. Victor Kosko (talk) 19:06, 21 September 2025 (UTC)

You may think that "Believing in each and every member of the set is the same as believing in the set", but finitists don't. JBW (talk) 21:14, 21 September 2025 (UTC)

Finitists OR ultrafinitists?

Also document what you just said Victor Kosko (talk) 20:49, 22 September 2025 (UTC)