Talk:Foundations of mathematics

Foundations and philosophy of mathematics

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The recently-added text has the bias that a foundation of mathematics is actually possible. Before that the text was balanced and ended with 'the matter remains controversial' without advocating any particular approach. The subject matter after that line belongs in philosophy of mathematics under whatever old school (or new school) believes in it.

The majority of modern mathematicians seem to believe that foundations are either non-existent or unnecessary. They accept neither the positions of the older schools, nor do they accept the cognitive or social foundations theories. Hilary Putnam pointed out that one can accept a sort of weak realism without accepting any form of Platonism tied to Plato's ontology <-- apologies for the lousy condition of that article, it was written by a local sacred idiot.

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Article now more complete. Papers at end are critically important to this field and deserve summary treatment as articles, each of them. Particularly as there are fools out there who think realist=Platonist which is just wrong. Calling Putnam in particular a Platonist would piss him off to no end, dude.

Foundations search itself may be scientism. Is that view too contoversial?

Scientism is defined on that page as:

Scientism is the acceptance of scientific theory and scientific methods as applicable in all fields of inquiry, including morality.

Personally I don't see the connection between a search for a Foundation for Mathematics, and say, morality; so I would call it controversial. Chas zzz brown 21:16 Jan 20, 2003 (UTC)

Fine, we'll leave it out. Various arguments about this connection are in Talk:Philosophy_of_mathematic, which I notice you've found, so discuss it there.

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Hilary Putnam raised an extremely important point in What Is Mathematical Truth? - which is, that you must consider foundations of mathematics as a quite separate issue from an ontology, which most philosophy of mathematics theories fail to do, as they confuse the two quite hopelessly. Specifically, Putnam said he was a realist but not a Platonist, and was clear that one could believe that mathematical ideas were 'real' without believing in any aspect of Plato's ontology. What he meant was foundation ontology or cosmology it seems, but that aspect or usage of an ontology is not in the ontology article due to it being authored by, as the above claims, a local sacred idiot. According to the existing article there, a foundation ontology is just some kind of comp sci thing, and cosmology is only about physics and religion.

At some point this sacred idiocy must be challenged and Putnam's disctinction accepted and noted in either or both of the ontology and philosophy of mathematics articles, plus here in foundations of mathematics. As it stands, this article is trapped in about 1980 and those are trapped in about 1959, without even acknowledging Wigner's points. This is a bit confusing for anyone who actually knows the topic, and may discourage intelligent contributors, leaving the articles sadly in the hands of local sacred idiots.

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Thank you Axel. Chas zzz brown 18:00 Feb 10, 2003 (UTC)

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Anyone, a mathematician especially, who appreciates the “unreasonable effectiveness of mathematics” and the “unreasonable ineffectiveness of philosophy" to scientific endeavors must recognize the dangers of letting "philosophy of math" ride roughshod over "foundations of math" and as a last line of defense, of letting "philosophy and foundations of math" ride roughshod over proper pure and applied maths.

Just look at the talk page for "philosophy of math"! What a mess. Note that some of these people actually believe the destiny of science can be mastered thru verbose semantics, concepts, schema, arguments, etc. The last time I looked, the language of science was still written in mathematics. Fortunately, bullshit had not yet taken over in the math journals.

Specialists in foundations and/or philosophy of math often over-estimate the importance of their work to those in other specialties. In fact, few mathematicians are typically concerned on a daily, working basis over logicism, formalism or any other philosophical position. Instead, their primary concern is that the mathematical enterprise as a whole always remains productive ... as evident by the work they are doing at the moment.

Typically, they see this as insured by remaining open-minded, practical and busy; as potentially threatened by becoming overly-ideological, fanatically reductionistic or lazy. I do not know of a name for this “philosophical position” but it may be the one most mathematicians adhere to more strongly than any traditional, philosophical position they also favor or agree with (if any).

I think it would be choice to boldly, truthfully publish that the nihilistic and productive "philosophical positions" are by far the most populus amongst modern mathematicians on the "philosophy of math" and "foundations of math" pages. Let the silent majority be heard at last. Let the fanatical minority, steeped with all of their dreaded, formidable philosophical arguments, just go nuts (as usual). We have nothing to fear from these people. Some of them are just "naked emperors". Some of them are qualified mathematicians but with a strange psychological affliction involving philosophy and/or religion.

OmegaMan

A working perspective

This section was moved here from the article. See above for a explanation why. :)

To give an example, in number theory there is a huge body of doctrine, a tiny fraction of which has been developed in a particular axiomatic system, say Peano arithmetic (PA). Most of this work could be developed in PA; as a famous example, the prime number theorem is provable in PRA (Sudac (2001)), a much weaker theory than PA. But the working number theorist is concerned with proving theorems from initial assumptions which are obviously true using proof methods which are obviously correct, not with any particular logical system. In fact, the "crisis"-causing assertions discovered by Gödel are assertions about Diophantine equations, one of the main avenues in number theory. It may or may not be the case that there is a fundamental limit to what humans can understand about numbers (i.e., there may be true number-theoretical principles which cannot be perceived as being true by any human), but Gödel's theorem does not tell us which of these is the case, and we have no way of knowing. It may or may not be that we are required to introduce principles which are not expressible in the language of first order arithmetic in order to decide questions which are (e.g. the consistency of PA), but Gödel's theorem does not tell us which of these is the case, and again we have no way of knowing. It is often asserted that in light of Gödel's theorem one must introduce set-theoretical principles in order to decide certain number theoretical questions, but this assertion is unjustified. Gödel's theorem does not put any such constraints on the nature of the principles involved (i.e. the language in which they must be expressed). The attitude of the working number theorist is thus a reasonable one: one does not spend time thinking about such things, as there is simply no way to know. Instead one continues to prove theorems, and true principles which may be outside this or that logical system will be appealed to as required. Such principles will be introduced by people thinking about and solving actual problems, on the front-line. The problems (assuming there is no limit to what humans can understand about numbers) will be solved by people carrying on in the same way as they did before.

Eudoxus

In the section on ancient Greek mathematics, we have the following:

The discrepancy between rationals and reals was finally resolved by Eudoxus of Cnidus (408–355 BC), a student of Plato, who reduced the comparison of irrational ratios to comparisons of multiples (rational ratios), thus anticipating the definition of real numbers by Richard Dedekind (1831-1916).

As I understand it from reading Eudoxus of Cnidus, what Eudoxus actually did was define equality of two ratios by way of a logical relation between comparisons of each ratio's components: given integers m and n, a/b = c/d if and only if for all R in {<, =, >}, ma R nb if and only if mc R nd. To use this definition to "reduce comparisons of irrational ratios to comparisons of rational ratios", we would need to choose m and n such that ma, nb, mc and nd are all integers. But this is impossible: if m and ma are integers, a = ma/m is rational, contradicting the assumption that it's irrational. Is the article Eudoxus of Cnidus mistaken in asserting that m and n must be integers? For this reduction to work, they must be allowed to be irrational - and possibly not even numbers, albeit still quantifiable, e.g. physical quantities, such as the lengths of two line segments. Hairy Dude (talk) 01:17, 6 September 2015 (UTC)

@Hairy Dude: This reads awkwardly for me, too, though it's important that m and n are integers. From reading Euclid, who uses Eudoxus, the Greeks were secure in constructing any whole number multiple of a given magnitude and in comparing magnitudes, so they could test for ma R nb. In modern terms, the condition can be expressed as a/b R n/m iff c/d R n/m (all quantities being positive), which starts looking more like Dedekind by placing a/b and c/d in the same cut. The key steps in both approaches have a lot of similarities, though I'm unsure if they're entirely equivalent, not least because they come from different start points. I wonder if there's an awkwardness in the text from trying to express ancient concepts in modern terms: neither a magnitude nor a ratio in Euclid seems quite the same as a modern Real number. NeilOnWiki (talk) 14:40, 3 May 2021 (UTC)

Definitions

Article jumps into talking about "ZF" and "ZFC" without explaining, or linking to, a definition of either. --Cholten99 (talk) 00:31, 2 May 2016 (UTC)

I've added definitions before the More paradoxes list — which seems to be about more general metamathematical results rather than paradoxes, so perhaps ought to be retitled. The article doesn't seem to include a section on set theory (where for example the Axiom of Choice could be explained more fully), which seems a strange omission in an article on foundations. NeilOnWiki (talk) 17:48, 30 May 2021 (UTC)

Ordinal Analysis and Stronger Arithmetics

Concerning the revision of the ordinal analytic perspective, the statment was meant as a closing view on the state of the crisis which is as yet unresolved; as no concensus exists on the foundational theory of mathematics in its many variations. It was not meant to be presenting a specific theory, rather pointing the reader in the direction where the foundations of mathematics may explore, in order to provide a more complete and consistent resolution to the modern crisis. Where ZFC is not unilaterally accepted as a foundation, and as it is well known to be a lacking basis for things such as category theories of sets, or proper classes, more infinitary logic systems such as the Tarski-Grothendieck formalism and its universe provide stronger ground for foundational mathematics. Not to mention theories which may have higher ordinals. Consider restoring the paragraph. 79.6.244.66 (talk) 11:29, 13 July 2021 (UTC)

The fact that the crisis is unresolved is your opinion, and, among mathematicians , there is a consensus on the contrary. "Pointing the reader in the direction where the foundations of mathematics may explore" is considered as WP:original research, and is forbidded, unless some recognized mathematician has asserted that this direction is worth to explore; in such a case, the opinion must be explicitely attributed its author, with an explicit quotation. Otherwise, Wikipedia is not a crystal ball, and the directions of research that are worth to explore have not their place here, at least when they have not yet been succesful. D.Lazard (talk) 12:45, 13 July 2021 (UTC)

@D.Lazard: The article itself makes the case that the crisis is only partially resolved, which is a known fact to any mathematician or logician, working in category or set theory and formal systems. The statement was made in order for lay and professional readers to explore the subject further on the wiki (which is no crystal balling but the very function of the platform), along lines of well established theories of higher order arithmetic; a subject which is fundamentally seminal to this article. There is no personal bias to promote any theory or particular speculation, but a statement made to connect further personal research on wikipedia for the reader; perhaps requiring rephrasing to this end. Possibly there may be bias for unregistered editing, which is understandable considering the plethora of crackpots lurking around fundamental scientific articles; not to cast any stones. The attempt was to enrich the explanation of a complicated topic. And, provide further reading for deep foundational subjects. If this seems misplaced, please provide some professional perspective on the relation of hilbert's program with godel's incompleteness theorems in light of inaccessible cardinals and their universes. Because ordinal analysis and its higher order arithmetics make some fundamental questions as to the foundations of mathematics. 5.90.127.84 (talk) 13:21, 13 July 2021 (UTC)

@D.Lazard: If one wishes to delete edits without proper cause; thereby controlling the state of an article, the least to do is to respond in its discussion. This type of behavior discouages people from investing time, and contributing to articles that need expansion or quality elevation. Especially, when they are specialists in the field, and wish to contribute freely to the beauty that is wikipedia. The point remains, this article needs a closing statement which elaborates on the current state of foundations, and higher order theories of arithmetic beyond the current controversial standard of ZFC. 79.6.244.66 (talk) 09:09, 14 July 2021 (UTC)

The success and "beauty" of Wikipedia result from its rules and its foundational policies. So, deleting edits that do not respect these policies is definitely a proper cause of deletion. The policies that are involved here are linked above. About a "possible closing statement": As there is no consensus on the content of such a closing statement, the Wikipedia policy is clear: either this statement does not belong to Wikipedia, or the different opinions are stated with clear attributions. The discussion between them does not belong to Wikipedia. D.Lazard (talk) 10:07, 14 July 2021 (UTC)

@D.Lazard: Well we seem to be at a conceptual impass. Providing a general perspective on the continuation of research is vital to this article, as it already suffers from fractured tone and conflicting conclusions. I do not see how the added content is within violation of policies, as its intent is none of the aforementioned. But in the end we must agree to disagree. Unfortunate. Perhaps someone can eventualy make a synthesis of these perspectives, and provide clarity to the closing of this article. My attempt was to point to deeper research, which in mathematics, is the whole point. 79.6.244.66 (talk) 10:41, 14 July 2021 (UTC)

Here from WP:3O. I guess you are talking about this edit.[1] I havent seen a reference at the end of the addition, so unless there is a WP:Reliable Source, there is no real case for adding this sentence. (not watching, please {{ping}})Cinadon36 14:55, 20 July 2021 (UTC)