Talk:Foundations of mathematics

message: 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)
replies: text: This reads awkwardly for me, too, though it's important that  and  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 . In modern terms, the condition can be expressed as  iff  (all quantities being positive), which starts looking more like Dedekind by placing  and  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)
thread_title:  Eudoxus 

message: 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)
replies: text: 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)
thread_title:  Definitions 

message: 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)
replies: text: 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)
replies: text:   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)
replies: text:   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)
replies: text: 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)
replies: text:   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)
thread_title:  Ordinal Analysis and Stronger Arithmetics