Talk:Existence theorem

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	Logic

What is the Existence Theorem, I wonder?

Existence theorems, are the archetypal example of ‘blackboard economics’, mathematical games yielding purely qualitative results that can be overturned with modest changes in assumptions. This understanding was extracted from:

http://www.kieranhealy.org/blog/archives/2003/09/07/existence-theorems-are-reductios/

Also see McCloskey's "Cassandra's open letter to her economist collegues" —Preceding unsigned comment added by 149.32.192.33 (talk) 16:03, 6 June 2008 (UTC)[reply]

Go and take a sleep. —Preceding unsigned comment added by 87.66.89.71 (talk) 00:09, 6 August 2008 (UTC)[reply]

pure?

Do we have a source for describing non-constructive existence results as "pure", or is this a case of OR ? Tkuvho (talk) 13:52, 31 October 2011 (UTC)[reply]

I think Qwertyus has just made a similar remark, and the question is still unsettled here. I don't have a reference at hand currently; however, the problematic issue is obvious: a theorem usually has several proofs, some of them might indicate a construction, while others might not. So, a theorem would be called "pure" if some of its proofs indicates a construction. (It is always possible to "blow up" a given constructive proof with irrelevant non-constructive details, so requiring that all proofs indicate a construction would be too restrictive.) Then it may happen that for some theorem, only non-constructive proofs are known as of today (so it would be qualified as "non-pure"), but a new constructive proof may be found eventually in the future (so the theorem should be re-classified as "pure"). In order to avoid such a status of a theorem as "currently non-pure", a proof of a meta-theorem would be necessary, viz. the no proof (in some fixed formal language) can ever be found that is constructive. Note that the latter (meta-)theorem is (like Godel's incompleteness theorems) about formal languages, even if the original theorem was e.g. about geometry. - Jochen Burghardt (talk) 14:15, 28 May 2015 (UTC)[reply]

This is less of an issue than it may appear. Certain theorems are intrinsically nonconstructive because they can be shown to be false in suitable constructive settings (when the law of excluded middle is not satisfied), for example the extreme value theorem. My question was purely with regard to terminology. I still find it odd to refer to nonconstructive results as "pure", and am still wondering if there is a source for this. Tkuvho (talk) 15:09, 28 May 2015 (UTC)[reply]

(edit conflict) Actually my direct concern was the word "problematic", which indicates WP:EDITORIALIZING if not sourced.

Btw., I think you're reversing the article's definition of purity. It asserts that "[a]n existence theorem may be called pure if the proof given of it doesn't also indicate a construction". So a theorem would be "pure existence" while only non-constructive proofs of it are known and any constructive proof would rob it of its purity. QVVERTYVS (hm?) 15:19, 28 May 2015 (UTC)[reply]

Most of this material was contributed by User:Charles Matthews so perhaps he can be asked to provide sources and/or deal with WP:EDITORIALIZING issues. Tkuvho (talk) 15:39, 28 May 2015 (UTC)[reply]

Yup, written in my first month on Wikipedia, before these guidelines existed (even in theory!) So I don't doubt something should be done to improve the text from a dozen years ago. Charles Matthews (talk) 15:47, 28 May 2015 (UTC)[reply]

I just realized that I, too, had overlooked the negation arising from "doesn't" in the definitional sentence. I guess (again, yet without a source), it intends to express "pure-existence results" rather than "pure existence-results". Or, slightly rephrased: "purly" refering as an adverb to "existential" (i.e. a "purely/merely/just existential theorem" is one that just guarantees the existence of e.g. a number, but nothing more, in particular not its value), rather than "pure" as an adjective to "theorem" (i.e. the notion of an "existential (and) pure/clean/clear/neat/... theorem" doesn't make sense).

What about changing the definitional sentence to: "A theorem may be called a purely existential one if none of its proofs indicates a construction ...", and changing "may be" to "is", or "is sometimes", after a source was found ? This way, we'd also avoid the misleading phrase "the proof given of it". - Jochen Burghardt (talk) 21:26, 28 May 2015 (UTC)[reply]