Wikipedia:Reference desk archive/Mathematics/2006 August 24

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[edit] Arithmetic of linear logic

Is there any theory of arithmetic within the framework of linear logic, analogous to how Heyting arithmetic is the arithmetic of intuitionistic or constructive logic? I'm particularly intrigued by the possibility of using both a "constructive" disjunction (plus) and a "classical" disjunction (par). For example, if P(d) is the statement that the digit d occurs infinitely often in the decimal expansion of pi, then:

  P(0) plus P(1) plus P(2) plus P(3) plus ... plus P(8) plus P(9)

would involve actually finding a specific digit that occurs infinitely often in pi, whereas:

  P(0) par P(1) par P(2) par P(3) par ... par P(8) par P(9)

would be trivially true. Thanks! -- Four Dog Night 01:39, 24 August 2006 (UTC)

Since there have been no prior answers, I'll simply mention that the 1992 book by Troelstra, Lectures on linear logic (ISBN 978-0-937073-77-3), says at the start of Chapter 5, "both classical and intuitionistic logic can be faithfully embedded into CLL" (meaning classical linear logic). Hope that helps. --KSmrqT 13:40, 28 August 2006 (UTC)
Thanks for the pointer. I will go take a look. --Four Dog Night 19:33, 29 August 2006 (UTC)

[edit] mathematics yesterday today tomorrow

what was mathematics yesterday? What is it today? what wil be tomorrow's prospect? The information is intended to educate the students of high school level.

mathematics today is obfuscated by the invention of the microcomputer. Mathematics yesterday was simply pencil scribbles in the footnotes of past "scientists". For tomorrow, mathematics will be discontinued, all logical functions in our lives will be handled by automatons. -Wjlkgnsfb 04:48, 24 August 2006 (UTC)
For the first two questions, see History of mathematics and Mathematics. We have no article "Future of mathematics", but the last articles may provide some inspiration. See also Unsolved problems in mathematics. For the record, I completely disagree with Wjlkgnsfb's sentiment expressed above. --LambiamTalk 05:35, 24 August 2006 (UTC)
i am sorry but the above comments by Wjlkgnsfb, if not made in jest, reflect the absence of even minimal understanding of mathematics. high school students should not be misled by such misconceptions. Mct mht 05:20, 25 August 2006 (UTC)
My apologies to Wjlkgnsfb for speaking on his behalf, but I strongly believe his comment was, indeed, made in jest. -- Meni Rosenfeld (talk) 18:41, 25 August 2006 (UTC)

[edit] Poincare Theorem?

Will the Poincare Conjecture shortly become the Poincare Theorem now that it has been proved, or will it keep its name out of habit?

Thanks, —Daniel (‽) 20:46, 24 August 2006 (UTC)

Well, sometimes the latter name means the Poincaré recurrence theorem... Melchoir 20:54, 24 August 2006 (UTC)
I guess what we need to do is start calling it the Poincare Theorem when that becomes the accepted name within the professional maths community. Madmath789 21:13, 24 August 2006 (UTC)

Yeah. "Fermat's Last Theorem" wasn't called a conjecture when it was. Likewise, the "Enormous theorem" didn't go from being a theorem, to being a conjecture, back to being a theorem in the past twenty years. The usage is conventional, and it will take some time for it to change, if it ever does. There is no rush. –Joke 21:26, 24 August 2006 (UTC)

Poincare certainly did not prove his conjecture. So it is possible that Poincare conjecture will become Hamilton-Perelman theorem. Also if a consensus will grow that Fermat did not prove his last theorem, it may become Wiles theorem. Yes, Wikipedia just has to follow the math community. (Igny 22:15, 24 August 2006 (UTC))
I think FLT is so ingrained that it's not going to be superseded. My (limited) understanding is that the Hamilton-Perelman theorem is significantly more general than its corollary, the Poincare Conjecture. It would be nice to have a name for the general n-dimenional version: Every simply connected closed n-manifold is homeomorphic to an n-sphere. "The Smale–Friedman–Hamilton–Perelman Theorem" sounds awkward. --LambiamTalk 00:24, 25 August 2006 (UTC)

I don't think there are very many people who seriously think Fermat proved the FLT. As for the Poincaré conjecture, what has been proved is the geometrization conjecture which will likely become known as the geometrization theorem. I imagine that, like the FLT, the theorem will forever be associated with Poincaré, even though Perelman provided the proof. –Joke 01:28, 25 August 2006 (UTC)

Another curious usage is Zorn's lemma, the axiom of choice and the well-ordering theorem (sometimes well-ordering principle). Why is one a lemma, another an axiom, and a third a theorem or principle when they are well known to be equivalent? The usage is historical. –Joke 01:32, 25 August 2006 (UTC)
Well, actually I think it's pretty natural. The axiom of choice is (at least to many people) "obvious", and therefore an axiom in the original sense of "self-evident truth". The wellordering principle is a somewhat deep theorem, requiring the axiom of choice for its proof. Then Zorn's lemma is a simple corollary of the wellordering principle. --Trovatore 05:48, 25 August 2006 (UTC)
I can't stop myself. Two other curious examples: the Riemann hypothesis and the continuum hypothesis, which are clearly "hypotheses" of totally different sorts (one is a conjecture, the other is a potential axiom). –Joke 01:35, 25 August 2006 (UTC)
I'm gonna have to disagree with that, too. The mere fact that CH and its negation are both consistent with ZFC does not mean it's an arbitrary choice. Work by W. Hugh Woodin suggests a reason to believe that CH is, in reality, false, and among set theorists who think the question is meaningful, I get the impression that many more think it's false than think it's true. I think Matt Foreman may possibly think it's true, but he doesn't say it directly and out loud. --Trovatore 06:02, 25 August 2006 (UTC)