July 2009 - mathfun@mailman.xmission.com

Re: [math-fun] Springer Verlag Publishes āProofā of Goldbachās Conjecture (& FLT)
by Dan Asimov 15 Jul '09

15 Jul '09

By FLT, Benschop definitely means Fermat's Last Theorem; see his paper here: < http://pc2.iam.fmph.uniba.sk/amuc/_vol74n2.html > ("Additive structure . . ."). --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture (& FLT)
by Dan Asimov 15 Jul '09

15 Jul '09

At the linked listing of the book by Springer, it says the book is "temporarily unavailable". --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: Springer Verlag Publishes âProofâ of Goldbachâs Conjecture (& FLT)
by James Propp 15 Jul '09

15 Jul '09

They mean Flt (Fermat's little theorem), not FLT (Fermat's Last Theorem). Jim

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[math-fun] Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture (& FLT)
by Fred lunnon 15 Jul '09

15 Jul '09

Recent item (June 30th) from "The n-Category Café" blogsite --- http://golem.ph.utexas.edu/category/2009/06/springer_verlag_publishes_proo.… I wonder should somebody tell Andrew Wiles? WFL

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Re: [math-fun] math, existence, and God
by James Propp 14 Jul '09

14 Jul '09

One doesn't need Category Theory in all its mind-bending glory in order to talk about projective limits. I think that all one needs for a projective-limit definition of God is a bunch of axioms that say that for any two beings there's a still greater being who subsumes them. By the way, when p-adic numbers were introduced, did anyone wonder whether they "really existed"? Jim Propp

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Re: [math-fun] math, existence, and God
by Dan Asimov 14 Jul '09

14 Jul '09

No, they weren't. If FLT had turned out to be false, it would have been provably false, by just doing the arithmetic for a counterexample. --Dan << Quoting Dan Asimov <dasimov(a)earthlink.net>: > I am assured by logicians that in any axiomatic system > rich enough to include number theory, there are > propositions that can be neither proved nor disproved, > regardless of whether they are "really" true or "really" false. > > A common example that may be of this type is the > twin primes conjecture (TWP). Not so long ago, people were saying this about Fermat's Last Theorem. >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] math, existence, and God
by Dan Asimov 14 Jul '09

14 Jul '09

I am assured by logicians that in any axiomatic system rich enough to include number theory, there are propositions that can be neither proved nor disproved, regardless of whether they are "really" true or "really" false. A common example that may be of this type is the twin primes conjecture (TWP). Now suppose that TWP is really of this kind -- that it can neither be proved nor disproved (with a conventional finite proof in number theory). Then we have the right to choose to add either TWP or its negation to the existing axioms and the resulting system will be consistent*. And so there exists a model for this new system. And so it (apparently) "exists". BUT -- what if a hypothetical infinite check of all possible twin primes shows that TWP is true . . . yet we choose to add as an additional axiom its negation ~TWP. Then there is a model of this system, despite the fact that it is WRONG. Would that system really exist? I.e., spiritually, it's inconsistent, but we could not prove it thus with a conventional finite proof. --Dan __________________________________________________ * If we assume Number Theory is consistent, which we shall do. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] category theory --- hmm...
by Fred lunnon 14 Jul '09

14 Jul '09

Having for the moment run out of steam as far as geometric algebra is concerned, I thought I might for a while resume banging my head against an ancient brick wall. It hasn't taken me long to stub my toe (or to mix my metaphors). Upon earnestly scanning the following introductory documents, http://en.wikipedia.org/wiki/Category_theory http://plato.stanford.edu/entries/category-theory/ http://math.ucr.edu/home/baez/rosetta.pdf each equipped with apparently impeccable credentials, I find that none of them can agree about whether the objects of a category belong to a set (Stanford), a class (Wikipedia), a "collection" (Baez & Stay), or an "aggregate" (anon --- mislaid this one). Furthermore, they are similarly contradictory (and in one case confused) about whether the morphisms thereof comprise a set or a class. I'm not going to insist that authors commit to some particular flavour of set theory, appreciating that category theory may well provide it with an alternative foundation. However, I feel entitled to expect that --- after nearly half a century --- the pedagogy of this discipline might have managed to reach agreement concerning fundamental definitions! Would anybody care to cast some light on this disturbing inconsistency? Fred Lunnon

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[math-fun] partitions into distinct part: spell this one out
by Joerg Arndt 14 Jul '09

14 Jul '09

Let P(x) = prod(n=1, oo, (1+x^n)). This is the generating function for partitions into distinct parts. We have P(x^2)^8 + 16*x*P(x^2)^16*P(x)^8 = P(x)^16 So the number of partitions of n into distinct parts of 16 kinds (rhs.) equals the number of partitions of n into even parts of 8 kinds (1st term lhs.) plus *WHAT*? P.S.: the equality really is k^2+k'^2=1 massaged to render a combinatorial relation.

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Re: [math-fun] math, existence, and God
by Steve Witham 12 Jul '09

12 Jul '09

There's a lot about avoiding "is" in religion already. "A journey and not a destination," The Buddhist "path of non-attachment," That bit in the Indigo Girls song, Theres more than one answer to these questions Pointing me in a crooked line And the less I seek my source for some definitive (the less I seek my source) The closer I am to fine." I always thought the word was "definity," which seemed illiterate, an apt coinage, and a good pun with "divinity." As "to fine" is with defined or divine. --Steve

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