Am I a crank?
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From: Herman Jurjus on 7 Sep 2006 05:38 Aatu Koskensilta wrote: > Dik T. Winter wrote: >> In article <tMfLg.4871$9c3.4603(a)reader1.news.jippii.net> Aatu >> Koskensilta <aatu.koskensilta(a)xortec.fi> writes: >> > So you would balk at asserting that it's true that whatever >> mathematical > property P is we have that >> > > if P(0) and for every natural n, P(n) implies P(n+1), then for >> every >> > natural n, P(n) >> > > is true, for example? >> >> Only if you accept the axiom of induction. If you do not accept that >> axiom >> it is probably false. > > How does your accepting or not accepting it affect its truth or falsity > in any way? In any case, surely if you accept the principle of induction > you will, trivially, accept its truth. > >> There is no *a priori* reason to either accept or >> reject it. > > Sure there is. It follows immediately from our mathematical picture of > the naturals. But there may be reasons to *want to develop* a form of mathematics in which induction fails. By the time such a mathematics has been made, it will make sense to talk of induction in the way Dik did: as something that depends on whether we 'choose' to accept it or not. Just like LEM, AC, Euclid's 5th postulate etc. -- Cheers, Herman Jurjus
From: John Schutkeker on 7 Sep 2006 11:03 "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote in news:87u03jykrq.fsf(a)phiwumbda.org: > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > >> John Schutkeker wrote: >> >>> "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote in >>> news:87pseakne6.fsf(a)phiwumbda.org: >>> >>>>John Schutkeker <jschutkeker(a)sbcglobal.net.nospam> writes: >>>> >>>>>Isn't Arxiv peer-reviewed? >>>> >>>>No. >>> How do they keep the cranks from posting garbage? >> >> There is no such guarantee in peer reviewed journals as well IMHO. > > No, but a respectable peer-reviewed journal is less likely to have > fallacious proofs of famous math theorems (i.e., crank output) than > arxiv.org. > > Peer review plays an overall positive role in the maintenance of > journals. A peer reviewed article is more likely to be a good article > than one that has been self-published. But either of them can be > garbage and either of them can be a great article, like a valid proof > of Poincare's theorem. Unfortunately, if it is true that Arxiv only accepts the work of the academically affiliated, then it is an absolute certainty that no correct work by amateurs, unemployed or underemployed professionals will ever be published in their venue. :(
From: Lester Zick on 7 Sep 2006 12:15 On Wed, 06 Sep 2006 17:13:32 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <cfjuf2l2jhb7kp275gif3gj054m77qh9ea(a)4ax.com>, > Lester Zick <dontbother(a)nowhere.net> wrote: > >> On Wed, 06 Sep 2006 13:00:00 -0600, Virgil <virgil(a)comcast.net> wrote: >> >> >In article <p8utf2t0ae4d19nktl1ro8mnq9frqfjhkt(a)4ax.com>, >> > Lester Zick <dontbother(a)nowhere.net> wrote: >> > >> >> On Tue, 05 Sep 2006 18:22:30 -0600, Virgil <virgil(a)comcast.net> wrote: >> >> >> >> >In article <agvrf21km3cu96aq3sn94hb65mb1ttc8lj(a)4ax.com>, >> >> > Lester Zick <dontbother(a)nowhere.net> wrote: >> >> > >> >> >> On Tue, 05 Sep 2006 15:51:32 -0600, Virgil <virgil(a)comcast.net> wrote: >> >> >> >> > >> >> >> >Better than truth because Zick says so" >> >> >> >> >> >> Hardly better than truth, Virgil, just better than fantasy land. >> >> > >> >> >Zick would know better that I about what goes on in the latter. >> >> >> >> You would certainly know better what goes on in neomathematiker >> >> fantasy land. >> > >> >As I am totally unfamiliar with any such thing, Zick is, as usual, wrong. >> >> Another trivium of truth assumed true in lieu of any demonstration. > >Zick is the one whose trivia is founded in the trivium. Math is a part >of the quadrivium. And modern math is founded, whatever that means, in the trivium and not in the quadrivium. ~v~~
From: Lester Zick on 7 Sep 2006 12:17 On Thu, 07 Sep 2006 07:32:41 -0400, "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote: >Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > >> John Schutkeker wrote: >> >>> "Jesse F. Hughes" <jesse(a)phiwumbda.org> wrote in >>> news:87pseakne6.fsf(a)phiwumbda.org: >>> >>>>John Schutkeker <jschutkeker(a)sbcglobal.net.nospam> writes: >>>> >>>>>Isn't Arxiv peer-reviewed? >>>> >>>>No. >>> How do they keep the cranks from posting garbage? >> >> There is no such guarantee in peer reviewed journals as well IMHO. > >No, but a respectable peer-reviewed journal is less likely to have >fallacious proofs of famous math theorems (i.e., crank output) than >arxiv.org. > >Peer review plays an overall positive role in the maintenance of >journals. A peer reviewed article is more likely to be a good article >than one that has been self-published. But either of them can be >garbage and either of them can be a great article, like a valid proof >of Poincare's theorem. I believe peer reviewed articles are also more likely to reflect established orthodoxy whether good, bad, or indifferent. ~v~~
From: Tony Orlow on 7 Sep 2006 12:59
Aatu Koskensilta wrote: > Tony Orlow wrote: >> You say here that induction follows from our mathematical picture of >> the naturals, but isn't our mathematical picture of the naturals based >> on the inductively defined set given by the Peano axioms? > > No. In whatever way, we develop the idea of a series of numbers obtained > from some starting point by repeatedly applying[1] the successor > operation (the "add one" operation). Abstracting further we realise that > this can be expressed by an induction principle, saying that whatever > property P is, then if 0 has P, and if n has P then n+1 has P, then > every number has P. These are informal principles and considerations, > and from them we may, for various purposes, attempt to devise e.g. > formal theories which capture them to some extent. One of the most > studied of these is Peano arithmetic, but this theory really has mostly > technical interest; number theorists, like other mathematicians, just > prove things to their satisfaction using whatever principles they find > acceptable. To understand stuff like functions, formal theories, sets, > and so forth we must already have an understanding of the naturals - or > inductively generated structures of similar complexity - obtained > informally in whatever way it is that we understand anything in general > - certainly not by means of formal or informal axioms! Oh, I agree. The axioms merely reflect an understanding we have, in syntactical form, for logical manipulation. Then, we have to translate back from the syntax to understanding, or we haven't gotten anything. I just meant that the naturals are a special case of an inductive set, albeit one that is about as simple as possible while still including some notion of measure, such that it's convenient to define mappings from this standard set to all inductive sets. > >> Secondly, I would like your opinion on inductive proof in the infinite >> case. I am aware that this concept is not compatible with transfinite >> cardinalities or limit ordinals, but independent of that, does the >> following make any sense to you? > > None whatsoever. > > [1] For some reason the phrase "finitely many times" is added here, as > if "applying the successor function infinitely many times" made any sense. > I don't think you mean to disagree with standard theory here, so I must be reading you wrong. In standard theory, aren't there infinitely many naturals in succession? If you mean "uncountably many" successions, I agree that's not standard, but I fail to see the problems it causes. I also don't see that as a barrier to understanding that if n>m and n>m->P(n) then P(n), even if n is greater than any finite. Thanks for your response. Tony |