Am I a crank?
in [Math]
|
Prev: Any coordinate system in GR?
Next: Euclidean Spaces
From: Virgil on 5 Sep 2006 22:31 In article <rnvrf2dhbfd5d87ht13c238o53vnvmfvqk(a)4ax.com>, Lester Zick <dontbother(a)nowhere.net> wrote: > On Tue, 05 Sep 2006 15:54:22 -0600, Virgil <virgil(a)comcast.net> wrote: > > >In article <vckrf21eftu9dec4bndsritu590geljcak(a)4ax.com>, > > Lester Zick <dontbother(a)nowhere.net> wrote: > > > > > >> I have no idea what > >> Virgil was trying to say but that's nothing unusual. > > > >As Zick usually does not even have any idea what Zick is trying to say, > >one is hardly surprised at his inability to understand what others say. > > Oh don't go getting all bent out of shape, Virgil. What sort of shape is Zick bending himself into, then? > Isn't > there anyone else you can talk to? Zick seems to be the one determined to talk to me.
From: Tony Orlow on 5 Sep 2006 23:08 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. > Hi Aatu - I just saw your conversation with MoeBlee concerning us cranks, and you seem to have a reasonable attitude. MoeBlee challenges you to talk some sense into me, and since I see you are discussing induction here, and that is central to my developing theory (or fetus thereof) I thought I would ask your opinion on this matter, and see if you think I need correction. 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? It seems to me that inductive proof is one with a loop of logical implication, such that one fact implies another, which in turn implies another, ad infinitum. The existence of a natural implying the existence of a next natural is but an example of this kind of logical construction. Does that seem like a wrong perspective to you? 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? I see any set size as a count, something in the same sequence as the naturals, albeit possibly infinitely far beyond them. It's a quantity. When we prove that, for instance, x^2>2x for all x>2, it seems to me that any infinite quantity is a number greater than 2, and that this inequality would hold for any infinite x as well. If we put the infinite numbers on the same line as the finites in this way, and extend inductive proof to also hold for the infinite case, then not only can we say that x < 2x < x^2 < 2^x < x^x for finite x>2, but also for all infinite x. In this way, we can formulaically order all sorts of infinite set sizes, in a manner which distinguishes a difference of a single element between infinite sets. Do you have any immediate reaction to that, besides that it contradicts finite set theory? Thanks, Tony
From: Han de Bruijn on 6 Sep 2006 04:05 Proginoskes wrote: > Jesse F. Hughes wrote: > >>John Schutkeker <jschutkeker(a)sbcglobal.net.nospam> writes: >> >>>Isn't Arxiv peer-reviewed? >> >>No. > > There are a couple of Four Color Theorem papers which have obvious > errors in them. (I found them in < 5 minutes.) [ ... ] My experience (from long ago) is that peer reviewing doesn't help much in finding errors. All of the "real" publications by myself about i.e. the Least Squares Finite Element Method are "peer reviewed", but there are some obvious errors in them. My only excuse is that I've been quite naive at that time. (Don't think I've been dishonest on purpose) Mea culpa, nevertheless. Han de Bruijn
From: Aatu Koskensilta on 6 Sep 2006 06:47 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! > 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. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: John Schutkeker on 6 Sep 2006 10:22
"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? |