Am I a crank?
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From: MoeBlee on 6 Sep 2006 17:52 Aatu Koskensilta wrote: > True indeed, but first order PA is entirely irrelevant to our > understanding of natural numbers. Maybe entirely irrrelevent to your understanding, but not to mine. MoeBlee
From: Aatu Koskensilta on 6 Sep 2006 17:59 MoeBlee wrote: > Aatu Koskensilta wrote: >> Yes. But it seems our understanding of such inductively generated sets >> is much more basic than our understanding of any set theoretic ideas. We >> readily understand natural numbers, various formal expressions in formal >> languages, and so forth, without any recourse to set theoretic thinking >> of these things as the smallest set closed under this or that rule. Such >> set theoretic reasoning and understanding only becomes important when we >> reach inductive definitions - such as arithmetical truth or being an >> ordinal notation in O - of more generalized nature. Taking these things >> to fall under the more generalized notion of inductive definition is >> mathematically fruitful, but does not reflect anything of conceptual or >> "foundational" significance. > > My remarks are to offer Orlow an appreciation that induction is more > general (and 'general' not 'foundational' is the word I used) in > mathematics than one particular (though, of course, most salient) axiom > in one particular theory. You did use the word 'general' while I used the words 'conceptual' and 'foundational'. This is intentional; your comments have, as far as I have followed, not been incorrect in any way, and I do not wish to imply they are. My comments have concerned only the wider conceptual significance of such observation. > And without claiming that this genearality is > necessarily intuitively or philosphically foundational, an > understanding of this generality does greatly enhance one's > mathematical understanding of any particular induction schemata. Indeed. And that is exactly why I said that the generalized notion of inductive definition is mathematically fruitful. In fact, I believe that this notion should be introduced explicitly in many books and expositions where we only find its particular instances. As an example, a great number of logic books ask their readers to prove something by "induction on the length of formulae" while it is much more natural to prove stuff by induction on the *complexity* of formulae. The distinction is most naturally appreciated in context of a more or less general settings of inductive definitions. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 6 Sep 2006 18:00 MoeBlee wrote: > Aatu Koskensilta wrote: >> True indeed, but first order PA is entirely irrelevant to our >> understanding of natural numbers. > > Maybe entirely irrrelevent to your understanding, but not to mine. How does first order PA ever affect your understanding of natural numbers? -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Chip Eastham on 6 Sep 2006 18:09 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? It is slightly moderated. You must have an email address from an academic institution of some stripe in order to post there, and submissions must be pro forma scholarly articles. There is certainly room for a determined crank to slip "garbage" in, but there is substantially greater tolerance for a broad range of viewpoints than you will find in peer-reviewed journals, and the tactic of last resort to deprecate largely crankish, self- aggrandizing papers is to pigeonhole them as miscellany. --c
From: Lester Zick on 6 Sep 2006 18:34
On Thu, 07 Sep 2006 00:26:30 +0300, Aatu Koskensilta <aatu.koskensilta(a)xortec.fi> wrote: >MoeBlee wrote: >> Tony Orlow wrote: >>> Hi Aatu - >>> You say here that induction follows from our mathematical picture of the >>> naturals, >> >> He's talking about a particular axiom of induction in PA. I agree that >> it essential to our understanding of the naturals. But you should >> understand that induction is even more general. Inductive sets are a >> certain kind of set. Roughly, an inductive set is a set that is the >> intersection of a class of sets each of which is closed under a given >> relation. > >Yes. But it seems our understanding of such inductively generated sets >is much more basic than our understanding of any set theoretic ideas. We >readily understand natural numbers, various formal expressions in formal >languages, and so forth, without any recourse to set theoretic thinking >of these things as the smallest set closed under this or that rule. Such >set theoretic reasoning and understanding only becomes important when we >reach inductive definitions - such as arithmetical truth or being an >ordinal notation in O - of more generalized nature. Taking these things >to fall under the more generalized notion of inductive definition is >mathematically fruitful, but does not reflect anything of conceptual or >"foundational" significance. > >> I don't speak for Koskensilta, but this may work in many directions - >> first order PA may be thought of as a formalization of our mathematical >> understanding, so that it is not required to reverse this as you are >> doing. > >In some sense we can take PA as a "formalization of our mathematical >understanding" - in that it does indeed incorporate a part of our >mathematical knowledge, or beliefs - and in others such a identification >would be horribly wrong. What is central to our understanding of the >naturals is that they are generated from a single element, 0, by >repeated application of the successor operation. I think there is a difficulty here however. Zero is not a natural number in my book or it would have been discovered long before it was. So in effect we have the generation of natural numbers by the addition of a natural number, 1, to something which is not a natural number. Curious indeed. > As said, abstracting on >this we get the informal induction principle, applying to any property >we regard as determinate. In particular, if we accept the language of >arithmetic as meaningful, induction will apply to properties definable >in that language, giving us PA. Of course, if we accept the language of >arithmetic as meaningful, we should accept as meaningful also properties >definable by quantification over properties thus definable, and so >forth. There is no formal theory we could recognize, only on basis of >our acceptance of the induction principle and the intelligibility of the >language of arithmetic, as correct, that would cover all principles that >are acceptable on such basis, for G?delian reasons. Pondering on such >things, we recognize that such is the case even in case of set theory, >and indeed any basis we can come up for mathematics. Formalized theories >can only capture a portion - perhaps the only interesting portion! - of >what our informal ideas lead us to accept, or what is acceptable on >basis of our informal ideas. > >> Generally, I don't see existence of naturals as arrived upon inductively. >> In first order PA, the natural numbers are not mentioned in the theory itself. > >True indeed, but first order PA is entirely irrelevant to our >understanding of natural numbers. > >> We do have transfinite induction in Z set theories. It just does not >> prove what you want it to prove here. So get some axioms that prove >> what you want them to prove. > >A silly request. People like Orlow are not interested in mathematical >proofs or derivations as usually understood, but in heroic resistance to >the evil establishment. ~v~~ |