```Date: Feb 12, 2013 11:59 AM
Author: Alan Smaill
Subject: Re: Matheology § 203

WM <mueckenh@rz.fh-augsburg.de> writes:> On 4 Feb., 13:35, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:>> WM <mueck...@rz.fh-augsburg.de> writes:>> > On 2 Feb., 02:56, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:>>>> >> "The logicist reduction of the concept of natural number met a>> >> difficulty on this point, since the definition of ?natural number?>> >> already given in the work of Frege and Dedekind is impredicative. More>> >> recently, it has been argued by Michael Dummett, the author, and Edward>> >> Nelson that more informal explanations of the concept of natural number>> >> are impredicative as well. That has the consequence that impredicativity>> >> is more pervasive in mathematics, and appears at lower levels, than the>> >> earlier debates about the issue generally presupposed.">>>> > I do not agree with these authors on this point.>>>> So, on what grounds do you suppose that the notion>> of natural number is predicative?>> The notion of every finite initial segment is predicative because we> need nothing but a number of 1's, that are counted by a number already> defined, and add another 1.Entirely beside the point.It's in the justification of the claim that induction yields a conclusionthat holds for *any* natural number where the impredicativity lies.You clearly have not read the article I cited --always easier to dismiss a position from a position of ignorance.In this case, you are dismissing an argument that is in factone that actually supports your position, insofar as you have one.Well, that's your choice.>> >> So, how on earth do you know that induction is a correct>> >> principle over the natural numbers?>>>> > If a theorem is valid for the number k, and if from its validity for n>> > + k the validity for n + k + 1 can be concluded with no doubt, then n>> > can be replaced by n + 1, and the validity for n + k + 2 is proven>> > too. This is the foundation of mathematics. To prove anything about>> > this principle is as useless as the proof that 1 + 1 = 2.>>>> This is justification by fiat, the last refuge of>> the Matheologists.  When in doubt, say that there is no doubt.>>>> So, WM take this as an axiom of WMathematics.>> There are no axioms required in mathematics. Mathematics has evolved> by counting and summing without any axioms, but by comparison with> reality. And similar to Haeckel's "ontogeny recapitulates phylogeny"> we can teach and apply mathematics on the same basis where it has> evolved.It's an axiom in the sense that you feel no need to explain or justifyit.Further, you take the conclusion to hold "no doubt":this is not empirical mathematics, where it doesn't matter howmany examples we have seen, it remains possible that somelarger number will break a conjecture.It's a mystery, isn't it?>> (1 + 1 = 2 is purely computational;>> and more is not necessary for the beginning.>>>> But the conclusion tells us that there is a property that holds for>> *every* natural number (not all) -- some of which by your account>> will never come into existence at all (otherwise we would>> then have all of them).>> The results holds for every natural number that can become existing.> There are many natural numbers (according to classical and current> mathematics) that will never become existing (since their Kolmogoroc> complexity surpasses the ressources of the universe).But nevertheless the conclusion by induction holds "no doubt".>> "for every natural number n, if n is odd then n^n is odd".>> There is in fact an unsolved question: We cannot name all natural> numbers between 1 and 10^10^100, as we cannot read 123123123123 from a> usual pocket calculator, but we can add them, their squares, their> cubes and so on. I find this surprising, as surprising as the fact> that it is dark at night.I do not find it surprising that you think there is no problemin computing n^n for an arbitrary number in your chosen range.>> Regards, WM-- Alan Smaill
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