> 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 conclusion that 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 fact one 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 justify it.
Further, you take the conclusion to hold "no doubt": this is not empirical mathematics, where it doesn't matter how many examples we have seen, it remains possible that some larger 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 problem in computing n^n for an arbitrary number in your chosen range.