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. > > >> 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. > > (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). > > "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.