In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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 number¹ > > >> 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.
Where did you get the first 1? > > . . .
> There are no axioms required in mathematics.
Even Euclid knew better than that.
> 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.
Except that there is considerable reason to doubt that "ontogeny recapitulates phylogeny".