> 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?
>> 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.
(1 + 1 = 2 is purely computational; induction over formulas of abritrary complexity, say with several quantifiers is a whole different affair)
> Compare Matheology § 205 here_ > http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf >> >> You only ever have finitely many of them, so you can never know >> what will happen when you look at a new one. > > The new one is finite and not more than 1 different from its > predecessor. And there are never more than finitely many. That's > enough to apply the above formalism.
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).
"for every natural number n, if n is odd then n^n is odd"
You will run out of ink to write down n^n pretty quickly. When n is available, but not n^n, you are lapsing into theology.
Nelson's attitude on exponentiation is different. But no doubt you disagree with him also.