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Re: Matheology § 203
Posted:
Feb 4, 2013 7:35 AM


WM <mueckenh@rz.fhaugsburg.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?
>> 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.hsaugsburg.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.
> Regards, WM
 Alan Smaill



