Date: Feb 4, 2013 7:35 AM
Author: Alan Smaill
Subject: Re: Matheology § 203
WM <email@example.com> 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_
>> 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