Date: Feb 4, 2013 7:35 AM Author: Alan Smaill Subject: Re: Matheology § 203 WM <mueckenh@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?

>> 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.

> Regards, WM

--

Alan Smaill