The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Matheology § 203
Replies: 11   Last Post: Feb 5, 2013 4:58 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Alan Smaill

Posts: 1,103
Registered: 1/29/05
Re: Matheology § 203
Posted: Feb 4, 2013 7:35 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

WM <> writes:

> On 2 Feb., 02:56, Alan Smaill <> 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

Alan Smaill

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.