Date: Feb 12, 2013 11:59 AM
Author: Alan Smaill
Subject: Re: Matheology § 203

WM <mueckenh@rz.fh-augsburg.de> writes:

> On 4 Feb., 13:35, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>> WM <mueck...@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?

>
> The notion of every finite initial segment is predicative because we
> need nothing but a number of 1's, that are counted by a number already
> defined, and add another 1.


Entirely beside the point.
It's in the justification of the claim that induction yields a conclusion
that holds for *any* natural number where the impredicativity lies.

You clearly have not read the article I cited --
always easier to dismiss a position from a position of ignorance.

In this case, you are dismissing an argument that is in fact
one that actually supports your position, insofar as you have one.
Well, that's your choice.

>> >> 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.

>
> There are no axioms required in mathematics. Mathematics has evolved
> by counting and summing without any axioms, but by comparison with
> reality. And similar to Haeckel's "ontogeny recapitulates phylogeny"
> we can teach and apply mathematics on the same basis where it has
> evolved.


It's an axiom in the sense that you feel no need to explain or justify
it.

Further, you take the conclusion to hold "no doubt":
this is not empirical mathematics, where it doesn't matter how
many examples we have seen, it remains possible that some
larger number will break a conjecture.

It's a mystery, isn't it?


>> (1 + 1 = 2 is purely computational;
>
> and more is not necessary for the beginning.

>>
>> 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).

>
> The results holds for every natural number that can become existing.
> There are many natural numbers (according to classical and current
> mathematics) that will never become existing (since their Kolmogoroc
> complexity surpasses the ressources of the universe).


But nevertheless the conclusion by induction holds "no doubt".

>> "for every natural number n, if n is odd then n^n is odd".
>
> There is in fact an unsolved question: We cannot name all natural
> numbers between 1 and 10^10^100, as we cannot read 123123123123 from a
> usual pocket calculator, but we can add them, their squares, their
> cubes and so on. I find this surprising, as surprising as the fact
> that it is dark at night.


I do not find it surprising that you think there is no problem
in computing n^n for an arbitrary number in your chosen range.

>
> Regards, WM


--
Alan Smaill