[FOM] To Vladimir Sazonov and others doubting the unambiguity of N

[Lucas Wiman]

On Sunday, June 15, 2003, at 10:40 AM, Vladimir Sazonov wrote:

> Aatu Koskensilta wrote:
>> Surely the notion of *non*-standard
>> model of arithmetic is much more illusive, as any such model must
>> necessarily be non-recursive?
>
> I have a serious problem
> with understanding when "standard" model of PA is mentioned
> in some ABSOLUTE, metaphysical, quasireligious sense (not
> RELATIVE to ZFC or the like). It seems you do not understand
> what I mean. If I would know what is unclear for you when I
> refer to ABSOLUTE and RELATIVE, I would try to explain.

Recursiveness is supposed to be an epistemological notion, though one of 
great mathematical significance.  Interestingly, it also seems to lead 
to nontrivial metaphysics in this case, since "the standard model of PA" 
becomes "the only one we can construct."  No religion.  Nothing like 
that.

It seems you must have a serious counterargument to Church's thesis or 
Turing's thesis, which would be a significant event for philosophy 
generally.

> Yes, we can continue further and further, but how further?
> Until we will get tired? What this AND SO ON really means?
> Can anybody explain? If not, then this is something indefinite,
> vague. Thus, the "resulting" N is also vague. Let us be honest
> before ourselves.

I agree with Bill, you're definitely an ultra-finitist.

> Again, what is the "length" of the resulting N? Intuitively,
> it is much more comfortable for me to think (together with
> Esenin-Volpin) about many (infinite) Ns of various "length",
> with various abilities to iterate the ability to iterate the
> operation x+1. It is intuitively plausible that the simple
> iteration of x+1 leads us only to feasible numbers where
> 2^1000 is non-feasible.

What?  A number you just named is non-feasible?  This makes absolutely 
no sense.  I ask that you give me a definition of non-feasibility.  For 
any definition you give me, I can name a larger number than that, and it 
is in that sense I think the idea of an "absolutely non-feasible number" 
makes no sense.

Let's think about the number 2^127-1.  This number was proved prime by 
Edouard Lucas in the late 19th century, by hand.  However, he used a 
method which did not check for all prime factors, in fact using 
elaborate calculations on a a group of size 2^127 (actually his proof 
was to show that this group does indeed have this size).  You seem to be 
saying that none of these results makes sense because he couldn't 
possibly write down a string of 2^127-1 marks.  Or am I misunderstanding 
you?

But I can tell you things about vastly larger numbers.  I can tell you, 
for example (without doing the calculation!) that 3^(2^127-2) has 1 as 
the remainder when divided by 2^127-1.  This is all basic number theory, 
yet it easily transcends the epistemological limitations you put on 
mathematics.  You're denying mathematical facts, and hence your 
epistemology is false.  There are certainly limits to my ability to name 
numbers (for example, there are numbers on the order of 3^2^127 which I 
cannot name), but there are gaps and in my ability to name them, I can 
still name much larger ones.  If someone could give this a precise 
formulation, it would be interesting, but it would not have the 
epistemological weight that you give it.

You'll (no doubt) say that I'm "really" working in PA above, though 
this, too, denies mathematical reality.  (I know, for example that 
addition is commutative and associative, but I don't know the proof of 
it in PA.)  If you explain the proof of Godel's first incompleteness 
theorem to most mathematicians, they'll say something like "yes, but 
then the Godel sentence has to be true."  What this tells you is not 
that mathematicians are working in second order number theory (because 
one can obtain a Godel sentence about that, and a mathematician will 
tell you the same thing), but that mathematicians aren't working in a 
formalized system at all.  If one thinks of logic as the "projection" of 
mathematical arguments, it's interesting to study, but it is objectively 
different from real mathematical arguments.  Gauss never needed PA to 
understand arithmetic.

It's interesting to note that Vladimir has had precisely the opposite 
reaction of some notable mathematicians to Godel's theorems.  Alain 
Connes (a Fields medalist), for example, said in a recent book that 
because of the ability to see that Godel's statement is true, 
mathematicians must in fact be accessing some kind of reality separate 
from formal systems.  No formal system can totally capture all the 
ability of mathematicians have to reason about mathematics.

- Lucas Wiman