[FOM] To Vladimir Sazonov and others doubting the unambiguity of N
wiman at uiuc.edu
Sun Jun 15 14:09:41 EDT 2003
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
It seems you must have a serious counterargument to Church's thesis or
Turing's thesis, which would be a significant event for philosophy
> 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
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
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
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