Hi All,

I have often try to explain what is mathematical realism.
May I quote the full section 24 of G. H. Hardy's "A Mathematician's
Apology" which explain so well what I try to say?  I will then make
some comments, offer an apology myself, and finish
by two conjectures.
Let us first listen to Hardy:

      "There is another remark which suggests itself here and
which physicists may find paradoxical, though the paradox
will probably seem a good deal less than it did eighteen years
ago. I will express it in much the same words which I used
in 1922 in an address to Section A of the British Association.
My audience then was composed almost entirely of physicists,
and I may have spoken a little provocatively on that account;
but I would still stand by the substance of what I said.
      I began by saying that there is probably less difference
between the positions of a mathematician and of a physicist
than is generally supposed, and that the most important seems
to me to be this, that the mathematician is in much more direct
contact with reality. This may seem a paradox, since it is the
physicist who deals with the subject-matter usually described
as 'real'; but a very little reflection is enough to show that the
physicist's reality, whatever it may be, has few or none of the
attributes which common sense ascribes instinctively to reality.
A chair may be a collection of whirling electrons, or an idea in
the mind of God : each of these accounts of it may have its
merits, but neither conforms at all closely to the suggestions of
common sense.
      I went on to say that neither physicists nor philosophers
have ever given any convincing account of what "physical reality"
is, or of how the physicist passes, from the confused mass of
fact or sensation with which he starts, to the construction of the
objects which he calls "real". Thus we cannot be said to know
what the subject-matter of physics is ; but this need not prevent
us from understanding roughly what a physicist is trying to do. It
is plain that he is trying to correlate the incoherent body of crude
facts confronting him with some definite and orderly scheme of
abstract relations, the kind of scheme which he can borrow
only from mathematics.
      A mathematician, on the other hand, is working with his own
mathematical reality. Of this reality, as I explained in  22, I take
a "realistic" and not an "idealistic" view. At any rate (and this was
my main point) this realistic view is much more plausible of
mathematical than of physical reality, because mathematical
objects are so much more what they seem. A chair or a star is not
in the least like what it seems to be ; the more we think of it, the
fuzzier its outlines become in the haze of sensations which surrounds
it ; but "2" or "317" has nothing to do with sensation, and its properties
stand out the more clearly the more closely we scrutinize it. It may
be that modern physics fits best into some framework of idealistic
philosophy---I don't believe it, but there are eminent physicists who
say so. Pure mathematics, on the other hand, seems to me a rock
on which all idealism founders: 317 is a prime, not because we think
so, or because our minds are shaped in one way rather than another,
but *because it is so*, because mathematical reality is built that way."

        G. H. Hardy,   "A Mathematician's Apology"
                                      Cambridge University Press, 1940.

Comment:
-Idealism is what we call solipsism or subjective idealism, as opposed
to objective idealism, or platonism and which Hardy called
mathematical realism.
Hardy is a well known Number Theorist, and he exemplified his "realism"
with numbers. It should be clear, from preceding posts, how much
I agree with him. Nevertheless I take such a realist position as obvious
only for number theory. I am less sure that such a realism is defendable
for set theory. For example no one would say "R or C is well-ordered not
because we think so, or because our minds are shaped in one way rather than another, but *because it is so*, because mathematical reality is built that way."
It follows from the axiom of choice that all sets can be well-ordered, indeed.
But we can manage to build different set concepts for which the axiom
of choice is natural or is not natural.
In other texts Hardy insists very much on the importance of "proof", and I
guess that is what makes, for him, mathematician closer to reality .
Consider Riemann Hypothesis. It tells that some function from C to C
annihilates itself only on a precise subset of C. It has been verified for *many*
(billions) values. But mathematicians will only believe it if a proof is given.
Once such a proof is given they will have a contact with "their reality" that
no physicist will ever have. But, by incompleteness, we can expect
that even "pure" number theory will be more and more experimental.


Now, Hardy lacks Church thesis and comp and so seems not aware that
the physical reality could be mathematical reality "seen from inside",
in which case, "empiricalness" can be justified by "incompleteness",
and this, of course makes the distinction between physics and mathematics,
still more fuzzier.

Apology: I have not finish my promised paper. I'm infinitely sorry.
Worst: I have begin a new one. The fact is that I begin to smell a shortcut
between "number theory" and physics. I have already mentioned
Watkins' website: http://www.maths.ex.ac.uk/~mwatkins/zeta/index.htm
Actually this goes against comp and the derivation of physics from "machine
psychology" (alias the logic of self-reference), unless there is a connection
between self-reference and prime number, which is what I'm looking for.

I am converging toward two quasi-opposite conjectures:
 -1) The prime numbers emulates (in some precise but technical way)
a universal  *quantum computer*.
 -2) Some variant of the self-reference logic emulates a universal
*quantum computer*.

"2)" is a consequence of comp (cf my thesis). "1)" is currently doubtful because
the physics which is currently extracted from the primes seem to be
irreversible (which is not good for quantum computing!).
Still, I believe that both "1)" and "2)" are partially true, and that the difference
between the two conjectures can lead to a way of using "pure" number theory
for measuring the degree of truth of the comp hypothesis.
Note that in any case where "1)" or "2)" is correct, some Pythagorism
would be assessed. "2)" gives a more important role of the observer/machine
in the "fabric of reality".


Bruno





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