FOM: FOM : Jurassic pebbles on the beaches of infinity

[Olivier Souan]

Kanovei wrote :
>The Largange theorem is a product of social activity,
>and as such it most likely did not exist before L
>proved it.

Lincoln Wallen wrote :
>if a
>Jurassic beach were *to all relevant intents and purposes* the same as
>a beach I can take you to now, the same things would hold.  Of course.
>That is the point of stating explicitly, or implictly "all relevant
>intents and purposes".

Let me develop your viewpoint, to which I do not subscribe.
The Lagrange theorem has not sprung from nothing. The reason of its
existence is the social context in which he had taken place, and more
peculiarly the "intents and purpose" of a social community. I have a
goal
(counting pebbles), so I use the Lagrange's theorem for that (tell me if
I
do distort your thoughts). Mathematics' reason of being is thus social
activity.
1) Let us say that social purposes are necessary reason of existence 
(one
should add as an example, the "production" of geometry from land
surveying
in Egypt in order to avoid the Nile floods, etc.). But are they
sufficient?
I do not think so. You have to reach sufficient levels of abstraction in
order to contemplate the situation and formulate a solution. You have to
make a model of reality (model not used in the technical sense), to find
a
scheme which works in all cases and in all situations. Those levels of
abstraction are not at all included as such in practical activities,
even if
those benefits from them. They are used by accident, not for their very
sake, not for themselves, but for their applications.
2) I do not think that practical purpose is even necessary. The Egyptian
who
invented land surveying or the Babylonian who invented comptability
indeed
were led by practical motives, but Pythagoras and Thales, the men who
invented (daresay discovered?) mathematics (AND philosophy at the same
time)
did so not for their practical usefulness but for the sake of knowledge
itself. The reason for which they were led to mathematics were the will
to
know the more abstract levels of reality for themselves, independently
of
any practical purpose.
This concerning mathematics as a whole. It is obvious indeed that most
theorems are not discovered for practical purpose; this purpose comes
after.
Lagrange never found its theorem because he wanted to count pebbles on a
beach or whatsoever. Else you confound the antecedent and the
consequent,
the condition and what it makes possible. All that debate reminds me an
older one on whether logic is a science or an art (similar to the
Hilbert
vs. Godel debate). But this is not yet the point here.

However, I do not disavow completely such a viewpoint, safe we rise to a
very different plane. Doing mathematics, we can say : in order to prove
a
theorem, we need such and such axioms to prove it. There is here also an
intent, a purpose, but it has nothing to do with practical purposes. One
could call them "theoretical purposes", because they are included in the
abstract levels of reality (as are the theoretical means) and not in the
material ones. It is also strinking to see to what extent are the
mathematics a whole. Lautman once underlined the fact of the unity of
mathematics and Jean Petitot (a French epistemologist) even used the
term
"holistic order". All happens as if all theories were "interexpressive"
(in
Leibnizian French one could say "s'entr'expriment"), imply each other.
The
more interesting theorems use a considerable amount of mathematical
knowledge (try to read Wiles' proof to convince yourself about it!).
Mathematics do not need common reality to exist. I would even say that
they
are the condition for common reality to exist.

=====
Indeed the departure of M. Feferman would be a great loss. I hope he
will endure and stay with us.

=====
Martin Davis wrote :
> NOTE: You could play the same game with the proposition that 2 + 3 = 5;
but
> then someone might ask: `in what axiomatic theory?'
with that comment from Vladimir Sazonov :
Yes, your explicit fixing the formal system ZFC was very appropriate.

It reminds me an old joke :
"Sir, allow me to ask you one question. If the Church should say to you,
"two and three make ten", what would you do? "Sir," said he, "I should
believe it, and I would count like this :one, two, three, four, ten." I
was
now fully satisfied"(James Boswell)



Olivier
Souan                                                                                            
Student of Philosophy, and
Logic                                                       
Université Paris IV et Paris I
Sorbonne                                            
Tél: [011 33 from the US/ 010 33 from the UK] +                     
           [0*]1.46.33.92.34   (*0 to be added if calling from France)
Areas of interest : Phenomenology, Greek philosophy,
Logic and Set theory, Mathematics and its foundations