Subject: Re: [HM] Kant and non-Euclidean geometry
From: Luigi Borzacchini (gibi@pascal.dm.uniba.it)
Date: Mon Mar 13 2000 - 03:47:06 EST
Dear HM-list members,
I think that the relationship between kantism and non-euclidean geometries
is the key to understand many aspects of the relationship between modern
mathematics and philosophy.
First, I agree with Gordon Fisher:
> The idea that Kant's views on euclidean geometry somehow undermined his
> whole system strikes me as a kind of academic "urban myth".
In addition, I want to deal with a broader connected theme: THE EMERGENCE IN
MODERN MATHEMATICS OF THE FRACTURE BETWEEN TRUTH AND PROVABILITY.
At the end this theme is the core of Goedel incompleteness theorems.
To be short I am going to give very sharp statements.
1) This fracture is completely absent in Greek mathematics.
2) This means that the axioms/postulates must be absolutely (conceptually)
evident and (objectively) true as well, whereas the theorems are
(rigirously) proved and (objectively) true.
3) To this extent the V postulate, being not absolutely conceptually evident
and being objectively true (even because strongly linked to other
objectively true propositions/constructions, as the sum of the interior
angles of a triangle and the construction of the square), had to be proved.
4) This line of thought is undisputed till Renaissance. Even Leibniz'
philosophy is thoroughly inside this line.
5) In the XVIII century something happens: I do not know what, but surely it
has no empirical ground. The result is the emergence I stated at the
beginning in capital letters. I want only to underline that such a fracture
seems necessary to build a 'true' science, whose base is not pure
'deduction' but 'experiment'.
6) The best philosophical side of this emergence is Kant's philosophy.
Aprioris are first and foremost something which can neither be proved, nor
considered analytic, but whose intersubjectivity is so deep to appear
indistinguishable from objectivity. And this merging of intersubjectivity
and objectivity is the background of the emergence of a new creative and
legislative role of the "subject of knowledge".
7) This allows a reshaping of the V postulate's question, inside a shift of
the center of mathematics from geometry to arithmetic (Gauss). In the XIX
century geometry will become more and more linked to physics (external
reality), whereas logic will emerge connected to algebra and arithmetic
(knowing subject).
Dear HM-members, please, forgive me if these seven points are a sort of
"short cultural history of mankind": I just hope them sharp enough to
express my thought.
Best wishes
Luigi Borzacchini
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