QUOTE:
"It is quite clear that there can be no intuition of so pervasive and
abstract a concept as that of magnitude (Größe). There is therefore a
noteworthy (bemerkenswerter) difference between geometry and arithmetic
concerning the way in which their basic laws are grounded. The elements of
all geometrical constructions are intuitions, and geometry refers to
intuition as the source of its axioms. Because the object of arithmetic is
not intuitable, it follows that its basic laws cannot be based on
intuition." (again, my translation)
The same basic point concerning the "unintuitedness" of the objects of
arithmetic is made in §105 of the _Grundlagen_, where Frege remarks that
"In arithmetic we are not concerned with objects which we come to know as
something alien from without through the medium of the senses, but with
objects given directly to our reason and, as its nearest kin, utterly
transparent to it."
:ENDQUOTE
I am aware that Kant used the word "intuition" (at least sometimes) with
the meaning of perception through the senses. This is almost the exact
opposite of the way we usually use the word, which is more like what Frege
means in the last paragraph of the quote above (something given directly to
our reason). For instance we talk about whether or not we can discover an
intuition about sets that will settle the question about the cardinality of
the continuum. So I understand that when Frege, using Kant's terminology,
says arithmetic is not intuitable, he means not receivable through the senses.
My question, then, is what does he mean when he says geometry IS
intuitable? It does not seem possible that he could mean "through the
senses" here, especially if the existence of non-Euclidean geometry was
crucial for him. Only one geometry is "intuitable" through the senses, and
that is the actual geometry of physical space, whatever that may be. It
was understood since Riemann that this is a question of physics, not
mathematics, and that it is difficult or impossible to decide, since being
exactly Euclidean would mean having exactly zero curvature, and one cannot
measure an exact zero.
When I first read your words about Frege's claim that geometry is based on
intuition, I unthinkingly took the word intuition in the modern sense (in
which we claim that the axioms for Euclidean, as well as the other,
geometries are not based on physical space experienced by the senses, but
rather on our intuition of what a particular kind of ideal space might be
like). Under this interpretation, Frege's claims about geometry made
sense, and his claims about arithmetic did not. As it gradually dawned on
me that you were consistently using intuition in Kant's sense, Frege's
claims about arithmetic began to make sense, but his claims about geometry
lost their sense. Frege seems to be claiming that there is a third
category, so that the senses (intuition in Kant's sense) are the basis for
physical laws, objects "directly given to reason" (intuition in the modern
sense) are the basis for arithmetic, and the third kind of intuition
(whatever it might be) is the basis for geometry. What is this third kind
of intuition?
..............................
Jim Murdock
Mathematics Dept.
Iowa State University
Ames, Iowa 50011