I have only skimmed your article "Platonism, Intuition and the Nature of
Mathematics", and I am printing out a copy to read in the old-fashioned
way. On the face of it, I see many points of agreement between your views
and mine. You have worked out and supported your views in much greater
detail than I have.
There is one thought that has struck me already which I will put forward
now, subject to later revision. I hope you will won't object to my
characterizing your overall views of the nature of mathematics as a
"model-building enterprise". Here "model" is to be taken in the sense of
a particular kind of abstraction, which are axiomatizable. The first
thought that strikes me (assuming I haven't distorted your views) as having
the desirable property of linking together "pure" and "applied"
mathematics. Also it seems that you are linking together what
mathematicians (often loosely) refer to as "intuition" and "formalization"
(or some term similar to "formalization").
The second thought that strikes me is due to a habit of mine of asking
myself how broad a characterization of this sort may be. For example, one
can ask whether or not the process you describe as leading to mathematical
thought can also be used to describe certain kinds of theological or
philosophical thought, as in, for example, St. Thomas Aquinas or Spinoza.
One can ask to what extent a process of the sort you describe can be
applied to legal thought, or political thought, and so on. It seems to me
at first sight that something about the content of what we commonly call
mathematical thought might be required to distinguish it from a more
general kind of thought.
Of course, one can argue that attempts to provide a kind of axiomatization
of say, democracy or totalitarianism or oligarchy, etc., these latter being
abstractions meant to fix certain political entities (imperfectly!) are a
kind of mathematical thought. One might say that any fixing of ideas into
a coherent system in which deductions can be made is "mathematical".
However these seem to go against traditional uses of "mathematical", such
as those found in present day academies where mathematicians qua
mathematicians are expected to content themselves with a content (double
meaning!) of numbers and shapes, i.e. arithmetics and algebras, and
geometry. I take it calculus and differential equations can be subsumed
under these -- if not, then add a third category to cover them. I wonder
too if formal systems of logic should be included here, or under the aegis
of philosophy, or both?
In any case, here are a couple of other random thoughts. Do you notice any
resemblance between your views about the inevitable imperfections of
mathematical models, and Aristotle's views, as contrasted with those of
Plato. I seem to detect certain similarities, although of course Aristotle
did not have the advantage of Hilbert's ideas, as amended by Goedel, Cohen,
etc.
Also, you say: "How far can we proceed in the axiomatic of some theory?
Complete elimination of intuition, i.e. full reduction to a list of axioms
and rules of inference. Is this possible? The work by Gottlob Frege,
Bertrand Russell, David Hilbert and their colleagues showed how this could
be done even with the most serious mathematical theories. All these
theories were reduced to axioms and rules of inference without any
admixture of intuition. Logical techniques developed by these men allow us
today to axiomatic any theory based on a fixed system of principles (i.e.
any mathematical theory)."
Does one eliminate "intuition" with axiomatization? If by "intuition" we
include the meaning one finds more evident in the German "Anschauung", it
seems to me that geometric visualization and its extensions continue to be
used in interpreting axiomatic systems, e.g. Hilbert's axioms for geometry.
(Russell and Whitehead never got to geometry, and I don't think Frege ever
tried to get to geometry.)
Also, you say: "Therefore, it is nonsense to speak about the limited
applicability of axiomatic: the limits of axiomatic coincide with the
limits of mathematics itself! Goedel's incompleteness theorem is an
argument against Platonism, not against formalism! Goedel's theorem
demonstrates that no fixed fantastic "world of ideas" can be perfect. Any
fixed (self-contained) "world of ideas" leads us either to contradictions
or to undecidable problems."
Here again, as I indicated above, it seems that you consider "mathematics"
to include all kinds of reasoning from fixed principles, presumably in some
logically coherent and consistent manner?
I would appreciate very much your comments on these thoughts of mine.
Among other things, they could guide me in reading your work.
Best regards
Gordon Fisher gfisher@shentel.net