I spent some time some years ago trying to answer questions similar to
yours. I was led to the work of Edmund Husserl, who, I take it, was also
trying to answer some questions of this kind. And you may recall that the
recently late Gian Carlo-Rota was also interested in the phenomenological
movements in philosophy. I mistakenly thought you were asking about how
historians of mathematics proceed in their everyday work, rather than about
philosophical foundations of mathematics. In connection with the latter,
after I wrote an article based on Husserl's work which was never published,
I turned to neuroscience to see what I could find, after having passed
through an indoctrination in artificial science which never quite took.
Among numerous other things, I concluded that as far as geometry is
concerned, there are very interesting facts about the way our eyes,
considered as parts of our brains, work in connection with our brains'
construction or conceptualization or abstraction of geometric shapes.
In connection with numbers, I will tell an anecdote related to your reply
to Luigi Borzacchini and Walter Felscher. When I was about 9 years old,
one day, while in the kitchen of my home, I was struck by the label on a
Land O'Lakes butter carton, on which there is a Native American lady
holding another Land O'Lakes butter carton, on which that same lady is
holding another Land O'Lakes butter carton ... and so on (the usual three
dots!). Great Scott, this could go on forever -- in theory, of course,
since our printing presses aren't up to the task. Shortly after that it
dawned on me that the natural numbers likewise could go on forever, since,
by George, you can always, given any positive integer, create another
positive integers by adding one to the given positive integer. My fate was
sealed, had I known it. I was to become a mathematician, and to some
extent, a philosopher. Eventually, in connection with my doctoral
dissertation, I was led also to become an historian of science, especially
mathematics and physics.
I don't think this is an unusual story. I recall an article in the
American Mathematical Monthly some years ago in which a number of
topologists were described as having had similar experiences in their youths.
-From this, and other data, I concluded that some function, inborn in most
human minds, and probably in some form in most living beings, leads us to
think, let's say, inductively. In particular, there is a kind of
constructive induction available to most of us in which we move in our minds
from a thing of one kind to a successor of the same kind. This is, I take
it, the original source of numeration. And, I might say, this isn't all
that far from saying mathematics is present when numbers are present.
This brings us to another point in your reply to Borzacchini and Felscher.
What you seem to me to be talking about there is mathematical *proof*,
rather than mathematics itself. Thus to prove certain theorems in number
theory, it is convenient to have as an axiom one that covers what we call
mathematical induction. Having that, we can recognize the necessity of
various theorems being, let's say, true, or at least valid deductions based
on an induction postulate and some accepted axioms of logic. I add the
latter clause in case you might have in mind some other sense of the term
"true".
But to mind "mathematics" precedes "mathematical necessity" or
"mathematical proof". Indeed, I expect that many historians would agree
that mathematical proof as we now understand it was invented (or
discovered) by the classical Greeks, and that earlier people who did
mathematics operated in different ways. The commercial people referred to
in your reply to Borzacchini and Felscher no doubt customarily operated in
a different way. I would ascribe this to inborn capacities of their
brains, as developed and altered by their experiences.
There are indeed in nature (which of course can be taken to include
ourselves) mathematical forms, and not surprisingly in view of the way we
appear to have evolved in the world, our brains have the capacity to
recognize them to some extent, although in doing so it appears that we have
to resort to abstractions of various kinds and degrees. In my view, the
kind of necessity which mathematical theorems usually appear to us to have
came relatively late in the game, speaking historically and
prehistorically. It's not that this necessity isn't, at least, usually,
based on mathematical forms and processes in nature. It's that humans have
had to develop sublanguages of their languages in which to express, as best
they can, necessities of these kinds. These kinds of necessity involve, as
far as mathematics in a present-day academic sense is concerned, numbers
and shapes. So again, this seems to indicate that historians of
mathematics aren't too misguided when or if they call that which has to do
with numbers and shapes "mathematics", even in the absence of explicit
knowledge of mathematical necessity and mathematical proofs.
For brevity, I've deleted your replies to Borzacchini, Felscher and Cabillon.
Gordon Fisher gfisher@shentel.net