One of the philosophers at my university was also interested in
moral theory as found in or inspired byPlato. Or perhaps it was
aesthetics. As I recall, in her dissertation she dealt especially
with the *Symposium*. Or maybe it was love she was after.
In any case, she asked me about certain mathematical matters as
found in Plato and Aristotle, which I studied to some extent
years ago, although the center of my attention was on what light
this could shed on works by Euclid, Apollonius, Archimedes and
suchlike people. I gave a couple of lectures in one of her
classes in which I dealt especially with how misleading or
cryptic many translations into modern languages were of the
mathematical material in Plato and Aristotle. My comments
were based partly on knowledge of the mathematics of Euclid,
et al, but also partly on my non-historical mathematical training.
This can lead to anachronism, but I tried my best to avoid that.
The immediate result on the philosopher was the next summer she
went somewhere to study the ancient Greek language. I thought
at the time it might have been more useful to her to sign up
for a course of some sort in Euclidean geometry. But I didn't
say that to her. I don't know what the eventual outcome was.
> I do read Greek (ancient) but I am a philosopher, not a
> mathematician, by training. I am wondering if I can
> legitimately pursue my research, which concerns the
> influence of mathematics on Plato's thought, specifically
> his views on moral theory. Most people remember that in the
> _Republic_ mathematics is a required step for the study of
> dialectic or philosophy. In later dialogues, mathematics
> does not have this role, but Plato often uses examples drawn
> from mathematics and claims in the Laws that a knowledge of
> the paradoxical fact of incommensurability should be
> required of every schoolchild. I have tried to give myself
> some grounding in ancient mathematics, but can only do so
> much. I wanted to have some sense of what it might be like
> to be thinking in the ways open to such people as Theaetetus
> that Plato knew. I am not sure if this idea even makes sense
> to mathematicians.
>
That incommensurability passage in the *Laws* and related matters
have inspired much commentary and theorizing among historians of
mathematics, and still do. In fact, there was some related
discussion on this this not long ago. I spent a lot of time many
years ago on the mathematical parts of the *Theaetetus* passage
is one which I spent a lot of time on many years ago, and so have
many other mathematicians. A favorite problem among historians
of mathematics has been to propose theories as to why Plato has
Theaetetus stop at what we would call the square root of 17 (having
started with square root of 2) when considering incommensurability.
I personally came to the conclusion that Plato had no *mathematical*
reason for stopping there, but rather was just leading in his way
up to epistemological matters. This doesn't mean, however, that
one can't invent ingenious mathematical schemes for explaining
why one might want to stop at 17, and indeed many have been suggested.
(NOTE: I remember Mark Kac choosing the number 17 in
a lecture once to illustrate something, and calling 17 "the smallest
*arbitrary* integer. That's an inside joke!)
I also can tell you that while there have been some purple passages
written about the genius of the young Theaetetus, which compare him
to Galois if not more so, I'm not convinced that Theaetetus was in
fact a mathematician at all, nor that the 10th book of Euclid was
written by him. As Kurt Rademacher remarked in a book of his about
history of mathematics, a close study can make one wonder if the
whole tradition about Theaetetus as a gifted mathematician isn't
a distortion derived from Plato's dialogue, who might have wanted
to commemorate Theaetetus's death, and built a story around
Theaetetus and his teacher Theodorus, who evidently *was* a
mathematician of some note. (Rademacher was a mathematician who
did a little history of mathematics on the side, when he got older,
as mathematicians sometimes do.)
> As philosophers, we have to ask questions about the role of
> mathematics in philosophy, and time does not allow all of us
> to be experts in everything. The result may be that views of
> mathematics that would never be accepted by 'real'
> mathematicians become used in the pursuit of philosophical
> dogma. I believe that it is important to gain some idea of
> the way that mathematics has been regarded by philosophers
> in history, as this has clearly influenced the way that they
> presented and understood the philosophical remarks that they
> made (the same seems to be true today, as ideas drawn from
> decision theory are increasingly used in thinking about
> moral and political philosophy, as in modern Contract
> theory). It might then seem that one should not do
> philosophy without a degree in mathematics, but this is
> demanding: one might not realize until much too late that
> this was required. There is no easy answer, but the question
> merits discussion. There is a need for philosophers to bring
> such questions into the discussion of the texts, and so we
> need material which is available to non-mathematicians (as
> provided currently by Ian Mueller for Greek philosophy). It
> seems true that (some) mathematicians need to let the rest
> of us know what is -or has been- going on.
>
Well, some mathematicians and historians of mathematics try,
but, as you indicate, it's not easy for a philosopher to
consider mathematical matters from this or that mathematician's
point of view, just as it's not easy for a mathematician to
consider philosophy from this or that philosopher's point of
view. One can but try.
> For instance, one should be cautious of the assertions of
> those who dogmatically present methods as mathematical.
> Worse, along with this goes an attitude: as mathematics is
> often conceived by non-mathematicians as attaining dogmatic
> certainty, so this supports a view of philosophy as capable
> of attaining equally dogmatic results. Did not some writers
> who tried to present the 'mathematics necessary for the
> study of Plato' misconceive the role of mathematics, as well
> as not understanding mathematics? Should I defer to them?
> Can I legitimately use the views of people like Lakatos to
> suggest that while Plato saw mathematicians achieving
> results, his real philosophical interests should (and in my
> view did)
> lie in the fact that they were prepared and able to canvass,
> develop and reject ideas in a communal way, not that they
> reached some otherwise unattainable certainty? Most of all,
> can I do any of these things without being a mathematician?
> and how otherwise can I argue against dogmatic
> interpretations? And I mean doing more than reading the
> excellent works of people like Ian Mueller, although this
> has helped immensely.
>
Well, Lakatos, as I recall, maintained, roughly speaking, that
mathematicians participated communally (to use your word) in
"research programs" (I think that was the term), and studied
how mathematicians gradually refined and generalized or
particularized results. If memory serves, his prime example
concerned the Euler characteristic of polyhedra, which he
studied from Euler through Cauchy and onward, commenting on
such matters as convex versus non-convex polyhedra, the
gradually refinement and extendion of definitions, etc. Did
he say, though, that the mathematicians involved weren't after
some kind of certainty, if only by making definitions precise
enough so that, given Aristotelian logic in some form, one
could be certain that certain theorems followed from given
geometric axioms, and these definitions? I don't recall him
making that point, but it's been many years since I studied
the question, and when I did, postmodernistic theories of
non-Truth were in their infancy.
> I am only too well aware that I am trying to do something
> which may well be beyond my powers: my excuse is that I
> think there are some interesting questions about method to
> be examined, that go deep into the heart of philosophy, and
> I do not think that one can simply leave them to the
> mathematicians, but we must try to discuss them in more
> accessible ways.
>
Ah, method! Is there a madness in their method(s)? Both those
of mathematicians and those of philosophers, not to mention
everyone else? Maybe so.
Gordon Fisher gfisher@shentel.net