On Mon, 9 Nov 1998 kermit@polaris.net wrote:
> >
> > 3. Has the mathematics of Greek descent enabled valuable or interesting
> > knowledge outside the field of mathematics that other imaginable or
> > existing "cultures of mathematics" could not enable?
>
>
> No. All cultures, sufficiently developed, lead to the same [
> meaning equivalent ] body of mathematical knowledge.
>
I disagree with both (1) the whole line of questioning and
(2) the specific answer.
(1) I think that talking about "all cultures" presupposes a broader
experience than we actually have. I can imagine Mr. Spock saying to
Captain Kirk, "Captain, we've visited 50 planets where intelligent life
has developed independently, and each of them has developed essentially
the same ideas of mathematics." But here in real life, we haven't had
clear knowledge of very many different cultures, and the few that we do
know about mostly did not develop entirely independently. So we
don't have much experience. All we can do is speculate, based on what
we believe are the limitations and non-limitations inherent in
being a sentient species.
Well, I have no objection to that, as long as it's understood what
is the basis of our speculation. I enjoy a good speculate as much
as the next person does.
(2) I find it plausible that Spock might instead say to Kirk, "Captain,
in visiting these 50 planets, we've found some startling blind spots,
all different. This culture had no grasp of infinite cardinals. That
culture had no grasp of geometry in dimensions higher than 2. And this
other culture knows some things that Chekhov and I have been unable to
fathom, as yet, but they seem to be truly brilliant."
I think that each culture has certain blind spots. There are
some concepts that, though not necessarily difficult, nevertheless
get overlooked by a culture for many years, simply because those
ideas are too different from the culture's usual way of seeing
things. I think a good example of this is the idea of evolution,
which is really not a complicated idea at all, but it came as
a great shock to our modern culture when Darwin formulated it.
I believe this is called a "paradigm shift." Einstein, too,
saw things that other people could not see, though perhaps in
his case one cannot say that the ideas were uncomplicated.
A visionary such as Darwin or Einstein removes the blind spot
and changes the culture.
Blind spots may be reinforced and perpetuated by certain other,
related aspects of our culture that do not change rapidly.
For instance, our perceptions are biased by our language, and
some aspects of our language do not change rapidly. (Admittedly,
*after* a discovery is made, the language changes rapidly --
e.g., look at all the technological words added recently to
our own vocabulary.)
I see two possibilities:
(i) Any culture, given enough time, can eventually overcome
all of its blind spots. Thus all cultures converge to the same
truths. This seems to be the opinion of kermit@polaris.net. Or,
(ii) Some blind spots are permanent, and cannot be removed
except by outside intervention. We've seen this "outside
intervention" many times on Star Trek, where the visitors from
the Enterprise somehow remove an obstacle and suddenly a
stagnant culture has new hope. :)
I guess there is no way to know which of these two possibilities
is correct. What really intrigues me is thinking: What might
be the blind spots of *our own* culture? But I guess there's
no way to answer that one, except to wait for the next Einstein
or Darwin.
**************************************************************
Eric Schechter *** http://www.math.vanderbilt.edu/~schectex/