Math & Language 2.
The controversy between Newton and Leibniz over the
"invention" of the calculus is interesting in this regard and
sheds some light on the subject. The three greatest mathe-
maticians of all time are generally considered to be Archimedes,
Newton and Gauss. The crown probably belongs to Newton although
he insisted that he "stood on the shoulders of giants" -- which
is correct. It is said that Newton worked out his proofs using
his newly invented?/discovered? calculus, but then restated or
translated these proofs into the language of Euclidean geometry.
Thus the great treatise called _... Principia Mathematica_(1687)
is incredibly obscure because calculus is carried out in the
language of Euclidean geometry. Why did Newton do such a thing?
He said he wanted to make it difficult in order to avoid
intellectual squabbling. However, I suspect that another more
important reason is that he was first and foremost concerned to
demonstrate without question the truth of his theorems. He
couldn't do this at that time with "calculus" because arithmetic
and calculus were not axiomatized until after centuries more
work (eventually in axiomatic set theory this century). However,
Euclidean geometry had been axiomatized by Euclid, was based on
five "transparent" axioms (except the fifth wasn't so trans-
parent) and hence Newton could demonstrate the truth of his
theorems by "translating" calculus into Euclidean geometry,
thereby creating an incredibly exact but obscure treatise.
Later on there was a huge intellectual dispute over who
"invented" the calculus, Newton or Leibniz. Leibniz was the one
who invented the language of the caluclus that we use today. He
took great pains in crafting the language. For the next hundred
years English mathematicians, out of loyalty to Newton (English
nationalism) attempted to develop the calculus along Newton's
lines and failed. Rather, further development of the calculus
was carried out on the continent because Leibniz had forged the
superior mathematical symbolism.
Point -- mathematical languages themselves undergo develop-
ment. What motivates this development? The ease in carrying out
proofs and performing calculations. However, such ease in one
direction (proofs and calculations) does not make for an easy
language to understand. Rather it makes for a new language to
learn. On the other hand, the deepest mathematical results are
very often most lucidly explained in ordinary language.
There was a linguistic progression in the development of the
symbolism (language) of mathematical logic also. The first work
on this subject by G. Frege came out in 1879. The symbolism was
hopelessly obscure. Thus Frege is obscure. Bertrand Russell
studied under the Italian mathematician G. Peano for a few years
and adopted Peano's symbolism as the symbolism of his _Principia
Mathematica_ -- like Newton's tome, another obscure work that had
to be gone over by scores of mathematicians and subsequent
generations thereof. Gerhard Gentzen reformulated Russell's
awkward "symbolic logic" into a system of "natural deduction,"
the way we (i.e., mathematicians) "naturally" reason, the mature
form of mathematical logic today.
However, this reasoning is not that "natural" to most of the
species, but must be learned, just as you have to learn French
if you're an English speaker. Math majors seem to be so natural-
ly adept at this language that they don't need or bother to study
it formally. It's sort of a sixth sense for them, even though it
took Bertrand Russell his entire early career to codify the
grammar. (Should this grammar be taught to everyone, like
English grammar in grade school? Is this possible or is the
subject too difficult?) But I have to get on to mathematics in
economics. I am hardly an expert on this subject, but I did take
a look at mathematical economics once in the 60s and formed an
opinion on it. Which may be of interest.
Curtis Moore <[EMAIL PROTECTED]>
