In India Madhava of Sangamagramma flourished around 1400 (many sources
give his dates as ca. 1350-1425). "Circa 1400 Madhava developed infinite
series" (specifically trigonometric series) seems a reasonable statement.
Gauss and Legendre independently invented the Method of Least Squares,
Gauss in 1794 and Legendre in 1805. (Yes, there was a minor but
occasionally nasty priority dispute between the two.) These two dates
average out to 1800 (well, 1799.5).
An important event in 1900 is that Karl Pearson introduced the chi-squared
test and thereby introduced the practice of formal hypothesis testing into
statistics.
On 99-05-18 23:32:34 EDT MANN@vms.huji.ac.il (Avinoam Mann) wrote:
>
> Lagrange thought that mathematics was practically finished at his age.
> This shows how futile it is to try to predict what will happen in a
> hundred years time. I cannot take such a preview seriously.
> [however]...
> each of us has some favourite problems he would like to see solved.
> I may mention the infinitude of perfect numbers and the (in)existence
> of odd perfect numbers, these being perhaps the oldest open problems
> in mathematics, going back to Pythagorean times.
Circa 2100 it is proven that there exist an infinite number of Mersenne
primes, and hence an infinitude of even perfect numbers. Number theorists
can now devote their energies to proving whether or not there exist
infinitely many Mersenne composites. Also, starting with the fact there
there is only one even prime, it is proven that there exists exactly ONE
odd perfect number. Unfortunately the proof does not show how to construct
this Holy Grail of a number.
In algebra the Borwein-Chudnovsky algorithm is finally perfected. This
algorithm, discovered independently by the Borwein brothers and the
Chudnovsky brothers during a friendly competition to compute the first
one billion digits of pi^e, is an improvement on the old Gauss-Seidel
iterative method for solving linear equations. The Borwein-Chudnovsky
iteration converges quadratically and all over the world supercomputers
find themselves unemployed as they are no longer needed to solve large
sets of linear equations. Some of these supercomputers are put to work
searching for the odd perfect number.
Geometry is in an uproar, because it has just been discovered that
Euclidean geometry is a special case of "Leblanc geometry" in which
through a point not on a line there is an arbitrary but finite number
of parallel lines. (This discovery is based on a suggestion made by
Sophie Germain in one of her "M. Leblanc" letters to Gauss.)
The fuss about Leblanc geometry extends to set theory, because the
existence of a countable infinity of geometries is shown to imply (if
you accept the Axiom of Non-Choice) the existence of a countable
infinity of infinities between aleph-null and C. As a result in
calculus it is found that there are logical flaws in the epsilon-delta
technique (Zeno's Paradoxes apply, with epsilon as Achilles and delta
as the tortoise). Similarly the method of infinitesimals is
questioned, and analysts have to fall back on Archimedes's Method of
Exhaustion, which set theory has now placed on a firm theoretical basis.
Just as elliptic functions was the glamour field of 19th Century
mathematics and topology that of the 20th Century, in the 21st Century
the glamour field is "Catalyst Theory", a discipline which subsumes all
of the 20th Century work on chaos, fractals, and Mandelbrot sets. In
Montevideo one Professor Agosto Gonzalez Cabillon writes a book examining
how Dr. Mandelbrot managed to overlook so much that was obvious.
Historians discover that Madhava of Sangamagramma was the first to work
with the square root of negative one. To the relief of mathematical
pedagogues everywhere, the American Mathematical Society declares the
phrase "imaginary numbers" to be politically incorrect, and 0 + bi is
now universally known as a "Madhava number".
In computer science, "natural language" programming finally ousts
C++++ and FORTRAN XIV. The reason for this belated switchover is the
discovery that of all the major languages, only Latin is well-suited
for natural language programming.
The teaching of mathematics in high school is quite different than in
the 20th Century. Students beginning analytic geometry are introduced
to the complex plane rather than the Cartesian plane (the HM mailing
list has become notorious for being the home of reactionaries who wish
to bring back Cartesian coordinates). From the complex plane they are
introduced to 4-dimensional space-time by way of quaternions. Calculus,
instead of being treated as a separate subject, is, uh, integrated into
the study of algebra. The usual method of introducing integral calculus
is to pose the question of what happens when you hang a triangular
weight on a lever.
Finally, in the history of mathematics, an Egyptian papyrus is discovered
that not only confirms the existence of Moses but also clarifies several
previously obscure passages in the Old Testament. Exodus 2:12 is mistaken;
Moses did not kill an Egyptian overseer. Instead he was a student priest
specializing in mathematics who was doing his thesis on cubic equations.
When his thesis adviser died of apoplexy after trying to solve a cubic,
Moses received the blame. Disgusted, he told the faculty, "For all I care
any more, pi is equal to 3", dropped out of school, became a hippie, and
joined the Israelites. The Mosaic influence can still be seen in I Kings
7:23 and II Chronicles 4:2.
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Greetings from foggy New Jersey, where baseball games suffer from fog delays.
James A. Landau
Systems Engineer
Federal Aviation Administration Technical Center (ACT-350/BCI)
Atlantic City Airport, New Jersey 08405 USA