Friday April 4, 2008
century can be characterized by one huge trend.
= move towards abstraction.
-moving on from late 19
century, from things like group, vector theory, which
were the beginnings of abstract math.
-started really taking off. Why??
-Were starting to get really positive results from abstract work
eg) Riemann working on the Manifold Thoery
-generalization of geometry. As abstract as it gets.
-no practical consequendces, no real use in the least, etc
1905- Einstein produced one of his first papers on the theory of relativity.
-to the astonishment of the mathematical world at the time, he described the entire
universe using the basis of manifold theory.
Ie space is curved.
. how to describe it? It’s a manifold. Space and time? It’s four
dimensions= a 4
-physicists took off, an enormous take off for this type of work
-quantum theory came about, locating position of electron by probability within
an electron cloud
These applications were unexpectedly quick and immediate.
-eg. X-Rays developed, atom bomb, utopian ideas of energy from nuclear
reactions… changing at an enormous pace.
1900, most didn’t even have electricity.
1920,30 Xrays were so well known, superman had xray vision.
Another trend- “we didn’t expect this, we weren’t looking for any application! We
were doing the rigorization of calculus.”
-highly practical, enormous results, rigorized it using SET theory.
Not a very
practical theory, it was very abstract.
SET theory was invented before Calculus needed it
for rigorization. (defining limits, functions, etc, using this theory to DO it)
=pattern that things were come up with not for need, but because ‘just for fun’
that ended up with practical results.
Therefore…. Since this has happened in the past, it will continue to
happen (abstract math eventually becoming practically useful)
-eg develop Groups, Vectors, Fields, Topology
1941 Hardy wrote a book for public use, called ‘A mathematcian’s Apology” saying his
field was so abstract, it couldn’t possibly be applicable.
Gauss: ….number theory is even