I think we all secretly hope that in the long run mathematics becomes easier, in that with advances of perspective, today's difficult results will seem easier to future mathematicians. If I were cryogenically frozen today, and thawed out in one hundred years, I would like to believe that by 2110 the Langlands program would be reduced to a 10-page pamphlet (with complete proofs) that I could read over breakfast.
Is this belief plausible? Are there results from a hundred years ago that have not appreciably simplified over the years? From the point of view of a modern mathematician, what is the hardest theorem proven a hundred years ago (or so)?
The hardest theorem I can think of is the Riemann Mapping Theorem, which was first proposed by Riemann in 1852 and (according to Wikipedia) first rigorously proven by Caratheodory in 1912. Are there harder ones?
Best Answer
Difficulty is not additive, and measuring the difficulty of proving a single result is not a good measure of the difficulty of understanding the body of work in a given field as a whole.
Suppose for instance that 100 years ago, there were ten important theorems in (say) complex analysis, each of which took 30 pages of elementary arguments to prove, with not much in common between these separate arguments. (These numbers are totally made up for the purposes of this discussion.) Nowadays, thanks to advances in understanding the "big picture", we can now describe the core theory of complex analysis in, say, 40 pages, but then each of the ten important theorems become one-page consequences of this theory. By doing so, we have actually made the total amount of pages required to prove each theorem longer (41 pages, instead of 30 pages); but the net amount of pages needed to comprehend the subject as a whole has shrunk dramatically (from 300 pages to 50). This is generally a worthwhile tradeoff (although knowing the "low tech" elementary proofs is still useful to round out one's understanding of the subject).
There are very slick and short proofs now of, say, the prime number theorem, but actually this is not the best measure of how well we understand such a result, and more importantly how it fits in with the rest of its field. The fact that we can incorporate the prime number theorem into a much more general story of L-functions, number fields, Euler products, etc. which then ties in with many other parts of number theory is a much stronger sign that we understand number theory as a whole.