“If I have seen farther it is because I have stood on the shoulders of giants,” is a quote that is often misattributed to Sir Isaac Newton, one of the two amazing minds that developed the calculus around the turn of the 18th century. While that quote is not Newton’s, it still is entirely true about his work. Without the work of the mathematicians of the previous two centuries Newton, and Leibniz, would not have been in a position to develop the calculus, an area of mathematics which can easily be argued to be the true engine behind the industrial revolution.
Mathematics, in fact all of science, is not done in a vacuum. All the work that is done today builds off of the work of generations of thinkers, inventors, and researchers. When a mathematician decides to shuck off the yoke of previous work they find themselves staring up from a hole that might as well go all the way to the center of the earth. One of the classic examples of this is the work Principia Mathematica by Bertrand Russel and Albert North Whitehead where they decided to tear down mathematics and build it back up from a foundation centered only upon the most elementary of logic. The Principia was never finished, in fact not two decades later it was proven that such a treatment of the subject could not be done, they ended up using the first 362 pages to prove the statement 1+1=2. Of course mathematicians can also find themselves indebted to someone well outside of their field. A wonderful example of this is the work of Duncan Watts and Steve Strogatz on the topic of Small World Networks which all started because Watts remembered his father once telling him that everyone is separated by only six handshakes, and idea popularized in 1929 by Frigyes Karinthy in his short story Chains.
This debt to what has come before is treated very differently by mathematicians, and while few would completely disavow what they owe none take it more seriously than Grigori Perelman. A Russian mathematician, Perelman gained international renown in November 26 for his proof of the Poincaré conjecture, one of the Clay Mathematics Millennium Problems which have a million dollar bounty on their solutions, which had stood unsolved since 1904. This work was so important and groundbreaking that in 2006 they earned Perelman the highest award that a mathematician can earn, a Fields Medal. An award that, with a mind at least partially thinking of the debt to the mathematical community that comes with accepting such a high honor, Perelman declined, citing that he did not want to displayed like a zoo animal. This was only the first award that Perelman declined, he also turned down the million dollar Millennium Prize bounty and this time his refusal was clearly to deal with debt. Perelman’s proof of the Poincaré conjecture came from following what is known as the Hamilton program, essentially a plan for producing a solution to the conjecture that was developed by American mathematician Richard Hamilton. Perelman thought that it was unjust that he alone was getting all of the accolades, and prizes, for the work and Richard Hamilton, a man to whom Perleman clearly feels a great debt, was languishing on the sidelines. There is hope that Perelman’s thoughts about the unjust nature of the mathematical community, which have cause him to withdraw from the community, may still be assuaged as Richard Hamilton was a co-award winner of the Shaw prize for the work he did towards the proof.
While Perelman’s empathy towards those whose work he used as stepping stool to new results is unusual, it is illustrative of just how important previous work is to those who really notice.
The ocd-logician in me must object a bit to the way you present the proof of 1+1=2 from Principia.
The way you wrote it is a little misleading implying that they had to prove so much stuff beforehand, taking them 362 pages.
Obviously (from your own description), PM was not designed to prove 1+1=2 and it was written to develop a solid foundation for a much larger scope. So the impression that it took them 362 pages is a bit like watching a skyscraper being built and then, when the first floor stands, saying “What, it took them 2 years to build a one floor building?”.
In short, I’m pretty sure they could have proved it on page 100 😉
I know that the way I characterized the 1+1=2 thing was hyperbolic, but only to try to hammer home the point.