Can we still learn from the old masters?

Question

So, let me first describe how my doubt originated: out of curiosity I started to study Newton's Opticks, a book written more than 300 years ago. I was doing some of the experiments described on it, and had some doubts, so I asked my teacher for help, and he helped me, but also said that he thought it was good for fun but it wasn't really worth the time to study from such an old book, and that I would benefit much more from studying the more recent books.

I thought that maybe (:D) he was right (for physics), but... what about mathematics? I mean, the theorems proved more than 2000 years ago are still valid and will always be... but maybe the old way is already a closed system and doesn't serve to inspire new mathematics...

My main question: Are there examples where someone continued an old (like more than 80 years), abandoned research (or program of research) in mathematics and succeeded in extracting new, exciting, and important results (for contemporary mathematicians)? If possible, talk of examples of similar events that already have happened (and please, telling the old work that was useful!).

Could we, living in the 21st century, still be beneficed from reading the mathematical writings of the old masters (e.g. Euclid, Archimedes, Newton, Fermat, Euler, Gauss, Poincaré, and others)? -- Of course, in parallel with the contemporary literature!!!

Answer 1

The work of Manjul Bhargava on "Higher Composition Laws" (2004) is said to be directly influenced by his study of "Disquisitiones Arithmeticae" (1801) by Gauss.'

Answer 2

There are some classics that are still worth reading. Certainly Gauss's Disquisitiones Arithmeticae should be on any number theorists's reading list.

But Newton provides a pretty interesting case. The problem with Newton is that he really pre-dates the time when math became rigorous like it is today. At the time Euclid was God and he tried to do mostly geometric proofs where today we would prefer algebraic methods. He worked with infinitesimals with a rather intuitive foundation, and not limits as we do today.

BUT, it was discovered hundreds of years after Newton that the infinitesimal approach can be made rigorous. That led to the development of a whole branch of math called non-standard analysis. I think that non-standard analysis provides an excellent example of exactly what you are talking about - something that was dead and buried but successfully resurrected by a later generation.

Answer 3

André Weil has written about the origin of the Weil conjectures: "In Chicago, in 1947, I felt bored and depressed, and not knowing what to do, I began reading Gauss's two memoirs on biquadratic residues [from 1828 and 1832], which I had never read before. (...) This led me in turn to conjectures about varieties over finite fields."

Answer 4

Yes, there are past examples, and it is likely still possible. Here are some recent examples that I've heard of:

• Newton's polygons (which Newton described in letters he wrote in the seventeenth century) have evolved into Newton's polytopes, which have interesting applications in algebraic geometry. The Bernstein–Kushnirenko theorem (a result from the twentieth century) is an example of this.

• V. A. Vasil'ev (a student of V. I. Arnol'd) generalized Newton's theorem about ovals in 2002 (a result from the Principia, which is from 1687!), and presented many conjectures inspired by it and by Newton's proof. He described how it also inspired Arnol'd into the investigation of bodies that are algebraically integrable (Vasil'ev's book Applied Picard–Lefschetz Theory, p. 112).

Maybe, if Newton is still popular among some mathematicians from Russia (I think that recently it was the country that most respected Newton as a mathematician), we will see more mathematical research inspired in those old works of Newton!