Talks about the human aspect of pursuing mathematical research, how they shape the attitude of the field towards a problem abs are crucial in progressing towards knowledge. Should be very readable for everyone; no formal math as such.
> "It was an interesting experience exchanging cultures. It became dramatically
clear how much proofs depend on the audience. We prove things in a social context
and address them to a certain audience. Parts of this proof I could communicate in
two minutes to the topologists, but the analysts would need an hour lecture before
they would begin to understand it. Similarly, there were some things that could be
said in two minutes to the analysts that would take an hour before the topologists
would begin to get it. And there were many other parts of the proof which should
take two minutes in the abstract, but that none of the audience at the time had
the mental infrastructure to get in less than an hour"
I wonder if we would ever get to a point where we would find an effective and desirable mental infrastructure such that this wouldn't happen.
In general better abstractions(similar ideas as in a recent discussion of Peter Naur's "Programming as theory building").
Erdos and Selberg eventually put out fully arithmetic proofs of the Prime Number Theorem. And generally the helicopter analogy from the article probably doesn't apply so well to mathematics because you can probably always reduce theories and encapsulate all the dependent proofs to arithmetic first principles, but of course you already have the map.
Recently the proofs of the Sensitivity Conjecture by Hao Huang and of the Bounded Gaps Between Primes by Yitang Zhang surprised mathematicians in how little new machinery these seemingly intractable problems required -- in the case of Zhang application of "hard work" on top of GPY and Hao Huang, a single clever insight.
That's just a little too late. It was proved in 1896 independently by Hadamard and de la Vallée Poussin.
Hadamard, J. "Sur la distribution des zéros de la fonction zeta(s) et ses conséquences arithmétiques (')." Bull. Soc. math. France 24, 199-220, 1896
de la Vallée Poussin, C.-J. "Recherches analytiques la théorie des nombres premiers." Ann. Soc. scient. Bruxelles 20, 183-256, 1896
"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."
You gotta go at least to the third level of abstraction to get the real meat.
Perhaps a result can be proven using a little known proposition in a completely different area of math, but it is hard to find that result in the literature.
That is one reason I came up with https://mathlore.org. It is a place to collect mathematical info (with links to articles for a deeper look) so you or others can find it later when you need it.
It supports of public collection of math info as well as allowing you to build your own private collection so you can keep track of what you have learned.
The hope is it will be useful to others to help learn math and prove new theorems.
I don't think a map for math techniques is feasible, but a map relating topics via mathematical steps is possible in Physics . (Disclaimer: I'm the author of that map for Physics.) I think the reason that a map in Physics is feasible is Physicists do not use math techniques in the way mathematicians do, and the objectives are different.
Education research tells us that what you actually want to do is the exact opposite: chunk as much as possible. You should learn algebra separately, and then use your preexisting knowledge of algebra to group f = 1/T and T = 1/f into one conceptual node. If you need 10 nodes every time something that basic is done, then your map will contain a vast amount of redundancy and be too large to use to get anywhere...
The chunking of atomic steps is what enables leaps in understanding. The mapping process starts with understanding each step.
Furthermore, this excessive mathematical structure hides the physical assumptions that really drive the validity of these equations. A real string doesn't actually obey the wave equation perfectly. The reason has to do with physical aspects of the string itself, not minutiae in the mathematical derivation of the wave equation. I can't think of an example where progress in physics was stalled because somebody tried to divide both sides of an equation by T and failed...
I recently asked a related question regarding proof maps and quantifying their similarity/distance, but didn't get any answers: https://math.stackexchange.com/questions/3482135/are-there-p...
Very poetic indeed.
Most of entrepreneurship is applying known models to new areas. Intellectually not nearly as stimulating or hard as theoretical math, but the shape and form looks similar - you do not know if a solution exists, you do not know if a problem really exists.
What's funny to me is that, since it's usually applications of engineering, the technology is almost always there. It's a matter of tinkering a collection of things the right way.
I ditched a career in Physics to start a company long back.
This post made me think I probably haven't lost much :-)
The only measures of goodness that I can come up with are logical consistency, elegance and lurking mystery.
Here's a neat BBC piece featuring Gell-man and Feynman on Strangeness -3 .