~ from "topοlogy: the secret ingredient in the latest theory of everything" @techreview
For example, certain quantum particles cannot form pairs but do form triplets called Efimov states. That's curious--surely the bonds that allow three particles to bond together should also allow two to become linked?~ from "you and your research" by richard hamming ('86) @paulgraham:
Actually, no and topοlogy explains why. The reason is that the mathematical connection between these quantum particles takes the form of a Borrοmean ring: three circles intertwined in such a way that cutting one releases the other two. Only three rings can be connected in this way, not two. Voila!
One of the characteristics of successful scientists is having courage. Once you get your courage up and believe that you can do important problems, then you can. If you think you can't, almost surely you are not going to. Courage is one of the things that Shannon had supremely. You have only to think of his major theorem. He wants to create a method of coding, but he doesn't know what to do so he makes a random code. Then he is stuck. And then he asks the impossible question, ``What would the average random code do?'' He then proves that the average code is arbitrarily good, and that therefore there must be at least one good code. Who but a man of infinite courage could have dared to think those thoughts? That is the characteristic of great scientists; they have courage. They will go forward under incredible circumstances; they think and continue to think.~ from "why does it take so long to learn mathematics" @tonysmathsblog
.. There's another trait on the side which I want to talk about; that trait is ambiguity. It took me a while to discover its importance. Most people like to believe something is or is not true. Great scientists tolerate ambiguity very well. They believe the theory enough to go ahead; they doubt it enough to notice the errors and faults so they can step forward and create the new replacement theory. If you believe too much you'll never notice the flaws; if you doubt too much you won't get started. It requires a lovely balance. But most great scientists are well aware of why their theories are true and they are also well aware of some slight misfits which don't quite fit and they don't forget it. Darwin writes in his autobiography that he found it necessary to write down every piece of evidence which appeared to contradict his beliefs because otherwise they would disappear from his mind.
I don't think I'm a better mathematician than I was 35 years ago. In terms of solving exam questions, I would not perform as I did when I was twenty. Even with practice, I am sure I could not get back to that level, and not only because I no longer value that kind of cleverness enough to put the effort in. I now have a much better general understanding of mathematics and how it all fits together, but I no longer have the ability to master detail that I once did.
Perhaps I am misremembering my difficulties as a student: perhaps I didn't find it as difficult as I now remember it. Certainly I had little understanding of how an area of mathematics fitted together: my learning at University consisted of reading strings of definitions and theorems, with little idea where it was all going, making sure I understood each result before going on to the next one, until, perhaps, in the last lecture of the course the lecturer would say something like "and so we have now classified all Lie algebras" and I would suddenly find out what the point of it all had been. I now feel that I would have been a much more effective mathematician if I had read more superficially, skipping proofs until I understood the context, but since got good marks as an undergraduate I had no incentive to adopt what I now feel would have been a much better strategy..