Which is the mark that the competent mathematician must have more than any other?

Question

I'm reading the excellent autobiography of Norbert Wiener on its couple of books (Ex-Prodigy: My Childhood and Youth and I am a mathematician). Certainly there is much material to discuss, but there is one paragraph that catch my attention entirely, it is on page 86 in the second book titled I am a mathematician:

Indeed, if there is any one quality which marks the competent mathematician more than any other, I think it is the power to operate with temporary emotional symbols and to organize out of them a semipermanent, recallable language. If one is not able to do this, one is likely to find that his ideas evaporate from the sheer difficulty of preserving them in an as yet unformulated shape.

It is a textual citation.

First of all I always though that the mark of any competent mathematician would have the capacity of focus on one problem and advance in a solution. I mean on my mind a mathematician must always know how to "attack" a problem. But this idea of Norbert confused me. As I understand this is that the mark of the competent mathematician is to operate with symbols ( I'm here ignoring the complete sentence used by him: "temporary emotional symbols" since the temporary emotional part confuse me) in order to preserve complex ideas, something like be ble to put a lot of information with a few symbols.

Is my interpretation correct? (I'm not a native english speaker) If that's the case, do you agree? why? there is some competent mathematician here to bring light in this matter?

Answer 1

An incompetent mathematician here, who wants to take part in the discussion :)

I don’t think any mathematician has the gift to precisely know how to attack mathematical problems, otherwise there wouldn‘t be millenium problems, which were not solved by some of the most ingenious mathematicians. However a gift which all of these folks had (and have) is seeing the essence of mathematical statements and properties (whatever essence means). This makes them find more things trivial than normal people would, not in a sloppy but in a precise way, if backed up with enough mathematical education (by which I mean learning the language, not memorizing proofs). I think Ramanujan is a good example for this.

Now regarding the text you cite I understand it like follows. Normally, when attacking a hard problem one tries many things, which might or might not lead to some results, but in any case increase the intuition for the problem. After a (possibly very long) while an intuitive (or emotional) path to the solution might come in sight, which needs to be turned into a rigorous mathematical argument. However, while making it rigorous, this path may get out of sight again, so it is important to have a way to note it in an unprecise manner for the sake of recovering it.

In the end one might argue that mathematical texts and proofs are essentially just symbolic support for recovering complex ideas and maybe this is what the author is talking about (I don’t think so since the adjective temporarily does not fit, but it’s another interpretation)

Answer 2

My interpretation of the quote is that a competent mathematician has the ability to memorise intuitions by translating them into some (not necessarily fully formal) form. As the important thing is to be able to recover the intuition later on, even a year later, you need appropriate notation, comments or drawing to be sure to remember the idea.

I don't quite follow you (and I agree on that with PrudiiArca) when you say that a competent mathematician always know how to "attack" a problem. This might be true for some problems, but really difficult ones may require to think outside the box. Important progress in mathematics has been made when a deep connection has been found between two seemingly disjointed areas. For instance, Galois theory provides a connection between field theory and group theory. For this to happen, you may need to be competent on several domains, or sometimes just be lucky: it might be enough for a colleague working in another field to say, "Hey, this is reminiscent of what I'm doing."