There are two interesting but quite different general discussions about "quality of maths" available on the web... There is the recent paper by Tao on "what is good mathematics?" and Serre's talk on "how to write mathematics badly"...
I strongly recommend to listen to Serre's talk which will result for sure in a definite improvement of the writing style of the listener. The talk is clear, funny, and makes a number of well taken points. As an example Serre explains the distinction between a proof and a "Bourbaki proof" (a term often used with a pejorative connotation): a proof is understandable by experts, a Bourbaki proof is understandable by non-experts (and of course that's much better).
It is hard to comment on Tao's paper, the second part on the specific case of Szemeredi's theorem is nice and entertaining, but the first part has this painful flavor of an artist trying to define beauty by giving a list of criteria. This type of judgement is so subjective that I really had the impression of learning nothing except the pretty obvious fact about arrogance and hubris...
I was asked last year by Tim Gowers to write some advise for beginner mathematicians and reluctantly made an attempt. My main point is that mathematicians are so "singular", (and behave like fermions as opposed to the physicists who behave like bosons) that making general statements about them often produces something obviously wrong or devoid of any content.
Haven't read Tao's paper, nor listened to Serre's talk (yet), but I did read your "Advice to beginners" some time ago. But rather than the fermion-style for mathematicians, to me (as a real beginner) one of the most helpful points was the one about keeping moving and looking for new sources of inspiration. I like to think of a mathematician like a shark, that has to keep constantly moving forward to help dying...
Is there another version of this video where I can hear what he is saying?
The video is fine. I downloaded and listened to it yesterday and had no problem.
I remember having a look at Tao's paper and tried to address his points BRIEFLY.
Anyway, I thought that the following responses should be given in regards to what is obviously progressive mathematics – mathematics in which each student is allowed to make progress, and also the notion of absolutely true and obviously good mathematics.
Most mathematicians (ie: all Cambridge Mathematicians, or MIT mathematicians, for example) should have this notion of being “born” with at least a substantial proportion of the mathematics that they know having been given to them – this is the idea that, independent of political opinion, ethnicity and all the rest, most mathematics seems natural and clear when it is finally understood. There is no need to mutilate the mind when accommodating good mathematics – it is usually a pleasure to understand some kernel of truth to what is being studied.
The experience of feeling that the solution to a problem existed before that solution was found is a common one to many mathematicians – they feel, rather than having solved a problem, that the solution to the problem existed the moment the problem was created.
These experiences/issues are in no ways confined just to mathematicians, but I digress.
I think one interesting area of study that is relevant to good mathematics is the notion of truthful mathematics. All good mathematics is true (ie: derivable from a set of axioms – like in ZFC). Not all true mathematics is 'good', though this depends upon your definition of 'good', some people think that truthful knowledge is the best type....
Anyhow, for some people, the idea that several mathematical theorems that are relied upon haven't been formalised within ZFC (except if such theorems comment upon physical phenomena that are not easily computed with) is quite annoying, and almost unethical - how do we KNOW that the proof for a theorem is true?
This is connected to your question of what good mathematics is, as we really do need to know what mathematics is at all to answer the question in totality.
Anyhow, enough of my ramblings.
What are some of the points that you fail to talk about or mention?
1)You have not mentioned that good mathematics requires GOOD MATHEMATICIANS to practice that mathematics. Surely we require those who are willing to FAIRLY compete with others on a common core of mathematics so that it can be seen that all mathematicians (say) in research tenure deserve their positions and posts. We need individuals who are capable of doing mathematics and willing to publically account their abilities to others. How else can we be sure that we are dealing with good mathematicians?
2) You have not mentioned the POWER of mathematics (which, of course, is the power of truth). Mathematical definitions have a precision at times that cannot be achieved by using thousands of words. This shows that mathematics has a precision that most other fields of knowledge do not.
3) STANDARDISED MATHEMATICS - Mathematicians are diverse - but they must have a common core of EXAMINED AND ACCREDITED MATHEMATICS so that we can be sure that they can communicate a given core of mathematics to one another. Otherwise each mathematician ends up on their own Island without having the ability to communicate to others, and without proving their merity within examined tests and or COMPETITIONS.
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