If a mathematician wishes to disparage the work of one of his colleagues, say, A, the most effective method he finds for doing this is to ask where the results can be applied. The hard pressed man, with his back against the wall, finally unearths the researches of another mathematician B as the locus of the application of his own results. If next B is plagued with a similar question, he will refer to another mathematician C. After a few steps of this kind we find ourselves referred back to the researches of A, and in this way the chain closes.

How happy is the lot of the mathematician! He is judged solely by his peers, and the standard is so high that no colleague or rival can ever win a reputation he does not deserve. No cashier writes a letter to the press complaining about the incomprehensibility of Modern Mathematics and comparing it unfavorably with the good old days when mathematicians were content to paper irregularly shaped rooms and fill bathtubs without closing the waste pipe.

Mathematical research currently relies on a complex system of mutual trust based on reputations. By the time Simpson's paper appeared, both Kapranov and I had strong reputations. Simpson's paper created doubts in our result, which led to it being unused by other researchers, but no one came forward and challenged us on it.

Mathematicians are coming up with s that they use in their actual mathematical research. And these, like , I think is the name of one of them ... these are actually like s that have been engineered in a way that they can actually be used by working mathematicians to check the work they're doing.

A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid indulging in thoughts and doubts of the kind I develop. He has learned to regard them as something contemptible and […] he has acquired a revulsion from them as infantile. That is to say, I trot out all the problems that a child learning arithmetic, etc., finds difficult, the problems that education represses without solving. I say to those repressed doubts: you are quite correct, go on asking, demand clarification!

The whole thing that makes a mathematician’s life worthwhile is that he gets the grudging admiration of three or four colleagues.

I am never forget the day I first meet the great Lobachevsky. In one word he told me secret of success in mathematics: Plagiarize! Plagiarize, Let no one else's work evade your eyes, Remember why the good Lord made your eyes, So don't shade your eyes, But plagiarize, plagiarize, plagiarize... Only be sure always to call it please "research".

If he is consistent a man of the mathematical school washes his hands of applications. To someone who wants them he would say that the ideal system runs parallel to the usual theory: 'If this is what you want, try it: it is not my business to justify application of the system; that can only be done by philosophizing; I am a mathematician'. In practice he is apt to say: 'try this; if it works that will justify it'. But now he is not merely philosophizing; he is committing the characteristic fallacy. Inductive experience that the system works is not evidence.

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.

The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.

People tend to think that mathematicians always work in sterile conditions, sitting around and staring at the screen of a computer, or at a ceiling, in a pristine office. But in fact, some of the best ideas come when you least expect them, possibly through annoying industrial noise.

I suppose that the applied mathematician whose theory had just passed one still more stringent test by observation ought not to feel satisfaction, but rather disappointment—"Foiled again! This time I had hoped to find a discordance which would throw light on the points where my model could be improved." ...I own that I have never felt very keenly a disappointment of this kind.

But how can I use a method to discredit that very method, if the method is discreditable?

There is much to be said for being a mathematician. To begin with, he has to be completely honest in his work, not from any superior morality, but because he simply cannot get away with a fake. ... A mathematician's normal day contains hours of close concentration, and leaves him jaded in the evening. ... This is why we tend to relax either on mild nonfiction like biographies, or - to be crude, and to the derision of arts people - on trash. There is, of course, good trash and bad trash. ... Minor depressions will occur, and most of a mathematician's life is spent in frustration, punctuated with rare inspirations. A beginner can't expect quick results; if they are quick they are pretty sure to be poor. ... When one has finished a substantial paper there is commonly a mood in which it seems that there is really nothing in it. Do not worry, later on you will be thinking 'At least I could do something good then.' At the end of a particularly long and exacting work there can be a strange melancholy. This, however, is romantic, and mildly pleasant, like some other melancholies.