[T]here is no mathematician so pure that he feels no interest at all in the physical world; but, in so far as he succumbs to this temptation, he will… - G.H. Hardy

[A] good deal of elementary mathematics... 'elementary' in the sense in which professional mathematicians use it... [e.g.,] knowledge of the differential and integral calculus—has considerable practical utility. These... are... rather dull... the parts which have least aesthetic value. The 'real' mathematics of the 'real' mathematicians... of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly 'useless' (...as true of 'applied' as of 'pure' mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of... 'utility'...

It is... astonishing how little practical value scientific knowledge has for ordinary men, how dull and commonplace such of it as has value is, and how its value seems almost to vary inversely to its reputed utility. ...We live either by or on other people's professional knowledge.

[M]athematical reality lies outside us ...our function is to discover or observe it, and ...the theorems ...we prove, and ...describe grandiloquently as our 'creations', are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards [...]