On Functions (Unsourced)

"Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing: you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea - combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques." (Gian-Carlo Rota, [interview]) 

"How could it be that the Riemann zeta function so convincingly mimics a quantum system without being one?" (Michael V Berry) [33] 

"I believe that numbers and functions of Analysis are not the arbitrary result of our minds; I think that they exist outside of us, with the same character of necessity as the things of objective reality, and we meet them or discover them, and study them, as do the physicists, the chemists and the zoologists." (Charles Hermite)

"I turn away with fright and horror from the lamentable evil of functions which do not have derivatives." (Charles Hermite, [letter to Thomas J Stieltjes])

"[...] in one of those unexpected connections that make theoretical physics so delightful, the quantum chorology of spectra turns out to be deeply connected to the arithmetic of prime numbers, through the celebrated zeros of the  Riemann zeta function: the zeros mimic quantum energy levels of a classically chaotic system. The connection is not only deep but also tantalizing, since its basis is still obscure - though it has been fruitful for both mathematics and physics." (Michael V Berry)

"It’s a whole beautiful subject and the Riemann zeta function is just the fi rst one of these, but it’s just the tip of the iceberg. They are just the most amazing objects, these L-functions - the fact that they exist, and have these incredible properties are tied up with all these arithmetical things - and it’s just a beautiful subject. Discovering these things is like discovering a gemstone or something. You’re amazed that this thing exists, has these properties and can do this." (J Brian Conrey)

"Reality is a wave function traveling both backward and forward in time." (John L Casti)

"The chief weakness of the machine is that it will not learn by its mistakes. The only way to improve its play is by improving the program. Some thought has been given to designing a program that would develop its own improvements in strategy with increasing experience in play. Although it appears to be theoretically possible, the methods thought of so far do not seem to be very practical. One possibility is to devise a program that would change the terms and coefficients involved in the evaluation function on the basis of the results of games the machine had already played. Small variations might be introduced in these terms, and the values would be selected to give the greatest percentage of wins." (Claude E Shannon)

"The theory of functions with one independent variable is very closely connected with the theory of algebraic curves. The geometry of such a curve becomes therefore of fundamental importance." (Heegaard)

"We shall consider the simplest maximum and minimum problem that points to a natural transition from functions of a finite number of variables to magnitudes that depend on an infinite number of variables." (Vito Volterra)

"When graphing a function, the width of the line should be inversely proportional to the precision of the data." (Marvin J Albinak)