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The only generally agreed upon definition of mathematics is "Mathematics is what mathematician's do." [...]
In the face of this difficulty [of defining "computer science"] many people, including myself at times, feel that we should ignore the discussion and get on with doing it. But as George Forsythe points out so well in a recent article*, it does matter what people in Washington D.C. think computer science is. According to him, they tend to feel that it is a part of applied mathematics and therefore turn to the mathematicians for advice in the granting of funds. And it is not greatly different elsewhere; in both industry and the universities you can often still see traces of where computing first started, whether in electrical engineering, physics, mathematics, or even business. Evidently the picture which people have of a subject can significantly affect its subsequent development. Therefore, although we cannot hope to settle the question definitively, we need frequently to examine and to air our views on what our subject is and should become.
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*Hamming cites Forsythe, G.E., "What to do until the computer scientist comes", Am. Math. Monthly 75 (5), May 1968, p. 454-461.

 

Richard Hamming

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Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true ... If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.

 

Bertrand Russell

 

Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i.e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but "logical relations" or "artificial manifolds". They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance.

 

Hans Reichenbach

 

Computer Algebra Systems are NOT the Devil but the new MESSIAH that will take us out of the current utterly trivial phase of human-made mathematics into the much deeper semi-trivial computer-generated phase of future mathematics. Even more important, Computer Algebra Systems will turn out to be much more than just a `tool', since the methodology of computer-assisted and computer-generated research will rule in the future, and will make past mathematics seem like alchemy and astrology, or, at best, theology.

 

Doron Zeilberger

 

Numerical analysis has begun to look a little square in the computer science setting, and numerical analysts are beginning to show signs of losing faith in themselves. Their sense of isolation is accentuated by the present trend towards abstraction in mathematics departments which makes for an uneasy relationship. How different things might have been if the computer revolution had taken place in the 19th century! [...] In any case "numerical analysts" may be likened to "The Establishment" in computer science and in all spheres it is fashionable to diagnose "rigor morris" in the Establishment.

 

James H. Wilkinson

 

I think that it's extraordinarily important that we in computer science keep fun in computing. When it started out, it was an awful lot of fun. Of course, the paying customers got shafted every now and then, and after a while we began to take their complaints seriously. We began to feel as if we really were responsible for the successful, error-free perfect use of these machines. I don't think we are. I think we're responsible for stretching them, setting them off in new directions, and keeping fun in the house. I hope the field of computer science never loses its sense of fun. Above all, I hope we don't become missionaries. Don't feel as if you're Bible salesmen. The world has too many of those already. What you know about computing other people will learn. Don't feel as if the key to successful computing is only in your hands. What's in your hands, I think and hope, is intelligence: the ability to see the machine as more than when you were first led up to it, that you can make it more.

 

Alan Perlis