Subjects — A
We cannot seek or attain health, wealth, learning, justice or kindness in general. Action is always specific, concrete, individualized, unique.
If you have a large number of unrelated ideas, you have to get quite a distance away from them to get a view of all of them, and this is the role of abstraction. If you look at each too closely you see too many details. If you get far away things may appear simpler because you can only see the large, broad outlines; you do not get lost in petty details.
— Mathematics: People, Problems and Results, “Rigor Versus Intuition in Mathematics ”
There is no abstract art. You must always start with some thing. Afterward you can remove all traces of reality.
Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics.
As soon as there is language, generality has entered the scene.
Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.
— The principles of Quantum Mechanics, preface, 1930
The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. This is only natural and to be expected. What however was not expected by the scientific workers of the last century was the particular form that the line of advancement of mathematics would take, namely it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation.
— Paper on Magnetic Monopoles, 1931
The abstract realm in which a mathematician works is by dint of prolonged intimacy more concrete to him than the chair he happens to sit on.
Mathematics is the art of giving the same name to different things.
… the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
— I Want To Be a Mathematician, 1985