Mathematics, in its operations with variable quantities, enters the field of dialectics; and it is significant that it was a dialectical philosopher, Descartes, who introduced this advance. The relation between the mathematics of variable and the mathematics of constant quantities is in general the same as the relation of dialectical to metaphysical thought. Engels, Anti-Duhring, Part I Philosophy, Chapter 12, Dialectics, Quantity and Quality . . . The primary subject matter of mathematics is still the variation of quantity in time and space, but that also this primacy is partly of the nature of a "first approximation", is reflected in the increasing importance of structures (not only of quantities) in the mathematics of the last 100 years. For example, a first approximation to a theory of a material situation involving three apples might be simply the number (constant quantity) 3. The idea of an abstract set of three elements is a somewhat more accurate theory. If one of the apples happens to be distinguished, for example, by being green, we may consider the simple structure of an abstract set with a distinguished element, a theoretical refinement which the quantity three does not really admit; the unique non-trivial automorphism of this structure is a theoretical operation which again the quantity itself does not admit. This simple example indicates that at least in some cases the idea of a structure (representing a quality) is a refinement of the idea of a quantity. The abstraction from structure/quality to quantity (obvious in the above simple example) is present and is significant also among variable structures and variable quantities. . . . Platonic idealism, while mistaken on the relation between consciousness and matter, contains an aspect of truth when applied within the realm of ideas. For example, the doctrine of algebraic theories begins with the affirmation that there exists as a definite mathematical object the "perfect idea of a group" of which the idea of any particular group is "merely an imperfect representation". Indeed, it is a simple further step to form as a mathematical object the "idea of an algebraic theory" of which any particular algebraic theory is a representation, as was carried out as a part of Benabou’s 1966 dissertation. Of course, even here Platonic idealism is wrong on the order of development, since the ideas of several particular groups were concentrated from practice well before the idea of a group in general was concentrated from the practice of mathematicians around 1800.

F. William Lawvere (1972) Perugia Notes: Theory of Categories over a Base Topos, Universita di Perugia, Perugia.