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Walking into Cartesian theater in the middle of a movie

September 27, 2012

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 unlike others, 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 auto-morphism 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 is a refinement of the idea of a quantity—but still constant.  But it is variable structures which are in a general way the subject matter of the theory of categories over a base topos.  Of course, most of our examples of base topos can be interpreted within mathematics over the base topos of abstract sets, but this does not trivialize our aim any more than the continuous is annihilated by the discrete or variable quantities “reduced” to constant quantities through the “construction” of the real numbers within the higher order theory of the natural numbers.  That the abstraction from structure to quantity (obvious in the above simple example) is present and is significant also among variable structures and variable quantities, is already exemplified by a flourishing branch of mathematics, K-theory.

–from Perugia Notes of Professor F. William Lawvere

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