I recently read Alexander Grothendieck’s “Equisse d’un Programme”- an extremely insightful summary of one of the most interesting mathematical research programs of the century. On page 29, he presents an extremely interesting general principle of mathematical research, which he calls the “Principle of Anodine Choices.” In his words, “When for the needs of some construction of a geometric object in terms of others, if we are led to make a certain number of arbitrary choices along the way, so that the final object appears to depend upon these choices, and is thus stained with a defect of canonicity, then this defect is indeed serious (and to be removed requires a more careful analysis of the situation, the notions used, and the data introduced). Whenever at least one of these choices is made in a space which is not “contractable” i.e. a space whose invariants are non-trivial, and that this defect is on the contrary merely apparent, and the construction itself is “essentially” canonical and will not bring along any troubles, whenever the choices made are all “anodine” i.e. made in contactable spaces.” (translated from french and hence a bit unnatural sounding)

To me, this gets at a very fundamental aesthetic point in mathematics- that the objects we construct should embody the structure of the objects we use to construct them. If we must introduce arbitrary extra data into our constructions, then these constructions are mathematically unnatural, and we must avoid them however we can. From an aesthetic point of view, we must express our concepts in a language which does not constrain them with more structure than is necessary to define whatever concepts we wish to examine.

I would like to consider an example-

Consider the notion of continuity of functions from R to R. A function f: R –> R is continuous if for every real number x, the limit as y approaches x of f(y) equals f(x). When we write out the definition of limit, we get a mess of deltas and epsilons telling us that for every interval of length epsilon around f(x), there is some interval of length delta around x that maps inside of it. Thus, our definition of continuity refers to a series of quantitative bounds on distance in the euclidean metric.

Despite this, the definition of continuity is really not fundamentally related to any quantitative notion of distance on the real numbers. We prove the following:

THEOREM: A function f: R —>R is continuous if and only if the preimage under f of every open set is an open set.

PROOF: Suppose f is continuous. Let S be an arbitrary open set in R. To show that the preimage of S is open, we must show that every point in the preimage of S is an interior point in the preimage of S. If the preimage of S has no points, it is the empty set and is thus defined to be open, so we can assume the preimage of S contains at least one point. Fix a point p in the preimage of S. Because f(p) is in S, and S is open, an interval I of some length epsilon about f(p) is contained in S. Because f is continuous, some interval of length delta about p is mapped into I and thus has its image contained in S, so p is an interior point. Thus, since S is arbitrary, we have shown that the preimage under f of every open set is open.

Conversely, suppose the preimage under f of every open set is open.  Fix a point P in the image of f and consider an interval K of length epsilon around P.  Since intervals are open sets, we know that the preimage of K is open.    Since P is in the image of f, some point z in R must map under f to P.  Since P is in K, z must be in the preimage of  K.  Since the preimage of K is open, some interval of length delta about z must be contained in the preimage of K, so we have an interval about z of length delta that maps into an interval of length epsilon about P.   Thus, f is continuous at z.  Since every point in the domain of x maps to some point in the image of f, and P is arbitrary, this shows that f is continuous.   QED

This proof shows that the quantitative bounds we find in terms of the euclidean metric to prove that a function is continuous are really not necessary.  Any other metric which induces the same open sets as the euclidean metric will also define exactly the same functions to be continuous.   Any map which puts the open sets of R in bijection with the open sets of some other space (called a homeomorphism) will form a natural identification between the continuous functions in R and the continuous functions in the other space.   It is only the structure of open sets (called topological structure) that matters in any notion of continuity.   Thus, it is much more natural to define a function f to be continuous if the preimage under f of every open set is open, and throw away our deltas and epsilons.  Their mentioning brings in a concept which fundamentally has nothing to do with the concept of continuity.   The choice of a specific metric is an “anodine” choice- its relation with continuity is an illusion which a better definition of continuity will destroy.