The following is an unfinished, obtuse, and probably bad post I wrote at the end of last semester. I hope to revise it soon—but the gist of the idea, that mathematics is not an exercise of “pure reason,” but it heavily relies on the mathematician’s “aesthetic” sense. This sense is still rational, but not explicitly logical.
I’ve often thought of mathematics as a mere exercise in necessity. Something about the pure logical foundations of higher math—the way that theorems and propositions of such significance were supported ineluctably on pillars of axioms reasoning that seemed so indubitable as to be tautological—something about that computerized, Spock-like ideal of ironclad reasoning attracted me. But I’ve been somewhat mistaken.
Now that I’ve tried (and more often than not failed) to do mathematics, I’ve begun to think it more proper to say that the material of mathematics is logic: without consistent, logical relations between concepts, mathematics would not exist. But a thing’s material reality is, at best, only half the picture. Logic can limit the number of things you can say that are useful in discovering or describing truth—by guiding the statement of valid claims and restricting the statement of invalid ones—but it does not tell you with necessity what should be said, what should be claimed. If logic were the only guiding principle of mathematics, Euclid had already finished his elements as soon as he formulated his axiomatic definitions of lines and points and planes. Since an ironclad and ineluctable chain of reasoning connects the definitions of Euclidian geometry with the Pythagorean theorem, to state the (reduced) former thing is, in a sense, to state everything. This level of reductionism is akin to thinking that you are a carpenter once you know what wood is, or that to look at a timber forest from 50 years ago is to have seen every boat, home, and kitschy bear statue that would be made from it.
But this is obviously not the case. Just as a kind of formal principle is needed for carpentry to be carpentry, Mathematics itself needs a formal principle, a principle of what should be said, not merely what can be said. And, in my limited experience with mathematics, these formal principles seem to be largely aesthetic. Why should points and lines be constructed so as to join three of each of these elements and make a triangle, and to further compute the relations between the side lengths of the figure? It is either by a principle in the mind of the mathematician to represent the triangles he sees in nature or, else, by an exploring, pioneering, experimenting principle in the geometer to sound the depths of what can be constructed within the possibility space that founded on Euclid’s axioms.
Not only this, but all development in mathematics seems to stem from exploration and attempts to build bridges to these explorations with the necessary building materials of consistent logical statements. It was the futile attempts of ancient geometers to square the circle that clued us into the existence of mathematical concepts inaccessible by geometric construction.
There is also an aesthetic principle of symmetry—in the sense of both true mathematical symmetry and of a parity and evenhandedness and fittingness—that rules mathematical development in addition to those of exploration and a drive for natural representation. The basis of most mathematical conjectures is that a theorem or principle that applies to one type of space or one region of consideration should hold for another space or region deemed similar enough. Even though logic in no way occasions or guarantees the extension of particular truth claims to other spaces, such a belief is the default position of the mathematician when conjecturing, and this jumping off point allows him some formal handhold onto the sheer wall of necessity. [To use another analogy, he can see whether the brick he has picked up actually fits into the wall he is attempting to build.] For example, in topology, the functioning of derived space topologies—in quotient-, product-, and subspaces—is distinct, and there is no necessary reason that these three spaces should function similarly, but what we do know about each of these derived spaces is based off their similarity and correspondence: that connectedness is preserved by projection mappings and identity mappings alike, [there are more examples here which I have not yet listed], etc.
My concept of aesthetics likely has a definition (or the beginnings of one) that begs the question: aesthetics are concerned with formal principles and relationships between things that are not merely necessary. Such relationships are much easier to see in terms of a superstructure instead of in single propositions itself. The content of 2+2=4 is mostly a matter of necessity. [I hope to develop this idea a bit more with revision.]