I was talking to someone last night, and they said they loved math because it was "beautiful". This maybe always struck me as a rather odd, if widespread, idea; it at least seemed odd then. My idea of "beautiful", aesthetically or artistically speaking, involves expressiveness, or at least something to do with human emotions, some human connection. Maybe that's *just* my idea, but in any case mathematics has nothing whatsoever *expressive* about it, so it doesn't fit at least my idea of beautiful in any artistic sense (yeah, a sunset is pretty, but, I wouldn't go see a movie that was just a sunset). This got me thinking about what I do find, if not exactly beautiful, at least appealing, about some mathematical work. I posted it because I think it's not the same thing a lot of my peers find appealing, and, I was curious as to their reaction.
Basically, I think of mathematics as a bag of tautologies--statements that are trivially true, and thus in a sense devoid of content, not really saying anything at all, like "If we're going to the movies, either we'll see 'Star Trek' or something else". (Or "If you take a bunch of apples and put them in 10 boxes, at most one to a box, you have 10 boxes you can put the first one in, and then only 9 boxes you can put the second one in. Because it's one to a box. Can't put the second one in the same box as the first one. Ten minus one is nine. Etc."). That's why I don't find mathematical theorems, as such, very interesting. Theorems in the physical sciences have content, they describe how the world works or might work. Mathematical theorems have no content, they're theorizing about nothing, a bag of tautologies; they're trivialities in disguise. Who cares?
That's (certainly) not to say they're trivially true to me, or to anyone else necessarily, but, to the extent I, or everyone else, *don't* think of a mathematical result as tautological, I believe it reflects on a failure of my personal, or our collective, understanding. Overall, mathematical language doesn't seem like something we're terribly well-suited to, as a species. We have quirky and limited human brains that are good at other things, like figuring out how to bite things or passing immediate judgment on the hotness of people we just met. So in one sense, mathematical results don't say anything; in another sense, they say something trivial that we can't recognize as trivial because we're idiots about that kind of thing. The interesting thing about the triviality in disguise is the disguise, not the triviality, and when I work on math, I'm much more motivated by frustration over not being able to see through the disguise than any particular interest in the triviality. This is a "problem-solver"'s attitude, I suppose, and a "theorist" would be more interested in the triviality (or, more likely, characterize the whole thing differently).
What I like in a mathematical argument, and maybe this is what people think of as "beautiful", is when it manages to make the essentially trivial or tautological nature of the result apparent to our quirky and limited human brains. That's why I don't like long or messy arguments and why I like visual proofs of combinatorial identities and why I feel compelled to redo certain proofs--if they "work", but don't make the essentially tautological nature of the result clear, then a "new" result may be established, but the basic failure of understanding--the recognition that the result *isn't saying anything*--remains, and it's that basic failure of understanding that interests me. But if the essentially tautological nature of the result *is* made clear, you can think of it as making our quirky and limited human brains a little more universal and a little less limited, which provides some human connection to the enterprise. It doesn't feed a single starving child, but, it has an appeal, as activities go. Is it something to spend 80 years doing until you die of old age? YMMV.
This conjured up lots of possible responses and this wasn't something meant to stir debate, but I felt like I had to be a voice for "beauty" in mathematics. This is a subject I have thought about for quite some time, but I eventually settled on the following response:
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Well articulated Paul, but I disagree on two primary issues:
1. Mathematics is not expressive -- I will argue that mathematics is the pinnacle of human expressiveness.
2. Mathematics does not connect with human emotion. -- Mathematics is both something to be appreciated passively and carried out actively. The first expresses satisfaction in certainty, elegance and other archetypical emotions. The second is the consequence of curiosity and mystery.
First I will address the second issue because it refers directly to your use of the word "expressive". Then I will move to broaden the definition.
There are several emotions that mathematics appeals to. The ones that come most readily to mind are: elegance, permanence, purity and certainty. Beauty is a multi-faceted sense but I would argue that several of the aforementioned components do account for a classic sense of beauty. The entire pursuit of art is to express feelings of forlorning, hope, despair, love, abdonment, achievement, failure and the whole palette of human experience.
What emotions are conjured when appreciating the marble statues of Greco-Roman past? Although their world bears little resemblance to our own they were familiar with much the same range of emotion that we are. Their art expresses archetypes and reflects universal stories or ideals that humans spread across millenia can appreciate. Art has the same purpose of philosophy -- it is meant to make explicit these emotions so that one may stand back and reflect. It is an attempt to purify a given aspect of human experience and abstract away irrelevant details. Art accomplishes this through physical instantiation, whereas philosophy and thus mathematics (since the two were always part of the same enterprise and only recently have grown separate) sets aside the object for study inside of one's own mind so that introspection may be applied. One cannot say that one method is better than another just because the philosopher and mathematician frames his or her art in words and symbols. If one can appreciate the marble rapture of two bodies intertwined, but not Aristotle's words "Love is composed of a single soul inhabiting two bodies" then I concede the issue at hand.
On a meta-level the age of statues and words alike -- their permanence -- evokes a sense of solidity and foundation -- a rock that even the most tortured souls can cling to. Mathematics is without a doubt the most permanent human enterprise (meaning: as long as there are humans to support its meme pool). When a theorem is proved it is acknowledged as true for the rest of time. It is changeless. It echoes Zeno and other members of the Parmenidean school's belief that reality is one, change is impossible, and existence is timeless, uniform, and unchanging.
This way of thinking is spawned out of a desire for certainty. This is perhaps the most primordial human emotion. It is the impetus for all mythology, all religions and all science. Religion and Science are manifestations of a desire to narrate human existence. They both advocate methodologies for how to construct this narrative, but ultimately they are just developments of the more basic role of mythology -- coherent story-telling. Mathematics is the most coherent form of story-telling yet devised by human beings. One might argue that this is like comparing apples to modules, but the distinction is superficial. In one, the scene and characters are put in place and the governing dynamics are given by the personalities and tempers. In the other, a background universe and cast of definitions are put in place with logic and inspiration for what might be true determining the dynamics at hand. The success of physics since the time of Archimedes has been the application of abstraction to human perception to create a model with initial conditions and letting mathematics provide a story that unfolds in logical, clock-work fashion. The simple models our ancestors had pale in comparison with our current (and probably still over-simplified) model of the universe as a fiber bundle with local sections equipped with SU(3)xSU(2)xU(1) (or E8) symmetry group. Perhaps faith provides the input but reason creates the output. The individual steps may be tautologies, but the unfolding pattern is the most beautiful, spectacular, psychadelic creation humankind has ever conceived.
What is it that makes this patterned way of thinking beautiful? It is the sucess of expression unparalled by any other human endeavor. The example of putting apples in boxes may lack expressiveness because the initial conditions and structure of the problem are transparent. The action of solving the problem is just a process of deduction (or of building a general algorithm and then deducing the answer). What really makes mathematics expressive is the process of identifying features of system or processes and then naming them. This resonates with the old Christian/Greek notion that naming something gives one power over that thing. The tautological statement that "plugging in numbers into this polynomial in different ways leaves the result unchanged" seems like a silly thing to say, but when the correct notion of a Galois group is harnessed suddenly problems of the ancients come tumbling down (trisecting the angle, doubling the cube, squaring the circle). The ability to transplant one problem from a completely unrelated domain into another is at the core of human intelligence. More fundamentally, meaning comes from the isomorphism discovered between two things. Reasoning by analogy -- a method perfected in mathematics -- is all that expression is and can ever be. It is all literature, psychology and art has ever done. One must take some personal internal state that is largely unspeakable, unnameable, undrawable, irrational and point to some isomorphic system sitting outside of ourselves and say "That is me!" We are ourselves manifolds, only locally similar to anything that is understood. Although antithetical in nature, I believe that Riemann would have agreed with Emerson, a contemporary on a different continent using different methods to grasp at transcendental truths, when he said
"Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth."
