Paul Lockhart starts his “Mathematician’s Lament” (later expanded into a book under the same title, but I’ll be discussing the shorter article here) by comparing math class to a misshapen music or art class. Suppose that in music class, enjoying actual music is supposed to be too advanced for children, so they are made to start with memorizing the circle of fifths and pointing the stems of quarter-notes the right way; or suppose that in art class, painting is postponed until after preparatory “Paint-by-Numbers” classes. This, Lockhart suggests, is how math class works; it stifles creativity and natural curiosity and therefore goes against the spirit of mathematics.
Obviously now, Lockhart is making some assumptions about the nature of mathematics, and he is explicit about this: mathematics is an art, like music and painting. This, he says goes against the common perception, where mathematics is connected to science:
“It’s not at all like science. There’s no experiment I can do with test tubes and equipment and whatnot that will tell me the truth about a figment of my imagination. The only way to get at the truth about our imaginations is to use our imaginations, and that is hard work.” (4)
“Mathematics is viewed by the culture as some sort of tool for science and technology. Everyone knows that poetry and music are for pure enjoyment and for uplifting and ennobling the human spirit (hence their virtual elimination from the public school curriculum) but no, math is important.” (6)
Math, as an art, has an aesthetic dimension that teachers need to be able to communicate; and it is creative and needs to be taught as such. This is a very compelling image – and, I’d like to state here before going on to some points of criticism – a much better rationale for teaching math in high school than applications of math in the sciences (which would render math redundant for most students).
Lockhart’s monologue is interrupted by short dialogues between Simplicio and Salviati, named after the two antagonists of Galilei’s Two Systems – very apt, I’d say, because of the radical and somewhat Platonic nature of Lockhart’s own intervention; and at the same time somewhat ironic given the role Galilei has played in linking mathematics to knowledge of the material world and thus, eventually, the sciences. In these interruptions Lockhart goes into the potential problems involved in framing math as an art.
“SIMPLICIO: But don’t you think that if math class were made more like art class that a lot of kids just wouldn’t learn anything?
SALVIATI: They’re not learning anything now!”
Though presented as a critical interrogation of Lockhart’s own perspective, ‘Salviati’ in practice answers Simplicio’s questions by simply returning fire. Some of Simplicio’s objections could have been made better use of by answering them in such a way that Lockhart’s own position would be fleshed out more. I hope to do this by raising some points of criticism of my own.
The two questions I will raise are, first, the extent to which Lockhart is justified in classifying math as an art not a science; and, second, the extent to which it is reasonable to wrap a large-scale school curriculum around an aesthetic understanding of math. This post is about the first question; a next will be about the second one.
Aesthetics and understanding
The comparison between math class and music or painting class is not just for comic effect; it is the first thing to understand about math, Lockhart says (3). Is there no difference? Yes, well, no. “The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.” (3)
Is this so? Or can we think of a few things that music and painting have in common with each other but not with math? I’d suggest that one point of distinction is that we need very little understanding to be impressed by things we see or hear, but that in order to be impressed by mathematical beauty or structure, we need to be prepared by an understanding of the structures involved.
I think Lockhart would agree to this point; I also think it somewhat undermines his analogy. For this suggests that while understanding and analysis are only accidentally relevant to the appreciation of music, they are essential to the appreciation of math. We can appreciate a certain chord progression without theoretical knowledge of the circle of fifths; but we cannot appreciate a proof that there are infinite prime numbers without theoretical knowledge of prime numbers, and therefore of natural numbers and division.
It is not completely evident to me that the distinction I made here can withstand attempts at deconstruction. Do analysis and understanding add nothing to our appreciation of music or painting? Surely, when we see a Rembrandt painting, our appreciation takes into account not only the visual experience per se, but also some background awareness of the skill involved in making such a painting; and of the artistic decisions made – which suggests that our appreciation depends in part on an appreciation of the ways in which the creation of this painting transcends our regular expectations. Lacking such understanding, we should be as impressed by modern copies of Rembrandt’s paintings, or even by photocopies.
Maybe aesthetic appreciation requires background knowledge, then; and maybe it is greatly improved by analytic understanding. Is there a limit to this, though? Does being impressed by sunsets or clouds or starry skies require ‘understanding’? To a certain extent, our aesthetic perception is undoubtedly influenced by background knowledge; seeing the night sky has never been just a matter of looking at shiny dots, and in our time it is hardly possible without being reminded of the vastness of the universe. Even the most immediately visually impressive experiences are very hard to disentangle from their equally immediate meanings.
But we have shifted our ground slightly in the last two paragraphs, moving from understanding to meaning. And I think there remains a case to be made that where some experiences can overwhelm us with their beauty with only very little preparation, some in fact require their subject to be pre-formed. Clouds, sunsets and the night sky seem to me to be on one end of the spectrum, where a proof of the Pythagorean theorem is on the other.
This does not establish the place of math relative to the arts; for it is true of other arts that at least some of their productions require such pre-formation. Not all paintings or compositions are readily accessible; many assume the development of some kind of taste. An even closer analogy might be to literature; in order to appreciate literature, you need to be able to read, and you often need an awful lot of background knowledge. If mathematical beauty needs a special mode of appreciation, this alone does not seem to distance it too much from the ‘other’ arts; it may, however, have repercussions for the question to what extent math education is simply the extension of natural creativity and curiosity – but more about that later.
Truth and Validity
A second thing that music and painting may have in common with each other but not with math, is that they are not in the business of truth. It would be a weird kind of injustice to Rembrandt’s Night Watch to dismiss it on the grounds of factual inaccuracy (“these men were not together here at this time”). But what to say of a proof that 1 = 0, such as this one?
This may be aesthetically pleasing to some, in its elegance and its surprising nature; it is also, of course, invalid. But what kind of art is this, in which the concept of validity – of the preservation of truth-value – plays such an important role? We don’t say that a piece of art is invalid; we may say that it is ‘true’, but then in only a very metaphorical and liberal sense.
This, it seems, limits the creativity of the mathematician. Lockhart says that mathematics “allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe)” (3), and in a purely quantitative sense he is, of course, right; there is nothing in our physical universe as infinite as, to name something dull, the real numbers; so literally speaking, the mathematician has an infinite advantage over artists of the physical world when it concerns the matter she can work with. But it seems her creativity is also limited in a much stricter way, for the possibilities for error and falsehood are also infinite. We may be playing with mental constructions of our own making, but usually we are supposed not to contradict ourselves in the process, which is a very stringent demand that composers or painters do not seem to face.
Again though, we need to be careful not to overestimate the gap. We already admitted that there is meaning in art too, so it’s not too big of a jump, actually, to say that there is a connection to truth, and that perhaps we may judge pieces of art for their implied perspective on the world. In literature, film and other media which allow for representations of verbal statements, this is perhaps easier than in purely visual media, but there is no clean break; photography is both an art and an instrument of journalism, and cartoons can be at the same time a work of art and an unambiguous statement of political or social fact, open to the criticism that they fail to do justice to the facts.
And on the other side: isn’t our invalid proof as much a part of ‘math’ as the Pythagorean theorem? Psychologically speaking, someone studying this invalid proof is doing math; someone believing this proof to be valid may be making an error, but she is still doing math. Why not say that an invalid proof is to math like a false note is to music? Not every field of art has a concept of validity, but then not every field has a concept of staying in tune.
Perhaps Lockhart’s claim that math is freer than music because it doesn’t require matter is slightly misleading; in practice, composers and musicians are ‘unfree’ not so much because the air refuses to do their bidding, but mainly because of rules internal to the art of music itself. The fugue is a human invention, but one that composers couldn’t simply circumvent; there are rules in music too. They may be less rigorously defined, and they are probably, on closer inspection, rooted in different domains of human activity – the fugue form being more of a traditional-conventional norm and the impossibility of dividing by zero more strictly implied by what we mean by division; but these scholastic distinctions, valid though they may be, do not seem to matter much psychologically to the artist. You are not supposed to make arithmetical errors, and you are not supposed to sing out of tune – the immediate consequences of doing those things are psychological shame and social disapproval; the immediate aesthetic consequence is that the artistic product, the edifice you were building, is tainted by them, sometimes in a major way but often in an innocent and harmless way, for small mistakes are easily corrected.
The comparison with staying in tune and honoring a tradition of composition by following the rules for composing a fugue also suggests that creativity does not rule out rule-boundedness. Rules in music can be navigated creatively and virtuously – just as they can in math.
Conclusions
It seems that my two most urgent intuitive objections to Lockhart’s thesis that math is an art – namely, that mathematics essentially requires understanding, and that mathematics involves validity – are not quite as damning as they could seem to be. However, answering them did involve a certain framing of the arts as well – namely, a view of aesthetics as not only natural and spontaneous but also assuming knowledge and the formation of taste; and a view of art as partially rule-bound rather than completely free. This may have repercussions for Lockhart’s recommendations for the math curriculum, in so far as it relies on analogy between math and the other arts. I will return to this question later.