Mathematics and Art
Let us investigate a possible analogy between mathematics and art—art as analyzed in Collingwood's Principles of Art. It may be useful to keep in mind the disclaimer in Collingwood's preface:
Everything written in this book has been written in the belief that it has a practical bearing, direct or indirect, upon the condition of art in England in 1937.
For Collingwood, art should be distinguished from what used to be called art, but is rather craft, or τέχνη (tekhnē) in Greek. Art is an expression, an exploration of our emotions, in order to find out what they are. Art as such does not arouse these emotions; we already have them. There is no technique for expressing emotions [pp. 109–111]. We do it by creating an imaginary experience [p. 151]. The imaginary experience is the real work of art: a painting on the wall, for example, is the residue of this experience and is perhaps a hint by which the viewer can recreate the experience for himself.
Ultimately, art is language [p. 273]. The struggle to say what is on one's mind is the struggle to make art. But by successfully saying something, one may create habits and by-products [p. 275]. These may then be used in craft, as means to an end. Once one has successfully expressed one's emotions, one may end up with some sentences, for example, which are observed to arouse emotions in other listeners. So one might use these sentences in order to achieve this effect. This arousing of emotions is not art; but art had to come first.
Now, mathematics is something that can be used to achieve certain effects: the construction of a house or a nuclear power plant perhaps, or an understanding of how the planets move in the heavens. Does the mathematics have to be there first, in order to be used?
The title of Penelope Maddy's article ‘How Applied Mathematics Became Pure’ (Review of Symbolic Logic, vol. 1, no 1, June 2008) suggests not. The title is a bit misleading though, since Maddy begins with Plato, for whom (it is suggested) the physical world is at best an approximation to what is known through mathematics. (So mathematics does not apply to the world; the world applies to mathematics.) Later, with Galileo and Newton for example, physics and mathematics are one. Only after that is it understood that mathematics provides only an approximation to the world. (When matter is discovered to be atomic, then differential equations cannot describe it exactly; but even use of statistical methods will sometimes require a discrete variable to be treated as continuous.)
Noneless, it has been understood by now that mathematics can be done independently of physical considerations. Either this was understood by the Ancients and forgotten, or else it is a completely new realization. In either case, it would appear that, at least in recent centuries, applied mathematics precedes pure. Whereas for Collingwood, art, or ‘pure’ art, precedes craft, or ‘applied’ art. (Note however that ‘fine art’ and ‘useful art’ are not two species of the genus art: this would imply the ‘technical’ theory of art, whereby art is craft [p. 36].)
The lack of parallelism here between art and mathematics is only apparent. By Collingwood's account, the meaning of the word art has changed, from craft—the making of things according to a plan—to the imaginative expression of emotion. There was not earlier any recognition that there was such an activity that we now call art. Likewise, mathematics as we now understand it was not recognized earlier. This doesn't mean it didn't exist. The physical world did not teach Newton the mathematics he needed for its description. Newton had to have the mathematics within himself.
But it is not that simple. For Collingwood, good art is art that succeeds in expressing an emotion; bad art fails. But expressing emotion is not the same as arousing it. Bad art may arouse useful emotions, such as patriotism. What is bad mathematics? Three possibilities are that it is
Mathematics in the Newtonian age is non-rigorous, because it finds its justification in its agreement with observation. (Maddy discusses this.) This mathematics can be made rigorous, by our standards; but this does not mean that Newton understood this possibility.
The border between the non-rigorous and the incorrect is vague.
What is interesting or uninteresting in mathematics is subjective. Still, one may find it useful to do uninteresting mathematics, as a Rembrandt may fill in the background of a portrait with dark paint,—or a great novelist might write bestsellers to pay the rent (example?).
Possibly something interesting quâ geometry or algebra or analysis is not that interesting quâ mathematics. Then some mathematics is interesting when it draws connections between parts of mathematics, or when it shows something in common among these parts. Here a parallel with Collingwood should be considered. An artist expresses what he feels. If he isolates himself, in order to avoid emotions unworthy of his art, then he makes bad art. If he knows which emotions to avoid, then he must have understood them (through art), but then disowned them. Whereas if one's life just happens to be somewhat circumscribed—like Jane Austen's perhaps—one may still produce great art.