# 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

- non-rigorous,
- incorrect,
- uninteresting.

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.