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Joseph William Peach

Mathematics is often thought to consist of absolute truths, beyond human creativity and experience. I disagree. Mathematics does not exist in an unchanging realm of objective, absolute truths, but in the relative, intersubjective world of human agreement and creativity.

I defend the view known as Game Formalism, henceforth formalism. If I am right, mathematics is like a game played with symbol-like characters such as equals signs, numerals and operators. The moves of a game may be valid or invalid, but are neither true nor false. Similarly, mathematical statements are neither true nor false. Like moves in chess, equations and transformations have no meaning.

Philosophers are puzzled by a variety of mathematics related questions. I will consider the application of formalism to just two: ‘what are numbers?’ and ‘how is mathematics applicable?’

Many philosophers have given answers to these questions which require manifold species of problematic metaphysical entities to support their claims. For example, Plato’s account of geometry requires mathematical Forms, mystical abstract entities beyond human experience. Of these things, one can plausibly ask ‘but how could these apply to the real world? What relation have these abstract Forms to the objects we wish to understand mathematically?’ The answers are not simple.

Numerals are the physical marks or signs used in mathematics. They include the particular ‘6’ on this page, and the chalk, graphite and ink characters drawn by mathematicians worldwide. Numbers are not numerals. I shall show this with an example from Shapiro (2000: 142).

Consider the equation ‘0 = 0’. It cannot be the case that the mark on the left is identical to the mark on the right. These are obviously two different marks. However, we commonly maintain that they are, or represent, the same number. The conventional philosopher of mathematics claims that numbers are a kind of abstract object above and beyond a type of symbol, that they exist independently of any written mathematics. These would be a very queer species of objects indeed, though it is beyond the scope of this essay to fully elucidate their queerness and unintelligibility. One can clearly see that philosophical strategies which avoid such problems are preferable.

I contend that formalism does indeed circumvent these problems. For the formalist, numerals are merely characters in a game. Strictly, they are not characters or symbols at all, because symbols have meaning. There are no such things as numbers, merely rules for manipulating numerals. These rules include rules such as ‘whatever one puts to the left of an equals sign one may put to its right.’

If mathematics has no content, how is it useful? Mathematics is useful and meaningful in areas from pricing to power plants, from protons to population growth. If mathematics is just the game-like following of arbitrary rules, ask formalism’s opponents, how can it be of practical use?

The formalist may reply that the application of mathematics is not mathematics, but a subject in itself. If one wishes to read numbers into numerals one may do so, but this is not mathematics. It is somewhere between accountancy and metaphysics. There are problems with this view. If we are to call applied mathematics by different names, say economics or mechanics, we will need an account of metaphysical problems in these (expanded) areas such as traditionally plague mathematics, and formalism will be unable to provide such an account. Also, though the formalist may wish to shirk his mathematical duties, handing his work over to the accountant or astronomer, we may end up in a situation where there are traditionally mathematical areas of study for which nobody is responsible.

It is better to retreat to the position that although mathematics is the following of rules, and mathematical knowledge is knowledge of such rules, these rules are not arbitrarily chosen. Whether or not a rule is a part of mathematics is not to be found in the world of absolutes, not in the subjective mind. Mathematical rules are chosen intersubjectively, by communities of mathematicians.

The a priori, necessary nature of mathematical deduction can be preserved even with this conception of mathematics, if we stipulate that these rules be truth preserving. One way of cashing out this notion is as follows: logical terminology, such as ‘is equal to’, ‘is not’ and so forth must retain its normal usage rules. Mathematical terminology, such as ‘line’, ‘point’ and ‘numeral’, has special rules governing its behaviour. These rules are useful in practice because mathematicians choose them for their epistemic usefulness. Necessary deductions can be made between them, because the logical terminology is intact, and one can stipulate that mathematical deduction be confined exclusively to rule-creation and its logical consequences. One could argue that the requirement for logical consistency is an objective component. However, an illogical rule is not a genuine rule at all, but a meaningless conjunction of symbols. Contradictory rules could never apply, so could never model any situation.

The nature of symbolic interpretation in formalist mathematics can be understood by an analogy with poker chips. Often, one buys chips with money when one enters into a poker game. During the game, the chips are moved around according to certain rules. One does not need to consider the monetary interpretation of the poker transactions to play and understand the game. After the game, one exchanges one’s chips for money. So it is with mathematics. One invents rules useful for modelling a real world situation. One then follows the logical consequences of these rules, thinking only of symbolic manipulation. After the symbol-crunching, one reinterprets the symbols as physical quantities again. The rigorous algebraic manipulation did not require thoughts of the situation while mathematics was being practiced.

Formalism is true; mathematics is the following of (truth-preserving) rules without thought for their application. These rules are chosen socially, and require no verification beyond social usefulness and truth-preserving logical consistency. Mathematics is intersubjective.

Joseph Peach is a second year philosophy undergraduate at the University of York.

References

Shapiro, S. 2000: Thinking about Mathematics: The Philosophy of Mathematics. New York: Oxford University Press.