Before Galileo, "natural philosophers" followed Aristotle's example in looking for causes and reasons in nature. Why do objects fall to the ground when dropped, and why does smoke rise? Aristotle said that everything has its natural resting-place - for heavy objects, the centre of the earth; for light ones, the sky. When objects are unimpeded they seek their natural homes.
Such explanations are offered as answers to "why?" questions of the kind we find it natural to ask. But Galileo changed the approach. He did not try to explain, only to describe. He did not try to explain why objects fall when dropped; instead he described the situation mathematically, by quantifying the relationship between the distance an object falls and the time it takes to do so.
As scientists from his time onward have discovered, it appears not just that physical phenomena are perfectly captured by mathematics, but that the world actually seems to be mathematical in character. Galileo himself called mathematics "the language of nature", and Sir Arthur Eddington remarked that a study of physics gives us a "knowledge of mathematical structures". The amazing fact is that the physical universe, for all its great diversity, can be described with precision and power in terms of a small number of equations. Is mathematics, in some sense, therefore, the fundamental reality?
This question immediately prompts others. Some thinkers plausibly argue that mathematics is the creation of mind. We define certain terms - number, or set - and we define certain operations that can be performed with them according to certain rules. All the truths that follow are the logical consequences of these definitions. A change of definitions produces a change in the resulting truths. If mathematics is the product of human thought, and if reality is mathematical in character, does this mean that mind is the ultimate reality?
Even if one takes the opposing view, namely, that mathematics is objective (that is, exists independently of mind, so that mind does not create mathematical truths but discovers them), the puzzle is still unresolved. For the things referred to in mathematics - numbers, sets, operations, functions - are abstract entities; if they exist at all, they do so in an eternal realm which is outside space and time. How do these transcendent entities constitute the ordinary solid objects of our world?
A possible solution is to argue that there is a misunderstanding here. In describing the world mathematically we deliberately take account only of its measurable aspects, only of what can be stated in quantitative terms. The rest - the qualitative dimensions of things: how they appear to subjective experience in their moral, aesthetic or social significance - is ignored. If we choose to look at the world through mathematical spectacles alone, of course we will see it as a fundamentally mathematical realm: but that leaves out too much.
This argument has much plausibility. But a stubborn fact remains: that the mathematical picture of nature, in the form of an insistent group of equations describing atoms and stars, makes possible such things as electric light, the exploration of space, and the destruction of Hiroshima. This powerfully suggests that, for both good and ill, mathematics contains a germ of truth about the world.
A C Grayling is a lecturer in Philosophy at Birkbeck College, University of London, and a supernumerary fellow of St Anne's College, Oxford.
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