As I read more discussions of the relationship between mathematics and physics, I find that what mathematics might reveal about how physical science progresses becomes an increasingly interesting question.
I recently found the text of a lecture given by Paul Dirac in 1939. It was reproduced on the occasion of the Dirac Centennial Celebration organized by the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge in 2002. The title of the lecture is “The Relation between Mathematics and Physics.” Dirac remarks, right away, that “there is no logical reason” why mathematical reasoning should succeed as one of the two methods used by the physicist to study natural phenomena (the other being experiment and observation). And, early in the talk, he says the following:
One might describe the mathematical quality in Nature by saying that the universe is so constituted that mathematics is a useful tool in its description. However, recent advances in physical science show that this statement of the case is too trivial. The connection between mathematics and the description of the universe goes far deeper than this, and one can get an appreciation of it only from a thorough examination of the various facts that make it up.
He goes on to survey the conceptual shifts that have happened in physics – from the equations that represent the laws of motion in Newtonian physics, to the geometry of Einstein’s space-time, and the non-commutative algebra of quantum mechanics. He made what I thought was an unexpected distinction between classical laws, governed by “a principle of simplicity,” and the mathematical beauty that makes the theory of relativity so compelling. About this Dirac says the following:
The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty…It often happens that the requirements of simplicity and of beauty are the same, but where they clash the latter must take precedence.
In this light he proposes that a powerful method of research for physicists may very well be to first choose a promising branch of mathematics “influenced very much in this choice by considerations of mathematical beauty,” and then proceed to develop it, keeping an eye on the way it lends itself to physical interpretation. String theorists are among those who seem to have chosen this route. And here’s an interesting statement:
One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.
‘The rules,’ of course, are the consequences of various observed relationships among the concepts captured in mathematical symbol. These may be relationships among numbers, or among operations and transformations, or among spatial properties and geometric structures. While there continues to be disagreement about whether the rules are invented or discovered , what Dirac may be emphasizing about mathematics is that the mathematician looks only at the mathematics, requiring no other validation of what mathematician Richard Courant once called ‘verifiable fact.’
Yet mathematics is an equal partner in the design of physical theories, what we consider a purely empirical science. This fact must say something about the nuances of what we mean by ’empirical,’ which are often reflected in disputes between rationalists and empiricists. For some, the senses are like detectors to which a rational device of the mind is applied. But this perspective has been consistently challenged by research in cognitive science. These studies indicate strongly that the senses are not easily distinguished from that rational device. And perhaps it is this that is reflected in the growing blend of mathematics and physics.
Stephen Hawking was one of the speakers at the Dirac Centennial Celebration. The title given his talk was “Gödel and the end of physics.” But he wasn’t predicting the end of physics. The view he presents is, to some extent, a critique of the standard positivist approach to science, where the mathematical treatment of sensory data is the only source of knowledge. In this light, mathematics is not considered a product of the mind. Intuition and introspection, after all, play no role in the acquisition of knowledge. But how could one divorce intuition and introspection from mathematics? Physics’ increasing reliance on mathematics must be pointing to a relationship between mathematics and perception. And Hawking sees another problem. Physical theories, or mathematical models of physical systems, are self-referencing.
…we are not angels, who view the universe from the outside. Instead, we and our models, are both part of the universe we are describing. Thus a physical theory, is self referencing, like in Gödel’s theorem. One might therefore expect it to be either inconsistent, or incomplete. The theories we have so far, are both inconsistent, and incomplete.
Like Gregory Chaitin, Hawking seems to find this incompleteness promising.
Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I’m now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery.