What is the relation between mathematics and the physical world? For example consider Newton's laws of motion. If we deduce results about mechanics from these laws, are we discovering properties of the physical world, or are we simply proving results in an abstract mathematical system? Does a mathematical model, no matter how good, only predict behaviour of the physical world or does it give us insight into the nature of that world? Does the belief that the world functions through simple mathematical relationships tell us something about the world, or does it only tell us something about the way humans think. In this article we explore a little of the history of the philosophy of science in order to look at differing views to the type of questions that we have just considered.

The most natural starting place historically for examining the relationship between mathematics and the physical world is through the views of Pythagoras. The views of Pythagoras are only known through the views of the Pythagorean School for Pythagoras himself left no written record of his views. However the views which one has to assume originated with Pythagoras were extremely influential and still underlie the today's science. Here we see for the first time the belief that the physical world may be understood through mathematics. Music, perhaps strangely, was the motivating factor for the Pythagoreans realised that musical harmonies were related to simple ratios. Moreover the same simple ratios hold for vibrating strings and for vibrating columns of air. The discovery of this general mathematical principle applying to many apparently different situations was seen to be of great significance. Pythagoreans then looked for similar mathematical harmonies in the universe in general, in particular the motions of the heavenly bodies. Their belief that the Earth is a sphere is almost certainly based on the belief that the sphere was the most perfect solid, so the Earth must be a sphere. The shadow of the Earth cast on the Moon during an eclipse added experimental evidence to the belief.

Plato followed these general principles of Pythagoras and looked for an understanding of the universe based on mathematics. In particular he identified the five elements, fire, earth, air, water and celestial matter with the five regular solids, the tetrahedron, cube, octahedron, icosahedron and the dodecahedron. On the one hand there is little merit in Plato's idea: of course Plato's elements are not the building blocks of matter, and anyway his identification of these with the regular solids had little scientific justification. On the other hand, at least he was seeking an explanation of the physical world using mathematical properties.

Another who followed the general approach of Pythagoras was Eudoxus. He produced a remarkable model to explain the movements of the heavenly bodies. It was a remarkable mathematical achievement based on 27 rotating spheres set up in such a way the retrograde motion of Mars, Jupiter and Saturn were modelled. Here we have the first major attempt to model the movements of the heavenly bodies, but it is far from clear that Eudoxus thought of his mathematical model as a physical entity. For example he made no attempt to describe the substance of the spheres nor on their mode of interconnection. So it would appear that he thought of his model as a purely geometric one but the difficulty with the model was that it did not allow the positions to be predicted with reasonable accuracy. Despite the fact that the model would not have passed the simplest of observational tests, Aristotle accepted the crystal spheres of Eudoxus as reality.

Aristotle proposed a scientific method which was highly influential for many centuries. His method, in broad terms, consisted of making observations of phenomena, using inductive arguments to deduce general principles which would explain the observations, then deducing facts about the phenomena by logical argument from the general principles. He saw this as leading from observations of a fact to an explanation of that fact. Although Aristotle saw the importance of numerical and geometrical relationships in the physical sciences, he made a very clear distinction between the sciences and pure mathematics which he saw as an abstract discipline.

One approach was to set up axioms, that is a list of self-evident truths, and from these deduce results which were far less obvious. Euclid set up geometry in this way but there were interesting aspects of this as far as physical science was concerned. On the one hand Euclid did not completely achieve his aim, for he did use methods of proof which went outside his axiom system. In other words he invoked further axioms without realising it. More worrying as far as physical science was concerned, is the fact that the objects of Euclid's geometry can have no physical existence. Points and lines as defined by Euclid could not be physical objects. How can axioms be considered as self-evident truths when the objects of the axioms have no physical existence? Archimedes too set up axioms to deduce properties of levers. In this he was very successful, for he was able to create wonderful machines through the understanding that he gained. However, again his axioms refer to objects having properties that no real world object will possess; rods with zero weight, levers that are perfectly rigid. As a consequence theoretical results deduced from the axioms will never fit experimental evidence exactly but Archimedes never discussed such points.

Archimedes certainly had developed an excellent mathematical model but never discusses its limitations in describing physical situations. Similarly, neither Eudoxus nor Aristotle, despite looking at the physical reality of the crystal spheres model differently, made clear the distinction between a mathematical model and reality. The first to think deeply about this particular problem seems to have been Geminus. He states clearly that there are two different approaches to modelling the motions of the heavenly bodies, that of the physicist who looks to explain the motions by the nature of the bodies themselves, and the astronomer or mathematician who says that:-

This approach became known as "saving the appearances", that is putting forward mathematical relationships which correspond to observation, without making any attempt to suggest a physical explanation for the relationships. The most famous of the ancient models of the heavenly bodies put forward to "save the appearances" was that by Ptolemy. His model was the epicycle-deferent model where the motion of the heavenly bodies was circular, but based on a number of circles whose centres travelled around circles. Ptolemy is quite clear in stating that his model is not intended to represent physical reality, rather it is only a mathematical model that will represent what is observed. He also states clearly that other mathematical models are equivalent and will lead to the same observed appearance.... it is no part of the business of the astronomer to know what is by nature suited to a position of rest, and what sort of bodies are apt to move, but he introduces hypotheses under which some bodies remain fixed, while others move, and then considers to which hypotheses the phenomena actually observed in the heavens will correspond.

The problem of whether a mathematical model represents reality became highly significant when Copernicus proposed his Sun centred system. The Christian Church had no problems with mathematical models, and were quite happy to allow publication of models to "save the appearances" based on a Sun centred model. However, this was a very different matter from stating that the Copernican system was more than a mathematical model, and did indeed represent reality. Clavius, for example, was happy to accept the Copernican model as a mathematical model, but he declared that Copernicus had saved the appearances by using axioms which were physically false. Copernicus, however, maintained that his Sun centred system was superior for it provided an explanation of the retrograde motion of the planets as opposed to Ptolemy's model which was devised to produce the observed effect.

The strongest supporter of the reality of the Copernican system was Galileo. He was a great believer in the mathematical approach to science which originated with the Pythagoreans. For Galileo the simplicity of the mathematics of the Copernican system over the complex mathematics of Ptolemy's system was a strong proof of the reality of the Copernican hypothesis. But it is not only Galileo's belief in the Copernican system which interests us here, for he made very significant advances in understanding the nature of mathematical models. He stressed that an important aspect in understanding physics is abstraction and idealisation. He could not conduct experiments to test objects falling in a vacuum, nor could he conduct experiments with a pendulum consisting of a point mass supported by a weightless string swinging without air resistance. However, clever experiments could lead a scientist towards the idealised situation. Working with the abstract mathematical model of the idealisation would enable results to be predicted which would be approximately true in reality, and approximate confirmation could be made. Here was a complete understanding of the relation between the idealised theory of levers produced by Archimedes so many centuries earlier and real levers. It was a remarkable achievement, but when Galileo was mislead it was often because he had not confirmed an attractive mathematical theory by experiment.

Like Galileo, Kepler believed in the Copernican system. He argued that the Sun had a driving force which propelled the planets in their orbits. This force diminished with distance from the Sun and so the outer planets moved more slowly. Now Kepler could claim that the Copernican system was real since it provided an explanation for the planetary motions while that of Ptolemy did not. In *Apologia* written in 1600, but unpublished, Kepler argues that accuracy in "saving the phenomena" cannot distinguish which mathematical theory might correspond to reality. The theory which corresponds to reality will provide a physical explanation for the appearances. It was a belief that a simple mathematical relationship must be physically significant which led Kepler to discover his third law of planetary motion. He tried various algebraic formulas to relate the velocity of a planet round the Sun with its distance from the Sun before he stumbled on:

The same approach also led him into error. For each of the planets he calculatedThe ratio of the squares of the periods of two planets is directly proportional to the ratio of the cubes of the radii of their orbits.

Another example of a mathematical relation which was thought to have physical meaning was Bode's law. This took the sequence

4, 4+3, 4+6, 4+12, 4+24, 4+48, 4+96, 4+192, ...

divided by 10 to get
0.4, 0.7, 1.0, 1.6, 2.8, 5.2, 10.0, 19.6, ...

Now the distances of the planets Mercury, Venus, Earth, Mars, Jupiter, Saturn from the Sun (taking the distance of the Earth as 1) are
0.39, 0.72, 1.0, 1.52, -, 5.2, 9.5

When Ceres and other asteroids were discovered at distance 2.8 it was firmly believed that the next planet would be at distance 19.6. When Uranus was discovered at distance 19.2 it was almost considered that Bode's law was verified by experiment. However, the next planets did not fit well at all into the law, though a few scientists still argue today that Bode's law must be more than a mathematical coincidence and result from a physical cause. [On this theory the outer planets have been disturbed since the system was created and there is certainly independent evidence that this has happened.]

The next axiomatic system we wish to examine is that given by Newton. He adopted the approach that (see the *Principia):-*

He set up an axiom system consisting of hard particles which were at rest or in motion, obeying three simple laws concerning motion and forces, and a universal law of gravitation. Newton was careful to distinguish between laws which he believed he had verified, and underlying reasons why the laws existed. For example he believed he had proved his law of gravitation, but he was clear that he put forward no explanation of why or how two bodies underwent mutual attraction in a vacuum. One could argue that Newton was "saving the appearances" again, putting forward a mathematical model of the world without any physical explanations. He did, however, make very clear the relationship between mathematical dynamical results proved from his axioms and the outcomes of experiments conducted in the real world.... particular propositions are inferred from the phenomena, and afterwards rendered general by induction. Thus it was that the impenetrability, the mobility, the impulsive force of bodies, and the laws of motion and of gravitation, were discovered.

As we have suggested there were problems with Newton's system despite the fact that it appeared to reduce the whole of nature to consequences of simple mathematical laws. Perhaps most significant was the fact that his theories required a postulate of absolute space and time. He was well aware of this and he put forward his rotating bucket experiment to try to prove that absolute space did exist. But there was a weakness here, namely he had introduced a concept of space independent of the material of the universe. Is space an independent concept, or are there simply relations between the material objects? Berkeley criticised Newton's absolute space by asking how spatial relationships could be meaningful in a world without matter. If there is only one particle in the universe, said Berkeley, how is it meaningful to say that it is at rest or, for that matter, what could it possibly mean to say that it was accelerating.

Although Newton had made a clear distinction between a mathematical theory and a physical reality, Berkeley argued that he had fallen into his own trap for he spoke of forces as physical entities, where Berkeley believed that they were nothing other than terms in the equations set up by Newton. Indeed Berkeley argued against abstract ideas in general for, in his view, they led to the mistaken belief in the reality of concepts such as force, absolute space, absolute time, and absolute motion. No idea, argued Berkeley, can exist unperceived and nothing exists except things which are perceived.

Poincaré put forward important ideas on mathematical models of the real world. If one set of axioms is preferred over another to model a physical situation then, Poincaré claimed, this was nothing more than a convention. Conditions such as simplicity, easy of use, and usefulness in future research, help to determine which will be the convention, while it is meaningless to ask which is correct. The question of whether physical space is euclidean is not a meaningful one to ask. The distinction, he argues, between mathematical theories and physical situations is that mathematics is a construction of the human mind, whereas nature is independent of the human mind. Here lies that problem; fitting a mathematical model to reality is to forcing a construct of the human mind onto nature which is ultimately independent of mind.

**Article by:** *J J O'Connor* and *E F Robertson*

**MacTutor History of Mathematics**

[http://www-history.mcs.st-andrews.ac.uk/HistTopics/World.html]