MonoRealism Philosophy Site

Mathematical Realities

Philosophical Reflections XXXIII

Part A: Realities

Mathematics has proved to be an extremely powerful tool for science. This power has resulted in disparate philosophical reactions ranging from puzzlement ("why does maths so successfully describe what happens?"), to mathematical realism ("if the maths works, then it represents reality"), to mathematical mysticism ("perhaps reality is mathematical equations!").

Thus the relationship between mathematics, science and reality deserves closer examination.

Description and Conception

That a mathematical equation gives a correct description of behaviour doesn't mean it actually describes what reality is. That is, the actual existents in nature, their valid conceptual hierarchy, and the chain of cause and effect may be quite different from a verbal expression of the maths.

This is most clear where a mathematical system is only an approximate match to reality. For example, the popularity of fractals has led to their adoption to describe many diverse systems, from coastlines to patterns on animals. But the underlying reality is clearly not fractal (whose essence is equivalence at all scales), because the apparent fractals must terminate at atoms at the small end of the scale and finite bounded entities (animals, planets) at the other end. Thus, such systems can be studied as fractals only approximately over a certain range.

More significantly, it is also true where the maths precisely describes reality. This is proved by the numerous cases where quite different mathematical formulations give the same results. For example, in Richard Feynman – A Life in Science (J & M Gribbin), it is noted that there are three quite distinct mathematical formulations of quantum mechanics: Schroedinger's wave equations, Heisenberg's particle-uncertainty descriptions, and Feynman's "path integral" approach. These three, while quite different conceptually and in terms of their mathematical formulae, are in fact exactly equivalent mathematically! Hence the choice of which to use is entirely a matter of convenience of calculation, as all three will always give the same answers.

The fact that all three work equally well means that we can't point to the assumptions or conceptual essence of any one model and say "the math works, therefore this is what quanta are." What they are has to be determined by other criteria than successfully predicting correct numerical results. Elsewhere I have argued that quanta are waves, whose "particle" properties are an illusion created by localised absorption of quanta of energy. That is a conceptual model of the nature of quanta, which does attempt to identify what they are, as opposed to providing a means of calculating their behaviour without identifying their nature.

Similar multiple possibilities are found in classical physics as well, one important class of which we'll turn to next. Indeed, quite bizarre mathematical mappings can be done, as in the model of the universe touted by the "Wizard" of Christchurch, in which the Earth surrounds the cosmos! That it can be done does not make it so, nor does it make the correct understanding a matter of arbitrary choice.

Least Action Principles

The lack of necessary correspondence between accurate mathematical descriptions and correct causal explanations is brought into sharp focus by comparing descriptions of the refraction of light.

Fermat's "Principle of Least Time" states that light always takes the path of shortest time, not distance, between two points. Thus, in air or water light travels in a straight line, but when moving from one to the other it bends by exactly the right amount to take the least time, given its faster speed through air. That is, the shortest time is achieved by travelling further through the air than the water, resulting in its path bending at the interface. The greater the difference in speed in any two media, the greater the bending.

However, this accurate description clearly is nonsensical as a causal explanation: the light would have to know where it was going to end up in order to back-calculate the quickest path. But light has no volition, let alone powers of calculation and prophecy. The correct causal description is that light is a wave whose direction bends according to its velocity changes. Indeed, Fermat's Principle is a mathematical consequence of this causal explanation (the slowing of light in water causes the exact bent path that results in the minimum time taken).

Such "least action" principles are common in physics, and many physical systems can be expressed that way, from Newtonian mechanics to electromagnetic waves (e.g. see the entry in Principles of Nature).

The Law of Mathematical Equivalence

Though such equivalence might seem mysterious, it is common both in physics and mathematics. One standard approach to solving problems that are intractable in one mathematical system is to show an equivalence, by mapping or transformation, to a more tractable one: which is then used to solve the original problem. Similar techniques are used in the maths of topology and knots, in which apparently quite different shapes have useful basic equivalence that can be exploited. Many mathematical proofs are based on such tricks.

My conclusion is that if different mathematical approaches give the same answers in physics, it is because at a deeper level (known or not) they are equivalent in just the same way. When looked at from that perspective, it is not at all surprising that they give identical results: they must.

Mathematical equivalence is a key clue to the relationship between description and conceptual understanding, and it is worth its own law. I call it The Law of Mathematical Equivalence:

Two or more distinct mathematical formulations can be equally correct numerical descriptions of reality, due to an underlying mathematical equivalence or interchangeability.

Following from this law is my Corollary of Conceptual Independence:

A numerically correct mathematical description of reality does not necessarily provide a conceptually or causally correct description of reality.

For example, Einstein's General Theory of Relativity has been astoundingly successful in its predictions – but how do the equations relate to what reality is? The simplest conceptual analysis indicates that there must be more to it than a naive verbalisation of the maths. For what does its centrepiece, "curved space", mean? If by space one means "the nothingness between objects" – you cannot curve "nothing". Curvature is a concept dependent on some existent that can be curved. But if space is "something", such as a substrate which vibrates in the transmission of electromagnetic and gravitational waves, then that is a concept of great significance – and perhaps the old "luminiferous ether" theory is closer to the truth than is generally appreciated!

The need for care in interpreting what successful mathematical systems actually mean is thrown into sharp relief by the reversal of causality involved in things like Fermat's Principle of Least Time (i.e., the principle is caused by the path light takes, itself caused by the nature of light wave propagation – rather than the principle causing the path light takes). Similarly, that equations of motion can be solved for negative time values doesn't mean that time can actually go backwards: it just means we can use the equations to calculate back into the past as well as forward into the future. Or consider Newton's formula F = ma (force equals mass times acceleration). The equation per se tells us nothing about causality: it works equally well whether acceleration causes force, mass derives from force and acceleration, or (correctly) force causes acceleration of mass, which is fundamental. The Corollary of Conceptual Independence must always be borne in mind when moving from equations to causal explanations.

Does it Matter?

If we have a mathematical system that churns out the right answers, do we need to worry about finding the correct conceptual understanding? That is, if it works, do we have to understand why?

As our example of General Relativity suggests, the answer is yes, for reasons of both principle and practicality.

In terms of principle, the purpose of science is to understand reality, which means, to understand both the entities that exist and the chains of cause and effect that link them. At a deeper level, I have noted before (Philosophical Reflections 25) that explanatory induction is far more powerful than descriptive induction. Thus, while it is certainly valuable and even necessary to have a set of equations that will give us the right answers, science fails in its deeper purpose if it gives up at that point.

In practical terms, we are beings of conceptual consciousness. That has two implications for this question.

First, our mode of consciousness requires conceptual understanding: to have any real understanding at all, to integrate it with our other knowledge and build on it with further understanding. To accept a tool of calculation without understanding why it works is like using a pocket calculator in the absence of understanding the principles of arithmetic: it works, but leads nowhere new. Blind acceptance without asking why is just that – blindness.

Second, in the absence of explicit conceptual understanding, we can't help conceptualising by default. Thus the temptation to accept the maths as "reality". But if the maths is not reality, then this leads us down blind alleys. Fundamentally, I think that is the origin of much nonsense in modern theoretical physics. One example is the egregious attempts to justify subjectivism on the basis of quantum mechanics – when that theory was and could only ever have been arrived at by the most exacting objective methods. Another is the acceptance of the existence of full-fledged black holes, with the consequent heartburn about inexplicable "singularities" and "loss of information" – when the very equations of Relativity that predict them also mean that, due to gravitational time dilation, an event horizon cannot form in the lifetime of the universe. (Nor does it form from the standpoint of the things falling in: from their perspective the universe will end, or the black hole evaporate, before they can get there).