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.
That a mathematical equation gives a correct description of behaviour, doesn’t mean it actually describes what reality is.
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.
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).
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 a 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.
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.
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 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).
Some people are puzzled that mathematics is so successful at describing reality. Others take its success to mean that fundamental reality might just be mathematical equations. Such conundrums vanish when we understand that all of mathematics is abstraction, and what that implies.
The concept of “number” is the fundamental underpinning of mathematics. But in reality there are just existents, each one distinct from all others. The very concept “number” is dependent upon conceptualisation. You cannot identify “two” of anything in the absence of the mental integration of multiple existents into one concept, based on their similarities. Thus, in order to count two oranges, you must first have identified “orange” as a particular kind of existent which you can distinguish from other existents; and the same is true of more general abstractions from “fruit” all the way to “thing.”
As the process of forming concepts is a process of abstraction, it follows that all of mathematics, being founded upon abstraction at its lowest levels, is an abstraction. Indeed, it only gets more abstract from there, as is reflected in the history of number “types,” characterised by increasing distance from the perceptual level.
The only numbers which are directly represented by concrete instances of concepts are the positive integers. You can have two oranges, 3 people, or 100 ants – but there are no “negative existents” or “nothings.” You will never see -1 people walking around.
Negative integers are a higher level of abstraction, dealing not with existents as such, but with actions done to existents (e.g., removal or destruction), and qualities of existents that cancel or oppose each other (e.g., directions or electrical charge). Referring to an absence, to a void, to literally nothing at all, is zero.
The most abstract numbers that are actualised in reality are the rational numbers, or ratios. We can directly refer to 3 out of 5 apples, or 99 out of 103 arrows.
Numbers more abstract than that have no exact representation in reality and thus are “purely abstract”. This is all of the irrational numbers – those with infinite, non-repeating decimals, that cannot be expressed as ratios – which include surds like the square root of two and transcendental numbers like pi. The reason such numbers are not exactly represented in reality is that reality is not the infinitely smooth canvas of mathematical abstraction. For example, any real instance of a circle has a circumference line of nonzero width. For any such real circle, “its” value of pi is simply the ratio of its actual measured circumference to its actual measured diameter. Depending on how perfect the circle and how precise the measurements, this number will be equal or close to the value of the abstraction pi to a certain number of decimal places, and no more. There is no circle in the universe whose circumference divided by its diameter is exactly pi: it has no exact value, by its definition as a non-terminating non-repeating decimal.
A telling illustration is how surprisingly few digits of pi apply to any real circle. The number of relevant digits is only about the number of zeroes in the ratio of the diameter’s length to its thickness (due to the difference in circumference between the “outside” and “inside” of the line). Imagine a perfect circle around our galaxy (diameter 100,000 light years, or 10^{21}m), “drawn” in a line only as thick as a hydrogen atom (10^{-10}m). That is a ratio of 10^{31}: so only about 30 decimal places of pi are needed to calculate the upper and lower bounds of the circumference at that diameter plus or minus half that width! And both increasing the diameter and decreasing the thickness by a factor of a billion each increases that by only another 18 digits! And this is ignoring the physical impossibility of actually having a perfect circle at such scales.
Even more abstract are the imaginary numbers, based on square roots of negative numbers. They don’t even have approximate representatives in reality.
Note that such numbers are no more invalid or useless than any other abstraction which omits consideration of irrelevant measurements. A prime example of that is zero, which, while by definition referring to nothing that exists, is an extremely powerful part of mathematics. Similarly, the abstractions of pi, square roots, and imaginary numbers give us tools for calculating dimensions in the real world to any required level of precision. In short, while we need to remember which numbers actually have exact representatives in the real world, purely abstract numbers can still be eminently useful for calculations on their approximate representatives, or for any other relevant application.
An unusual mathematical concept which is worth special mention is infinity. Philosophically, actual infinities cannot exist, because by the law of identity any entity must have a specific nature so cannot be “indefinite”. One can have a “potential” infinity (e.g. of time, as in “the universe will never end”), but not an “actual” one (e.g. a universe that is infinite in volume).
As we’ve seen before, this is one of the few philosophical principles that would seem to have a direct implication for science: if there can be no infinities, then the universe must be finite, which restricts the possible cosmologies to finite ones (for example, a “closed universe” which is “finite but unbounded”).
As with other mathematical abstractions, this is not to say that the concept (as a mathematical one) is invalid. Both infinity and its inverse, infinitesimals, have mathematical value. But just as infinitesimals are abstractions which have value for calculation but no referent in reality (because reality isn’t infinitely divisible), so is infinity.
Given that, I do consider it meaningless to speak of different “sizes” of infinity, and I have yet to see a justification of that which did not reek of logical fallacies or sleight-of-definition. I believe that to attempt to extend the abstraction of infinity in such a way is to totally divorce oneself from relevance to reality in any form. Infinity is infinity, and one abstraction of endlessness can’t exceed another.
Tracing the conceptual hierarchy of mathematics in this way allows us to easily dispose of the notion sometimes encountered that as the universe is so well described by mathematics, perhaps reality is at base just mathematical equations.
Such a Platonic notion is a clear example of the fallacy of the “stolen concept.” Mathematics describes the relationships between existents. To drop the existents yet attempt to retain mathematics is conceptually invalid because it is cut off from its own roots and validation. Mathematics is an abstraction: it has no independent existence outside of the concretes from which it is abstracted, any more than the concept “number” has an existence separate from things that can be counted, or the concept “apple” has an existence separate from the particular individual fruits it refers to. It is not that things exist because of mathematics; mathematics exists because of things.
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