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).

Part B: Abstractions

Abstract Maths

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 1021m), “drawn” in a line only as thick as a hydrogen atom (10-10m). That is a ratio of 1031: 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.

To Infinity & Beyond

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.

Mystic Maths

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 thingsthat 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.

The question of why maths works at all leads us down more interesting paths, and that’s what we’ll turn to next.

Part C: Validity

Real Maths

We can distinguish between different types of maths depending on what kind of things they apply to.

Statistical or probabilistic maths is concerned with the average behaviour of multiple entities that are roughly the same, where individual items can’t be tracked or don’t have to be. Examples range from the odds of throwing a double six in dice, to statistical mechanics such as heat and gas laws.

At the other end of the scale is deterministic maths, in which the behaviour of individual entities can be described and predicted more or less exactly by general equations. Examples are Newton’s laws of motion and the equations of general relativity.

The recent TV series “Numb3rs” also provides some interesting “real world” examples of both kinds of maths.

Despite appearing at the opposite ends of a spectrum, these are united by what links them to reality – what makes them actually work. Each depends on the fact of reality which underpins the validity of concepts: multiple entities share qualities which can unite them. Those common denominators both allow us to form valid concepts that subsume multiple entities – and allow abstract mathematical equations to describe and predict the behaviour of individual or collective items of the same type.

Let us examine both these cases in more depth.

Taking Chances

One can derive equations based on the abstraction of perfect randomness, just as one can derive an equation for the circumference of an abstract “perfect circle”, without either having to exist in reality – yet with both applicable to those entities in reality that approximate them, to whatever degree of precision is involved.

For example, consider a bag of black and white balls, all alike except for their colour. If you mix them in a cement mixer, the position of any individual ball rapidly becomes impossible to predict. Even though the path taken and final position of each ball is deterministically caused by its nature and that of everything it interacts with, the system is chaotic. That is, its next state is sensitive to tiny variations in its current state, so any attempted calculation rapidly breaks down due to exponential amplification of unmeasurable uncertainties in position and velocity.

The result is an assortment of balls which is effectively random. Because the balls all share salient qualities to the required degree and the system is effectively unpredictable, the abstraction of “randomness” (whose essence is “completely the same and completely unpredictable”) applies. Hence, one can accurately predict things such as the odds of picking out 5 black balls in a row.

Of course, the balls are not identical: there will always be subtle variations in qualities that affect their paths, such as mass and shape. However in the context we are talking about, the resulting non-randomness in distribution is undetectable, being either too subtle or taking too long to become measurable.

Notice how this corresponds to the nature of concepts. While the entities we group into concepts are not identical (usually), all that matters is that their differences don’t matter in the context of the use of the concept.

These considerations apply to any system in which the distribution of entities approximates randomness, not only simple physical systems such as tossing coins or rolling dice, but complex biological systems such as the distribution and spread of genes in populations or even the average behaviour of human beings. Note again how this links back to the nature of concepts: we can ignore extraneous details such as the physical nature of the system in order to focus (validly) on the relevant factor they share in common, namely “random” behaviour.

Of course, in some cases the natures of the existents will cause observable deviations from randomness. This in itself tells us something more about that nature. Indeed, it is a foundation of statistics, whose equations compare actual to random outcomes: calculating how likely it is that any differences are themselves merely random fluctuations, or due to causation in a particular direction.

Being Determined

Why can one derive equations that accurately describe the free trajectory of any object, from a brick to a boulder, from glass to steel? Because the nature of matter is to attract other matter according to specific rules: and therefore, that is also the nature of all things made of matter.

Thus again, we see that maths works for the same reason that concepts work: the differences in size, shape and composition are irrelevant to gravitational attraction and therefore to free trajectories.

Again, the variations from the equations are instructive. Continuing with the example of trajectories, air resistance obviously is important. But again, this follows from the nature of the existents: in this case, all matter obeys Newton’s laws of motion. As a result, we can calculate the effects of density and shape on how the trajectory will be altered by air resistance. If we look closely enough, we can even find chaotic effects, due to things like fluctuations in air pressure, wind direction and even absorption, reflection and radiation of energy – though these are negligible in most contexts.

More interesting are cases where behaviour deviates from equations in the absence of external influences. Then, as with deviations from randomness in our earlier example, we learn something new about the entities” nature, which differs in some way from the assumptions of the maths. But for the same reasons, that too will be amenable to yet other mathematical formulae.

The Law of Mathematical Validity

From the above we can distil what I’ll call the Law of Mathematical Validity:

Mathematics is an abstraction which is valid and effective for the same reasons that all other abstractions are valid and effective.

That is, concepts work because they are valid, and they are valid because things in reality do, in fact, share common qualities that allow them to be grouped naturally and thought about – in relevant contexts – as a single mental unit. And the same is true of mathematics, replacing “thought about as a single mental unit” with “described by a single set of equations.”

To view this from another perspective: existents affect each other according to laws of cause and effect, which simply means according to their respective natures, and being able to describe that mathematically is a natural consequence. That is, concepts are valid because entities do share the fundamental qualities by which they are grouped into concepts. Therefore, the instances of concepts will behave the same way in response to the same causes, and numerical regularities and relationships will be observed in nature accordingly.

We look more deeply into this in Part D.

Part D: Identity

Deeper Waters

Perhaps the question still remains, not why maths works at all, but why so often the equations are “smooth” or regular. The simple answer, following from Part C, is that the entities actually behave in that regular manner. But why do they? Why, for example, should trajectories follow a parabola and not some loop-the-loop or spiky curve?

My first answer relates to conservation laws. As noted in Philosophical Reflections 32, the law of identity requires that there be conservation laws of some kind. Consider the implications of conservation laws. Mass and energy are conserved (each separately under “normal” conditions, but more precisely, the two together). Given that conservation, that mass has inertia, and that velocity imparts kinetic energy, there has to be some kind of regular relationship between mass, velocity and energy, and therefore between force, mass and velocity – which is reflected in the regularity of the equations describing them, such as Newton’s Laws of Motion. If you can’t get something from nothing, which you can’t, then outputs have to be smoothly related to inputs.

Thus, conservation laws lead to smooth, regular equations describing the resulting behaviours of affected entities.

Note that outputs being smoothly related to inputs does not necessarily imply proportionality. For example, while helium is “made of” two fused deuterium atoms, it takes somewhat less force to accelerate one molecule of helium than two of deuterium, because some mass was “lost” as energy in the nuclear fusion that created it. More subtly, no matter how much force is applied, an object can’t be accelerated to or beyond the speed of light, because of the relativistic increase of mass (toward infinity) as light speed is approached. These examples highlight why one can’t rationalistically deduce physics from a priori principles, but must look to nature to discover how entities actually act.

An even deeper reason behind “smooth” maths can be discerned, which will require a brief but interesting digression.

Causality & Identity

A fundamental metaphysical fact of existence is the Law of Identity – A is A – a thing is itself and therefore, it acts according to its nature. This implies causality: nothing a thing does can happen unless it is the nature of the thing itself, or caused by another entity acting upon it in accordance with both their natures.

Consider a ball lying on the floor. It is lying still, because the nature of matter includes inertia: in the absence of external forces, it will continue at rest or in uniform motion. So the ball stays where it is, unless I kick it or otherwise apply an external force. In other words, if I do nothing, the ball stays where it is: according to its nature. If I kick it, it moves: again, precisely according to its nature and mine (regarding the latter, consider the difference in what the ball would do between being hit with a bubble, my foot, or a sledgehammer).

Alert readers will have noted that this considers only my local frame of reference. In fact I, the ball and indeed our local frame of reference are all stuck on the Earth’s surface spinning at about 1,000 miles per hour, while the Earth is revolving around the Sun, which is revolving around the centre of the galaxy, which is itself in a gravitational dance with other galaxies. Thus far from being “at rest”, it is following a high-speed, complex curve. Nevertheless, the point remains. There can only be a local frame of reference because of the natures of the entities involved, and those natures also cause what they do both within and beyond whatever frames of reference we choose.

That all things act in accordance with their nature might not help us much, given the billions of things in our world, were it not for the fact that everything comes from only a few basic building blocks such as protons, neutrons, electrons and radiation¹. Given that, identity and causality imply that the atoms made of protons, neutrons and electrons, and the molecules made of atoms, and the macroscopic things made of those, inherit and derive their natures from those few progenitors. It is that which has lead to all the infinite variety around us, from stars to kittens, and which links all those things by the same “laws”.

As we saw in Philosophical Reflections 24B it is this complex flowering from a few types of entities whose representatives are identical, which makes understanding the world through concepts possible. Now we see that this same thing is what makes understanding the world through mathematics possible as well.

Identity & Mathematics

Thus the Law of Identity implies causality. But causality implies that the same cause acting on the same entity will have the same result. And as the basic constituents of matter (for example) are identical, if it takes a certain amount A of a cause to produce an amount of effect B on a single molecule of hydrogen, then it will take 2A to produce 2B on one such molecule, or 2A to produce B on two linked molecules. So it is not surprising that you need twice the force to accelerate twice the mass to the same degree, or that gravitational attraction is proportional to mass. (Again, bear in mind that such “proportionalities” might be limited or enhanced by other factors).

I summarise this in the Causality Principle: Smooth, regular mathematical descriptions of the behaviour of entities are a necessary consequence of identity and causality.

We can follow this trail even further. Consider a molecule of gas. Its trajectory can be understood by the laws of motion: we can predict its path until it hits another gas molecule or a solid wall, and how its path will change when it does so. The very same identity/causality factors that cause that behaviour also cause, at the larger scale of macroscopic amounts of gas, regular “statistical” laws such as the gas laws that relate the pressure, temperature and volume of a gas. And in the open world, it is the very same identity/causality factors that result in the atmospheric movements comprising the weather to be a poster-child of chaotic mathematics.

Thus even complex or chaotic systems derive ultimately from smooth causal relations between entities. This returns us to a deeper understanding of the Law of Mathematical Validity. Allmathematical treatments that accurately portray an aspect of reality, whether deterministic, probabilistic or chaotic, can do so because and only because of causality and therefore the law of identity.

Mathematics & Reality

It is worth stressing again that these considerations allow us to understand and predict the basic link between mathematics and reality. It is not a licence for rationalism (the belief that one can deduce the facts of reality from a priori abstract principles). The actual equations that describe reality can only be determined, like everything else, by induction from the reality we observe.

Similarly, it is a mistake to think that things in reality “obey natural laws” in the sense that natural laws have some kind of independent existence above and ruling the entities that exist. On the contrary, “natural laws” are merely a description of how the things in reality behave. Those laws are made possible by the deep links in identity and causality between the different things that exist, such that those different things end up behaving in related ways. Of course, things do obey natural laws in the sense that given their nature we know they will behave in the way those natural laws describe – we just need to remember that it is the latter that is primary. That is, natural laws are descriptive not prescriptive: they derive from the nature of the things that exist, they do not determine it.

Proof of Concept

I have noted before that science is the pinnacle, and thereby an unanswerable validation of, our basic inquiry method of senses, memory, reason and experiment.

Now we can see that mathematics holds a similar position in regard to our basic mental process of abstraction into concepts. Maths is basically abstraction par excellence, effectively an abstraction of concepts themselves. Concepts strip away the irrelevant and focus on the essential – hence the “measurement omission” and “conceptual common denominators” that characterise them. Mathematics ruthlessly strips away everything except the numerical relationships. Hence the same equations can describe phenomena that – by other criteria – are completely disparate: from gambling to genetics, from the fall of a feather to the orbits of galaxies.

Mathematics allows an engineer to calculate the structure of a bridge to straddle miles while withstanding traffic, waves and weather; and a physicist to calculate the trajectory of a probe to intercept a speeding comet; and is physically embodied in computer chips and the source of their efficacy. That power and success is a testament to the power and validity of the conceptual method on which it is based, and is its unanswerable validation.

 


Footnote:

  1. That these entities are not necessarily truly fundamental, e.g. protons and such are made of quarks, doesn’t matter. All that matters is that at some basic level, all entities of the same type are identical. That even lower, simpler levels might exist merely strengthens the case.
© 2005, 2006 Robin Craig: first published in TableAus in four parts.