Is Mathematics Just an Intellectual Game? — Part II
Real Mathematics
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 that underpins the validity of concepts[i]: 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.
Mathematics allows an engineer to calculate the structure of a bridge to straddle miles while withstanding traffic, waves, and weather; 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 success is a testament to the power of the conceptual method on which it is based, and is its unanswerable validation.
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
Maths is basically abstraction par excellence, effectively an abstraction of concepts themselves.
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.
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 why mathematics is valid in the first place, 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 When Science Meets Philosophy, 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 such 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 myself, 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 radiation[ii]. 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 on the validity of concepts, 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. All mathematical 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.
[i] A concept is a single mental unit that unites more than one separate existent on the basis of shared fundamental qualities that may vary in magnitude. All concepts are abstractions (it is only the existents that exist in the world) identified by words, and form a hierarchy of concepts. For example, the concept of “cat” groups together all instances of cats, who all share essential features but vary in size, colour, shape etc; the more general concept “mammal” unites cats, dogs, elephants etc. by more fundamental similarities whilst ignoring even more differences; the yet more general “vertebrate” brings in reptiles and fish by the same process; and so on. For more on the nature and validity of concepts see Philosophical Reflections 24.
[ii] 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: original version first published in TableAus
About the Author: Robin Craig
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