MonoRealism Philosophy Site

Mathematical Realities

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. 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.

 


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.