Naturalizing Mathematics

1,700 words

Sign up for the Blank Horizons mailing list here.

Header image taken from here.

Are numbers real? There is clearly a sense in which the existence of numbers is very different in nature from the existence of ordinary, everyday objects, such as a coffee cup. One coffee cup occupies space and time and can be perceived directly by the senses, whereas “oneness” is an abstract concept that lacks both of those attributes.

In the philosophy of math, those who deny the objective reality of numbers and all other abstract concepts are known as “nominalists.” In contrast, “mathematical platonists” believe that numbers, along with other mathematical objects, exist “independently of intelligent agents and their language, thought and practices,” according to philosopher Øysten Linnebo. Numbers dwell in the Platonic realm of forms, which are universalized abstractions of worldly objects that exist outside of space and time.   

Both nominalists and mathematical platonists assume that numbers are abstract. But could there be a third view that rejects this premise and instead characterizes numbers as concrete entities – just as concrete as a coffee cup? In this essay, I will be arguing for this third view, which I will refer to as the “naturalization of mathematics.” 

What does it mean for something to exist concretely? For me, the division of the world into “concrete” and “abstract” things is merely the result of confused metaphysics, lack of clarity on the meaning of existence in general. I am staunchly opposed to the dualist notion that the physical universe is fundamentally separate from the world of abstract forms, which to me exists as nothing more than a fanciful product of human imagination. A more rigorous justification of this idea will have to wait for another essay. But for now, let us say that there can be no things which enjoy a privileged existence that is somehow of a fundamentally different nature than the other things that exist. There are no “concrete” and “abstract” things that are irreducible to one another; there are only the things that are.

Over two years ago, I argued that what it means to have an identity, which is inextricably bound up with the question of what it means to be, is to persist through the changes that are brought on by time. This is not a new idea. The more original claim in the essay, however, was that identity is a structural feature of reality, rather than a feature of any one particular object contained within that structure; and that, furthermore, these structures are mathematical phenomena known as symmetries, which are, broadly, aspects of a system that remain unchanged when a transformation is applied to the system. So, what truly exists is symmetry, and symmetry upon symmetry.

What are symmetries? They are purely mathematical objects known as groups: sets containing elements that refer to transformations that preserve an invariance in the system. For instance, the symmetries of an equilateral triangle are described by the group D6, which contains the elements:

{r, r2, s, rs, r2s}

where r refers to rotation of the triangle by 120o, s to reflection, rs to rotation followed by reflection, and r2s to two rotations and a reflection.

Furthermore, numbers arise from “representations” of symmetries. Representations are a very challenging concept for those who don’t have a background in abstract math, such as myself. At the risk of oversimplification, representations are matrices that represent the operations of groups. One representation of D6 is:

since these rotations are used to perform rotations in 2D Cartesian coordinate systems. For instance, when an equilateral triangle determined by the vectors 

is rotated 120o, each of the vectors is simply multiplied by matrix A:

These vectors produce exactly the same triangle as the original one, confirming that the triangle is invariant under the transformation encoded by the rotation matrices. In other words, the rotation matrices yield a symmetry.

Fig 1. Visual demonstration of the rotations, as encoded by the matrix multiplication above.

Now that we’ve squared away the mathematical details, let’s return to philosophy. I suggested earlier that, contrary to traditional assumptions, numbers are just as real and concrete as ordinary objects like coffee cups. However, numbers that “arise from [matrix] representations of symmetries” appear to be even more abstract than the numbers that we are ordinarily familiar with.

Yet symmetries, as I have stated above, are the most concrete phenomena that exist. In previous blogposts, I have quoted physicists who have claimed that the elementary particles of the universe are actually “irreducible” representations of symmetries. In general, symmetry is arguably the most fundamental concept in physics, as basic principles like conversation laws are associated with symmetries. [1] If symmetry does indeed underpin the ontology of the universe, then it naturally follows that even macroscopic phenomena like coffee cups are nothing more than representations of symmetries, extraordinarily high-dimensional vectors or matrices that encode rotations, reflections, or other transformations of a geometric structure that is too complex for our small human minds to fathom.

This statement may seem like wild speculation at worst and highly abstract philosophizing at best. Symmetry is very well-understood in relation to particle physics and quantum mechanics but has not been applied as much to the understanding of macroscopic (especially biological) phenomena, i.e. the ordinary, everyday objects that populate our perceptible environments. Besides, what does it mean for a coffee cup to be a rotation of a geometric structure?

However, one area of research on symmetry has shown promise for revealing its applicability to macroscopic phenomena. In particular, symmetry can ground our understanding of the physics of self-organizing systems. Self-organization refers to the emergence of globally coherent patterns out of simple, local principles, and it governs everything from the formation of planets to the dynamics of traffic. Physicists have shown that symmetries underlie the physics of self-organizing systems near their critical points. [2] When a system exceeds its critical point, it undergoes a phase transition. One familiar example of a critical point is the boiling temperature for water. When water is heated up to 100o C, the global structure of its molecules suddenly and dramatically changes, such that water is transformed from liquid to gas. Another well-studied example of criticality is the magnetization of iron. As I wrote in the previous blogpost, “when the temperature of iron is above its critical point, the magnetic spins of its atoms point in random directions, such that the net magnetization is zero. At the critical point, altering the spin of a single atom can trigger a change in the spins of all the other atoms, due to scale invariance. Hence, below the critical point, the majority of the spins are aligned in the same direction.” 

The magnetization of iron serves as an order parameter ψ, or a measure of how ordered the iron atoms are, and it is related to the temperature T of the iron by the following equation: 

ψ ~ |TTc|β

where Tc is the critical temperature and β is something known as a universal exponent (I’ll have more to say on the latter shortly). Crucially, the physics of a self-organizing system near criticality is determined by the symmetry group associated with the order parameter. In particular, as physicist Harold Stokes explains, the order parameter is an irreducible representation of the “space group” of the disordered phase. The space group describes all the transformations under which a 3D physical configuration remains invariant. These transformations include the rotations and reflections that we have already seen, as well as glides, screws, and rotoinversions. For example, in the case of iron, the disordered phase occurs below the critical temperature, when the spins of the iron molecules are all pointing in different directions. The space group consists of all the transformations that could be applied to the iron molecules without changing their overall configuration. 

If the configuration of iron molecules were 2-dimensional, then the symmetry group of the order parameter would be isomorphic, or mathematically equivalent, to D6, the group I wrote about earlier. Hence, changes in the order parameter can be construed geometrically as rotations or reflections of an equilateral triangle. 

So, in conclusion, the naturalization of mathematics entails that numbers emerge from representations of the symmetry group corresponding to the order parameters of the self-organizing systems that occur throughout nature. What a mouthful! The underlying metaphysical premise is that symmetries are the ultimate reality of the universe. So long as we accept this premise, then it follows that a certain set of numbers, namely the representations of those symmetries, must also be real. Numbers that lie outside that set, according to this view, would not be real. Hence, the entire number line is a mental construct, but particular points on the number line are real.

An even more radical implication of the naturalization of mathematics is that it collapses the ordinary distinctions we make between physical phenomena. As I said above, the evolution of the order parameter is determined mathematically by a number known as the universal exponent. Fascinatingly, two systems can belong to the same universality class – that is, they can be characterized by the same universal exponent – even if, physically, they appear to have nothing in common with one another. The universality class for ferromagnetism, the process by which iron becomes magnetized, is the same for dynamic recrystallization, image processing, cell division, and even some neural networks. The shared universality class implies that all of these phenomena behave the same way as they approach criticality; near the critical threshold, the corresponding order parameter increases at the same rate. The naturalization of mathematics would claim that all systems that share the same universality class are fundamentally the same thing, since their order parameters are representations of the same symmetry. In other words, there is, metaphysically, no difference between recrystallization, pixelation of images, and so on; they are all manifestations of the same symmetry.

[1] The Qualia Research Institute (QRI), where I interned last summer, also believes that symmetry underpins the science of consciousness. Their “Symmetry Theory of Valence” argues that symmetry in the mathematical object of consciousness encodes the quality of conscious experience, i.e. how good or bad someone feels. Additionally, Mike Johnson, CEO of QRI, has suggested that Noether’s Theorem, which claims that each conversation law is associated with a symmetry, applies just as much to physics as it does to the science of mental phenomena.

[2] Criticality is also an important element of QRI’s research agenda. In particular, QRI has built on research demonstrating that taking psychedelics and engaging in other consciousness-raising activities pushes the brain closer to criticality.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s