Why the physical Universe is a mathematical Structure

And why that’s a good thing

Recently on Discord I had a discussion with some analytics about the nature of mathematics. Now, while I call myself an analytic Hegelian I have a mostly one-sided battle with the analytics (as well as one with the Hegelians), and while my disagreement starts with their rejection of Hegel it by no means ends there. At least since Popper analytics have mostly been physicalists, which they treat as some kind of refuge from the uncertainties of thought. We might not know whether one plus one is two, but at least we can measure physics! Thus, it goes, we can reduce a mathematical statement to a physical one, and thus reason about it through physics, though we might use shorthands to simplify our understanding. Nonsense, I claim! And I have an argument I’m willing to bet my chips on. Bear with me, because it is a somewhat subtle point. Here it goes:

One defense of Platonism in whatever variety has been our ability to observe mathematical laws. If one and one don’t exist, the argument would be, then how come we can reason about them? Clearly we have to have some special sense, as Gödel assumed, that allows us to traverse the mathematical landscape, or, more traditionally, we have to have been born with its knowledge. A physicalist would have not much trouble countering that argument however, by saying that these mathematical laws can be based on observations of natural tendencies whose understanding has been evolutionarily advantageous to the point that structures in our brains evolved whose task was to apply them in a variety of situations. Sticking further with the example we might say that the statement that one plus one equals two is actually evolved from the observation that objects follow the axioms of addition, or perhaps, even more basic, that whenever we have a number of items characterized by some property if we obtain another one we reach a number higher than the one before, and there are no skips as we climb that ladder: we have one stone, then we have two, then we have three, and so on, and if we have three stones and take one away we don’t land at one stone but at two. Of course, there is a chance, however slight, that some day we have one stone, then another, and then another stone simply appears due to a quantum fluctuation or similar, but that doesn’t threaten our theory: the statement just has to be true often enough for it to be evolutionarily advantageous to simply assume and reason from it.

So far, so good. However, for such an event to be true often enough, it has to be a natural tendency. Furthermore, from the fact that our brains haven’t changed all that much since we became human, and our mathematical observations still map onto reality, we can conclude that this tendency has been fairly stable for at least that time interval. However, there is no a priori reason to assume that they are constant. For instance, the expansion speed of the universe has been fairly stable since that point in time, say, 100.000 years, give or take a zero, but we know it is not really constant, this is just a fairly short time interval on a cosmic scale. In fact, our mathematical laws might stop working tomorrow for all we know. More precisely, just because they were constant enough to give us an evolutionary advantage in the past doesn’t mean they would have to stay that way going in the future: our adaptation speed might start lagging behind their change rate and our evolutionary advantage turn into a disadvantage in this new environment.

Before I continue let me explain what “a change to the mathematical laws” would mean, lest I be accused of obfuscating vocabulary or faulty reasoning. Over the course of human development it has often been the case that we constructed some mathematical model to describe a situation in reality only to then find out that said mathematical model does not apply. This is usually not a problem: we tweak the model or come up with a new one. Perhaps we need to use a new kind of mathematics, as has been the case when we transitioned from mechanics to field theory, or to quantum mechanics, or quantum field theory. However, in this process, we are at each step reliant on logical reasoning: my theory says x is the case, but x didn’t happen in experiment, therefore my theory has to be wrong and I have to tweak it, and though we can significantly tweak the logic we apply when reasoning on a base level (some of the most interesting developments of current mathematical logic are about this), we always have to have some level at which a logic works that corresponds to some reasoning capacity in the human mind. A failure of logic/mathematics (which we can treat as equivalent for the purposes of our discussion) would be one on any level. It wouldn’t just be that, whenever two items with some common property come together a third one magically appears, it would be that the physical context has changed to one in which this happens with the same naturality with which it is currently the case that one item and one item yield two items. This however, per definition, wouldn’t correspond to any formal system that could be humanly thought of. We might look at this changed state of affairs with some wonder, then try to find a different formal system that would correspond to it, but the endeavor would be impossible as, on a fundamental level, the map between our thinking and reality was severed. Similarly, a failure of logic would undermine the very logic that we use to conduct experiments, so we could not use them anymore to understand reality. We might still try, but no formal framework we use to make sense of the results would be able to yield predictions that conform to subsequent experiments, as no such formal framework could be applicable.

This would obviously be a very counterintuitive state of affairs and it was understandable that the people on Discord were repeatedly asking for an example for a concrete description for how it would look. However, since such a system could not be understood by a human being and I happen to be one it would be categorically impossible for me to describe it, even if it happened, much less without it being before my eyes. However, as has often been noted by scientists and writers of cosmic horror alike, just because something is counterintuitive does not mean that it is impossible, and if mathematics is to be treated as a physical phenomenon that we are evolutionarily adapted to comprehending there is no reason to assume that it would not change over time or that its change rate, so to speak, would not at some point exceed the evolutionary adaptiveness of the human species. Yet, it might simply be that this hasn’t been the case to this date, that mathematics is changing continuously but only to a very small degree or that it is only changing in small, discrete jumps and the changes from the beginning of human civilization to this point have been so small as to not be noticeable. This would be a possibility.

If we did not have vast swaths of space and eons of time that we can look back at at this point. Looking out into space we can observe from the very earliest stages of the universe depicted in cosmic background radiation its evolution through spacetime in faraway, (and therefore early) galaxies to those right in our neighborhood, and no matter where or when we look, we don’t encounter any states of affairs that would categorically defy a mathematical description. If mathematics were indeed to vary, one would think there should be some signs in the 13+ billion years we can now observe, a vastly larger time frame than the one that would be formative to our cognition.

There is one way out of this dilemma for the physicalist, which is to assert that the laws of mathematics are physical laws, on par with those of, say, thermodynamics, but no different in essential character. However, this apparent refuge is only a trapping in disguise, as, if the laws of mathematics are physical ones, they have to be treated as the same as other physical laws. Yet, while the analytics were all too happy to forfeit the quest for elegance, beauty through simplicity, in favor of what they might call truth, thankfully the actual physicists, nevermind the mathematicians, were not, and finding the commonality between the apparently different physical laws that have been observed through experiment has been the driving motor for physics at least since Faraday revealed the apparently different phenomena of electricity and magnetism as two sides of the same coin, electromagnetism, through the subsequent, already experimentally verified, unification of electromagnetism with the weak nuclear force, the, still somewhat conjectural, unification of the electroweak force with the strong nuclear force, all the way up to the longed-for “theory of everything”, which would unify all fundamental forces of physics. Yet, even such an apparent “theory of everything” would not unify mathematical and physical laws, those two would stand strangely apart, a physical theory giving a uniform description of physics on the one side and the laws of mathematics in our universe on the other. Yet, as physics has progressed through these stages, it had become necessary to develop mathematical tools of increasing sophistication and abstraction, much to the chagrin of conservative physicists and even mathematicians, as mechanical bodies became fields and atoms became strings, but even internally to what are essentially already mathematical theories, as it is becoming increasingly obvious that strings have to be treated on the same footing as higher branes. What a weird state of affairs…

Thus, to assert that mathematical laws are physical laws also means to assert the imperative to unify them with actual physical laws. But this cannot be done if physical laws are simply treated as the foundational properties of the only existing structure, for which there is apparently no further justification: the two would forever remain apart, the physical laws, which might, perhaps soon, be unified into one essentially mathematical framework, and the laws of mathematics, which are to be taken as shorthand for laws governing physical objects in the universe but are also necessary for the framework that is meant to explain physics, and which might be unified using a framework such as category theory which however the physicalist would have to take as a formal shorthand and thus not a real unification but only a mental framework we use to explain what are actually separate phenomena. Thus, it is not only conceptually clarifying but even necessary for the development of science to take one step further and treat not the laws of mathematics as physical laws but the universe as a mathematical object and its laws as mathematical structure. This does achieve a unification of mathematics and physics, and it explains quite a few things: the logic that holds in our physical universe holds in it because it is induced by the mathematical environment it inhabits, which is probably something like an infinitesimally cohesive ∞-topos, and our ability to think logically might well be evolutionarily derived but for it to even be possible to be so without running into the problems sketched above is only possible because logic itself actually exists, as the internal logic of a category. This, of course, is nothing but a thorough application of the Yoneda lemma, the statement, roughly, that internal structure is mirrored in external one and external structure manifests itself in the multiplicities of internal structure, applied here to the physical universe instead of some other mathematical object. Note that I’m saying “something like” an ∞-topos, not an ∞-topos per se. I think it is still too early to exactly determine the structure we inhabit, and perhaps we will only ever be able to establish minimal requirements, but it would seem that this is the least of the structure that is required. It also takes care of a variety of problems in physical theories, as stressed both by Lawvere and Schreiber, such as allowing for a uniform and much simpler treatment of field theories in terms of cohomology, which can, roughly, be described as structure on an object that is induced through a map into another. In general it seems reasonable given the astonishing effectiveness of cohomology in the description of field theories to assume that all field theories are cohomological in nature. All of this fits very well into Sati’s Hypothesis H, which he arrived at on purely physical grounds, that the “fundamental fundamental force” of physics, which underlies the four apparently fundamental forces gravity, strong nuclear force, weak nuclear force and electromagnetism, is induced through something like a map into the four-dimensional sphere. However, though I am an avid follower of Schreiber’s and Sati’s research program, my argument is of course independent of such details. In any case, it also explains the “unreasonable effectiveness of mathematics in the natural sciences”, to use Wigner’s phrase: this is simply us getting to know our mathematical neighborhood better and learning how it shapes our physical world. More particularly it would explain the confluence of physics and type theory that is happening within higher category theory: basically, the logic that we use to structure our programs is the same logic that structures our universe.

I want to close this post by pre-empting a few objections and then getting to their actual root. A conceivable one would be that this still defers the question, as there wouldn’t be any reason we couldn’t ask why logical laws hold in the mathematical environment we inhabit, applying the Yoneda lemma to it. This is indeed the case, but not problematic, as the structure in our topos or whatever it is is induced by the categorical structure it inhabits, and this is induced by the categorical structure of that environment and so on, until we end up in the ∞-category of ∞-categories, which would be an object of itself if not for Russell’s paradox, which forces us to distinguish large from small objects, so that the ∞-category of small ∞-categories really inhabits the ∞-category of 1-large ∞-categories and so on, all the way up. This hierarchy can’t have a fixed point, but for our purposes this is almost completely irrelevant, it makes perfect sense to treat the ∞-category of ∞-categories as an object of itself as long as one abstains from constructions that would require large objects.

Another possible objection would be that, since it is impossible for us to concretely conceptualize a state of changed mathematics it would be impossible to reason about it. This however is a fallacy: it is absolutely possible to abstractly reason about things for which no concrete example has been found or perhaps even can be found, as any logician worth their salt knows, as in fact this is done all the time in mathematical logic. It is not necessary to concretely provide a sentence that cannot be proven in a formal system to prove that Gödel’s first incompleteness theorem holds in that system, for instance.

Yet one more might pertain to the somewhat oversimplified example I’ve chosen to illustrate my argument: it might be argued that two is definitionally the number of objects that are present when one is added to another, thus this could actually not change even if we were to assume that mathematics were malleable. This is a fine point which can be true depending on how you define two, but, as I already hinted at when I introduced the thought experiment, there natural numbers are structured by several operations in a fine interaction with each other, an order-theoretic one, an additive one and a multiplicative one, and no matter which one is chosen to define exactly what “one” and “two” mean, it follows logically that they interact in precisely the way they do but not definitionally, thus if logic is allowed to vary what one has to allow the possibility for is that equalities that follow logically from the definitions in place no longer hold. So “two” might be definitionally the number one arrives at when adding one to one, but one could no longer assume that subtracting one from two would again yield one, since logic is used to arrive at that conclusion.

Other than that, I don’t see any reasonable objections. Others were hurled at me by the Discordians mentioned at the beginning of the article, but those should largely have been pre-empted by the clarified exposition in this post. Now I would like to address what I think is the actual root cause of these objections, which is not rational but emotional. As I can see it, there are two directions from which resistance to such mathematicism can come from, a romantic one and a materialist one.

The romantic one is easy enough to explain: we don’t like to be of the same substance as a stone. This is deeply engrained in the human psyche, as can be seen by the fact that small children tend towards vitalism as an explanation for the behavior of living things as opposed to reductivist explanations, and it is part of the fundamental unease that underpins our despiritualized age. From this perspective, mathematicism might be seen as the ultimate capitulation, a quantification of everything to a number, the death of emotion. To be clear, I can explain physics as mathematics to a degree that is conceptually satisfying enough for me, but I cannot do the same for consciousness. However, this battle is not about consciousness, it is about the nature of the physical universe, and connected to consciousness only insofar as it concerns the question of how any consciousness can make sense of the world at all. In other words, it cannot explain consciousness, but it can explain reason.

The other direction of objections, and the one I was actually faced with, is the one coming from metaphysical materialists, such as physicalists, who were voicing their objections with much more leeriness than could rationally be justified, and again I think the reason for this is actually emotional: it comes at some psychic cost to make the leap to materialism, to accept a world of cold, dead matter, an essentially inscrutable enigma and an utterly hostile place, but once one took the leap, one at least obtains the comfort of being the truthful one, not accepting any false consolation but looking reality squarely in its face, a superiority that is only expressed by the blatant but otherwise enjoyed without needing to be stated, perhaps even to oneself. Subsequently, anything going in the direction of decreased misery and increased meaning is looked at with suspicion or outright hostility, not just because it is seen as a concession to a false idyll, but, more significantly, since it threatens one’s sense of self-perception. In fact, in said Discord discussion one participant actually admitted to not liking physicalism but said it was “the only safe option”. Well, here I am to shatter their safety and superiority, and as I, unlike those rational deceivers, wear my convictions on my sleeve I will readily confess to preferring to a universe of cold, dead matter one of mathematics, which might be cold but is at least very beautiful. Yet, this doesn’t change the strength of my argument, which I am quite confident in, and as such I have no need to convince them, or anyone really, as history will prove me right.