Why the Universe is Not Math
A response to Alexander Prähauser
Alexander Prähauser wrote a response to my essay “What is Materialism?” which begins by suggesting that the entire idea of metaphysical materialism is incoherent from the beginning. He makes this claim with reference to the mathematical nature of the universe, that if physics is founded upon mathematical structures, this basically means that ideas and “reason” exist out there in the world, not just within our minds. As he says: “mathematical ideas are what even allows physical reality to exist”. There is some allure to this idea, after all, we use mathematical equations and logical reasoning to describe the most fundamental nature of our world. However, the universe is not math. This is something that I concluded in my critique of rationalist notions of intelligence, which really, truly, take this idea to its farthest logical conclusions. To explain precisely why this is not the case, we’ll need to go through the following points:
Computation & mathematics requires differentiation between objects
Differentiation between objects can only exist representationally
The totality of nature (the universe) cannot be represented in its entirety
Therefore the universe cannot be math
Computation & mathematics requires differentiation between objects
This, I feel, should be obvious, but in case it is not, let's run through the fundamentals of computation and mathematics. A mathematical equation is made up of more than one symbol, say 2+3=4, that’s 5 symbols right there. When you select a point on a number line, you’re differentiating that number from every other point on that number line. That also goes to show that whether we’re talking about discrete or continuous objects, differentiation is still required to produce a meaningful symbol. When we look at set theory for what exactly numbers are, they start with the empty set (0), 1 is the set that contains the empty set, and then 2 is a set that contains a set that contains the empty set and so on. This notion of the empty set is very illustrative. With physical objects we see and can count it is obvious what they are and aren’t, apple number 1 is clearly not apple number 2, ect. But the empty set isn’t supposed to stand in for anything right? Well, a set which contains nothing does draw attention to something: the fact that it's a set. And the empty set can necessarily be differentiated from every other set, sets which contain something. The empty set points to the nature of mathematics, which, even at its most abstract, does have its own object (indeed a material object!) which it crafts meaning from. Abstract mathematics starts with abstraction itself, after all, what we are left with in the empty set is the abstraction of “set”. What I will show, I hope somewhat rigorously, is that abstraction, and therefore computation and mathematics necessarily requires differentiation.
We know for a fact from cybernetics that variety, in terms of unique or distinguishable items within a set, are required to meaningfully communicate information. William Ross Ashby gives the example of a prison warden considering how to prevent the wife of a convict from transmitting a message to her husband by gifting him a cup of coffee. Assuming that the wife and husband had agreed upon a code beforehand, it could be almost anything: if the coffee is sweet or unsweetened, whether it's hot or cold, has cream or not, ect. The warden considers that he can eliminate any possible message by specifying the sweetness, temperature, cream, ect himself and telling the husband as much. All measures the warden can take involve reducing the possible states the cup of coffee could be in when it's presented to the prisoner, such that, all that remains is one possibility.
All computation and mathematics require sets with variety greater than 1. In the case of computation this is obvious. If you have a machine with only one possible input and one possible output, and the input/output are the same, you don’t have a machine at all, as a machine by definition is something which transforms its input. Similarly, while there are many definitions of mathematics, I struggle to think of one which is commensurate with there only being one mathematical symbol or number. Such a discipline would not be meaningful. If we used the “1” to symbolically represent physical processes, it could simultaneously stand for every conceivable process, as well as processes which simply don’t exist. “1” alone is a stand in for totality, with all its contradictions, but without a rule which transforms one thing to another, or which differentiates one thing from another, you could never represent, or recreate a specific process, reality or universe.
Differentiation between objects can only exist representationally
People always ask “why is there something rather than nothing”, really they should be asking “why are there things instead of one thing?”. How is it, after all, that we consider a river to be something different from the river-bank, and the river-bank different from the hiking trail? Why is the electron not the same as the neutron or proton? Why is it that we split up the continent of Europe into places like France and Germany? Why even have a continent of Europe at all? How are these things all not the one big thing?
It is a truism that “the map is not the territory”, that physical reality is not the same as our representation of it, that, for example, the idea of a circle is not the same as the circle as a real object. I’d like to point out that things like “atoms” and “France” only exist in the map, rather than the territory. It may be objected that the heterogeneous features of real objects lend themselves to natural divisions within themselves, for example, differences in the number of protons in atoms creating different elements. Certainly, an intelligent alien civilization would be capable of coming up with a concept equivalent to chemical elements, but that is only because they would have the ability of abstraction at their service, just as humans.
As I said in my essay on rationality and intelligence:
“This is the crux of the issue: the process of signification, of creating signs and codes, necessarily means cutting up reality into chunks, or sets or categories. Without this division, we couldn't have meaningful units of information. But these divisions do not exist in reality except in the way the representation is materially encoded onto a smaller part of reality. Even when we speak of a quark, why should we speak of it as a specific entity when we could speak of the quantum wave function, and why speak of one particular wave function when all that really exists is the wave function for the whole universe. If we said that the universe stood for itself as a symbol it wouldn't be a meaningful one, there is nothing you could compare it to which would be “not-universe”. In order to tell the truth, you have to be able to lie, and all lies are things which “don't exist out there”.”
Signification and representation are, of course, physical processes, at least in all the examples we know of. The map, as a physical phenomena, necessarily exists as a set of contingent codes within the territory. For this purpose, the heterogeneous nature of reality matters a great deal. Without switches in a computer motherboard being capable of taking on different physical states, the computer wouldn’t work. Within a local area of the territory, it is possible to use bits and pieces of it to stand in for other things, to create signs. This locality is crucial, and is what makes the differences differences, for as Ashby says, variety refers to different objects within a set.
Let’s say we have a big block of marble, one that has different levels of calcium and magnesium carbonate in different regions of the block. Could we distinguish the two chemical elements from each other if we only observed the whole block and took in its properties all together? No, we certainly couldn’t. We would have to observe some difference between the regions of the marble that would indicate different compositions, something that would allow us to create two different categories. The emergent properties of the block of marble hid the underlying variety within. We can also go the opposite direction, for example, we are unable to get a variety of different possible states for temperature or pressure while looking at a subatomic particle, at least, in the sense we’re used to.
Heterogeneity, difference, only exists within a certain frame of reference. It requires the confines of a code, a set, an arbitrary section of reality in order to exist. If we zoom out far enough, all the variety of objects on Earth disappears. If we zoom out to the level of the whole universe, would it be possible to see variety, to see difference of some sort? No. If we had access to the whole 4d, or however many d, shape of the universe, including everything going forward and backwards in time, we would be unable to make any symbolic representation of it without first focusing onto a section of it. Without that focusing, there is only “1”.
The totality of nature (the universe) cannot be represented in its entirety
When Prähauser says that the universe is math, he’s specifically referencing a theory like string theory, which supposedly describes the structures which underlie reality. Setting aside that string theory is still just speculation, it should be first acknowledged that even if it were not speculation, if it really was a totally correct theory for the universe, it is still only a representation for reality. Writing the equation on a chalkboard, after all, wouldn’t suddenly cause a new universe to burst into being.
Some people speculate, however, that there is a way to turn math into a “real” universe via computational simulation, often going further to suggest that we’re living in such a universe. The Wolfram Physics project, while agnostic about the question of whether we specifically live in a simulation, is premised on the idea that simple rules expressed on hypergraphs can be used to model extremely complex phenomena like the ones we encounter in reality. Present results include phenomena which at least somewhat can approximate general relativity. Let’s imagine, for a moment, that the universe is initiated by the infinite self-execution of a rule like the ones Wolfram suggest. Could we recreate that rule symbolically and execute in such a way to reconstruct the whole universe? Almost certainly not, for the simple reason that the speed the rule is being executed in the real world is the speed of causality, the speed of light, whereas the simulated version will be executed at some fraction of that speed. And here it is impossible to take shortcuts in a way that saves us time, due to what Stephen Wolfram calls computational irreducibility, which simply means that there are some programs for which the fastest way to see what they do is to run them. Which is to say, there are pockets of reality which cannot be described neatly by equations, at least, not any better than the most fundamental laws of reality working themselves out.
One might object that, sure, the universe may have pockets of computational irreducibility, but that doesn’t mean it’s not represented by math. After all, it seems possible we could one day have the rule which governs all reality. Is that not enough? Well, the universe is not just that rule, it’s that rule in motion, applying itself, or perhaps already applied infinitely, depending on your perspective, that’s what separates the rule written down from the universe itself. When you get to the most fundamental level of reality, you arrive at the level where there is nothing which is not the expression of that rule. But if there is nothing aside from that rule, can we say that it is abstract, like math is? Fundamentally we cannot, and the reason why is well known to philosophy.
Let’s say that nature, the universe, the totality of existence, whatever you want to call it, is defined as the set of all objects. Let’s say an object is a set that doesn’t contain itself, such that, just like the way sets can be used to build up real numbers, we can label each and every thing in the universe, for example a matrix of values for energy in space and time, as a collection of unique sets. But if nature is the set of all objects, the universal set, then does it contain itself? This leads us into Russel’s Paradox. If the universal set does not contain itself, then it is an object. But if it is an object, then it can go inside itself. This logical contradiction makes the universal set incompatible with most versions of set theory. The versions that do allow for a universal set make it so that either some sub-classes of sets are not sets, or they make it so the universal object is not a set at all.
This contradiction, I believe, results from the very nature of abstraction. Abstraction, at its limit, requires something outside of itself to exist. It needs a 1 to split into 2. The only way to truly accept everything that exists is to make this other thing, outside of the system of abstraction and codes.
Therefore the universe cannot be math
Prähauser began by suggesting that the mathematical nature of the universe was in direct contradiction to metaphysical materialism. He did so in a way which precisely followed my framework of materialism as ideas about ideas, and idealism as ideas about reality. Prähauser believes that math is the structure of the world, as a set of ideas, and reason, which exists “out there”. What I have shown is precisely that abstraction allows us to create ideas which are more or less real in terms of their description of reality, but reality itself is not an abstraction. In order for abstraction to exist, there must be something beyond it which cannot be represented in its entirety. This is the “1”, the totality, the universe, which only simply is.
It is not pseudoscience to not put our faith in math merely because it is logically consistent. The trouble with Prähauser’s reasoning that we should, for example, embrace M-theory/string theory because of its mathematical coherence, is simply that there are a lot of things which are logically coherent. Essentially, it’s an operation which is merely browsing the library of Babel and looking for books with an actual plot. You can, indeed, find an extremely large number, infinitely many if the length of the books isn’t bounded.
What cybernetics, semiotics, logic have all shown is precisely that Hegel and Plato were wrong, we do not begin from the concept, from the idea, and that realness emanates from the true concept and idea. But rather that all abstraction begins from something that is not abstraction, that all signs, ideas and concepts begin from the undivided being and existence.
To be honest, I think Russell's paradox is, well, over-hyped. It's not that hard to avoid, just don't allow the forming of arbitrary subsets of classes.
This isn't enough for a whole new post, but the main error you're committing is in the step of your reasoning where you argue that the entire universe could not be represented because representation requires differentiation. There are several components to this error:
- Representation does not require differentiation, only representiation of parts of a non-trivial system does. If we assume the surrounding context of the one-element set, every inhabitant of it can be represented with only one symbol and without being differentiated from one another. It's just not very interesting to do so.
- You're conflating the physical universe with a more general mathematical universe (which you have to assume exists to disprove my point from its premises, or simply to have a coherent physical theory). The physical universe is represented by one symbol all the time by physicists, usually in the form of a sentence like "Let X be spacetime…".
- Russell's paradox is only valid in regards to sets. In Von Neumann–Bernays–Gödel set theory, for instance, the "set of all sets" is not a set but a class, for which not all subsets can be taken. In such a context, it is not problematic whatsoever to assign to this universe a symbol, say U, and treat it just like a set, with only a slight number of additional restrictions of what can be done with it (namely not taking arbitrary subsets). The fact that ZFC is still used as the standard set theory, and not NBG or something better, is mostly due to institutional inertia. However, even in set theory using ZFC, the universe of sets is regularly signified by set theorists, though, depending on their metaphysics, if you were to press them they would say that they are using it as a shorthand. Lastly on that point, if we assume the existence of a class of all sets U, this class can still be differentiated from any other set or class X simply by the fact that X is contained within U but U is not contained within X. So simply because something contains everything that does not mean that it cannot be distinguished. It is also not clear how valid Russell's paradox remains if a paraconsistent logic is used, but I wouldn't hang my argument on that, it is perfectly plausible that no paraconsistent logic can be devised in which Russell's paradox could occur and which would still be rich enough to allow for the existence of the physical universe.
- It is not necessary to assume one final mathematical universe to treat the physical universe as a mathematical object, it can just be one node in an infinitely increasing network of connected nodes. Among category theorists, usually a framework of "Grothendieck Universes" U_i is assumed, where U_i contains all sets one size smaller and is in turn contained in U_{i + 1}. So U_0 contains all small sets, U_1 contains U_0 along with all small sets and all sets one size larger, which would appear as classes from the perspective of U_0, and so on. In fact, I do assume the existence of an "∞-category of ∞-categories", but only as shorthand for a pattern in such a hierarchy: there is an ∞-category of small ∞-categories, an ∞-category of next-larger ∞-categories and so on. In praxis however, one finds that very little changes as this size increases, the only thing to look out for is to not allow for constructions that would allow for something like Russell's paradox to occur.
I will also say that I don't find any of the variants of the Quine-Putnam indispensability argument addressed satisfactorily. Now, you are quite right in saying that I'm drawing from "string theory" (more precisely M-theory), which is not experimentally verified, which matters very little to me but does to you. What is experimentally verified (though not true) is the standard model however, as well as Einstein's relativity theory (whether special or general), which are gauge theories (to be completely precise, the standard model is actually a model of a gauge theory, namely Yang-Mills theory). What these gauge theories have in common is that they are invariant under the action of a symmetry group. In fact, the entire notion of a gauge theory, which is of fundamental importance in modern physics, whether M-theory or not, requires the notion of a symmetry group, one for every gauge theory (there might be some slight simplification here), so, again, symmetry groups have to be assumed to exist for gauge theory to make any sense.
On a more philosophical level, you are already assigning yourself to at least dualism when you write that "it seems possible we could one day have the rule which governs all reality. Is that not enough? Well, the universe is not just that rule, it’s that rule in motion, applying itself, or perhaps already applied infinitely, depending on your perspective, that’s what separates the rule written down from the universe itself." You now have two "substances": one, whatever the physical universe consists of, and one, the rule which governs how the first substance interacts, let's call it R. Now, there has to be some distinction between the "matter" of the universe and the rule governing its interactions, but to not be dualist those two would have to be brought together in some way, which can be done perfectly well if we assume the physical universe is a mathematical structure, because the axioms of one mathematical structure are themselves mathematical structures on a slightly higher level, and, while one one level distinct from the structure they regulate, they are fundamentally of the same substance. If meanwhile we do not make that assumption, we would need to introduce further logico-mathematical rules governing the consequences of the application of R in terms of logic/mathematics, otherwise the rule would be void and everything would be allowed since nothing could be deduced from it. Thus, you would have in fact a whole hierarchy of distinct substances: "matter"(/energy/whatever), the rule that governs the interactions of matter and rules governing the consequences of that rule as well as an open number of further rules giving coherence to those rules. It would be a nightmare!