# How can the physical world be an abstract mathematical structure a la Tegmark?

This is Tegmark's short formulation of the "mathematical universe" (paraphrased by detractors as "reality made of math"), and he goes out of his way to stress that he means the "is" literally:"Whereas the customary terminology in physics textbooks is that the external reality is described by mathematics, the MUH [mathematical universe hypothesis] states that it is mathematics (more specifically, a mathematical structure)". Deutsche gives a related physical Church-Turing thesis, roughly "every physical process is realizable on a Turing machine", although he is a bit more cautious.

This rings all sorts of Kantian alarm bells for me. The reason for "described by" in textbooks is that "mathematical structure" is a representation, while "physical world" is not, so one can not literally "be" the other for conceptual reasons. Representation by itself is not a representation of anything, it can only represent something else through a correspondence scheme, just like a book without a 'reader' (possibly inanimate) is only an object combining ink and paper. In the case of correspondence to something physical the scheme itself would normally consist of some physical procedures that relate "forces" to forces, "masses" to masses, "motion" to motion, etc. This is how "such and such is described by mathematics" is usually interpreted.

Tegmark's expansive formulation though seems to leave no room for such an interpretation. It would not help to say that the physical procedures involved are themselves mathematical structures, or realizable on a Turing machine, because what we are trying to understand is exactly what it means for the physical to be so structured, or so realizable. We'd be back to the same question, only now asked for the physical procedures that do the corresponding. It would not help to say that in place of "mathematical structure" it means some physical realization of it either, for the same reason, both set off infinite regress.

So what does it mean? If we put "described" back in, then "physical world is described by an abstract mathematical structure" makes sense, but I think that Tegmark wants more, like "fully described". I do not see how to make sense of anything like that though, how does one "animate" idealities without recourse to physical, or to supernatural? Philosophers of old invoked God's powers (sub specie aeternitatis?), but that would hardly work for Tegmark, and it does not explain.

• At first glance, it looks like Tegmark has simply committed a gross category mistake here. I'll read the paper more carefully and report back.
– user5172
Commented Feb 19, 2016 at 19:40
• To paraphrase the old joke: Biologists think they are chemists, chemists think they are physicists, physicists think they are God, and God thinks he is a Turing Machine. Commented Feb 19, 2016 at 19:44
• If horses had gods, Xenophanes observed, they would look like horses. The same goes for mathematicians, it seems. But I don't see why Tegmark's position excludes the physical correspondence. It only reverses the "description" relation to "prescription." A mathematical ontology grants existence to all that is mathematically definable or "not impossible." The physical just becomes a reduction of all mathematical possibilities to something like Kant's space-time "form of sensible intuition," which is already a "representation." Isn't this just Kant with "categorical" mathematics? Commented Feb 20, 2016 at 21:20
• @nelson Alexander: doesn't mind-stuff, for Kant, come beforehand; as well as the noumenal? Commented Feb 21, 2016 at 1:21

## 7 Answers

Douglas Hofstadter would call this a strange loop. If one believes mathematics can "fully describe" reality, one can make a pitch to claim that reality is a subset of mathematics. Empirically, these two would look identical. Tegmark is arguing that you can choose to put reality inside mathematics instead of putting mathematics inside reality. Like all ontologies, it is very hard to challenge. If you say that reality is not mathematics, and he says reality is mathematics, how can we really decide which one is "right?"

His theory does include some interesting threads to tug on. In CUH (his Computable Universe Hypothesis), he argues that the entire world is computable. Non-deciable things, such as the issue with band gaps being non-decidable are resolved by stating that only the description of things must be computable, not the actual time evolution of it. This implies that he considers reality and the description of reality to be one and the same. He also readily admits that this means that our universe can contain questions which cannot be answered within the universe. Whether this is acceptable in one's ontology or not is one's own business. However, it does give insight into how he would view things. If something in reality existed which was not fully made from a mathematical structure, he would be able to treat it as though it is something which can be described using mathematics but which is not decidable. You would be unable to come up with a logical process to disprove his claim, because his claim is that one cannot prove nor disprove his claim within this universe.

Also, paradoxically, you wouldn't be able to point our that non-mathematical real thing either. If you could successfully point it out to him in terms he would recognize as identifying an object, you would have to do so in a formal language (he would not accept anything else). By doing so, you provide a mathematical description for the thing (you used a formal language to do it), and he would be able to stand on his claim that it is merely an undecidable time-evolution, literally until the end of time.

In the end, I'd call his theory testable but not falsifiable. He makes the argument that he provides testable hypotheses that we will find more mathematical structures, but there's nothing in the theory which permits Popperian falsification. This puts it in a category alongside many Asian concepts such as Traditional Chinese Medicine, which permit testing but not falisification. Thus, his theory must find its use the same way TCM does. It gets picked up by people who feel their lives are improved by picking it up, but it is rejected by science because it does not conform to the strict rules science uses today.

• I think Tegmark's CUH is stronger than that: He argues that all computable structures exist. An odd, if plausible, combination of Everett's multiverse and Computational Platonism. Commented Feb 20, 2016 at 16:12
• My issue is not disbelief, if I said "truth is an organ" I'd be asked what I mean by "is" before it even gets to that, and answering that I embed truth into body won't cut it. This is different from disbelieving in God say, where the meaning is clear. I also suspect that spelling out "is" might make MUH incoherent for the same reasons that "world" of rational metaphysics was incoherent, as Kant showed in the first antinomy. Commented Feb 21, 2016 at 1:33
• Mathematical universes were proposed before, e.g. by Plato, Leibniz or recently Penrose, but they had extras, fantastic ones, to make them work. Replacing extras with unintelligible "is" is not a solution. If all MUH amounts to is that we will discover more structure it is not testable, almost every classical epistemology "predicts" that, including Kant's. These are legitimate concerns, arguments and difference of opinion can only happen after opinions are made intelligible, coherent (or paraconsistent at least, if one wants to go there), and non-vacuous. Commented Feb 21, 2016 at 1:34
• "This puts it in a category alongside many Asian concepts such as Traditional Chinese Medicine, which permit testing but not falisification." AFAICT TCM HAS been tested AND proven false, so this claim is also false. Whether its practitioners want to accept the falsification or not is another matter. Commented Apr 9, 2017 at 10:31
• On a tangent, I'm also curious about something else: If the universe contains non-computable elements, are they harnessable to do any sort of "non-algorithmic computation", e.g. to build a non-computer "computer" that can solve the halting problem? (E.g. we can't build a computer to decide the phenomenon, but we could use the phenomenon itself as an "engine" of computation beyond finite Turing computers' capabilities?) Commented Apr 9, 2017 at 10:34

If mathematics is the repository of what all humans can intuitively conceive, then whatever lies behind the material world may or may not be a mathematical structure, but the entirety of what we can ever understand about the external world would be.

Given that theory of what mathematics is, the question then becomes, in stages of progressive aggressiveness:

1. why we assume there is a remainder,
2. whether we can even know of the existence of a remainder, and
3. whether, if we are sure we can't know, it is more logical to just assume there is none.

As I see it Tegmark is just proposing the exact opposite of the Kantian notion of noumena in an indirect form. Kind of by definition, even for Kant, the nature of noumena is an unresolvable question. If we relied logically upon noumena for any real purpose other than inspiration, our inability to access them would contradict the idea that we are fully capable of becoming intelligences and doing things like acting morally.

The only real way out that leaves noumena intact at all seems to be the Hegelian response is that we perpetually 'move toward' them. But math itself does not work that way: what is entailed is resolved, even if you are not a Platonist. So a mathematical model of dialectic would be a single mathematical structure, whether or not it allows for everything to be resolved with a single pass through any given evolutionary process, or requires infinitely many reversals, or whether it can even be navigated by any possible mind. It is still the closure of some set of entailments that we cannot, by our given nature, question. (Presumably more than countably many basic notions are involved, since otherwise the nature of language keeps us from getting to closure.) The terminal point of the whole of dialectic has to be part of the model. Whether or not anyone can get there, Hegel's endpoint where "We, as God, know God," is in the model.

Math has gotten beyond topology, and limit points don't make us worry anymore. So we can't avoid the question by ledger-de-main. Noumena either are or are not part of our model. But if we have them, we still can't know anything about them -- even whether or not they are really there.

In that case, what is the difference? Whatever transcends those forms is lost to us, and we will not be able to understand, much less prove, anything about it. Either side of a proposition independent of your system is open for adoption into truth with no loss.

So Tegmark's is, at the very least, an un-disprovable assertion, and one that sets the very frame of Occam's razor. To doubt it not only requires we create unnecessary entities, but that we acknowledge that those entities are utterly unhelpful to us, since they are necessarily unknowable and beyond consideration.

What is the risk in presuming it is true, since we can never know anything about why it might be false?

The extension by Deutsche is unwarranted and almost unrelated. No Turing Machine can compute randomness. There are only so many states, and we will be drawn back into them. So ideas like those behind classical Quantum Theory, etc. are not consistent with Deutsch. In effect, Deutsch is making the same error Nietzsche makes in deducing the Eternal Recurrence, only about non-computability rather than chaotic dynamics. Being arbitrarily close is not being right, and in endless time, eventually the gap with show some effect.

• But "all we can understand" and "all there is" have very different semantic and logical properties, not the least because of indeterminacy about expressive means. There is no "shared reality" as a finished thing. Ignoring the difference, in Kantian or some other form, leads to exercises in lack of imagination, about both reality and understanding, and makes it harder to recognize limitations of existing means, and develop new ones. Most of what we know in biology, psychology, history is non-mathematical for a reason. But subtlety aside, Tegmark explicitly rules out your interpretation. Commented Feb 25, 2016 at 1:31
• Evolution is not teleological, there is multiple branching and no Omega point, so there is no back projected "proper perspective" on anything. Multiple ones are "proper" for different purposes even in physics, let alone in biology. Today's concepts, like "individual things", do not work even for some existing physical theories, and are likely to grow more useless in the future ones. Attempts to construct mathematical semantics of even natural language did not go far due to inherent pluralism and ambiguities, mathematics does not mix well with ambiguity. Commented Feb 25, 2016 at 23:30
• I am not asking what can possibly be a "mathematical" structure, or prescribing what "mathematics" (of the future) can or can not handle, neither I nor Tegmark know what it will "allow". But he is advancing his hypothesis today, and it is fair to ask what it is beyond empty words. Possible world talk and Platonist completions often blow up known bubble to the size of "all there is", on the contrary I am asking for more options, insight how mathematics can be made more real, not pronouncements. Incompleteness enters because it tells us how not to do it, given that we have no Platonic access. Commented Feb 28, 2016 at 0:21
• This strikes me as evasion via nitpicking. You need some pretty stark dualism to make any difference between the two questions "What are the bounds on what can be mathematical?" and "Can all of (physical) reality be mathematical?". As I see it, Tegmark is really just proposing the opposite of Kant -- that phenomena (which are mathematical, if you accept the definition of math from the first line of the answer) do not require noumena to back them up. If you just reject the opening premise, you don't have to be abrasively condescending about the rest of the post.
– user9166
Commented Feb 29, 2016 at 16:57
• @PeterJ Your individual dissonance is not a concern of philosophy. I would need proof that these "many people' actually exist, and are not crackpots.
– user9166
Commented May 19, 2018 at 18:19

Don't you agree with Alexander's nice joke?

Mathematical universe means that our "real" world is a "virtual" reality, designed and executed by mathematical algorithms.

It is difficult to find arguments against this view. A possible counter argument: We should register more anomalies due to rounding errors of the computation with finite precision.

• Universe as a simulation may work for Deutsche but not for Tegmark, the problem simply shifts to what it means for the simulator to be a structure, there's no need to refute it. I like the joke but I have a different interpretation of it: what they all think of is a "reduction", which upon reflection is either unintelligible or incoherent :) Similar to naive realism and "rational metaphysics" that Kant dismantled in the antinomies. Commented Feb 21, 2016 at 1:33

I have always understood consciousness to do a lot of the heavy lifting in his saying the universe is mathematical, and reality being very far removed from the pictures our senses provide us.

His chapter on the conscious illusion of the flow of time provides a template for how large of a role conscious experiences play. And because there is such a large gap to understanding conscious experience, a lot can be stuffed in there for later scientists. He routinely writes most of the work left will be for psychologists and biologists.

It is only in this near unfathomable gap between external reality and subjective awareness that Tegmark's phrasing makes sense. What bestows these mental properties of pain or time flowing? Presumably they supervene on something. Is it onto the computational structure of the brain? Unto unknown physical processes or objects? Or could it even be onto mathematical structures? Since the explanatory gap for something giving rise to mental states is so great ("How pulses of water in pipes might give rise to toothaches is indeed entirely incomprehensible, but no less so than how electro-chemical impulses along neurons can." -- Tim Mauldin https://doi.org/10.2307/2026650 ), we should not be so quick to dispel of a mathematical structural origin for them.

From within these mental states we have to probe the outside world. And since we can't absolutely trust the ontologies they have presented us until now are the "true" ontologies, Tegmark can say a tree really is mathematical. That is my thesis here, and I believe what Tegmark had in mind. We have to really take seriously we may be experiencing a heavily transduced perspective of reality. What we think of a tree existing classically is actually closer to something like Mario thinking he really exists in Mushroom Kingdom, and thinking it really exists on some fundamental level. That is the level of delusion we must be living under for Tegmark's remarks to make sense.

Somehow a mathematical structure can give rise to something believing it is experiencing time flow, colors, and trees existing. If you think circuits can lead to an AI experiencing, I think this argument is not hard to swallow. If the flow of time is not fundamental, likewise it shares similar logic. You just have to trust that you should distrust your own senses more than you probably do.

• "It is only in this near unfathomable gap between external reality and subjective awareness that Tegmark's phrasing makes sense." To me that is exactly where Tegmark's theory makes no sense at all. We have enough trouble bridging the gap between physics and consciousness. (And that is a huge understatement!!!) There is no imaginable way to bridge the gap between mathematics and consciousness. Commented Sep 3, 2021 at 17:22
• @DanielAsimov But see the Maudlin quote. He thinks there is no imaginable way to bridge the gap between physicalism and consciousness either. For scientists and philosophers who likewise see the gap as "huge", why not pick the most global, universal theory (math) and build up from that. Rather than from computation or physicalism, which are more assumptive and require an equally long bridge to join the gap Commented Sep 3, 2021 at 17:39

Our physical world, is not an abstract mathematical structure. It existed for a long time before mathematics was ever invented! It just so happens that mathematics can be used by us (humans) to understand (make sense of) some portions of it. Mathematics is only a "tool" created by (inside) our imagination. How can something inside our brain (thoughts) "create" physical matter? There's no way!

• 1. You're bringing too much preconceived notion into this. Philosophy is about suspending disbelief and exploring other people's ideas, not saying "THERE'S NO WAY!" Without providing reasoning as to why this is. Given your previous points, it seems you have some strong preconceived notions about the nature of mathematics as well. 2. Tegmark argues that math is not "invented" or "a tool" but is actually the fundamental building block or conduit of reality. Commented Feb 26, 2016 at 17:33
• It may appear that mathematics was "invented" but many people (including me) believe that in fact mathematics is discovered, not invented. Any true theorem is true independent of time and space and anything else. Commented Sep 3, 2021 at 17:27

Tegmark makes a distinction between mathematical notation and mathematical structures. Model theory makes exactly the same distinction between the concepts of a theory and a model of that theory. A mathematical structure is not a representation but more like a Platonic Form.

In Tegmark's MUH, it claims there's really a single ontological world originated from math entities, like a computer stack, final screen outputs are ultimately originated by its CPU instructions. MUH is a form of classic Platonism's "ideal forms world" while focusing on abstract math structures instead of old more concrete Platonic "form of goodness" ethics. MUH must conceive of Math structures like space/time as real substance so that it's possible the physical can be born from the ideal. He's a good theoretical physicist, so like many other modern physicists' metaphysical philosophy, QM String-theoretic world is increasingly becoming an application of Algebraic Geometry which is hoped to contain the final Theory of Everything.

However, a much more important and interesting question from my perspective is how either MUH or classic Platonism can explain and deduce "consciousness". For me no matter how clever in MUH math can be employed to construct some "integrated information" metric borrowed from neuroscience (Tononi, et al), ultimately it's just a notion only understood in a conscious rational mind. So MUH essentially claims Math creates consciousness while traditional idealists claim vice versa, like the egg-chicken trap. Similar to Searle's Chinese Room Argument or Leibniz's Mill Argument, a computer or any entity processing math has no true understanding of math itself. So I personally favor traditional idealist view that mind is the ultimate real ontological existence. As Renaissance Cardinal and philosopher Nicholas of Cusa put it, the ontological world is nothing but the perfect Oneness conceives with the supervenience of all its images reflecting with actions...