## RS101 - Max Tegmark on the Mathematical Universe Hypothesis

Release date: February 9, 2014

Those among us who loathed high school calculus might feel some trepidation at the premise in this week's episode of Rationally Speaking. MIT Physicist *Max Tegmark* joins us to talk about his book *"Our Mathematical Universe: My Quest for the Ultimate Nature of Reality"* in which he explains the controversial argument that everything around us is made of math. Max, Massimo and Julia explore the arguments for such a theory, how it could be tested, and what it even means.

Max's pick: "'Surely You're Joking, Mr. Feynman!': Adventures of a Curious Character"

## Reader Comments (25)

The ultimate nature of reality is defined mathematically by a single simple equation: =

Equal is.

=

Interview a bit frustrating, as the answers to Julia's questions were not answered crisply. Max did say but not emphasize enough that the universe is not described by mathematics at all, cause objectively speaking, it ain't there. Using his address and house analogy, there is no house, only an address. One can substitute any subset of 'universe' as well - so you don't have a microphone, you have only a description of a microphone, and that description is mathematical.

I am sympathetic to this, although I feel it s safer to stay with information rather than mathematics (or Ladyman/Ross' structure) because like reality, a structure or mathematical formulation implies reproducibility, and not quite sure we have this.

The way to understand the "description of something vs. the something" question better (it IS difficult if you haven't sent a lot of time on this) is to say "OK, there IS a territory, not just a map, now what is the nature of this territory? Matter? Energy? Today's quantum physics says "No. there are only assumptions". But even more convincing to the non-scientist is the turtles argument - there are turtles, below that, turtles, below that, more turtles, forever and ever. Therefore all we will ever know is the map and not the territory.

The infinity argument is always interestiing, and a reasonable conclusion is that any

implementationof a concept containing an infinite amount is by definitiion finite.I am not at all convinced by your guest's arguments about mathematics being all there is. I agree with Julia, that math can be used to describe the universe. But math itself is not the universe. I believe that the material world actually exists and nothing I heard on the show convinces me otherwise. Math is an abstraction. Abstractions are massively useful for understanding the world, but they are a step removed from the world. We may one day discover the formula that describes everything, the grand unified field theory. But it will still be a description of the behavior of matter. It will not be matter itself.

Some of your guest's arguments were downright ridiculous: He said that maybe a quantum computer can be built that would break modern cryptography, and that this would demonstrate the existence of the multiverse. "Maybe" is not an argument, and in any case, such a computer would be evidence of no such thing. It would simply demonstrate that quantum computers are really powerful. But in fact, quantum computers are purely speculative. We have no evidence that they can actually be built, or that they would perform as speculated. Then he said something truly silly: That if you played Russian Roulette and got a feeling of immortality, it would be evidence of the multiverse. Say what????? It would be no such thing. It would be evidence of mental illness if it was evidence of anything. How can a "feeling" be seriously regarded as evidence of anything other than the emotional or mental state of the individual involved?

However, I do agree wholeheartedly with your guest's comments on infinity. There is no such thing as infinity. "Infinite" is an adjective, not a noun. Infinite means unbounded, without limit, endless. The counting numbers, for example, are infinite because there is no end to them. There is no largest counting number because there is no end to the counting numbers. But for that very reason, there can be no such thing as "all" counting numbers. There is no end to them, so there is no "all" of them. Infinity is a meaningless concept. As your guest correctly pointed out, arguments that incorporate supposed infinities lead to meaningless or nonsensical conclusions.

Thanks David and Daniel for raising this important point about the distinction between the description and what is described, which I explore in great detail in chapters 11 and 12 of the book ( http://mathematicaluniverse.org ). This distinction is crucial *both* in physics and in mathematics.

Our *language* for describing the planet Neptune (which we obviously invent - we invented a different word for it in Swedish) is of course distinct from the planet itself. Similarly, as I mentioned above we humans invent the *language* of mathematics (the symbols, our human names for the symbols, etc.), but it’s important not to confuse this language with the *structure* of mathematics. For example, any civilization interested in Platonic solids would discover that there are precisely 5 of them (the tetrahedron, cube, octahedron, dodecahedron and icosahedron). Whereas they’re free to invent whatever names they want for them, they’re *not* free to invent a 6th one – it simply doesn’t exist. It's in the same sense that the mathematical structures that are popular in modern physics are discovered rather than invented, from 3+1-dimensional pseudo-Riemannian manifolds to Hilbert spaces. The possibility that I explore in the book is that one of the *structures* of mathematics (which we can discover but not invent) corresponds to the physical world (which we also discover rather than invent).

Daniel, your statement about quantum computers being purely speculative is highly controversial, and most of the quantum physicists I know would disagree with it - which is millions of dollars are currently being invested in trying to build them. I explain the quantum immortality argument in detail in the book (and not in the podcast!), and also why I think it doesn't work.

Max, thanks for your reply. I'm glad to hear that you do not buy the quantum immortality argument, which just seems silly to me. As for quantum computers, I am willing to drop the word "purely" and just assert that they are speculative. If the possibility of building them is controversial, then I think it's fair to call them speculative, and certainly, until someone does build one, any argument beginning with "Maybe someone will build one" is extremely weak. And even if someone does build one, which I admit might be possible, though I think it unlikely (just my opinion there) the mere existence of a quantum computer would not be proof, or even evidence of a multiverse.

Until we understand more about the Big Bang, I am very open to speculation that there may be more than one universe. Perhaps a mind-boggling number of them. But I think there is little if any grounds to speculate that there are analogs of all of us in different universes. The notion that everything that can possibly happen does happen in some universe strikes me as groundless barroom philosophy rooted in the misconception concerning the concept of infinity, which I reject as noted in my earlier post. The counting numbers are unbounded (infinite) but it makes more sense to me to think that the world of stuff (matter and energy, both dark and light) is bounded. The hypothesis of an infinite (unbounded) number of universes is an extraordinary one, and therefore should require extraordinary evidence. But in fact, we have zero evidence for any universes other than ours, other than groundless philosophizing.

I believe that there are a large number of intelligent, technological civilizations in the universe, and an even larger number of planets with non-technological life, but that the distances between stars and the Einsteinian limit to travel speed make it vanishingly unlikely that we will ever make contact with them. By the same token, I think it vanishingly unlikely that if there are other universes we will ever be able to find reasonable evidence of them, leaving such ideas purely speculative and of no actual use. Hypotheses which are not falsifiable belong to the realm of religion, not science. If someone wants to have a religion based on the belief in multiverses, that's neither more nor less valid than Christianity, Hinduism, or Voodoo. But it doesn't particularly interest me.

It is incredible how mathematically educated people can remain attached to this day to the sort of pre-cantorian intuitions and arguments like the comments above. Infinity is defined in a perfectly definite sense a property of classes and even an useful one. Infinite ordinal play a large role in abstract algrebra and and analysis, and even if one can sensibly doubt the truth of speculative mathematics, doubt about its coherence are not justified.

Hi Dedek: I'm not sure whose comments you're referring to. Just in case it wasn't clear: what I'm referring to in the book is infinity in *physics*, not infinity in *mathematics* (which is of course perfectly well-defined).

No, of course, your position was quite clear and even a sensible one. For what is worth, I share your hunch about infinity and even more continuity (wich is more than countability) being theoretically dispensable in physics. That said it seems to me like precisely those remarks at the end of the podcast are in deep tension with the mathematical universe hypothesis. If our use of mathematics in physical theorizing gave support for reduction to a mathematical structure it seem like it would point more to an infinite dimensional vectors space like those dealt with in functional analysis and quantum theory or a smooth continuous manifold as spoken of in general relativity than any finite structure. In any case one could use your same strategy against your hypothesis: mathematical theories are convenient fictions or at best simplifications but do not describe anything. Structuralism looks like a better suited name for professor Tegmark position. I was surprised not to hear Massimo refer to James Ladyman and Don Ross in this connection.

Anyway, about the hard work spoken by professor Tegmark, for those who are well-read in mathematics, physics, mathematical logic or a subset of the above I can point to very interesting work already done to try to dispense with continuous mathematics or to evaluate the question "what mathematics is needed to do physics?". I wonder what our guest would say about this attempts.

1) People in reverse mathematics have worked on subsystems of arithmetic trying to find the weakest set of axioms needed to prove mathematical results in ordinary uncountable mathematics like real analysis or topology or again algebra. Turns out much of it can be coded up as statement about countable stuff and proved in weak systems like Pano Arithmetic + König Lemma. Stephen Simpson spoke of all "undergraduate mathematics". Friedman conjectured Fermat's last theorem is provable in elementary arithmetic. The locus classicus is the book by Simpson "Subsystems of second order arithmetic".

2) Edward Nelson is a nominalist and ultrafinitist at Princeton. He has developed a "modified Hilbert program" to prove stuff on an even meager finitistic system of arithmetic and has proved we can do a lot coding nonstandard analysis.

3) Hartrey Field in philosophy wrote a book in the eighties called "Science without numbers". He tried to eliminate real numbers, functions and vectors from classical mechanics by using a purely geometric formulation and proving adding the math does not generate any new result. His program has been recently extended in a paper by Cian Dorr and Frank Artzenius in wich they try to do the same for more recent theories.

I like the metaphor of "breathing life" into equations. We can use this metaphor to build an analogy:

People used to be puzzled why some matter---like animals---was alive and some other matter---like rocks---was dead. People kept taking apart both living and non-living matter, but couldn't find any fundamental differences. Thus, they were forced to conclude that living beings are just atoms put together in an interesting way and non-living beings are the same atoms, put together differently.

Analogously, Tegmark's claim is that equations that "exist" don't seem to be any different, fundamentally, than equations that "don't exist". Thus, "existence", is not an ontologically fundamental state. Instead, existence is an epistemic experience of observers residing in certain funky equations.

To carry the analogy forward, in the case of life, after much work in evolution and biology, people explained how non-living things arranged themselves to form living things. Similarly, I think there is a major task facing the Mathematical Universes Hypothesis: in what way are mathematical structures in which "existence" can be experienced different from mathematical structures in which there is no concept of "existence". Basically: what universes admit observers? And what other kinds of observers can there be?

And just like living matter can manipulate itself, as well as manipulate other living and non-living matter, is there any possibility of interactions between mathematical universes? Can we manipulate the laws of our own universe?

I realize that these are all confused and ill-formed questions, but there you have it.

Thanks Dedek for these interesting points and references! I mention Ladyman's work in the paper and have a long discussion of description versus equivalence in the book. I'm a great fan of the mathematics research directions you mention above. Norman Wildberger is another mathematician with such goals, who's written a nice book about doing geometry without real numbers.

I had the pleasure of meeting Edward Nelson not long ago. Amazon shows Hartrey Field's book as out-of-print, so I think I'm going to look for it in our library - do you have any links to this or the more recent work of Dorr & Artzenius that you refer to?

I hope everyone realizes that math too is a conceptual contrivance, be it at a primary and primitive level of conceptualization. To say the universe is math is to say it is a concept and there's nothing new in that.

Sid: I like your metaphor!

Thanks Dalton for raising this important question about whether mathematics is invented or discovered, a "contrivance" or not - a famous controversy among mathematicians and philosophers. You’re quite right we humans invent the *language* of mathematics (the symbols, our human names for the symbols, etc.), but it’s important not to confuse this language with the *structure* of mathematics that I focus on in the book ( http://mathematicaluniverse.org ). For example, any civilization interested in Platonic solids would discover that there are precisely 5 of them (the tetrahedron, cube, octahedron, dodecahedron and icosahedron). Whereas they’re free to invent whatever names they want for them, they’re *not* free to invent a 6th one – it simply doesn’t exist. It's in the same sense that the mathematical structures that are popular in modern physics are discovered rather than invented, from 3+1-dimensional pseudo-Riemannian manifolds to Hilbert spaces.

Max, I'm going to second the sentiment that your presentation here doesn't really express strongly enough the basic notion, which (unless I'm misunderstanding it) is that the basic substrate of the universe isn't

matter, whatever "matter" might mean.Perhaps the following might be helpful for Dalton, who said

I disagree. For simplicity, let's pretend that we're in a Newtonian universe, i.e. governed by deterministic differential equation and in which particles can be located in a real (i.e. continuous) vector space.

So in this hypothetical, we assume that there really is something, which we call "matter", which is made up of particles, each of which has a location which can vary continuously. (It doesn't make a real difference, but let's also assume that there are only finitely many particles in the universe.) Particles can exert forces on each other, and will accelerate based on these forces in accordance with Newton's laws. Since we're in a Newtonian universe, a particle can only ever be one kind of particle; a proton is and will forever be a proton.

OK: now, let's create a huge vector, with components for each particle: x-position, x-velocity, y-position, y-velocity, z-position, z-velocity. In other words, if there are N particles, this vector will be a "point" in 6N-dimensional space. The state of the universe at any point in time is completely described by this vector; moreover, since the universe is Newtonian, the state vector of the universe will trace out a path through 6N-dimensional space, and for any point (corresponding to any state of the universe) there is a unique future path that a universe with that state would trace out (assuming that we also have determined the mass of each particle).

So far, so good: all we've done is assert that the universe that we're imagining evolves according to a mathematical rule.

OK, now here's where it gets weird. Remember how we supposed that there was actual stuff, "matter", underlying "the universe" above? Let's get rid of that. What if, instead, "the universe"

isthe curve through 6N-dimensional space, and youaresome very complicated pattern of values for several quintillion of the coordinates of that vector at any point on the curve? How would you tell the difference between that curvedescribingthe positions of "actual" particles, versus that curvebeingall those particles?It might be helpful just to distinguish some words. When we mean with "x exists" that x exists in this very universe, we obviousely talk about something different than Max did in the interview.

What I missed in the interview was the answer to Julias question, that I rephrase here in the hope to add cripsness: What would it mean for something in the universe to be not mathematical? (I expect, if we achieve to construct candiates of answers to this question, we can build math to handle them. If this worked out, the affirmation of mathematicity would be trivial.)

Well. Then I have a problem of understanding (still without heaving read the book). I like the finite attitute, but don't see how it works out, because of the assumtion of existence for all structures:

When we start with the set of finite mathematical structures, we provably have an at least countable infinite set. (As, say the finite groups, e.g., are a subset.) So the multiverse Max talks about is infinite, and, when we want to talk about these structures, our language (our logic) better be infinite too, and, e.g., we have our Gödel problem back.

Am I mistaken?

Thanks Falko for these interesting questions!

> What would it mean for something in the universe to be not mathematical? (I expect, if we achieve to construct candiates of answers to > this question, we can build math to handle them. If this worked out, the affirmation of mathematicity would be trivial.)

Don't you think that many philosophers would disagree strongly with your claim here, which seems to imply that any form of soul or deity must by necessity be mathematical?

> we have our Gödel problem back

You seem to be implying that we had a "Gödel problem" to start with, but what's your argument for this? If you are claiming that the

mathematical universe hypothesis is ruled out by Gödel's incompleteness theorem, I'd love to hear what your argument is.

Given any sufficiently powerful formal system, Gödel showed that we cannot use it to prove its own consistency, but his doesn't mean that it is inconsistent or that we have a problem. Indeed, our cosmos doesn't show any signs of being inconsistent or ill-defined, despite showing hints that it may be a mathematical structure. Moreover, what were we hoping for? If a mathematical system could be used to prove its own consistency, we'd remain unconvinced that it actually was consistent, since an inconsistent system can prove anything. We'd only be somewhat convinced if a simpler system that we have better reason to trust the consistency of could prove the consistency of a more powerful system - unsurprisingly, that's impossible, as Gödel also proved. Of the many mathematicians with whom I'm friends, I've never heard anyone suggest that the mathematical structures that dominate modern physics (pseudo-Riemannian manifolds, Calabi-Yau manifolds, Hilbert spaces, etc.) are actually inconsistent or ill-defined.

(Please note that I'm using "mathematical system" to refer to "formal system", as distinct from "mathematical structure": the former can describe the latter and the latter can be a set-theoretic model of the former.)

Hello Max,

thank you for the response.

1. What does it mean to not be mathematical?: Ah, right, I once read about those philosophers :-)

So, I rephrase, still capturing what I want to say, but getting rid of the ontology: What does it mean to

describesomething as not mathematical?Even if there exists any god (like, we live in a computer game, and there is a programmer, or stuff) - if we start describing it, we can do the description in math. So, what we can't describe in math, we can't describe at all. (There is a bit young Wittgenstein in this view.) And whether we call it existing at all is a matter of taste.

I actually belive this is the gist of one of Julias questions. So nothing original here :)

2. "Gödel problem": I mean with it that we cannot complete our models, as there are always questions our axioms don't give an answer to. So I did not refer to the unprovability of consistency, but to incompleteness. I admit to only remember you talking about the unprovability of consistency in the interview, not about incompleteness, now you mention it.

So I understand you clarification as: You try to go with finitism in the mathematical structures you deem to exist, but not in the formal systems you use to talk about them. This sounds sound to me. (I don't think that Gödel is a danger to the mathematical universe.)

So your multiverse is the set of all finite mathematical structures (and is thus not finite).

Many thanks for the clarifications!

Falko

Thank you, professor Tegmark, for your reference. I will look it up.

Hartry Field's book is a classic but unfortunately out of print. It is even hard to find in university libraries. There was a joke circulating in philosophy of mathematic's seminar I attended along the lines of "everybody talks about it,but did they read it ? Does it even exist ?". But it does. Frank Artzenius has published his paper in his book "Time, matter and stuff". It's called "Calculus as geometry". Cian Dorr has a penultimate draft on his website. http://users.ox.ac.uk/~sfop0257/

I have a suggestion for a precise definition of structuralism which might cover also the mathematical universe hypothesis (especially in light of some of the comments at the end of the podcast). It is perhaps too strong and somewhat problematic in its formulation but interesting. If it is an overkill it will at least contribute to temperate exaggerate structuralist talk. In a slogan it might be summarized thus : structural properties fix physical properties. With structural properties we understand simply those properties preserved under isomorphism. Isomorphism is understood in the usual algebraic or model theoretic sense.

The universe is mathematical iff

(1) there is a fundamental final theory of physics such that all its (set theoretic) models isomorphic to the "standard model" are physical universes qualitatively and physically identical to ours. (God only had to fix how many points and how to connect them with his pencil to fix how the world was to be like in all its gory details.).

Massimo and Julia,

I don't mind if you do not approve this comment for general viewing, but I just wanted to plead with you to please encourage guests in future to not play with loud desk items (what was that clicking, a retractable click eraser?) during the interview. Am I the only one who was repeatedly distracted by this? (Gaslight away)

David

David,

I also found the background noise hugely annoying and distracting. This is why you need a sound engineer, they'll hear it and can politely ask, "Who's making that noise?"

Marty

The universe is mathematics, however the author indicates that there is a problem with infinity, and I think there is another one with continuity.

If we take the collection of real numbers, we can always find another real number between any two real numbers. This means that the reals numbers is a collection of numbers perfectly continuous.

But physics always shows that the universe is discrete, made of elementary particles, governed by quanta's.

Therefore, we could argue that mathematics describes more than the actual universe, or that it has properties that are not found in the universe.

And therefore mathematics is different from the universe.

I find the guest's hypothesis as essentially meaningless, since he never was able to draw a distinction between the the claims that the universe was composed of math vs being described by math. Julia asked twice and never got an answer.

And the suggestion that you could disprove his theory by running into a roadblock in describing the universe by math is just a “God of the Gaps” argument. Just because at a point in time we were unable to describe the universe with math would not prove that we would forever be unable to do so. Therefore, his theory could never be falsified.

The composition of the Universe can be described or written by One single equation: =

Thanks Greg for raising this important point about the distinction between the description and what is described, which I explore in great detail in chapters 11 and 12 of the book ( http://mathematicaluniverse.org ). This distinction is crucial *both* in physics and in mathematics.

Our *language* for describing the planet Neptune (which we obviously invent - we invented a different word for it in Swedish) is of course distinct from the planet itself.

Similarly, as I mentioned above we humans invent the *language* of mathematics (the symbols, our human names for the symbols, etc.), but it’s important not to confuse this language with the *structures* of mathematics. For example, any civilization interested in Platonic solids would discover that there are precisely 5 such structures (the tetrahedron, cube, octahedron, dodecahedron and icosahedron). Whereas they’re free to invent whatever names they want for them, they’re *not* free to invent a 6th one – it simply doesn’t exist. It's in the same sense that the mathematical structures that are popular in modern physics are discovered rather than invented, from 3+1-dimensional pseudo-Riemannian manifolds to Hilbert spaces. The possibility that I explore in the book is that one of the *structures* of mathematics (which we can discover but not invent) corresponds to the physical world (which we also discover rather than invent).

"The possibility that I explore in the book is that one of the *structures* of mathematics...corresponds to the physical world"

The fact that mathematics is useful for physics makes it self-evidently true that there is a correspondence, but that still doesn't provide support for the idea that math is all that exists. I probably could describe all of physics using the language of fairies, but that doesn't mean that only fairies exist. :-) Again, you didn't provide any real way to falsify your hypothesis, so that kinda puts it into the semi-religion category, IMO. Don't get me wrong, it's a fun idea to toy with and I think it's great that you wrote the book and floated the idea about.

That the Universe is completely described by mathematics is indeed an old idea, however Pythagoras and Galileo did not provide enough arguments why it is so! It is more like a postulate in their philosophy. It is easy to say that the Universe is mathematical but we need epistemological and ontological basis for such claims. I saw in your website the link to ontic structural realism (OSR) so I guess you are familiar with it. OSR indeed provides good arguments about the underlying invariant structure of all our theories.

I think Immanuel Kant is the first philosopher to provide the strong arguments why the Universe is described by mathematics. If you are familiar with history of philosophy, Kant reacted to the famous debate between Rationalists (Leibniz, Descartes, Spinoza) and Empiricists (Locke and especially David Hume). Rationalists claim that the source of knowledge is reason and innate ideas, while empiricist claim that the source of knowledge is experience through the senses. Both are right from their perspective. Kant said that to speak about innate ideas in our mind which ground mathematics, metaphysics (a priori knowledge) as rationalists did is lazy business. David Hume has shown that everything comes from experience but he had problems with establishing mathematics on firm ground because maths speak of experience a priori. He could not explain how mathematics is possible! Kant tried to defend this a priori knowledge (mathematics, theoretical physics) and so-called synthetic a priori judgments. That's why I have used Kant to model our cognitive framework (and the Universe as it appears to us) as a quantum computer defined on a grid of cells. I claim that this grid is invariant structure within which all our thoughts, knowledge and theories originate. The structure OSR seeks.

Kant had influenced such mathematicians as Henri Poincaré and David Hilbert. In philosophy of mathematics Kant belongs to intuitionist school. It is also interesting to study the logicist school, that is Frege, Russell. I know that you are involved with FQXi. I claim that we will not understand ultimate reality unless we view everything as a system of mathematics, theoretical physics, philosophy of science and cognitive science. Cognitive science is of absolute importance in understanding ultimate reality because all our thoughts about the world originate in our brain. I know that you come from strictly scientific background but philosophy of science, philosophy of mind cannot be left out if you want to understand the ultimate reality.

It seems that you have buried the philosophy of corporeal nature. This is the true purpose of proper metaphysics of corporeal nature - to assist mathematics and physics. They should go together. It does not matter that people did not know about the Higgs boson or the mathematical description of general relativity 200 years ago. What Kant and Hegel knew is fundamentals - how our knowledge about the world in general is possible. If you know the roots of your knowledge, the epistemological basis of mathematics and physics, everything else is just details. To understand ultimate reality we must understand how we understand things in the first place! That is, we must have the picture of our cognitive faculties in general. This yields the big picture of the Universe how it appears to us.

That's why I took Kant who asked and provided answers in his work to the questions: ''how is mathematics possible?''. ''How is physics possible?''. ''How is metaphysics as science possible?''.

https://www.academia.edu/7347240/Our_Cognitive_Framework_as_Quantum_Computer_Leibnizs_Theory_of_Monads_under_Kants_Epistemology_and_Hegelian_Dialectic