My God, It\'s Full of Clones: Living in a Mathematical Universe by Marc Séguin

Dear Marc,

Indeed I might also describe my view as a monism, more precisely a mental one, since, as I wrote, consciousness can understand mathematics, but mathematics cannot describe consciousness. However, mathematics is then a remarkably stable part of this mental realm. And we (I mean, some people) have straightforward abilities to describe mathematical systems, that cannot be applied with such a success to non-mathematical ones.

"I am fully aware of this, but what can I do? Suppose it is true that capital-E Existence ("All that exists") does indeed follow from the most simple rule imaginable, "Everything Exists","

What can be done: notice that science is no more in its infancy, and that we do already have some effective materials to consider. Namely : is existence described by a probability law or not ? We did find one important probability law, that is the Born rule. So we know that a probability law exists, and we can already describe it. Then, in case we feel lost in our speculations on the nature of existence, we should not forget that there is this thing we know, that needs to be taken into account, so that our ideas remain compatible with it. Now what I criticize about Moravec's explanation, is its incoherent way of presenting arguments that are actually based on probabilistic assumptions (outside which no such arguments can make sense), while trying to deny having any clue on probabilities. My suggestion is to work on clarifying the structure of the probabilistic assumptions that these arguments are implicitly based on. Then, are such probabilistic assumptions compatible with the Born rule ? If you admit a non-probabilistic distribution of existence (i.e. not described by a measure), then in such a framework, how can any probabilistic law such as the Born rule make sense at all ?

About the flow of time, I wrote that we have an available analogy with the time of the foundations of mathematics, which I described in my site. So time has a sort of structure for which we have a mathematical analogy (better than the pure geometric view of linear order), yet it is only an analogy which does not suffice to describe the effective contents of consciousness and its time flow.

"qualia and the flow of time are not structures. What are they then?"

What sort of answer do you expect ? If a mathematical description was given, it would make these things mathematical. However it does not mean that it is a complete mystery, since, anyway they are things we somehow know by personal experience (unlike the concept of "nature" put forwards by materialists).

Dear Marc,

If I were to deal with every (from my point of view) objectionable idea, it would simply take too long. Instead, I selected three sections from your essay that I think might be representative of our differences. After each quote, I have added my response to it.

After these three sections, I continue with a general discussion of the issue, and explain (necessarily without much detail) my own views of these things.

"Imagine there's only math -- physics is nothing more than mathematics, we are self-aware mathematical substructures, and our physical universe is nothing more than a mathematical structure "seen from the inside.""

But this starting assumption is just that - an assumption. I don't deny that if you start with this assumption, then your conclusions (and likely a few more) can be said to follow.

"According to the MUH, physical reality is a web of relationships between entities that are themselves purely abstract: it's "all structure, no stuff," a view that Jim Holt [4] calls cosmic structuralism."

Only "stuff" can have real-world structure. Abstract structures (represented by drawings, computer simulations, or mathematics) contain no "stuff," and therefore cannot give rise to the (material) universe.

"But if you accept that a living being can be thought of as nothing more than a complex arrangement of atoms obeying the laws of physics, is it really that hard to accept that a physical universe can be thought of as nothing more than a complex mathematical structure?"

I think the implied equivalency is extremely weak.

We are free to define words like "existence" any way we like, but it is most useful to stick with accepted dictionary (usage) definitions. Ambiguity in philosophy or physics is not desirable, so I define "existence" as follows. Only those things that we can detect with our senses (even in principle) or those that we can detect with an enhancement of our senses (with instruments etc., again, even in principle) should be said to exist. If we included abstractions in "existence," it would be mixing very different things. (Of course, without abstractions, we would be leading a very primitive life.) Mathematics, per se, does not exist under this definition (what you see on paper or a computer screen is just ink or "pixels," the rest is happening inside your head).

Let me try a real world analogy. Consider an ordinary pocket calculator. It has a certain organization of atoms arranged so that when you press certain keys, you get a "desired" display as a consequence. We "say" that it has an internal logic, but what we actually did (in building it) is to utilize how the universe behaves on its own (in this case how silicon and electricity, etc. behave) and then used those behaviors to get the functioning we wanted. We are "exploiting" already built-in behaviors of the universe, but we think that we imputed logic into the calculator (and in an abstract way this is how we speak) while in fact we are only riding on the universe's coattails (to borrow an idiom from another discipline). The calculator is not doing any "mathematics" (although we call it that, and it is useful) - it is simply streaming electricity along different paths, depending on the keys you press.

The MUH is essentially saying that the implicit rules guiding the various electron streams (we abstract those rules and call them "mathematics") is the "cause" of the calculator itself (or better yet, is the calculator itself). The rules (which don't have a physical existence) cannot create the calculator. And the calculator is not doing any more "math" than a bicycle chain interacting with the gear it is engaged in. The bicycle chain only needs to have the right dimensions to fit the gear, and the rest is done "via that very fact" (no calculations are performed as you ride the bicycle).

My view is diametrically opposed to the MUH. The universe does no calculations of any kind. It just behaves that way without any mathematics. To put it more colloquially: "the universe does not even know what mathematics is." Mathematics just helps physics to describe (and economically codify) how the universe behaves. Of course, math has applications everywhere, and is not "subservient" to any other discipline.

Marc, I will give you a high rating on your essay. I like it. I don't feel any need to convince anyone about my views, and I don't think that a belief in what you said in your essay is in any way detrimental to science. I am sure you can think of many counterarguments to what I said, and after all, you could be right.

En

Dear Marc,

thanks a lot for these very interesting and in a sense maximally extreme thoughts. I read through many comments and your replies as well, but haven't found an answer yet to the following questions:

- How come the F-clone? Assuming I am a mathematical structure, what does it mean that there is a clone of me? I would say a mathematical structure is a set of equations, inequalities, or of axioms, and possibly the full set of theorems that follow from them. So what is a clone of that supposed to be? The same set of equations, axioms, theories and so on once more? I would think, one equation A=B is enough.

Or do you think of them defining universes in each of which a F-clone of me would possibly live? If so, then some more information is required that defines me in each universe, I could incorporate that information in the definition of "me", and then ask again: how come the F-clone?

- Is the number of F-clones countably infinite, or which infinity of the many different ones?

In fact, in writing my own essay, I initially intended to find out the proportion of the set of physical theories in the set of all mathematical theories, the latter corresponding probably to your Maxiverse. But it appears to me that this set is ill-defined, as set theory is a mathematical structure, and if applied to itself runs into logical problems. Do you avoid these problems by adding the qualifier "imaginable"? But who imagines? Is it some mathematical structure that imagines another (in which case the definition is cyclic and difficult to accept), or is it me or you? Then the whole thing becomes very anthropocentric and is difficult to accept as well.

More generally, I believe that mathematics as we know it is more of a human creation than we might want to believe - already judging from the fact that it uses Boolean logic in which logical variables have a well defined value 0 or 1, and at least the math established from finite axiomatic systems can be obtained from a classical computer program - as opposed to a quantum one. Will a new "quantum math" arise now that we are quite sure that not all observables have well defined value 0 or 1 at all times and that quantum computers will soon be on sale at Macy's? :-)

Of course you might argue that if so, this just extends the Maxiverse to so far unthought levels. But it makes me suspect that math has a human component to it - which would make the idea of the Maxiverse be too anthropocentric to be acceptable.

I would be happy to read your thoughts on this. In my own essay I finally settled down to the much more humble and down-to-Earth question what the "size" (cardinality, really) of the set of possible physical theories is, resting with the structure of those that we know. It is infinite as well. But which infinity? If you want to find out you will have to read it :-)

Daniel

Your essay might be described as MUH nothing else, but with your clear way of expressing these ideas you take it a lot further.

My own ideas about the MUH which I called the Theory of Theories was developed independently at the saem time as Tegmark. The unique ingredient in my version which goes further is the answer to the question "Why is our world so lawful and simple" The answer has to be that in the complex system of all mathematically logical possibilities there is a principle of universality that determines the underlying meta-laws of the universe. It is an emergent principle of self-organisation that is as important to the nature of mathematics as it is to the nature of physics. Anthropic principles only come into play when looking at the possible solutions to those meta-laws.

The latter parts of your essay follow thoughts that I have been through many times myself which lead towards ideas about conciousness. I hope a future FQXi contest will be brave enough to ask us about conciouness and free will so that I can write my own ideas about what we should learn from "F-clones" It was not something I had space for this time.

When I see how clearly you express the philosophical ideas that I agree with, I find myself surprised that there are so few of us who see it that way. I hope your exposition will generate a few more converts.

Marc,

Yes my head is spinning in the vortex of my disbelief in the Maxiverse hypothesis, but due to your rather fetching argument, I find it impossible to dispute your argument with anything but humdrum arguments.Love your methodological reductionism argument at beginning and your devil's advocate approach.

I too have used the Hut argument that as a type 0 civilization we have too little understanding of math, physics and consciousness to perceive the concepts of a type 2 civilization, for example. That is my excuse as well.

My essay seems pedestrian in arguing the use of math to represent and model a physical world that has brought epiphanies regarding our universe in terms of quantum biology, DNA and our early universe.

Jim

Dear Marc,

thank you very much for your reply. I think I understand now better what you call an F-clone. Is it correct to say that if you define "me" as a collection of information, e.g. complete information (the full many-body quantum state really) about all my atoms at a given time, F-clones would be a possible time-evolved state?

If so, coming from quantum mechanics, one wonders how relevant such a "mathematical substructure" that only pertains to "me" can be: the vastly overwhelming majority of states of "me" and universe are entangled, and so one can almost never describe "me" as an independent mathematical substructure of a larger mathematical structure. In the end, there would be only one big mathematical structure, the big [math]$|\psi\rangle$[/math] and subsystems don't have an independent existence.

On the other hand, with this definition of F-clones, the set of F-clones of "me" would have cardinality [math]\aleph_1,[/math] I suppose: a 1-parameter continuous manifold [math]|\psi(t)\rangle.[/math] The same would hold for the cardinality of the set of F-clones of the entire universe. Even for multi-dimensional time, it would still be [math]\aleph_1.[/math]

Does this make sense?

Daniel

Dear Marc,

I appreciate your lighthearted and relatively even-handed presentation of the maxiverse hypothesis. Although it is very far beyond what I'd be prepared to believe, your essay was fun to read. The idea of cosmic structuralism reminds me of the von Neumann universe, or cumulative hierarchy of ZFC. Let me throw a couple challenges your way:

1. The contrast between our 'boring' universe and the more exciting one seems like a false dichotomy to me, in the sense that one would expect there to be a spectrum from 'boring' to 'exciting', and I am not sure that anthropic considerations are equipped to handle this: It does not seem all that difficult to imagine more exciting universes than ours which are equally compatible with our existence. If so, then for each copy or our world there are is a copy each of an infinite number of more exciting universes compatible with our existence. Why don't we live in one of those?

2. If the maxiverse includes absolutely every imaginable mathematical structure, it also contains every imaginable inconsistent one. But any inconsistency in a mathematical structure infects the entire structure, so is it not the case the maxiverse is inconsistent?

3. What kind of mathematical structures are feelings, perceptions of color, dreams (I do not mean the neural correlates, I mean the qualia themselves)?

I hope you enjoy these,

Best wishes,

Armin

Dear Armin,

You are right: the Maxiverse has a lot in common with V, von Neumann's hierarchy of all sets. My first acquaintance with V came in graduate school, when I read Rudy Ricker's book "Infinity and the Mind", which had an important influence on the way I see the world.

The three questions you ask are very interesting.

1. I agree with you that a major challenge to the Maxiverse hypothesis is to explain why we live in a universe which obeys laws that are so regular and relatively simple (what Moravec calls a "boring" world). In my conversation with Philip Gibbs (two threads above this one), I hint at a possible solution that has recently come to my attention. Could it be that all the universes that contain versions of ourselves that are more or less similar "interfere" with each other like the different paths in Feynman's path integral formulation of Quantum Electrodynamics, "evening out" in the process to yield fairly regular and "boring" outcomes, in the same way that Fermat's principle averages out the behavior of the wavefunction of a photon to yield a behavior consistent with the laws of classical optics? This seems to me an idea well worth pursuing!

2. Gödel's incompleteness means that there are mathematical structures that are true but cannot be proven in a finite number of steps by a finite mathematician, but I don't think it means that there are inconsistent structures within mathematics. The way I see it, to be mathematical, a structure has to be consistent by definition (even if it is unprovable in Gödel's sense). In my opinion, if incompleteness could "infect" the Maxiverse and make it inconsistent, then nothing would exist and we wouldn't be here to argue about it! :)

3. I agree with you that qualia (perceptions, feelings) can seem awfully un-mathematical, but whatever they are, I believe they are "structures" of some kind... and if mathematics is the general study of structures, then they are ultimately mathematical, like everything else. But then again, I believe it is only because the structures that correspond to our universe are observed "from the inside" by self-aware substructures (us) that they acquire physical existence. (It would make no sense to say that a mathematical structure that doesn't contain self-aware substructure exists physically, because there wouldn't be anyone to "feel" its putative physicality.) Therefore, from a certain point of view, qualia could be said to make up the fundamental level of existence. So, depending on your point of view, "all is math" or "all is mind"...

Thank you for the questions... I hope you enjoyed the answers!

I will study your essay and post comments on your page, hopefully within a couple of days.

Marc

FYI:My Essay 2408 error corrections @

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Dear Marc,

In your comment to our essay you expressed agreement with some of our statements, and gave us your high compliments; thank you. In turn I appreciate your reflection on the Mathematical Universe Hypothesis (MUH), with your stressing its general philosophical aspects.

You noted in that comment: "Your arguments convinced me (contrary to what I state in my essay) that the Mathematical Universe Hypothesis (in its simplest form) does make predictions, and can be considered a scientific hypothesis." Indeed, in our essay we showed that the MUH in a form of the "mathematical democracy" of Tegmark (we called that 'chaosogenesis') is clearly refuted on the ground of high range and extreme precision of the discovered laws of nature. We showed that theoretizability of the universe by no means could be a consequence of the anthropic principle; it requires a special selection. It is important that this additional selection cannot be a law, since in that case the question of John A. Wheeler 'why this law, not something else?' would remain unanswered. That is why any sort of 'measure', being a law superimposed on this full-blown multiverse, cannot be the solution.

Good luck!

Alexey Burov.

Dear Marc,

Thanks for the response. A quick comment:

"The way I see it, to be mathematical, a structure has to be consistent by definition..."

Then I take it you are not aware of paraconsistent logic and mathematics? The online Stanford Encyclopedia of philosophy has some good introductory articles on these, calling the latter "inconsistent mathematics". The last sentence of section 6 expresses my second challenge in more general terms. I believe this issue is separate from Goedel's theorem. I wonder whether Max Tegmark is aware of this? If not, somebody should point this out to him.

Incidentally, I appreciate your offer to study my paper. I look forward to your challenges, although due to the incompleteness of the framework, it is probably not so difficult to come up with some but every bit of constructive criticism helps.

Best wishes,

Armin

Marc, our thought processes are very closely aligned.

Although I would not be surprised to be labeled as a platonist I do accept some of the criticisms of the anti-platonists. I think if mathematics is seen as a realm that exists in a physical sense as platonists like to see it then the problem of existence has been pushed back rather than solved.

I now prefer to think of mathematics as a description of possibilities for existence rather than of things which exist in a platonic realm. This avoids the criticism. I like your explanation that physics is mathematics plus nothing else in the same way that biology is chemistry plus nothing else. There is no missing vital spark needed for life to emerge from chemistry or for reality to emerge from mathematics. This is a powerful analogy (except to those who still think there is a vital spark or soul required for life) It expresses exactly how I see the mystery of existence resolved.

I am glad I brought some references to your attention. You also had some good references that I was not aware of.

Philip,

Your comment above (in particular: "I now prefer to think of mathematics as a description of possibilities for existence rather than of things which exist in a platonic realm") is eminently reasonable. But some of your "brethren" might consider you an apostate.

Btw, I observed your comments (giving wise advice to some of the more, shall we say, enthusiastic commenters) and, if you are willing to accept a compliment without thinking it is in some way patronizing, I would say that "you have got it together."

En

En, thanks.

I dont think I have converted over to the other side. I still see mathematics as prior to physics. The naturalists see it the other way round. I am merely taking my definition of "reality" to not include mathematical abstractions. Perhaps the difference is just semantic, but it matches my understanding better.