Monday, 15 October 2007

Is ultimate reality a mathematical structure?

I have always liked the style of Max Tegmark. He combines his expertise and witty language with courage to pick pretty unusual (for mainstream science) topics. In a couple of papers Shut up and calculate and The Mathematical Universe, one short and dedicated even to laymen, the other much longer and technical exposition) he presents a rather radical hypothesis: the universe itself is a mathematical structure, or rather multiple mathematical structures.
Although he openly admits that he is not the first to propose the ideas, the way they are presented is quite interesting. The proposal is based on two hypotheses. The first of these is close to my heart, while the other one, is, hmmm, courageous. What are the two hypotheses?

External Reality Hypothesis (ERH):
There exists an external physical reality completely independent of us humans.
Mathematical Universe Hypothesis (MUH):
Our external physical reality is a mathematical structure.

ERH seems to be quite widely accepted, although in the light of Quantum Mechanics results we must remember that the external physical reality might be very different from our intuitions. It might be nonlocal, unreal, quantum - pick the name you prefer. The key point, which I hold dear, is independent of us humans.

Now, the aim of the papers is to argue for the necessity of MUH. There are great reasons for this: it would solve the mystery of why is the Universe, in so many of its aspects, so well describable by mathematics.
It could also change our perspective on the Theory of Everything and science:
If the mathematical universe hypothesis is true, then it is great news for science, allowing the possibility that an elegant unification of physics and mathematics will one day allow us to understand reality more deeply than most dreamed possible. Indeed, I think the mathematical cosmos with its multiverse is the best theory of everything that we could hope for, because it would mean that no aspect of reality is off-limits from our scientific quest to uncover regularities and make quantitative predictions.

However, it is even more difficult to break the bounds of our limited imagination and intuitions and perceive our universe as some mathematical structure, which is by definition an abstract, immutable entity existing outside of space and time. What is `mathematical structure' anyway? Set of abstract objects and rules that connect them? How complex should that structure be to describe the seemingly infinite variety of our observations, the probable complexity of the Universe? Most people, even physicists and mathematicians, would not venture into his abstract space at all.

But - why not? Our human perspective is rather limited and inadequate. Even for `almost normal' phenomena. It suggests that Sun circles the Earth when we observe it moving across the sky. It does not help us in understanding how a light bulb works, or a hard disk in our computer. Why should our intuitions be more usable when we talk about the question of ultimate reality? Tegmark even uses our inadequacy as a `proof' of the Mathematical Universe hypothesis (although I detect some measure of tongue-in-cheek there):
Ultimately, why should we believe the mathematical universe hypothesis? Perhaps the most compelling objection is that it feels counter-intuitive and disturbing. I personally dismiss this as a failure to appreciate Darwinian evolution. Evolution endowed us with intuition only for those aspects of physics that had survival value for our distant ancestors, such as the parabolic trajectories of flying rocks. Darwin’s theory thus makes the testable prediction that whenever we look beyond the human scale, our evolved intuition should break down.

We have repeatedly tested this prediction, and the results overwhelmingly support it: our intuition breaks down at high speeds, where time slows down; on small scales, where particles can be in two places at once; and at high temperatures, where colliding particles change identity. To me, an electron colliding with a positron and turning into a Z-boson feels about as intuitive as two colliding cars turning into a cruise ship. The point is that if we dismiss seemingly weird theories out of hand, we risk dismissing the correct theory of everything, whatever it may turn out to be.

It is like saying `it must be true because I do not understand it'. Well, perhaps it is worthwhile to remember our limitations and to be sure that we do not dismiss some possible solutions because of these limitations.

But the papers are certainly worth reading.


xawer said...

Ex-experimentalist's comments, based (once again) on affection towards Lower Austria.

I esteem Niederösterreich not only for a soil to grow my favourite wines, but also as it gave a soil for my favourite philosophers: from Ludwig Wittgenstein, through Wiener Kreis to Karl Popper.
So I must take Tegmark's Mathematical Universe Hypothesis as either:
a) Definition rather than hypothesis introducing new mathematical concept of physical reality with its meaning limited to some kind of mathematical structure;
b) yet another vote in "problem of universals" discussion, placed close to Plato's. I'am really surprised that 800 years after Abelard and 80 years after Wittgenstein we can find some embers still burning in those ashes;
c) metaphysical gibbering, manipulating your minds to accept some misty implicit ideas ambushed in words like 'is' and 'exists'. I won't be so brave to compete against Stanisław Lem in the field of demonstrating the sterility and lack of meaning in such issues: read Ijon Tichy's Journey to Tairia from "Star Diaries" ('potato IS spry').

Only the last option deserves for a bit more elaborated critique. God save me from stepping into metaphysical discussion: I rather prefer to accept temporarily Plato's realism to point out weaknesses in Tegmark's argumentation.
1. Tegmark introduces (Math.Univ. II.D) the form 'is' instead of 'is described by', using arguments like isomorphism. I really like it. The mathematics is based on such simplifications, it couldn't exist without it. So we are now at option a). But all the rest of the paper is devoted to fishing the philosophical consequences of implicit platonian meanings of the word 'is', the meanings, which are not included in original 'is described by' nor in 'is equivalent to' nor in 'is isomorphic', nor... Once again (the problem of indo-european languages) the copula 'is' inflates too easily into metaphysic essences. There paper presents no justification for such interpretation.
2. "Testable predictions" of MUH: we should expect to find ourselves in a rather typical Level IV Universe. It is not a testable hypothesis! What does it mean 'typical Level IV Universe'? How to test it?
Maybe Karl Popper had more luck than we have. He had had opportunity to listen to Albert Einstein's lecture on GTR, precisely predicting light deflection, and a year later got Eddington's photos on eclipse. That was testable prediction!
2a. Even if the 'how to test' problem is solvable, the fundamental question remains: why do we expect to live in 'typical' Universe? Due to statistics? We have just one experimental result of infinite set. Even strongest anomalies may be explained by generalised anthropic principle in accordance with MUH (implying that continuum of equivalent Level IV universes must exist).
3. Tegmark's idea about 'pure mathematics with no baggage' (Math.Univ. II.B) is just a Hilbert's dream. There is no 'pure mathematics'. Even most formalized systems, like Euclidean geometry or number theory base on some relations to real world and some baggage of explanations addressed to our intuitions. Euclidean 'straight line' is only a simple idealisation of a trace made with a stick in wet sand on Alexandria beach. Στοιχεῖα would be meaningless without its definitions.
4. Amount of information in our Universe. Tegmark assumes as obvious it is really big - like googol of bits. Why this assumption? Is the amount of true information so high? Or maybe we are not smart enough to distinguish 3141592654... from truly random googol of digits? Maybe informational entropy is quite small, just to be printed on Tegmark's t-shirt? And all our science should be focused on finding the very simple rule generating this infinite stream of digits, passing (so far) all our randomness tests?

Anyway, I have one Austrian point for Tegmark (Upper Austria this time, but there they also have some good wines) - Anton Zeilinger, and his postulate: 'papers should be sexy'. No complaints.

Wanderer in the country of blindfolded said...

When you say
"Even most formalized systems, like Euclidean geometry or number theory base on some relations to real world and some baggage of explanations addressed to our intuitions." what is the baggage - apart from the language we use, to communicate / record in an accessible form these notions and constructs? As Goedel has proven it is possible to transform any mathematical structure into string of 1's and 0's. Of cource one may argue that even a notion of a bit is a human laden baggage, but I guess this would be stretching the argument a bit too far. While the euclidean geometry or number theory written down in series of (rather long) 0/1 strings would be certainly totally incomprehensible to me, it would be no less rich in terms of relationships between the elements. And, in the Ithaca Interpretation of Quantum Mechanics ( a lovely paper by David N Mermin, now roughly 20 years old) the topic of physics is postulated to be "just a set of relationships".

As for the information content: perhaps is is true that the world is discrete, that real numbers are only a construct not applicable to reality. Can anyone PROVE that any physical function/property is by necessity continuous? This would obviously help in reducing the information content. As Chaitin has shown, to encode a "general" real number one not only needs an infinite string of bits, but also there are infinitely many real numbers that require an infinitely long computer program to encode. So, if the Universe is continuous (in the sense of real numbers) the information content is infinite!

I defend Tegmark's right to be provocative: his papers (not just the two mentioned) are sexy, indeed. Even when the topics seem to be barely warm ashes.

xawer said...

Gödel's numerical coding for mathematics is such baggage. Baggage you've brought from the book written in human language.
I wonder if anyone even tried to express Gödel's Theorem in Gödel's notation? If so - we would just see a really big natural number, absolutely meaningless without Gödel's explanation.
Like The Meaning of Universe is included in number 42 only if we have proper interpretation tool.
There is no particular coding obviously obvious as 'natural to mathematics'. Gödel's coding is not much better than Knuth's coding: ASCII string containing TeX representation of the mathematical theorem. In any case we need either Gödel or Knuth to teach us how should we interprete those plain numbers as structures of relations. Without commiting the sin of self-reference the interpretation rule cannot be embeded in such number itself.

Provocative Tegmark: I really appreciate such provocations. We need more such guys. I like him much more than most of philosophers from philosophy departments. Wrong (or void) idea expressed with good style in a clear language is uncomparably better than Hegel. At least I am able to discuss with it.
But I would like him even more, if his papers would be found on a shelf 'philosophy/metaphysics' rather than 'science/physics'. On the later I would expect something a little bit more conforming to popperian falsifiability.

Wanderer in the country of blindfolded said...

Just one question: what is wrong with big natural numbers? Especially bit-coded? Such number may be (and in fact Goedel number IS) instructions to be decoded. Letting go of the baggage is absolutely unnatural for us, because the baggage IS us. But are we the absolute measuring stick of the Uni(Multi)verse?

xawer said...

Absolutely nothing is wrong. I love big natural numbers (especially primes).
My point is that the number itself has no meaning nor carry any structure. The meaning is not inherited in the number itself. We must assign the meaning by applying some interpretation rules. The coding scheme must be provided externally and arbitrary, while there is a continuum of different, equally valid decoders.
Does the number 42 mean:
- The Ultimate Meaning of the Universe (Douglas Adams's decoder)?
- the abstraction class of the sets bijective to { {}, {{}}, ... 41 left-braces 41 right-braces } (Cantor's decoder)?
- * [asterisk] (ASCII decoder)?
- invalid statement (Gödel's decoder)?
- Czech republic (telephone prefix decoder)?
- r8 <- r8-m8 (Intel Pentium instruction set coding)?
- holes in a paper tape in Turing machine (I've no idea what
Turing's machine should do with a tape containing ..*.*.*.*.)?
- whatever else you wish (I'll provide appropriate decoder)?

Wanderer in the country of blindfolded said...

Ahh, finally we touch the fundamental meaning of the Universe! And of Tegmark's proposal. As far as I understand it, he claims that our Universe is a specific example of the mathematical structure, by structure meaning not only values but the decoding scheme as well.

Consider a parallel (slightly twisted) physical example: why does a stone fall when you throw it? There are many possible explanations: devils push it down, angels lose interest in pushing it up, it has natural tendency to unite with Earth, it obeys Newtonian gravity laws, it obeys Einstein's GR laws, it obeys some yet unknown QM-GR combined laws...
We have moved from one `explanation' to others because of the predictive powers and accuracy.

Now let's move to the `42' example. Some of the decoders are `obviously' human baggage. (Unless there is some truth that the prefix for Hungary is a fundamental Universe property). But some might not be, for example Goedel decoding (based on Cantor?), or Turing machine (which?). But the point is - just as with the falling stone we would not deal with a single instance. The ToE has to explain all (or nearly all) instances. Then the weaning of the decoders could be done - as has been in traditional physics.

The aliens of Douglas Adams, asking their computer the ultimate question without preparing the communication process ar emuch less intelligent than Stanislaw Lem humanity in Golem XIV. Which was as `decoupled' from humanity as the alien computer, but at least some `translation' service has been provided.

The language of not-so-modern physics: particles, fields, quanta, still used in most of its branches changes when you go to the most abstract disciplines, trying to combine QM and GR. Then the mathematical focus becomes more and more visible. Of course it is because we lack any intuitions dealing with M-Theory landscape. But could we not, for a moment, assume that we may be getting nearer the underlying structure of the universe?