The answer of Lenny Susskind was probably one of the most entertaining ones: Lenny's answer was a conversation with a stupid (or "slow", as he says) student of him who wanted Lenny to explain rigorously what probability is. The student wants clear statements: if something has a 50% probability, it will happen exactly in 500 cases out of 1000. Lenny tries to refine his viewpoint. Lenny knows that very unlikely things won't happen, but he can't prove it. Well, it's because it's not exactly true, is it?
The beliefs of many other people are even more bizarre: Carlo Rovelli is convinced that time does not exist, but he can't prove it. Similarly, Lee Smolin believes that quantum mechanics is not the final theory, but he can't prove it. I will discuss these two opinions below. Also, Philip Anderson believes that string theory is a "futile exercise" in physics, but he can't prove it.
Global warming
Stephen Schneider - the co-author of models of global cooling, nuclear winter, and global warming - is convinced that there exists global warming, but he can't prove it. Incidentally, 5200 years ago there was probably an abrupt climate change. According to the global warming theory, the major climate variations in the presence of humans are not natural (for example, effects of solar variations), but they're rather caused by the humans. Well, then it's not hard to deduce that Krishna together with the Egyptians who just discovered the first hieroglyphs in 3200 BC started to produce too many SUVs and power plants. This would normally lead to a destruction of the humankind, but fortunately the Minoan civilization established Greenpeace and saved the day.
Steve Giddings is more reasonable and says that the black holes conserve information, but he can't prove it. Anton Zeilinger says that the real lesson of quantum mechanics is to abandon the difference between reality and information, but he can't prove it. Lawrence Krauss believes that there are many universes. John Barrow adds that the universe must be infinite. Martin Rees believes that intelligent life has the power to spread throughout the galaxy. Paul Steinhardt believes that our universe is not coincidental. Richard Dawkins says that all "design" in the universe follows from the Darwinian selection, but he can't prove it.
Brain and consciousness
Roger Shank believes that people can't decide rationally about important things. Steve Pinker, on the other hand, believes that the brain contains circuits that are able to do much more than what has been useful for the humans so far. (Incidentally, I liked his lecture about the evolution theory behind religion.) Also, Christine Finn believes that the modern humans use their brains very efficiently. Alison Gopnik claims that the children have more consciousness than adults.
Rupert Shaldrake, the inventor of the morphogenetic field, claims that there are no fixed laws of nature - everything is just about the "memory" of the Universe. The things we believe are the natural laws are actually just "habits" that survived the natural selection. Unfortunately he does not say what are the physical laws that govern the natural selection of the habits, and therefore I predict that his theory is just a habit that will not survive the process of natural selection between the candidate theories. ;-)
And so on, and so on. You should look at it.
Neither of these beliefs is tremendously new, but in my opinion most of them are strange. I am an "atheist" concerning most of these belief systems.
If I were asked the religious question, I would probably answer that
- All continuous dimensionless parameters in fundamental physics may be calculated from a theory that does not need any dimensionless continuous parameters as input. The future scientists will view this theory of everything as refined string theory, but I can't prove it.
Lee Smolin vs. quantum mechanics
This also answers what I think about Lee Smolin's belief. Well, I believe that Lee is completely wrong, but I can't quite prove it. I should try anyway. Physics has undergone many revolutions in the 20th century - but the quantum revolution remains the most brutal one. It's just hard for most people - which, of course, includes the physicists in the most radical cases - to accept the big lessons of these important discoveries. But these 20th century insights just seem to be completely true and we will never "undo" them - but I can't prove it.
The Lorentz symmetry will remain as important as the rotational symmetry, for example, and any deep theory that deals with high velocities will have to explain why. There is no indication whatsoever that special relativity - as a constraint on physical laws in small enough regions of spacetime in which the space looks flat - is wrong. There is no indication of a violation of basic principles of quantum mechanics - by which I mean roughly the following:
- only the probabilities of various events may be calculated from the fundamental theory
- the probabilities must be calculated as the squared modulus of some "complex amplitudes", or as a sum of such terms (to allow for the density matrix)
- the complex amplitudes are matrix elements of some linear operators
- observables - such as the momentum - are given as linear operators on a Hilbert space
- the evolution operator is what determines the evolution, and it is another linear operator
- because the sum of all probabilities equals one, the evolution operators (or the S-matrix) must be unitary
I think that there is an overwhelming evidence that the theories based on these principles - quantum theories - are excellent in describing the real world. There exists a circumstantial evidence that no "deformation" of quantum theory may lead to a meaningful theory. It is possible that a particular physical theory only allows you to work with a subset of the concepts that appear in another quantum theory - for example, it only has the S-matrix or only the squared amplitudes i.e. the probabilities are directly expressed using density matrices. But in all these cases, the objects in the truncated theory must exactly satisfy the same rules that also follow from the quantum postulates above.
It seems to me that this picture may describe
- all ordinary experiments we've ever seen
- all mind-blogging experiments involving entanglement (David Goss has pointed out that these experiments used to be called "mind-boggling" in the pre-blog era, but you should not misunderstimate these experiments because of the typo haha)
- all quantum gravity experiments that we have not seen yet - string theory allows the black hole to be described within the postulates explained above
Because of these reasons, I think that insisting on the viewpoint that this basic quantum structure is inaccurate, approximate, incomplete is a kind of misconception. A result of prejudices from classical physics. And I don't think that the models and ideas proposed by various physicists today - e.g. Gerard 't Hooft or Lee Smolin - are better in any sense than the old failed attempts of Schrödinger, de Broglie, Einstein, Bohm, Bell, and others. It's still the same classical prejudice that drives this machinery. No pretty mathematics. No signs of an agreement with reality. Useless superconstruction trying to supersede quantum mechanics which is what really works. Aether.
Note that I tried to formulate the postulates in such a way that they should be acceptable by all physicists who work on actual physics as opposed to philosophy, regardless of their preferred interpretation of quantum mechanics. These "allowed" interpretations include:
- Feynman's neutral interpretation "shut up and calculate"
- the Copenhagen interpretation, regardless how you exactly answer the unphysical question "what is happening with the wavefunction before you measure and how should you interpret it"
- the Copenhagen done right, i.e. the Consistent Histories - which is the interpretation I choose
- the Many-Worlds-Interpretation
Of course, the forbidden interpretations are those that actually want to modify physics:
- de Broglie-Bohm's interpretation that adds new degrees of freedom (classical positions of particles) and tries to argue that some observables (such as position) are more fundamental than others (such as momentum). This interpretation has too many other problems and I don't have enough space to list them
- Penrose's interpretation of the wavefunction collapse in the brain induced by quantum gravity ;-). I am not sure whether it is constructive to comment on this idea
- Other attempts to imagine that the wavefunction is real and there are additional "nonlinear" effects that sometime induce a wavefunction collapse. I think it's extremely easy to prove that all these pictures are incompatible with physics we know, but I can't prove it right now
- Transactional representation in which one travels back and forth in time
- Wolfram's interpretation of quantum mechanics that says that we should forget not only quantum mechanics, but also other physics that we've learned since the 16th century - for example all physics that deals with continuous numbers. Instead, we should return to some trivial combinatorial games with stones - the cellular automata (CA) - and believe that these games have something to do with physics because Wolfram's book shows that they generate pictures reminiscent of a piece of tiger's skin, and therefore they undoubtedly must include the whole biology and physics
Lee Smolin repeats many times that quantum mechanics "makes no sense" to him regardless of the interpretation one chooses, and he has tried many. I apologize, but I see no other way to interpret these sentences than to say that Lee Smolin does not quite understand quantum mechanics and with this limited understanding he probably thinks that it's inconsistent. I am deeply convinced that it is not a religious question whether quantum mechanics is an internally consistent theory that moreover agrees with reality. It is a completely well-defined, scientific question and the answer is, of course, "it is", and Lee Smolin must be doing something seriously wrong if his answer is different.
Does time exist?
Carlo Rovelli believes that time does not exist. That's a kind of vague statement. One would have to define what "time" means in the most general theory, and what does it mean for this time to exist. I might agree if he said that time is not an exact concept after all. There is a simple yet powerful argument composed of two parts supporting the viewpoint that time is not an exact concept:
- the progress in string theory shows that space is emergent, approximate concept - it is just the manifestation of some particular light modes in your theory, but there can be many other light and heavier modes. T-duality is able, in fact, to exchange the translations in space with a phase redefinition of fields depending on their winding number. Whether something is space or not is a matter of convention, especially if this space is small
- special relativity that is still exactly preserved by local physics - at least if the space is large - says that "whatever holds for space, holds for time as well". This means that if space is emergent and approximate, time should be emergent and approximate, too
This is a nice exercise in logic, but it may be a futile one. Unlike Carlo Rovelli, I think that all theories that will ever be accepted to have something to do with physics will have to show that time emerges from them - because we simply know that time exists. Perhaps, time won't be an exact observable - and there won't probably be any general exact Hamiltonians in quantum gravity. On the other hand, a Hamiltonian could always exist with an appropriate gauge-fixing. But even if you work with the S-matrix that does not allow you to study the evolution "moment by moment", you can still exactly distinguish the future from the past. Time will always have to "reappear" in one way or another. Physics only emerges if we have time, at least some time. There is no physics without time.
When Rovelli tries to clarify his point of view in detail, I think that this clarification makes his proposal even more obsolete. He literally compares spacetime to the "surface of the water". The latter is eventually made of atoms, and so forth. Jesus Christ, this is not just analogous - it is exactly the 19th century idea of luminiferous aether. Everything that can carry waves must be made of "atoms", they thought. Einstein with his special relativity had to throw away this garbage of luminiferous aether - and it's exactly what makes Einstein so special. I believe that in the 22nd century, only a few very specialized historians of science will remember these specific "discrete" attempts to revolutionize physics, and they will consider these physicists as late 20th century followers of the wrong 19th century ideas of aether - who had nothing new to offer. But I can't prove it.
Smolin, Rovelli, and relationism
Both Rovelli as well as Smolin also talk about "relations" - everything must be given in terms of "relations". Such ambitious philosophical statements escape from my left ear as soon as they enter the right ear. I don't understand how these statements can mean anything new. One must have a Hilbert space (or whatever replaces it in your framework) and some degrees of freedom, and the physical Hilbert space may be constructed from a larger one, imposing some gauge symmetries. Maybe, instead, we will have some "bootstrap" mechanism to define the degrees of freedom - but Rovelli and Smolin don't seem to be talking about this approach. Until you say which gauge symmetries you want to include and which global symmetries should survive, you have not said anything. Some guesses can lead you somewhere, other guesses are useless. But it's not a scientific approach to convert one conceivable symmetry into a "religion".
Rovelli and Smolin obviously talk about the independence of physical laws on some large set of operations - a symmetry. But which symmetry it exactly is?
They seem to talk both about some ambitious "large" symmetry between all possible objects, but at the same moment it also seems that they are speaking about something as trivial and discredited as Mach's principle. Those who want to return us before general relativity to the age of Mach's principle in which "space" (or the metric tensor) did not exist without the presence of "objects", also seem to misunderstand the steps that Einstein had to do before 1916 in order to convert some vague philosophical ideas to a physical theory. Some of these steps have definitely invalidated many details of Mach's principle. These insights cannot be undone either: the metric tensor, in one approximation or another, will always exist as a legitimate physical entity even in vacuum, although its short-distance structure can become more complicated. Do they really want to revive Mach's principle? I think that the 20th century has brought us no new evidence - neither experimental nor theoretical - that would indicate that physics should return to the 19th century when Mach was promoting a similar idea with a similar lack of evidence.
Cumrun and Edward on foundations of QM
There were some comments about well-known physicists who believe that we still don't understand the real logic behind the foundations of quantum mechanics. I guess that the person talked about the Chapter 15 of The Elegant Universe, right? I don't actually think that Edward Witten and Cumrun Vafa would seriously question that the probabilities will be calculated as the squared complex amplitudes - which are matrix elements of linear operators. In Cumrun's recent attempts to re-explain Bell's inequalities using a "classically" looking framework that allows for negative probabilities, he still works with the partition sum of the black holes that equals Z_{top}^2 where Z_{top} is the topological string partition sum.
From a calculational point of view, even these radical approaches seem to satisfy the principles above. My feeling is that neither Cumrun nor Edward want to deform the structure of quantum mechanics to something inequivalent. My impression is that they want something that I also want - to find an equivalent way of looking at quantum mechanics that will also include the emergence of all other conceivable classical limits in different contexts. You know, the quantum postulates are not yet a theory of everything. They're just a framework and you must add a definition of your Hilbert space and your Hamiltonian or the S-matrix. String theory requires all postulates of quantum mechanics to be taken seriously, but the reverse does not hold: you cannot derive string theory from the postulates. It would be, of course, nicer to find a generalized form of the quantum postulates that would actually imply the whole string theory - or at least that would imply some unification of quantum mechanics and geometry.
Freeman Dyson and math
Freeman Dyson interpreted the question "what do you believe but can't prove" as a real mathematician, and he wanted to offer something even more interesting than Kurt Gödel. Kurt Gödel was able to construct a statement that is true, but cannot be proved using the pre-determined axiomatic system as long as this system contains the axioms about integers. Gödel's unprovable statement is essentially a convoluted, encoded version of the statement
- I am a happy statement that cannot be proved within the system you talk about.
Is this statement true, or false? Obviously, if it were false, then its opposite would have to hold:
- The statement can be proved within the system.
But if it were so and if the statement were provable, then it would have to be true - only true statements can be proved in a consistent system - and therefore you get a contradiction with the assumption that the statement was false. By assuming the consistency of the system, we proved that the statement can't be false. So it must be true. It's true and it says that it's unprovable within the system. Well, so it means that it is true and it is not provable within the original axiomatic system. Of course, it is however provable in a system that transcends the original axiomatic system - any system. What is this stronger system that transcends any older system? Well, it's called Luboš Motl's reference frame - because we just proved that the statement was correct. Thanks to my fellow, German Czech (these words are combined just like in "African American") Kurt Gödel for helping me with the technical part. ;-)
Back to Dyson. He wants to be better than Gödel. Dyson believes that Gödel's particular unprovable statement is not comprehensible to normal people. Well, maybe it is, but it is certainly not a useful statement - rather a sophisticated version of the liar's paradox.
Dyson's example is simple indeed:
- Write down all powers of two - 1,2,4,8,16, ... 1024 ... and write them backwards in the decimal system - 1,2,4,8,61, ... 4201 ... You will never find a power of five (5,25,125,625...) among these numbers.
Of course, there is a trivial possible error in Dyson's statement: you must eliminate "1" from the powers of two because it is also a power of five. ;-) But I don't want to be picky. Dyson argues that his statement should be true because the probability that a large random number - such as a large inverted power of two - decreases rapidly with the exponent. Because the few first examples are not powers of five, none of them will be a power of five assuming that the reverted numbers are kind of random.
For a mathematician, Dyson suddenly adds a very non-mathematically looking step:
- you see that I used some probabilistic argument which is not rigorous, and therefore there is no proof
Freeman Dyson apparently neglected one fact - namely that he is not the only mathematician in the world. If Dyson can't prove it, it does not imply that no one can prove it! If there is no proof based on the divisibility by 3,7, and 11, it does not mean that there is no proof whatsoever: but I can't prove it right now. Concerning Dyson's example, my belief has always been just the opposite:
- any well-defined and interesting statement in mathematics that deals with the "real" finite, countable objects - for example the objects that you can write on the paper (such as Dyson's integers) - can either be proved, or its negation can be proved, using "legitimate" mathematical logic, not necessarily constrained to some possibly insufficient axiomatic system
This would also mean that someone can either find a counterexample that is a power of two that reads like a power of five backwards, or someone can find a rigorous proof that such a thing can't happen. I believe that imagining that the human intelect may be insufficient to deal with some well-defined questions is just a defeatist prejudice without any justification, but I can't prove it. ;-) As far as I know, it's not possible to show that my religion about provability is incorrect.
I also believe, much like CIP, that Dyson's family got a little bit too much room in this happening, but I can't prove it.
