Conway's look-and-say sequence looks as follows:
If you started with "22", it would be a fixed point: "two 2's", "two 2's", and so forth.
Otherwise you can prove that only the digits 1,2,3 will appear anywhere in the sequence. More interestingly, the sequences may contain one of 92 different "audioactive atoms" composed out of 1,2,3 that don't interact with their neighbors. Numerologist Conway has jokingly identified them with chemical elements. For each digit larger than 3, if you allow them at the beginning, there also exist two "transuranic" elements.
The strings increase in size. You can prove that a new entry is asymptotically 1.303577... times longer than the previous one. This Conway's constant is actually the only real positive root of a known and "simple" polynomial of degree 71. ;-)
"Game of Life" and other fun things for adult children
Conway's inventions in recreational mathematics and his popularization pieces and books are far too numerous to enumerate here. He can calculate the day of the week in two seconds or so, using his Doomsday Al Gore Rhythm. Also, Conway's "Game of Life" remains the most popular cellular automaton although the suggestion that the game contains all essentials of life is surely and dearly exaggerated.
John Conway, a worker in the classification of finite groups, is also a big shot in the sporadic groups, the father of the Conway groups, and a co-father of monstrous moonshine involving the monster group.
Free will theorem
Recreational mathematics may be fun but it is somewhat orthogonal to the main interests of this blog. And we have already spent quite some time with finite groups (and their application to gravity in AdS3) on this blog anyway.
Instead, I want to mention the free will theorem, a robust result about the foundations of quantum mechanics that Conway proved in 2004, together with Simon Kochen, a colleague from the Princeton math department. It is a fascinating and crisp modern version of the no-go theorem for hidden variables that shows that Conway really understood what quantum mechanics is all about and why you should never mess up with its probabilistic character and with the apparent fact that the individual random results of microscopic measurements cannot be explained by any mechanism.
The no-hidden-variables theorems usually show that there can't be any hidden variables that would "materialistically explain" the otherwise random outcomes of quantum experiments as long as you agree that it is impossible for the actual physical information to propagate at arbitrary speeds. Because Bell's inequalities (or, alternatively, assumptions that would lead to "paradoxes" via the Kochen-Specker theorem) are violated in Nature, it follows that a hidden variable theory has to be non-local. You have heard me saying that such a non-local theory would inevitably be acausal and Lorentz-breaking - which really means unacceptable - and Conway and his collaborator have indeed proved this implication.
The free will theorem is a very cute sharpened reformulation of the hidden variables no-go theorems that can be phrased in the following way:
If experimenters have free will, then so do elementary particles.Because this statement may sound too entertaining, let me emphasize that it is not a caricature of the theorem. They actually prove nothing else than the exact, most obvious interpretation of the sentence above.
So if you accept that an experimenter's decision to press particular buttons during his experiments has no calculable "materialist" explanation - if the experimenter's decisions during the experiment are not functions of his history, heritage, or environment, allowing us to say that his decisions resulted from the "free will" - we must also accept that there can't be any "materialist" explanation of a particular behavior of elementary particles: particles have the "free will", too. The outcomes of events involving microscopic physical systems must be truly random and unexplainable by the environment, too, which may be interpreted as a "free random decision" of the particular particles. There can't be any hidden variables that would dictate particles how they should be measured in particular experiments as long as you accept that there are no hidden variables that would dictate experimenters what buttons they should press or what buttons they are allowed to press under certain circumstances.
quantum xerox no-go theorem a month ago), and the cute old Kochen-Specker combinatorial proof that one can't define 101 functions i.e. assign consistent colors to certain 33 axes in space. They show that hidden variable theories and also a bizarre scenario by Ghirardi, Rimini, and Weber (GRW) are impossible. If you don't remember The Fabric of the Cosmos by Brian Greene, the GRW theory suggests that the classical character of macroscopic physics is achieved by "stochastic hits", occassional spontaneous "collapses" of the wave function of some kind, induced by weird non-linearities added to the Schrödinger equation. Incidentally, GRW has over 750 citations - it is a whole industry of pseudoscience. The result by Kochen and Conway is thus stronger than Bell's theorem because it falsifies two unprofessional (or "unromantic", using Bell's jargon) classes of interpretations of quantum mechanics instead of just one.
Conway & Kochen 2006, original theorem (31 pages), published in Foundations of Physics
Bassi & Ghirardi 2006, confused critics (16 pages)
Tumulka 2006, another confused critic (10 pages)
Conway & Kochen 2007, a reply to critics (5 pages)
Conway and Kochen show that the only way how GRW might be right is that the stochastic hits not only control the particles' behavior but also all experimenters' actions which seems as a fantastic enough possibility for us to say that it is impossible. They emphasize that the problem with the hidden-variable and GRW theories is not that randomness is needed to reproduce the real observations. What they really prove is that there can't be any underlying reason - deterministic or otherwise - behind the particles' behavior in a given situation or experiment: you really have to admit that their random behavior is not a function of any data in their past cone: it depends on their "free will". With this language, Conway and Kochen put a positive spin on this philosophical conclusion: it is all about freedom and the hidden-variable theorists as well as GRW are really freedom-haters. ;-)
Bassi, Ghirardi, and Tumulka didn't really understand anything but their criticism has convinced Kochen and Conway to make their proof much more efficient. In the last preprint, they only assume a simple "MIN axiom" about the experimenters' free will. And they only spend half a page in proving their theorem. It is this minimal version of their proof that I have always had in mind - and independently found - when I argued that the "unexplainable" probabilistic interpretation of quantum mechanics is both necessary and sufficient to preserve special relativity.
Much like in the case of the arrow of time, the issue of "free will" in quantum mechanics, if you allow me to use this jargon, seems to reveal that some people simply can't get rid of a certain religious zeal about these matters. They just believe that some explanation that denies the "free will" and has a "materialist" description of the real state of the physical system must be possible. They believe that if one particular model of this kind fails, there must exist a slightly modified model which is fine. However, people like Kochen and Conway prove that this whole huge class of models - the whole way of thinking and all possible modifications of the hidden-variables or GRW models that hundreds of generations of physicists and philosophers could be constructing for thousands of years - can be safely ruled out by a half-a-page proof.
Ghirardi, Bassi, Tumulka et al. just don't seem to be able to swallow such a result. See Tumulka's habilitation thesis posted in November 2007 that shows no progress whatsoever. Another recent confused paper of this kind was written by Daniel Bedingham. One year ago, Gerard 't Hooft has also offered his own analysis whether John Conway had the free will to throw his coffee across the room. However, I am afraid that 't Hooft has only discussed how the free will would look like in the old-fashioned, philosophically trivial deterministic world rather than the real world around us.
But there is also a difference between Ghirardi and Bassi on one side and Tumulka on the other side. Tumulka believes that a GRW-like theory may be fully Lorentz-invariant which is clearly wrong: his GRW-like "counterexample" to the Kochen-Conway theorem has no interactions which is a huge defect because interactions are essential for a "measurement" to occur and for us to have any problem to solve and data to explain at all. On the other hand, Ghirardi and Bassi agree that such a theory can't be fully Lorentz-invariant. Instead, they hope that their future theory's predictions will be effectively Lorentz-invariant in a stochastic sense but one reference frame is fundamentally privileged.
Kochen and Conway have not yet been able to show that such a stochastic agreement assuming only one fundamentally privileged reference frame is impossible - and it is quite likely that it is logically possible (just imagine a "simulator" of a real quantum world with some random changes designed not to change the statistics) - but they defend a principle first articulated by Pierre Curie that a theory should have all the exact symmetries that seem to be exact experimentally. I agree with this principle. For example, if there is no experimental way, not even in principle, to find a privileged reference frame, our theories shouldn't have one either.
Can I rigorously prove Curie's principle? Not really. Why do I believe it? Because the experimental validity of a symmetry is an important and general insight and a powerful constraint that should be taken seriously, much like (and perhaps more than) other forms of empirical data, while all attempts to show that a known symmetry is an approximate coincidence is nothing else than a belief that a "miracle" will cure something that otherwise looks like a contradiction between a theory and experiments.
But Occam's razor tells us not to believe such "miracles" and to eliminate non-trivial constructions predicting new otherwise unnecessary phenomena that seem to be absent in the empirical world - unless we have a good reason. Also, Gell-Mann's totalitarian principle teaches us that a theory that fundamentally disrespects a symmetry is likely to violate it substantially, not just by a tiny and unmeasurable amount. It is this combination of empirical reasoning, Occam's razor, and Gell-Mann's principle that leads me to accept Curie's assumption, too.
Now, this assumption can't be a universal dogma: indeed, we know various symmetries (such as the baryon number conservation) whose violation hasn't been seen experimentally so far but seems likely anyway. But in the case of the baryon number, its violation is likely because one can't construct satisfactory theories of high-energy phenomena (or quantum gravity) that would exactly preserve the baryon number (except for theories where it is a gauge symmetry with an unnaturally small coupling constant). The baryon number violation seems to be a derivable assertion. On the other hand, we don't seem to gain any explanatory power from a hypothetical GRW-like violation of the Lorentz symmetry. The probabilistic explanation involving the "free will" just seems to be a more successful description of known experiments than the awkward fine-tuned theories with effective symmetries that are not symmetries at the fundamental level.
There is another, complementary difference between the accidental baryon number symmetry and the accidental Lorentz symmetry in theories of the GRW type: the former can be proven to approximately hold in the Standard Model at low energies - this accidental symmetry is thus not a real "miracle" - while the latter has no known derivation, except for one based upon wishful thinking and philosophical prejudices.
Proper science vs dogmas: once again
We should believe the theories and philosophical principles that actually seem consistent with the observed phenomena, whether their philosophical background is psychologically convenient to us or not. In science, the question such as "does a particle have the free will?" is not and cannot be answered a priori. A physicist must be a priori open-minded about the answer. When he properly and logically analyzes the results of experiments involving entanglement, he finds out that the particle simply does have and must have the "free will", whether this answer was pleasing or expected or not: an underlying explanation of a particular random outcome of a microscopic experiment simply cannot exist, not even in principle, whether you like this conclusion or not.
The wrong assumption of a hidden "explanation" beneath every microscopic event could have been compatible with classical physics but quantum physics clearly shows that it was a wrong intuition based on our limited, approximate, classical everyday experience. Quantum mechanically, particles do have the free will!
Science is about a fair and balanced treatment of possible answers, about logical analysis and experimental verification. It is not about preconceived dogmas about the answers and ad hoc definitions of "science" or "good science" that might implicitly include incorrect assertions about reality.
And that's the memo.