No half a bit of info

I argue below that A photon is not a math object. I elaborate with a simple analogy.

Say I have a piece of paper with a 0 and 1 on it. I tear it in half, and mail the pieces to London and Australia, so one gets a 0 and one gets a 1. How many bits of info are in the London envelope? One, of course, as it has a 0 or a 1. How many in the Australiz envelope? Also one, but now it gets tricky because we don't have two bits of info overall.

So there is just one bit of info overall, but it does not make much mathematical sense to say that each envelope has half a bit of info. What is half a bit? It seems like a whole bit to the guy in London.

The spin of an entangled electron pair is similar. You can separate the electrons and find that each has a spin that is easily understandable by itself. But somehow the pair has a collective spin that is not so easily divided into two spins.

When the guy in London opens the envelope, he does not have any spooky nonlocal effect on the Australia envelope. There is nothing mysterious at all. You could explain it to a child. It only gets tricky when you try to apply certain mathematical quantifications, such as asking how many bits of info are in the London envelope, or asking what the probability is of a 1 in that envelope. (It seems like a probability of 0.5, but the answer seems to change as the Australian envelope gets opened.)

Likewise, the physics of entangled electrons is not tricky or mysterious. It only gets tricky when you try to match up the physics with mathematical quantifications of spin.

I say that the best explanation is that the electron does not have an adequate mathematical quantification. Just like the London envelope.

This would all be fairly trivial except for physicists who insist on saying all sorts of crazy things about the entangled electron. They say that the electron has a spooky effect on distant electrons, or that there is no reality. And they say those crazy things because they assume that the electron has a state that is completely characterized by mathematical quantifications.

Quantum crypto on aeroplane

NewScientist reports:

AN AEROPLANE has beamed quantum encryption keys to a station on the ground, paving the way for an ultra-secure global communications network.

Quantum key distribution (QKD) uses photons polarised in two different ways to encode the 0s and 1s of an encryption key. The laws of quantum mechanics ensure the transmission is secure, as any attempt to intercept the key disturbs the polarisation - a tip-off to the receiver that the key has been seen and should be discarded. ...

Back on the ground, researchers at the European Space Agency's Optical Ground Station in Spain's Canary Islands broke the distance record for quantum teleportation, which uses quantum entanglement to make message transmission more secure (Nature, doi.org/jbs). They teleported quantum states over 143 km, breaking a record of 97 km reported earlier this year.

I have criticized Quantum cryptography on this blog many times, such as in Aug 2011, Jun 2011, May 2011, Apr 2011, Sep 2010, May 2010, Jan 2010, and Oct 2008. It is the biggest scam in modern physics.

There are three big problems with quantum cryptography: (1) it assumes equipment idealizations that can never be met in practice; (2) the laws of quantum mechanics do not ensure any security; and (3) even if the theory and practice could do everything promised, the method has no practical utility.

Mathematicians care about proofs

In my FQXi essay, I argue that mathematics is the same as what is provable in Zermelo–Fraenkel set theory (ZFC), and hence subject to the limitations of such proofs. This view is conventional wisdom, so I did not give much support for it. But I just saw another view on another blog.

Economist and mathematician Steve Landsburg writes:

The (really really) big news in the math world today is that Shin Mochizuki has (plausibly) claimed to have solved the ABC problem, which in turn suffices to settle many of the most vexing outstanding problems in arithmetic. ...

No mathematician would consider rejecting Mochizuki’s proof just because it relies on new axiomatic foundations. That’s because mathematicians (or at least the sort of mathematicians who study arithmetic) don’t particularly care about axioms; they care about truth. ...

Mathematicians care about what’s true, not about what’s provable; if a truth isn’t provable, we’re fine with changing the rules of the game to make it provable.

That in turn implies that there is such a thing as mathematical truth, independent of what we can prove.

No, this is contrary to two millennia of mathematical wisdom. Mathematicians only care about what is provable. For them, the truth is what is provable. Nobody changes the rules of the game.

Since Euclid axiomatized geometry, a geometry truth meant a theorem with a proof from the axioms. Sure, people have invented other geometries with other axioms, but that has not changed fundamental ideas about mathematical truth or the necessity of proof.

Mochizuki's proposed proof of the abc conjecture is being taken seriously here, here, and here.

Mochizuki's own papers (pdf) say:

In the following discussion, we shall work with various models — consisting of “sets” and a relation “∈” — of the standard ZFC axioms of axiomatic set theory [i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice — cf., e.g., [Drk], Chapter 1, §3]. We shall refer to such models as ZFC-models. ...

We shall refer to a ZFC-model that also satisfies this additional axiom of the Grothendieck school as a ZFCG-model. ...

Although we shall not discuss in detail here the quite difficult issue of whether or not there actually exist ZFCG-models, we remark in passing that one may justify the stance of ignoring such issues — at least from the point of view of establishing the validity of various “final results” that may be formulated in ZFC-models — by invoking a result of Feferman [cf. [Ffmn], §2.3] concerning the “conservative exten- sionality” of ZFCG relative to ZFC, i.e., roughly speaking, that “any proposition that may be formulated in a ZFC-model and, moreover, holds in a ZFCG-model in fact holds in the original ZFC-model”.

So in fact Mochizuki cares very much about proving his theorems in ZFC, even if he likes to use tools that go outside ZFC. Mathematical truth is what is provable in ZFC.

Update: In the comments on his blog, Landsburg says that algebraic geometers cheerfully cite the work of Grothendieck without worrying about foundational issues. Sure, mathematicians like to specialize, and most of them do not study foundations. Algebraic geometers especially have a cult-like reputation of ignoring others. The Italian school of algebraic geometry is famous for making sloppy assumptions that are sometimes wrong. But even still, today's algebraic geometers do not condone changing the rules of the game because some (alleged) truth is not provable.

Saying that algebraic geometers cite Grothendieck without worrying about foundations is about like saying that they don't worry about differential equations. Yeah, they would rather let someone else worry about the differential equations. But that does not mean that they don't care about what is provable.

Update: Wikipedia says:

Grothendieck originally developed étale cohomology in an extremely general setting, working with concepts such as Grothendieck toposes and Grothendieck universes. With hindsight, much of this machinery proved unnecessary for most practical applications of the étale theory, and Deligne (1977) gave a simplified exposition of étale cohomology theory. Grothendieck's use of these universes (whose existence cannot be proved in ZFC) led to some uninformed speculation that étale cohomology and its applications (such as the proof of Fermat's last theorem) needed axioms beyond ZFC. In practice étale cohomology is used mainly for constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with a little care it can be constructed in this case without using any uncountable sets, and this can easily be done in ZFC (and even in much weaker theories).

So ZFC is adequate for mathematicians.

Update: For another contrary view, John Horgan once wrote a SciAm article on The Death of Proof. It was very unpopular among mathematicians.

Update: The above explanation is not entirely correct, according to these matheticians:

Ulrik Buchholtz on September 7, 2012 said:
It seems to me that Mochizuki defines species, mutation and morphism-of-mutation as being whatever defines a category, functor or natural transformation in any model of ZFC. Thus, by the completeness theorem, these are nothing but formalizable-in-ZFC categories, functors, and natural transformations. ...

On another note, on p. 43 Mochizuki seems to claim that Grothendieck set theory is a conservative extension of ZFC, which it certainly is not. What the Feferman-paper he refers to shows, is that an extension of ZFC with a constant for a universe satisfying schematic reflection, ZFC/s, is a conservative extension of ZFC, and that’s obviously because every model of ZFC can be extended to a model of ZFC/s (using a Löwenheim-Skolem construction), but a model of ZFC/s need not be a model of Grothendieck set theory.

Terence Tao on September 8, 2012 said:
On reading the Mochizuki paper, it seems to me (if I am not mistaken) that all the set theoretic and model theoretic stuff is really only used in Section 3, whereas the ABC conjecture is proven (at least allegedly) by Section 2. As far as I can tell, there are no forward references to Section 3 in previous sections other than side remarks. So perhaps all the set and model theory here is in fact something of a red herring as far as the application to ABC is concerned, and are primarily relevant for further development of Mochizuki’s inter-universal geometry instead? (Among other things, this would render the issue of the non-conservative nature of Grothendieck set theory somewhat moot.)

The point remains that all these mathematicians are interested in proving the abc conjecture in ZFC. If mathematicians were really willing to lower their standards of proof, then they would accept this heuristic argument. More info on the proposed proof is here.

A photon is not a math object

I posted below on the limits of math, and now I want to give a physical example.

Consider a simple photon. It would seem to be completely describable by its direction (velocity), frequency (energy, wavelength, color), and spin (polarization). The photon can then be considered synonymous with a few numbers. This is not really possible because of the uncertainty principle, but physicists like to pretend that there is an underlying mathematical reality.

Now suppose you have two entangled photons that are emitted in some process that assures that they are equal and opposite. So their.y frequencies are the same and their velocities and spins are opposite. Measuring one tells you that you will get the same outcome if you later do the opposite measurement on the other.

The trouble is that even if the photons are miles apart, there is no way to give separate mathematical descriptions of them.

These entangled photon experiments are sometimes cited as proof of nonlocality. But there is no physical nonlocality. There is nothing that you can do to one photon that will have any physical effect on the other photon. The apparent nonlocality is just an artifact of the mathematical models.

The problem is that when a quantum of energy splits into two opposing photons, the mathematical description of that quantum does not split into separate mathematical descriptions of those photons. The photons are physically separate, but not mathematically separate.

My conclusion from this is that the mathematical description of the photon is inadequate. Others conclude that there no such thing as causality, or that there is no reality, or that physics is spooky. These latter conclusions seem crazy to me.

The current RadioLab podcast explains how locality is essential to our everyday experiences. Life is like being at the end of a slinky, in the words of the show. Any locality violation would be huge news.

Usually when mathematical models fail, the solution is to look for more accurate models. But the failure to describe a photon is not an accuracy problem.

The uncertainty principle says that you cannot measure the position and momentum of a particle at the same time. A particle cannot be defined by its position and momentum. Your first reaction to this principle was probably that it was just a statement of our lack of cleverness in making measurements. But it is not. The principle is a statement that physical reality does not match our mathematical descriptions.

Likewise, entanglement shows that the physics does not match the mathematics.

There are physicists who argue that the whole universe has a completely mathematical description. In fact we have no such description for even a single photon. Or a single electron. And much of 20th century physics says that there can be no such description.

Limits of math

My FQXi essay is now being publicly discussed and rated, as contest submissions are closed. I am not sure I addressed these objections fully enough. The argument is:

Using mathematics to model physical reality cannot be a limitation because mathematics allows arbitrary structures, operations, and theories to be defined.

No matter what new phenomena we discover, it will always be possible to describe our observations mathematically. It doesn’t matter how crazy nature is because mathematics is just a language for expressing our intuitions, and we can always add words to the language.

This is an incorrect view of math.

Math has been axiomatized as Zermelo–Fraenkel set theory (ZFC). The content of math consists of the logical theorems of ZFC. It is not just some language or some arbitrarily expandible set of intuitions.

I wonder whether many physicists understand this point. I doubt it.

I thought that everyone understood that math has limits, and the point of my essay was to argue that there might be no ultimate theory of physics within those limits. But if someone does not even accept that math has limits, then the rest of the argument is hopeless.

If you agree that math has limits and that they may not include possible physical theories, then it ought to be obvious that nature may not have a faithful mathematical representation. Saying that there is such a representation is an assumption that may be unwarranted.

So I guess I should have explained the ZFC issue better to the physicists who will be judging my essay.

An example of a limit of math is the unsolvability of quintic polynomials by radicals. That is what keeps us from having something like the quadratic formula for more complicated equations. This fact does not necessarily stop us from solving the equations, but certain kinds of formulas just won't work.

My essay "public rating" is not very high, but my "community rating" must be much higher. If you look at the list of essays sorted by community rating, then you will see that mine is near the top.

Update: Bob Jones argues below that ZFC is not enough because we need mathematical constructs like the Grothendieck universe that are outside ZFC. That is an interesting example, as noted here:

Colin Mclarty says that the big modern cohomological number theory theorems, including Fermat’s Last Theorem, were all proved using Grothendieck’s tools, making use of an axiom stronger than Zermelo–Fraenkel set theory (ZFC). He says that there is a belief that these theorems can be proved in ZFC, but no one has done it.

I would really be surprised if cohomological number theory needs something more than ZFC. And it would be even more amazing if theoretical physics found some mathematization of the universe that could be formalized in an extension of ZFC, but not ZFC.

Born rule is an interpretation

It is common to say that quantum mechanics is fundamentally based on the probability amplitude:

The principal use of probability amplitudes is as the physical meaning of the wavefunction, a link first proposed by Max Born and a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the wave function were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation was offered. Born was awarded half of the 1954 Nobel Prize in physics for this understanding,[1] though it was vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein.

Probability was just Born's interpretation, and not essential to the Copenhagen interpretation.

I have two simple reasons for saying that quantum probability is just an interpretation.

If an electron wave function tells you the probability of the electron's location, then you are implicitly assuming that the election is a particle. But the electron is not like a classical particle, and thinking of it as being like a particle is just an interpretation.

If the theory only predicts probabilities, then it cannot predict individual events. We cannot observe probabilities directly, so we can only confirm them by redoing experiments many times. Such an interpretation is weaker than one which predicts individual events, such as the positivist interpretation.

So I say that quantum mechanics needs no probabilities, not even for the double slit.

Polanyi against positivism

One of the leaders of the 20th century philosophical attack on positivism was the Hungarian Michael Polanyi. He is known mainly for being the one to convince Thomas Kuhn that science was not really scientific, and operates by paradigm shifts instead. Here is how they used Einstein to refute positivism:

But Polanyi devotes only a few pages to these matters, for his main proof depends on what he calls “the story of Relativity.” That theory was indeed taken by the positivists to show that through instrumentalist thinking Einstein had freed l9th-century physics from its metaphysical underpinnings, and thereby made the breakthrough to modern science. Polanyi correctly points out that every textbook of physics tried to present the rise of relativity as the necessary response to an experimental situation, namely the supposed null result of the Michelson-Morley experiment searching for an ether drift in l887 -- fully in accord with the sensationist or positivist view of how theories must proceed. (As well, we should add, the easiest pedagogic method of convincing students that they must take seriously what otherwise would be so counter-intuitive.) But, Polanyi declares, “the historical facts are different.”54 He noted that Einstein, in his publication, had not mentioned the Michelson-Morley experiment at all, and concludes from it that this theory was proposed “on the basis of pure speculation, rationally intuited by Einstein before he had ever heard about it.”55

No, the facts are not different. The textbooks are correct that the rise of relativity was indeed the necessary response to an experimental situation. See my book for details.

Nevertheless, Polanyi eventually convinced other philosophers and Einstein historians that Einstein invented special relativity with pure thought, while ignoring experiments like Michelson-Morley.

This is crazy. Relativity pioneers FitzGerald, Lorentz, Poincare, and Minkowski all said that the Michelson-Morley experiment was crudial to their work. Einstein himself said in 1909 essay that the experiment was crucial:

The Michelson and Morley interference experiment showed that, in a special case, second-order terms also cannot be detected, although they were expected from the standpoint of the ether-at-rest theory. To include this experiment in the theory, Lorentz and FitzGerald introduced the postulate that all objects, including the parts of Michelson and Morley's experimental set-up, changed their form in a certain way, if they moved relative to the ether. ...

Michelson's experiment suggests the axiom that all phenomena obey the same laws relative to the Earth's reference frame or, more generally, relative to any reference frame in unaccelerated motion. For brevity, let us call this postulate the relativity principle.

Einstein did later acknowledge that the experiment had little or no direct influence on his famous 1905 paper. That is plain to see from reading the paper. Einstein gave an exposition of Lorentz's theory, without giving Lorentz's reasoning. As Lorentz later said, Einstein postulated what had previously been proved from experiment an analysis.

I draw attention to the history of relativity because it is so frequently used to sell some completely wrong picture of what science is all about.

A comment on my FQXi essay argues that Michelson-Morley was not crucial because it is consistent with emission (particle) theories of light. But there was a lot of evidence at the time that light was an electromagnetic wave, and so the emission theories were not popular. It is a historical fact that Michelson-Morley was crucial to relativity.