Medieval positivist minister

Suppose that a scientific book was published, and the publisher added a forward written by someone else, saying:

To the Reader
Concerning the Hypotheses of this Work

There have already been widespread reports about the novel hypotheses of this work, ...
they will find that the author of this work has done nothing blameworthy. ... Since he cannot in any way attain to the true causes, he will adopt whatever suppositions enable the motions to be computed correctly from the principles of geometry for the future as well as for the past. The present author has performed both these duties excellently. For these hypotheses need not be true nor even probable. On the contrary, if they provide a calculus consistent with the observations, that alone is enough. ...

And if any causes are devised by the imagination, as indeed very many are, they are not put forward to convince anyone that are true, but merely to provide a reliable basis for computation. However, since different hypotheses are sometimes offered ... [one] will take as his first choice that hypothesis which is the easiest to grasp. ...

Therefore alongside the ancient hypotheses, which are no more probable, let us permit these new hypotheses also to become known, especially since they are admirable as well as simple and bring with them a huge treasure of very skillful observations. ...

Would this support or undermine the book? Would it be pro-science or anti-science?

The statement is positivist. It praises the book's theory for its ability to predict observations. This is the core of science.

It is the famous Osiander forward to De revolutionibus orbium coelestium, by Copernicus. Many people have criticized this forward as being anti-science. An encyclopedia said that it "clearly contradicted the body of the work." Arthur Koestler called it a "scandal".

The most anti-science part is this, by the Lutheran minister:

the astronomer will take as his first choice that hypothesis which is the easiest to grasp. The philosopher will perhaps rather seek the semblance of the truth. But neither of them will understand or state anything certain, unless it has been divinely revealed to him.

This suggests that Truth is in the domain of philosophers and Bible-readers, not scientists.

But Osiander is correct. Copermnicus had no reason to be certain of his hypothesis. He had an argument that his model fit the data, and that it was easier to grasp. The model was later replaced by Kepler's and by general relativity, so it was not certain or True.

Arthur Koestler, in The Sleepwalkers, confirms that Osiander meant what he said, citing earlier letters:

Two years earlier, when Copernicus was still hesitating whether to publish the book, he had written to Osiander to pour out his anxieties and to ask for advice. 60 Osiander had replied:

“For my part I have always felt about hypotheses that they are not articles of faith but bases of computation, so that even if they are false, it does not matter, provided that they exactly represent the phenomena… It would therefore be a good thing if you could say something on this subject in your preface, for you would thus placate the Aristotelians and the theologians whose contradictions you fear.” 61

On the same day, Osiander had written on the same lines to Rheticus, who was then in Frauenburg:

“The Aristotelians and theologians will easily be placated if they are told that several hypotheses can be used to explain the same apparent motions; and that the present hypotheses are not proposed because they are in reality true, but because they are the most convenient to calculate the apparent composite motions.”

Prefatory remarks of this kind would induce in the opponents a more gentle and conciliatory mood; their antagonism will disappear “and eventually they will go over to the opinion of the author.” 62

Rheticus has published in support of heliocentrism before Copernicus, and thereby encouraged Copernicus that his book would get a favorable reception.

I think that Osiander's attitude is more scientific than today's academics, who all say that positivism is dead. They make fun of Bible readers, but they have their own ideas about believing in Truths that are independent of observation.

Here is a description of the controversy:

Copernicus was unhappy with this since it had always been his aim to show the true structure of the planetary system rather than some mathematical fiction. Kepler, who saw Copenicus's reply to Osiander said that the astronomer had had no intention of complying with his idea and had decided to maintain his own opinion 'even though the science should be damaged'.

In the event however Osiander did publish a unsigned preface suggesting the heliocentric theory was a hypothesis for computation. Rheticus was furious at this and tried to publish a corrected version without success. Bishop Giese was also outraged, describing the preface as a 'fraud' and writing to Rheticus with the aim of having the preface denounced to the Senate of Nuremberg.

In the event however, Osiander's motives were far from malicious. Osiander was following the custom of the Middle ages which was to propound new theories as hypotheses whose truth remained to be tested in the public forum; this reflected a tradition of instrumentalism that had been applied to astronomy since the time of Ptolemy. He was also making an honest attempt to disarm the defenders of the Aristotelian position and at most he was guilty of an error of judgement. In the event, the label 'hypothesis' applied to the heliocentric theory did open a way for the wide dissemination of its main principles and prepared the ground for the acceptance of the new cosmology.

Stephen Hawking used to call himself a positivist, but his latest book advocates model-dependent realism. It says that "a well-constructed model creates a reality of its own." He ends up arguing for M-theory, which is the dominant paradigm among theoretical physicists but has no connection with observations.

Most people (in ancient and modern times) assume that either Sun goes around the Earth or the Earth goes around the Sun. But it was a gigantic unwarranted assumption to say that one of these is true and one is false. There was no evidence for such a belief. I say that the better scientific attitude is to identify that as an assumption and to admit that it may be false. So Osiander had the better philosophy than Copernicus.

I argue for positivism in my FQXi essay, but admit that is a minority view. But even if I cannot convince you of positivism, I ought to be about to convince you that the sort of positivism shown by Osiander is a completely legitimate scientific view.

Hoping for a mathematical understanding

The Standard Model of particle physics is a very successful mathematical model for all of physics. But it has not truly reduced physics to math. Here is what a couple of our leading mathematical physicists, Arthur Jaffe and Edward Witten, say:

Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. ...

2 Quest For Mathematical Understanding ... On the other hand, one does not yet have a mathematical understanding of the quantum behavior of four-dimensional gauge theory, or even a precise definition of quantum gauge theory in four dimensions. Will this change in the twenty-first century? We hope so!

Okay, they hope so. But we may never have a purely mathematical understanding of quantum reality.

Laplace's Demon is absurd

Jared Lorince writes:

Instead we are built to operate with limited time, processing power, and available information. To imagine a truly rational human being, one unbounded by these constraints, is to imagine nothing short of a Laplacean demon. For those unfamiliar with the term, it references the hypothetical super-intelligence proposed by French mathematician Pierre-Simon Laplace, capable of knowing the state of every atom in the universe, and associated with the first published description of scientific determinism.

He refers to Laplace’s demon, a fanciful idea that is surprisingly widely believed. Lorince is just attacking the idea that humans can be all-knowing, but I attack the idea that such knowledge can even exist in a mathematically defined form.

Physicists might argue that if you only knew the quantum state function of the universe at some time in the past, then you could apply some unitary operator to get the state at future times. To believe anything else is contrary to a scientific worldview, they say. Occasionally this sort of thinking will drive them to deny free will or to accept multiple universes.

I say that Laplace's Demon is absurd. It is a huge unwarranted assumption. Nobody has ever explained how it would work. It seems contrary to what we know about quantum mechanics. My FQXi essay explains why I think it is wrong.

Possible prize for teleportation

Someone predicts that next month's physics Nobel prize will go for quantum teleportation. Lumo is skeptical. The obvious choice is Higgs, but Anderson says he had that idea first.

Quantum teleportation is an overhyped idea with no practical application. The main use is supposed to be Quantum key distribution:

Quantum key distribution (QKD) uses quantum mechanics to guarantee secure communication. It enables two parties to produce a shared random secret key known only to them, which can then be used to encrypt and decrypt messages. It is often incorrectly called quantum cryptography, as it is the most well known example of the group of quantum cryptographic tasks.

An important and unique property of quantum distribution is the ability of the two communicating users to detect the presence of any third party trying to gain knowledge of the key. This results from a fundamental aspect of quantum mechanics: the process of measuring a quantum system in general disturbs the system. A third party trying to eavesdrop on the key must in some way measure it, thus introducing detectable anomalies. By using quantum superpositions or quantum entanglement and transmitting information in quantum states, a communication system can be implemented which detects eavesdropping. If the level of eavesdropping is below a certain threshold, a key can be produced that is guaranteed to be secure (i.e. the eavesdropper has no information about), otherwise no secure key is possible and communication is aborted.

Transmitting a message securely means that the recipient gets a message that could have only come from the sender, and no eavesdropper can read or interfere with the message. Conventional cryptography solves this problem. The QKD method has no way to autheniticate the sender, and can never be secure. At best it has a statistical method for probably detecting eavesdroppers.

A physics blog explains quantum teleportation. That allows someone to send a quantum bit (qubit) to another.

If individual photons had mathematical representations, then they could be teleported by just transmitting those numbers and reconstructing the photon state at the other end. Without such representations, the so-called teleportation is a little trickier, but not big deal. I think someone else will get the Nobel prize.

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.