Foolish and absurd in Philosophy

When Galileo appeared before the Inquisition in 1616, consultants were hired to appraise his heliocentrism, reporting:

The sun is the center of the world, and entirely immobile by local motion.

Appraisal: All have said the stated proposition to be foolish and absurd in Philosophy; and formally heretical, since it expressly contradicts the sense of holy scripture in many places, according to the quality of the words, and according to the common exposition, and understanding, of the Holy Fathers and the learned Theologians.

The earth is not the center of the world, and not immobile, but is moved along Whole itself, and also by diurnal motion.

Appraisal: All have said, this proposition to receive the same appraisal in Philosophy; and regarding Theological truth, at least to be erroneous in faith.

This translation is from a new paper, The Inquisition's Semicolon: Punctuation, Translation, and Science in the 1616 Condemnation of the Copernican System, by Christopher M. Graney:

This paper presents high-resolution images of the original document of the 24 February 1616 condemnation of the Copernican system as being "foolish and absurd in philosophy", by a team of consultants for the Roman Inquisition. Secondary sources have disagreed as to the punctuation of the document. The paper includes a brief analysis of the punctuation and the possible effects of that punctuation on meaning. The original document and its punctuation may also have relevance to public perception of science and to science education.

Nearly everyone omits the first semicolon above, thereby giving the impression that a scientific idea was "foolish and absurd" just because it was contrary to the Bible. The consultants only used the Bible as a reason for saying that the Earth's motion is "formally heretical".

There were legitimate scientific arguments against Galileo's Copernicanism, such as those listed here, here, here, and here. There were also theological arguments on both sides.

Wikipedia recently deleted this statement about Galileo's telescopic discoveries:

None of these findings, which were difficult at first for other astronomers to verify, proved that the Earth moved, or directly contradicted either Aristotle's model or Christian doctrine.

The statement is correct. The sources all agree that Galileo did not have the proof that Cardinal Bellarmine wanted, but the editors deleted the statement anyway because of an argument about "how science works". They say it was unreasonable to expect proof.

Owen Gingerich points out that decades later the great scientist Robert Hooke was also asking for proof, just like Bellarmine. Gingerich argues that stellar parallax by itself may not have convinced Bellarmine or Hooke. Scientists were convinced by the combination of the Newtonian framework with certain crucial experiments like stellar parallax. He does not mention that the relativistic framework of the 20th century showed that motion was relative and both arguments are valid.

It is impossible for an experiment to prove the Earth's motion because the laws of physics are valid in any frame of reference. Poincare made this argument, and later Einstein did also. Galileo gave a version of the argument, saying that we might not notice motion in the lower deck of a ship, but he also claimed that his goofy theory of tides proved the motion of the Earth. Bellarmine's position was closer to the modern view than Galileo's.

Rigorous math v physical heuristics

Israeli Rutgers University math professor Doron Zeilberger writes in the AMS Notices:

What Frenkel and Ross did not tell us is that the “math” that led to the discovery of the Higgs boson is not their kind of (pure-and-rigorous) math, but the much more effective, and efficient, nonrigorous mathematics practiced by theoretical physicists called quantum field theory. This highly successful (and precise!) mathematical theory would not be considered mathematics by most members of the American Mathematical Society, since it is completely nonrigorous.

I would not phrase it that way. I agree that the nonrigorous math is not really math, and that this is a dividing issue between mathematicians and physicists. Most physicists are sloppy in their math and do not really appreciate mathematical rigor.

But there is a lot of legitimate math behind quantum field theory, even if most physicists ignore it.

He appears to have a legitimate gripe about how the math community devalues computational math.

Proof is the only way to determine mathematical truth, and experiment (or observation) is the only way to determine scientific truth. When you have neither, then you have abominations like string theory.

He doubles down:

Traditional “rigorous proof” is yet another religious dogma, which did some good for a long time (as did the belief in God). Of course, it is not surprising that people can get deeply offended when someone denies the existence of their “God”.

But the God of (alleged!) rigorous proof is dead (well, not yet, but it should be!), and we should allow diversity. Rigorous proofs should still be tolerated, but they should lose their dominance, and the Annals of Mathematics should mostly accept articles with mathematics that has only semi-rigorous or non-rigorous proofs (of course, aided by our much more powerful and superior silicon brethren), because this way the horizon of mathematical knowledge (and mathematical insight!), broadly defined, would grow exponentially wider.

The Annals of Mathematics does not prohibit computer-assisted proofs. It published this paper of mine. It was a rigorous proof, not a heuristic or experiment.

Yes, rigorous has been the standard for a long time. Ever since Euclid's Elements in 300 BC.

His web site has many opinions, starting with this:

First Published (in Hebrew): ... The history of science supplied us with many examples of true "geniuses" that were kicked out of high school because of poor achievements, and one of the most prominent examples is the greatest scientist of our time, the physicist Albert Einstein. But Einstein was the greatest scientist of all times, and hence he managed somehow to find his way in life (but it wasn't easy, even for him).

No, Einstein was not kicked out of school or anything like that. He had very good grades and progressed thru a fairly rigid system to get a doctoral degree in physics.

Einstein did not succeed in doing any rigorous math. I am not sure he ever proved anything. Relativity is very mathematical, and he needed mathematicians Poincare and Minkowski to figure out the special theory, and Grossmann and Hilbert for the general theory.

For an example of math that seems sloppy but can actually be made rigorous, see
this NY Times account of how the natural numbers sum to -1/12. Or read Terry Tao's rigorous explanation or others.

Kepler used epicycles

A new paper says:

I discuss the problem of secular inequalities in Kepler by giving account of a manuscript note that has not been published until 1860. In his note Kepler points out the need for a model, clearly inspired by the method of epicycles, that describes the secular inequalities as periodic ones. I bring attention to this point, that seems to have been underestimated, since the references to Kepler's work usually report only that he observed a decreasing mean motion for Saturn and an increasing one for Jupiter.

Kepler is widely credited with abolishing the evils of epicycles in favor of elliptical orbits, but I did not know that he used epicycles himself. The paper starts:

The discovery by Kepler of the elliptic shape of the planetary orbits is often considered
as a discontinuity with the traditional models of Classical Astronomy, based on geometrical tools such as circles, epicycles and equants. The enthusiastic announcement of the discovery in chapter LVI of Astronomia Nova [6], where Kepler says that he was “quasi e somno expergefactus, et novam lucem intuitus” (like suddenly awakened from sleep, and seeing a new light) seems to confirm that this was his feeling, too. But, as often happens, reality turned out to be complex enough to escape our theories: the orbits of the planets are not exactly elliptic. This is well known today, but it seems that the circumstancy that long term deviations from the elliptic motion have been investigated in great detail by Kepler himself, with an explicit conjecture that these deviations should be periodic, is not so known, even among astronomers.

Epicycles are widely mocked as being contrary to science and rational thought, but they are legitimate tools for representing orbits.

You might say that the planetary orbits, as seen from the Earth, are circles. The first order correction is the Ptolemy epicycles, related to the Earth's orbit around the Sun. The second order correction is the Copernican epicycles, related to the elliptical orbits, The third order correction is the Kepler epicycles, related to the gravity of other planets. Modern representations might use a Fourier series with hundreds of terms.

Counterfactuals: Law

Many court actions are contingent on counterfactual reasoning. Tort law is based on the idea that if someone commits a wrong act against you, then you can sue for damages. For example, if someone crashes into the rear end of your car, you can sue for the cost of restoring your car to its condition before the accident.

The court has to somehow figure out what would have happened if the tort had never happened. The counterfactual is the plaintiff's life without the tort. If someone slanders you and you sue for damage to your reputation, then the court has to estimate the dollar value of the difference between your actual reputation and your counterfactual reputation if you had not been slandered.

More common are lawsuits over contract law, and these also depend on counterfactual reasoning. This is one of the most misunderstood points in all of law. If you sign a contract to rent a house from someone for $1000, and you break the contract, then he can sue you for the $1000. But the court will only award the counterfactual damages. If he could have easily found another tenant to pay the $1000, then you may not owe anything. Or you might even owe more than $1000, if he has lost business because of your vacancy.

The debate about Trayvon Martin asked questions like: What if Martin had been white? What if Zimmerman stayed in his car?

At the Zimmerman trial, the prosecutor misunderstood the role of the expert:

Prosecutor Bernie de la Rionda: In order to give an opinion, when someone gives you a hypothetical, it has to based on facts that are accurate and truthful, correct?
Di Maio: A hypothetical doesn't have to be true. A hypothetical is just "suppose this and this happened ...".
Prosecutor Bernie de la Rionda: So we would be speculating, I guess, or potentially speculating?
Di Maio: It is not even speculating. You are giving a presentation and asking what it is.

Di Maio is correct. An excellent expert witness could testify entirely by answering hypothetical questions, and never even look at the facts of the case. For example, a witness might be useful just answering these hypothetical questions: "What does a gunshot would look like if the gun is pressed against the skin? If the gun is 3 inches away? 12 inches? 24 inches? How is that known?" The jury could then apply his knowledge to the facts of the case.

Most witnesses in court are not allowed to speculate, or offer counterfactual reasoning. That is restricted to lawyers making arguments (not under oarh), and expert testimony. A expert must follow the Daubert standard, as codified in US federal court as FRE 702:

A witness who is qualified as an expert by knowledge, skill, experience, training, or education may testify in the form of an opinion or otherwise if:

(a) the expert’s scientific, technical, or other specialized knowledge will help the trier of fact to understand the evidence or to determine a fact in issue;
(b) the testimony is based on sufficient facts or data;
(c) the testimony is the product of reliable principles and methods; and
(d) the expert has reliably applied the principles and methods to the facts of the case.

When the expert describes those reliable principles and methods, he often tells how they apply to hypothetical situations. As the jury (or maybe the judge) determines the facts, testimony about counterfactuals in the most useful. Often there is no need for the expert to offer any opinion about the facts of the case, if the jury is taught how to apply the generally accepted knowledge.

Thus counterfactuals are essential to the law.

Update: A 2010 US Supreme Court decision said that a lawsuit "does not require precise proof of what the Board's policies might have been in that counterfactual world." I mention this because it is understood that lawsuits are all about proving scenarios in counterfactual worlds, even tho that terminology is uncommon.

Nearby star is oldest in the universe

A new research announcement says:

The Oldest Star in the Known Universe

A team led by astronomers at The Australian National University has discovered the oldest known star in the Universe, which formed shortly after the Big Bang 13.7 billion years ago.

The discovery has allowed astronomers for the first time to study the chemistry of the first stars, giving scientists a clearer idea of what the Universe was like in its infancy.

"This is the first time that we've been able to unambiguously say that we've found the chemical fingerprint of a first star," said lead researcher, Dr Stefan Keller of the ANU Research School of Astronomy and Astrophysics. ...

The ancient star is around 6,000 light years from Earth, which Dr Keller says is relatively close in astronomical terms. It is one of the 60 million stars photographed by SkyMapper in its first year.

I am surprised that the oldest star is so close. I would have guessed that the oldest known stars would be outside the Milky Way. After all:

In physical cosmology, the Copernican principle, named after Nicolaus Copernicus, states that the Earth is not in a central, specially favored position in the universe. ...

Michael Rowan-Robinson emphasizes the Copernican principle as the threshold test for modern thought, asserting that: "It is evident that in the post-Copernican era of human history, no well-informed and rational person can imagine that the Earth occupies a unique position in the universe."

So if we are in the oldest galaxy, then maybe we are in the center of the universe!

Perhaps the researchers only looked at stars in our galaxy. If so, they should have explained that.

Arguing that photons are massless

Physics professor Brian Greene was on Comedy Channel Colbert Nation plugging worldscienceu.com, and spent most of the time arguing that light has energy, but not mass. Even Colbert knew that E=mc², so he was confused. So were most viewers, I am sure. Greene was there to explain science, and he did a lousy job.

Textbook terminology has changed over the decades on this point. Here is how Wikipedia explains mass in special relativity:

A so-called massless particle (such as a photon, or a theoretical graviton) moves at the speed of light in every frame of reference. In this case there is no transformation that will bring the particle to rest. The total energy of such particles becomes smaller and smaller in frames which move faster and faster in the same direction. As such, they have no rest mass, because they can never be measured in a frame where they are at rest. This property of having no rest mass is what causes these particles to be termed "massless." However, even massless particles have a relativistic mass, which varies with their observed energy in various frames of reference,

John Baez explains:

Photons are traditionally said to be massless. This is a figure of speech that physicists use to describe something about how a photon's particle-like properties are described by the language of special relativity.

A figure of speech? Yes, you can say that photons have mass, or do not have mass, as long as you define your terms carefully.

Here is a modern opinion:

For many years it was conventional to enter the discussion of dynamics through derivation of the relativistic mass, that is the mass–velocity relation, and this is probably still the dominant mode in textbooks. More recently, however, it has been increasingly recognized that relativistic mass is a troublesome and dubious concept.

Obviously Greene was concerned that he might be embarrassed before his colleagues using some antiquated terminology.

It would have been better to say:

Yes, photons have mass and momentum. Some physicists like to say that the rest mass is zero, but the photon is never at rest anyway.

Greene is teaching two new courses on "Einstein's special theory of relativity". It is funny that he always has to say that it is Einstein's theory. Everything he mentions was discovered by someone else, not Einstein.

Counterfactuals: Grammar

Consider the sentence, "if pigs had wings, they could fly." This is a very strange thing to say, as pigs do not have wings. And yet the sentence is readily understandable to children.

The English language has a construction called the "subjunctive conditional" for counterfactuals like this. Statements that describe reality are in the indicative mood, such as "pigs do not have wings" or "pigs do not fly". So are straightforward conditional statements, like "if pigs have food, they eat." If you want to suggest some sort of imaginary world in which pigs fly, you need the verb to be in the subjunctive mood.

Consider this illustration of the English subjunctive:

If he was in class yesterday, he learned it.
If he were in class today, he would be learning it.

The first is a simple statement about a boy learning in class. The second suggests that he was not in class, but it says something about some false scenario anyway.

Here is another example:

If Oswald did not shoot Kennedy, someone else did.
If Oswald hadn’t shot Kennedy, someone else would have.

The first is a true statement because it is indicative conditional with a false antecedent. Oswald did shoot Kennedy. Even if you are not sure that Oswald shot Kennedy but know that Kennedy was shot, then you will still accept the statement as true. It is only an assertion about actual historical events.

The second statement is much more complex. It uses the subjunctive mood to alert you that it is hypothesizing some counterfactual world and trying to reason about that world as if it were real. It is hard to say whether the sentence is true or false because you have to make assumptions about what else might be funny about that unreal world, and speculate how events might play out.

The counterfactual conditional is a false history scenario. In other cases, the counterfactual is a future hypothetical. For example, you could say that "if the Antarctic ice melts, New York would go under water."

The subjunctive is commonly misunderstood. In the famous 1968 zombie movie, Night of the Living Dead, Barbara says, "Johnny asked me if I were afraid." No, Johnny wanted to know whether she was actually afraid. He was not trying to create some counterfactual scenario. The correct grammar is "Johnny asked me if I was afraid."

The subjunctive "were" would be appropriate if Johnny said "I would run if I were you" or "I would be angry if I were you." Obviously Johnny cannot be someone else, and the subjunctive mood alerts the listener that she is getting a superficially nonsensical scenario.

Here is another example:

If he was in line, give him a ticket.
If he were in line, he would be eating by now.

The first is indicative and factual. The second is subjunctive and counterfactual.

The subjunctive is not used often in English, unless there is some counterfactual being expressed. The verb "be" is subjunctive in "I suggest that they be removed." Then sentence does not say that anything has been or will be removed. It creates a hypothetical scenario where something is removed.

The counterfactual concept is essential, and so is the English subjunctive mood for expressing it.

Classics scholar Phuc Tran argues in a TED talk that his Vietnamese relatives are unable to ponder counterfactuals because their language lacks a subjunctive mood.

I am not sure how other languages deal with these concepts. Chinese does not even have past and future tenses, so it has trouble with expressing time as needed for counterfactuals. (I am not sure about all Chinese, as there are at least 300 different varieties of Mandarin that were more or less mutually unintelligible.)

Frenkel on Platonism, Tegmark, and simulation

Math professor Edward Frenkel writes in the NY Times:

Indeed, there may be. In a recent paper, “Constraints on the Universe as a Numerical Simulation,” the physicists Silas R. Beane, Zohreh Davoudi and Martin J. Savage outline a possible method for detecting that our world is actually a computer simulation. Physicists have been creating their own computer simulations of the forces of nature for years — on a tiny scale, the size of an atomic nucleus. They use a three-dimensional grid to model a little chunk of the universe; then they run the program to see what happens. This way, they have been able to simulate the motion and collisions of elementary particles.

But these computer simulations, Professor Beane and his colleagues observe, generate slight but distinctive anomalies — certain kinds of asymmetries. Might we be able to detect these same distinctive anomalies in the actual universe, they wondered? In their paper, they suggest that a closer look at cosmic rays, those high-energy particles coming to Earth’s atmosphere from outside the solar system, may reveal similar asymmetries. If so, this would indicate that we might — just might — ourselves be in someone else’s computer simulation.

This is science fiction. The fact is that we do not have the know-how to mathematically simulate the universe. We can simulate particular experiments, but any simulation we would produce would have some differences with reality. If we lived in a simulation with the same shortcomings, then they would be detectable.

Part of the belief in quantum computing is based on the fact that we cannot simulate the quantum world. Feynman once gave a lecture where he speculated that we might need a quantum computer for an effective simulation.

An argument can also be made that mathematical ideas are objective and exist independently from the human mind — many mathematicians adhere to this view, called mathematical Platonism.

Yet Tegmark’s take is quite different from — and in some ways inconsistent with — Platonism. Math is so effective in describing the world, he says, because physical reality is a mathematical structure. He calls it the Mathematical Universe Hypothesis (M.U.H.). What exactly this means is a big question, which is never fully answered. Mr. Tegmark’s argument is that all physical properties of an electron, say, can be described mathematically; therefore, to him, an electron is itself a mathematical structure — as is everything else, including us.

Mathematical Platonism is reasonable, in my opinion, but Tegmark is saying something different.

Yes, our theories are mathematical, and they predict experiments well, but Tegmark wants to make an ontological argument that the physics is the math. If two distant electrons are entangled, then it is impossible to give a mathematical description of one electron independent of the other one. The usual way out of this paradox is to day that there is a mathematical action-at-a-distance, but not a physical one. To me, that says that the physics is not the math.

There are alternative explanations, such as saying that there is a physical nonlocality that is not directly observable. To me, this is unsatisfactory as it is like believing in psychic powers that can never be demonstrated in a controlled way, or believing in one of the unobservable multiverses.

Thus I disagree with Tegmark than an individual electron can be described as a mathematical structure. It cannot. We have a good mathematical description of the measurement possibilities of two entangled electrons jointly, but we cannot identify one of them individually with a mathematical structure. Attempts to do so have led to hidden variable theories, and they have all failed.

Update: Mad Max comments below that Frenkel has misrepresented his views, and concedes the point that entanglement issues prevent an individual electron from being equated with a mathematical structure.

I think that this is an important point. Some of Tegmark's critics have questioned whether there is any content to the claim that the universe is mathematical, because no one knows what a non-mathematical universe would look like. If he is now conceding that an isolated but entangled electron is non-mathematical, then that is an example of a physical object that is not a mathematical structure. It seems to me that if an electron can be a non-mathematical object, then so can the universe. Tegmark would say that the electrons and other objects fit together to make a giant mathematical structure, even if some of the physical components are not.

Time magazine on quantum computer

Lev Grossman, best known for novels on magicians, wrote the Time cover story on quantum computers

The Quantum Quest for a Revolutionary Computer
Quantum computing uses strange subatomic behavior to exponentially speed up processing. It could be a revolution, or it could be wishful thinking

The article is behind a paywall, but you can read it here, and read criticism.
At least the article does cite skeptics:

Another challenge rose and company face is there is a small but nonzero number of academic physicists and computer scientists who think they are partly or completely full of shit.

The print edition says "sh-t", which must be unusual for that magazine.

The term "quantum computer" does not just mean a computer that uses quantum mechanics. All electronic computers use quantum mechanics. A quantum computer must has some quantum speedup, where computations on entangled bits can scale up to a faster-than-Turing computer.

Claiming a quantum computer is like claiming to use dark energy for a perpetual motion machine. You can argue that experiment has shown dark energy to be there, and that theory says that it is inexhaustible, and that exploitation is just a matter of developing the technology, but you have to face the fact that no one has ever violated energy conservation or extracted any dark energy.

Likewise, no one has ever succeeded in getting any quantum speedup in a calculation. All attempts have failed.

I do'nt blame Time magazine for this silly hype. The author tried to write a balanced story. I blame the physicists for pretending that they can do the impossible.

Aeon trashes quantum interpretations

Aeon magazine reports:

The Copenhagen interpretation was very much in line with the scientific philosophy of logical positivism that caught on at around the same time. In particular, it rests on something like logical positivism’s principle of verification, according to which a scientific statement is meaningful only if we have some means of verifying its truth. To some of the founders of quantum theory, as well as to later adherents of the Copenhagen interpretation, this came to seem an almost self-evident description of the scientific process. Even after philosophers largely abandoned logical positivism – not least because the principle of verification fails its own test for meaningful statements – many physicists trained in the Copenhagen tradition insisted that their stance was no more than common sense.

However, its consequences are far from commonsensical. If you take this position seriously, then you have to accept that the Higgs boson wasn’t actually discovered at the Large Hadron Collider, since no one has ever directly detected a Higgs boson, and we have no direct evidence to support the claim that the Higgs boson is a real particle. Insofar as we learnt anything about nature from the Large Hadron Collider, it was merely what sort of records you get in your detectors when you build something like the Large Hadron Collider. It’s hard to imagine the scientists who work on it, or the citizens who funded them, being very enthusiastic about this justification, but on a strict Copenhagen view it’s the best we can do.

This starts out reasonable, and then descends into gibberish. Yes it is true that logical positivists and the founders of quantum mechanic were following a self-evident description of the scientific process.

It is also true that philosophers abandoned logical positivism, and along with it they abandoned scientific objectivity and progress towards truth. It is not true that logical postivism fails its own test.

But then the paragraph about the Higgs boson is nonsense. You could say that the Higgs was not detected directly, but you could say the same about electrons and photons. Quantum mechanics teaches that there is no such thing as a real particle, in the classical sense. There are fields and waves that have particle properties in certain observations. The only difference with the Higgs is that a lot of other particles are detected at the same time, so finding the Higgs is technically very difficult. But it is still an observation, just like all the others in the Copenhagen.

After attacking Copenhagen, the article moves on to other interpretations but finds them unsatisfactory also.

If we cannot get a coherent story about physical reality from the Copenhagen interpretation of quantum theory and we cannot get a scientifically adequate one from many-worlds theory, where do we turn? We could, as some physicists suggest, simply give up on the hope of finding any description of an objective external reality. But it is very hard to see how to do this without also giving up on science. ...

Bell was one of the last century’s deepest thinkers about science. As he put it, quantum theory ‘carries in itself the seeds of its own destruction’: it undermines the account of reality that it needs in order to make any sense as a physical theory. On this view, which was once as close to heresy as a scientific argument can be but is now widely held among scientists who work on the foundations of physics, the reality problem is just not solvable within quantum theory as it stands.

The problem here is that philosophers, Bell, Einstein, and physicists working on quantum foundations have abandoned scientific common sense in favor of a 19th century view of science. The founders of quantum mechanics thought that science was all about observation, and were happy with the theory. They have been replaced by fools who think science is all about postulating unobservable multiverses, hidden variables, and other fictions.

Tegmark explains to philosophers

I listened to this Max Tegmark interview, hoping that he would express his philosophical ideas better to a couple of profession science philosophers:

Those among us who loathed high school calculus might feel some trepidation at the premise in this week's episode of Rationally Speaking. MIT Physicist Max Tegmark joins us to talk about his book "Our Mathematical Universe: My Quest for the Ultimate Nature of Reality" in which he explains the controversial argument that everything around us is "made of math."

Max, Massimo and Julia explore the arguments for such a theory, how it could be tested, and what it even means.

Unfortunately, there was little substance.

He explained Gödel's incompleteness theorem as disproving Hilbert, and making it impossible to disprove 0=1. He claims to avoid this problem by avoiding infinity. He gets this wrong. It is possible to prove the consistency of arithmetic, if you use a larger system to prove it. It is not clear why any of this has any physical significance, or why infinity matters.

He made a big deal out of using real numbers to describe nature, such as the constant that is about 1/137. But then his rejection of infinity causes him to reject the continuum of real numbers, so it is not clear that any physics can be described by math the way he wants it.

He repudiated his earlier proposal to test many-worlds by playing Russian roulette. Now he argues that the best evidence is a quantum computer, presumably using David Deutsch's argument that quantum computing is so mysterious that it must be taking place in other universes.

He gave an argument about how mathematicians think of the number 5, but it sounded more like physicist thinking to me. Mathematically, the ordinal 5 is often defined as {0,{0},{0,{0}},{0,{0},{0,{0}}},{0,{0},{0,{0}},{0,{0},{0,{0}}}}}. See Von Neumann definition of ordinals or Von Neumann cardinal assignment for details.

He did not give a coherent explanation of what his math multiverse hypothesis means.

I also criticized Tegmark's new book here, here, here, and here.

Update: Lumo explains:

Laymen (e.g. postmodern philosophers) interested in spirituality and physics (...) often talk about things like the "influence of Gödel's theorems about incompleteness on physics" and similar things. They usually want to believe that this theorem must imply that mathematics and science must be limited, leaving the bulk of the human knowledge to witches, alternative doctors, ESP experts, dragons, priests, and global warming alarmists, among related groups of unscientific charlatans.

With their restricted resolution, "Gödel's theorem on imncompleteness" seems to be the same thing as the "Heisenberg uncertainty principle". However, the truth is very different. The mathematical insight by Gödel has no relationship to the Heisenberg uncertainty principle and none of the two imply that the laws of Nature cannot be pinpointed precisely, anyway.

When it comes to the irrelevance of Gödels theorems for physics, the truth is actually much more far-reaching. None of the major developments in the post-Cantor efforts to axiomatize mathematics and set theory has any implication for physics. ...

If I were a politically correct opportunist, I could say that the culture of mathematicians (with their cardinals, ordinals, theorems on incompleteness, disrespect for continuity, semi-bans on integration, and the proliferation of inseparable Hilbert spaces that follows from that etc.) and the culture of physics (with their operators boasting discrete, continuous, or mixed spectra, complete embrace of integration, continuity, unified treatment of operators with discrete and continuous spectra etc.) are simply two different, inequivalent ways to formalize certain (or superficially related) mathematical concepts and to define rules they have to follow.

But I am not a politically correct opportunist so I will tell you the actual truth. It is the physicists' perspective on these issues that is vastly superior and more profound even from a mathematical viewpoint – simply because it's the perspective that has already undergone some actual tests ...

It's the physicists' approach to the notion of infinity, infinite sums, infinite bases etc. that is the deeper one, more likely to be related with future important discoveries.

He is mostly correct about non-mathematicians talking nonsense about Godel and infinities. And yes, as long as you can do a physical experiment to test your formulas, then you do not need to worry about the math subtleties. But when there is some doubt about the correctness of the math, there is no substitute for the axiomatic approach, and the nonrigorous physicist approach gives wrong answers all the time.

Counterfactuals: Introduction

Counterfactual reasoning is essential to science. It could also be called hypothetical reasoning. It is at the root of many scientific confusions.

Hypothetical reasoning means formulating some hypothesis, and deducing the consequences. For example, a hypothesis might be that CO2 concentrations in the atmosphere might double in the next century. A deduction might be some global warming. The scientific method could be summarized as formulating a hypothesis, deducing consequences, testing those consequences, and ultimately making an inference about the validity of the original hypothesis.

The German philosopher Ludwig Wittgenstein once asked why anyone would think that it logical to assume that the Sun revolves around the Earth. When told that the Sun appears to be revolving, he asked how it would look if the Earth were rotating instead. The point of that anecdote is that science does not just find explanations consistent with observations. It compares the consequences of different hypotheses that may not be true.

Hypothetical reasoning is usually based on the hope that the hypothesis in going to turn out to be true. Counterfactual reasoning is similar, except that the hypotheses are likely to be false. The purpose is not to prove the hypothesis at all, but to elaborate on some artificial scenario.

If you are giving an hypothesis as a scientific explanation for some observed facts, then you are implicitly saying that alternative hypotheses are inconsistent with the facts. Thus even if you want to stick to truth, you must do some counterfactual analysis.

In the 1865 Lewis Carroll book, Alice in Wonderland says, “There is no use trying; one can’t believe impossible things.” The Queen replies, “I daresay you haven't had much practice. When I was your age, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast.” Carroll was a mathematician, writing for children.

Believing in counterfactuals does take practice, as it is barely distinguishable from believing in nonsense. It is also essential to quantum mechanics.

Aristotle is commonly quoted (incorrectly) as saying, "It is the mark of an educated mind to be able to entertain a thought without accepting it." An educated mind, or the open mind of a child. Or someone who accepts counterfactuals.

A counterfactual argument often starts with a what-if. Someone might ask, “what if the Axis Powers had won World War II?” It is a historical fact that they did not, and any such discussion seems like nonsense. Nevertheless, it is necessary to consider such questions if we are to come to conclusions about whether the war was worth fighting.

Comic books sometimes have counterfactual plots, such as the 1977 comic “What if Spider-Man joined the Fantastic Four?”. These are entirely fictional characters, but the comic was intended for readers who had accepted the story line that Spider-Man was not part of the Fantastic Four, and it was not trying to convince anyone that Spider-Man was joining the Fantastic Four. It was entirely counterfactual, but still made logical sense for its readers. It was like a dream within a dream. Disney recently tweeted that it was banning counterfactual Star Wars stories.

In case you are still wondering what "counterfactual" means, it is an adjective that means contrary to fact. When used as a noun, it is an abbreviation of "counterfactual conditional". But it does not just mean a false conditional, as false conditionals are meaningless. A philosophy site defines it as:

A conditional statement whose antecedent is known (or, at least, believed) to be contrary to fact. Thus, for example, "If George W. Bush had been born in Idaho, then he would never have become President." Unlike material implications, counterfactuals are not made true by the falsity of their antecedents. Although they are not truth-functional statements, counterfactuals may be significant for the analysis of scientific hypotheses.

I have become convinced that children understand counterfactuals better than adults, and that misunderstandings about them pervade many disciplines. I will be posting more on this topic. I will eventually relate this to relativity and quantum mechanics, but first I want to make sure that the concept is crystal clear in other contexts.

Falsifying the Mad Max multiverse

MIT cosmologist Max “Mad Max” Tegmark writes in SciAm:

Although we don’t know whether parallel universes exist, we know something else about them with certainty: many people instinctively dislike them, and whenever a physicist writes a book about them, the Web erupts with claims that they are unscientific nonsense.

Yes, I have criticized such multiverse proposals on this blog, and so has Peter Woit.

Many physicists have explored various types of parallel universes in recent books, including Sean Carroll, David Deutsch, Brian Greene, Michio Kaku, Martin Rees, Leonard Susskind and Alexander Vilenkin. Interestingly, not a single one of these books (my own included) makes any outright claims that parallel universes exist. Instead, all their arguments involve what logicians know as “modus ponens”: that if X implies Y and X is true, then Y must also be true.

It is true that none of those books have any real evidence for the multiverse. It is all speculative conditionals.

As a warm-up example, let’s consider Einstein’s theory of General Relativity. It’s widely considered a scientific theory worthy of taking seriously, because it has made countless correct predictions – from the gravitational bending of light to the time dilation measured by our GPS phones. This means that we must also take seriously its prediction for what happens inside black holes, even though this is something we can never observe and report on in Scientific American. If someone doesn’t like these black hole predictions, they can’t simply opt out of them and dismiss them as unscientific: instead, they need to come up with a different mathematical theory that matches every single successful prediction that general relativity has made – yet doesn’t give the disagreeable black hole predictions.

That someone can simply accept General Relativity outside the event horizons. That is a perfectly good theory that matches every successful prediction.

I agree that this is a good warm-up example for thinking about unobservable universes. The theory pretty clearly predicts a singularity at the center of the black hole, and yet it also pretty clearly says that nothing inside the event horizon is observable. So do you believe in the singularity or not? My guess is that the big majority of cosmologists do believe in the singularity, but I think that they should admit that it is outside the scope of observable science. Tegmark is apparently one of the exceptional cosmologists, as he now says that he does not believe in infinities, so I don't see how he can believe in the black hole singularity.

Since the Level III multiverse is implied by the (collapse-free) Schrödinger equation of quantum mechanics, it can be demolished with a type-B attack: an experimental demonstration of a violation of the Schrödinger equation. For example, if the current multi-million dollar attempts to build quantum computers fail and the cause is determined to be that the Schrödinger equation is violated by some form of wavefunction collapse process, then there are no Level III parallel universes.

The current multi-million dollar attempts to build quantum computers have failed, and I believe that they will continue to fail. The many worlds (MWI) fans like the collapse-free equations, but most people do quantum mechanics with collapse of the wave functions.

Tegmark goes on to argue that the math multiverse if falsifiable, and hence scientific:

The Level IV multiverse is also vulnerable to a type-B attack: we can simply reject the notion that there’s an external reality completely independent of us humans, for example in the spirit of Niels Bohr’s famous dictum, “no reality without observation”. A second type-B attack option is to falsify the mathematical universe hypothesis by demonstrating that there’s some physical phenomenon that has no mathematical description.

I say that there is such a falsification -- the electron itself has no complete mathematical description.

Quantum mechanics gives an effective way of making predictions about observations on electrons. But to assume that the wave function is a true description of the electron requires a psi-ontic view that many physicists have rejected because of nonlocality paradoxes. I elaborate on this argument in my FQXi essay. My essay did not win the contest, but I have seen so serious rebuttal of it.

History of the fifth dimension

String theory guru Ed Witten writes:

This note is devoted to a historical detail concerning the paper of Albert Einstein and Peter Bergmann, published in 1938, about unified theories of electromagnetism and gravitation derived from five dimensions [1]. ...

A major influence in Einstein's efforts to unify electromagnetism and gravitation was the proposal made by Theodore Kaluza [2] around 1921, later independently discovered and extended by Oskar Klein [3] and commonly called Kaluza-Klein theory. In this proposal, in addition to the four dimensions of conventional relativity theory (three space dimensions and a fourth dimension of time) there is a fifth dimension; electromagnetism results from a gravitational field that is "polarized" in the fifth dimension. ...

The main novelty of Einstein and Bergmann was to take the fifth dimension seriously as a physical entity, not just an excuse to combine the metric tensor and the electromagnetic potential as different components of a 5x5 matrix.

Wikipedia explains Kaluza–Klein theory:

A splitting of five-dimensional spacetime into the Einstein equations and Maxwell equations in four dimensions was first discovered by Gunnar Nordström in 1914, in the context of his theory of gravity, but subsequently forgotten. Kaluza published his derivation in 1921 as an attempt to unify electromagnetism with Einstein's general relativity.

In 1926, Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of a very small radius, so that a particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set, and the phenomenon of having a space-time with compact dimensions is referred to as compactification.

In modern geometry, the extra fifth dimension can be understood to be the circle group U(1), as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle, the circle bundle, with gauge group U(1). In Kaluza–Klein theory this group suggests that gauge symmetry is the symmetry of circular compact dimensions.

Hermann Weyl constructed a similar theory in 1918, and that was the foundation of gauge theory. Gauge theory is now used for all the forces in the Standard Model.

Thus you can think of the modern theory of gravity and electromagnetism as a unified geometrical theory on a 5-dimensional manifold, as was understood by experts around 1920 or so. The 5th dimension is not something that we can spatially measure with a meter stick, but it is physical in the sense that we need it to understand light. You can think of it as an electromagnetic phase that is only indirectly observable.

Einstein wanted to find an extra physical meaning to the 5th dimension in 1938, and Witten wants to find more meaning to more dimensions today. None of this research has ever amounted to anything.

Mocking a unified theory of quantum gravity

I identify with Brewster here. A unified theory of quantum gravity would not be a scientific breakthrough.

Philosopher Massimo Pigliucci partially explains what is wrong with Sean M. Carroll's justification of untestable theories.

He is way too easy on Carroll. String theory and the multiverse are pseudoscientific because they do not make testable predictions. They are not falsifiable. He says that they are "speculations somewhat grounded in well established physics" but they are metaphysical speculations, and the well established physics gives no reason to believe the speculations. The Popper notion of falsifiability works fine for dispensing with these ideas as unscientific.

Update: A comment on a new relativity book says:

“Maybe there’s a lesson here for some of today’s string-theory sceptics?”

It doesn’t matter how empty an argument is, you can always count with someone comparing it to Einstein and relativity. That’s a rule that works both with cranks and people with reputation.

Yes, that is right, and that is why this blog pays so much attention to the history of relativity. If the story is going to be used to justify all sorts of other endeavors, then it is important to get the story right.

Emergence of Lorentz transformations

A new paper on The Case for Lorentzian Relativity gives a history of the Lorentz transformation (LT):

The problem addressed by Lorentz and subsequently Einstein was the speed of light. This emerged as a constant in Maxwell's equations, but if as was generally supposed, light is wave-like, it seemed reasonable to assume that it must be carried by some medium (the "luminiferous aether") at a velocity characteristic of that medium. Thus its velocity relative to an observer should have varied with the motion of the observer through the medium. Experiments of increasing sophistication failed to reveal any trace of that variation.

Several explanations were put forward. It was proposed that the Earth must carry the local aether with it, but a more fruitful suggestion made independently by Fitzgerald (10) and Lorentz (11) was that objects moving through the aether must be somehow shortened along their direction of travel, thereby disguising relative changes in the velocity of light. It was supposed that intermolecular forces must be transmitted at the same velocity as electromagnetic waves, so that movement through the aether would influence the degree of attraction between molecules and thus the separation of those molecules. To effect a reconciliation with Maxwell's equations, it was necessary to assume changes not only of length, but also of time, and thus the LT, ...

The LT was already reasonably well known by 1905. There had been significant contributions to its development, not only from Lorentz and Fitzgerald, but also by (among others) Heaviside, Larmor and Poincaré. It was Heaviside's analysis of the disposition of fields accompanying a charged particle (the "Heaviside ellipsoid") that had suggested to FitzGerald the idea of length contraction (12). Larmor had described an early form of the LT and discussed the necessity of time dilation (13). Poincaré had recognized the relativity of simultaneity and had studied the group theoretic properties that form the basis for the covariance of the transformation (14). ...

Lorentz commented, presumably with some chagrin, that,

Einstein simply postulates what we have deduced, with some difficulty, and not altogether satisfactorily, from the fundamental equations of the electromagnetic field (17).

In what follows, the distinction drawn will be between Einstein's SR (ESR) and what we will call Lorentzian SR (LSR). This is not to diminish the contributions of others, but it was Lorentz in particular who sought to explain SR from underlying physical processes, as will be the objective below. Once the form of the LT was known, all else in SR then followed, including the composition of velocities, the group theoretic properties of the transformation, and the invariance of Maxwell's equations. It may be argued that with these refinements (largely due to Einstein and Poincaré), ESR and LSR are essentially equivalent. They cannot be distinguished, mathematically or empirically, through the privileged frame that was supposed by Lorentz, but declared "superfluous" by Einstein (1). It would seem that any such frame is rendered undetectable by the covariance of the LT. Nor can ESR and LSR be distinguished by supposing that in ESR, though not in LSR, the LT describes a transformation of spacetime. As we have seen, the LT must be explained in either case by changes occurring in matter as it is accelerated from one inertial frame to another.

This is mostly correct. Lorentz's program was more ambitious because he tried to deduce relativity from experiment and accepted principles, and because he looked for physical explanations. Einstein later described the importance of the Michelson-Morley experiment and how he looked for a more constructive theory, but acknowledged that he failed to get these for his famous 1905 paper. Little did he realize that his failure would someday be regarded as a sign of great genius!

Poincare (and later Minkowski) showed the covariance of the LT, and hence the equivalence of frames. Einstein also tried to apply Poincare's relativity principle, but only showed a correspondence of frames in the same sense that Lorentz had previously proved.

It is completely legitimate to say that the FitzGerald contraction of a solid object is caused by a distortion in the electromagnetic fields of the molecules, and that the distortion is predicted by Maxwells equations for motion thru a suitable coordinate frame. If you calculate the electromagnetic fields that hold the atoms and molecules together, apply the quantum field theory to get the atomic distances, and redo the calculations with the distortions from motion, then you will find that solid objects are contracted according to the FitzGerald contraction formula.

The modern view is that the contraction is a byproduct of the non-Euclidean geometry of spacetime developed by Poincare and Minkowski, and adopted several years later by Einstein. It is often useful to have more than one view in physics. Both views are completely correct and based on accepted physics.

I am not sure that Lorentz ever said that any frame was privileged. His theory was that the laws of electromagnetism would be valid in any frame. Einstein said more or less the same thing. The key step in showing that the theory did not need a privileged frame was Poincare's 1905 proof that the Lorentz transformations formed a group, and then that the group is a symmetry group for the theory. Lorentz and Einstein both missed this crucial concept.