The "Sin" of Galileo

Now I was being told that another deep aspect of nature was also unified with space and time -- the fact that there are fermions and bosons. My friends told me this, and the equations said the same thing. But neither friends nor equations told me what it meant. I was missing the idea, the conception of the thing. Something in my understanding of space and time, of gravity and of what it meant to be a fermion or boson, should deepen as a result of this unification. It should not just be math -- my very conception of nature should change...

Whereas the math worked, it didn't lead to any conceptual leaps.

Lee Smolin, The Trouble with Physics, The Rise of String Theory, the Fall of a Science, and What Comes Next [Houghton Mifflin Company, 2006 pp.94, 96]

[John von Neumann] never claimed he could not explain something to someone who did not understand the math.

George Dyson, Turing's Cathedral, The Origins of the Digital Universe [Pantheon Books, 2012, p.46]

The scientist who says, "The only way to explain this is to show you the math," either doesn't want to explain the question, and so is brushing you off, or he cannot explain the question. If he doesn't want to explain the question, either he cannot because he doesn't actually understand it, or he is a Positivist who doesn't think that it needs to be or can ever be explained. Either way, if he seems annoyed, rude, or hostile, one's suspicions are reasonable aroused.

Τηλεπατητικός (Telepateticus)

For many young people who aspire to be scientists, the great bugbear is mathematics. Without advanced math, how can you do serious work in the sciences? Well, I have a professional secret to share:  Many of the most successful scientists in the world today are mathematically no more than semiliterate...

Fortunately, exceptional mathematical fluency is required in only a few disciplines, such as particle physics, astrophysics and information theory. Far more important throughout the rest of science is the ability to form concepts, during which the researcher conjures images and processes by intuition.

E.O. Wilson, "Great Scientist Good at Math," The Wall Street Journal, April 6-7, 2013, C2

What then must be done about the shortcomings of quantum mechanics? One reasonable response is contained in the legendary advice to inquiring students: "Shut up and calculate!"

Steven Weinberg, "The Trouble with Quantum Mechanics," The New York Review of Books, January 19, 2017, p.53

In the Pentagon of Power: The Myth of the Machine [2 vol., 1967-70], architect, historian, and critic Lewis Mumford (1895-1990) coined the expression "the sin of Galileo" to refer to the manner in which the world had become merely an abstract mathematical object through Galileo's application of mathematics to physics. To Mumford this change was dehumanizing and ultimately productive of the alienation of modern life. As a "sin," of course, this kind of thing had nothing to do with Galileo's problems with the Church.

Mumford's idea about the alienation of modern life is really a dangerous nostalgia for mediaeval society. The meaning of life may have been clearer in past centuries than now, but it went along with pervasive poverty and an authoritarian political and religious hierarchy. The "alienation" of modern life follows largely from the wealth, leisure, and autonomy that technology and a consumer economy make possible. People are left to figure out or decide on the meaning of life for themselves. Those with the most leisure -- intellectuals and teenagers -- suffer from such alienation the most. The disapproval of intellectuals for what most people enjoy -- television, sports, drinking, smoking, sex, and violence -- is the same immemorial moralism of mandarins, priests, and aristocrats that always disapproved of vulgar, i.e. popular, pleasures, the same yearning for the day that political authority, namely them, can once again govern meaning and morality in life for everyone. At the same time, the reaction of a fully Mediaeval sense of life against modernity is now evident in radical Islâm, enforced with the violence of terrorism. This may not be the kind of remedy that intellectuals like Lewis Mumford had in mind.

The sense that we are alienated from nature is also a dangerous nostalgia:  No people absolutely vulnerable to famine, disease, insects, wild animals, nomadic invasions, etc. are going to complain much about conditions that limit or erase the danger of such things. Only those who have forgotten how hard and merciless life used to be are going to feel "alienated" by the culture that protects them from those things. For them, outlawing DDT and forbidding the draining of "wetlands" are important steps in protecting nature, whether or not people begin to die again around the world because of the malaria that is spread by mosquitoes who breed in the "wetland" swamps and who can no longer be effectively killed once produced. Those who live in comfort in Europe or the United States don't have to worry about people in Sri Lanka dying of malaria, though there now actually are places in Europe and the United States that feel some of these effects -- just not at major universities or newspapers yet. We also find the extraordinary and offensive phenomenon of Western environmental activists telling Africans, for instance, that deaths from malaria are a natural part of the beauty of their environment and that the United States has few problems with malaria because is had never (!) existed there -- just as Global Warming alarmists like to say that mosquitoes never existed at high latitudes, until recent anthropogenic warming. These are all falsehoods which can only be uttered because of ignorance or mendacity.

Nevertheless, there is an important sense in which we can apply the idea of the "sin of Galileo": Galileo represents an important shift in how mathematics is seen. With him, and with even more peculiar characters like Johannes Kepler and Isaac Newton, the Platonic-Pythagorean notion that mathematics reveals the inner structure of reality returns.

Mediaeval understanding had mostly followed Aristotle, seeing mathematics as no more than an device for calculation, something made up by us that had no essential connection to reality. Thus, where Plato had seen the four elements as consisting, atom-like, of four of the five Platonic solids, tetrahedrons for fire, octahedrons for air, icosahedrons for water, and cubes for earth, Aristotle ignored this completely and saw the four elements in Presocratic terms as distinguished by two sets of opposites -- hot and dry for fire, cold and wet for water, hot and wet for air, and cold and dry for earth. Both Plato and Aristotle were, of course, wrong; but Plato was not far off the mark: The Platonic solids do occur with the packing of atoms in crystals. Common table salt, for instance, the mineral Halite (NaCl), occurs in cubic crystals. Aristotle's opposites, on the other hand, have no modern form, unless we want to reach for the analogy of the presence or absence of sub-atomic properties like strangeness, charm, etc., which nevertheless have nothing to do with macroscopic qualia like hot and cold or wet and dry.

Both Plato and Pythagoras thought that mathematics would reveal the inner nature of things, a conviction preserved in Mediaeval Romania and brought to Western Europe by Greek refugees from the Ottomans. With Copernicus, Galileo, and Newton, this expectation seemed to be born out, although, curiously, modern philosophers of mathematics tend to prefer the idea, again, that we have made it all up ourselves. "God made the integers, all the rest is the work of man," is a very famous and often quoted statement by Leopold Kronecker (1823-1891). Great scientists themselves, from Einstein to Hawking, still think of mathematics as revealing the Thoughts of God (although Hawking is not consistent about this). From Kronecker's statement we can conclude that he was not interested in the Thoughts of God; and, perhaps not surprisingly, he does not seem to have substantively advanced physics himself.

The Platonic-Pythagorean-Galilean view of mathematics, however, is clearly limited. The "sin" of Galileo in a new sense must be the belief that because we have a mathematically successful theory, this means that we understand what is going on. This is clearly not true. Newton's theory of gravity was one of the most successful theories of all time. It was the paradigmatic mathematical theory of nature. But even at the beginning there were unanswered questions about it, especially in that it postulated action at a distance -- that two bodies would affect each other gravitationally even without being in contact and without anything whatsoever mediating that contact. Gravity was something that was nothing in itself that nevertheless exerted a force invisibly across a complete Void. The mystery of this was highlighted by Newton's own belief that it was the Will of God. This paradox and mystery, intense at first, slowly lost its power as the success of the theory silenced opposition.

Newtonian mechanics did not fall because of philosophical objections to action at a distance; but it fell to theories that, serendipitously, actually did not postulate action at a distance. These were, at first, Einstein's general relativity, in which the curvature of space-time eliminated the need for "forces" altogether, and then quantum mechanics, which postulated an exchange of virtual particles to mediate forces. These alternate explanations, between Einstein and quantum mechanics, now accentuate the circumstance that successful mathematical theories, as such, do not enable us to understand reality. Since Einstein's theory works for gravity, and quantum mechanics works for the other forces of nature, one might perhaps be tempted to say that gravity involves a curvature of space-time and the other forces involve an exchange of virtual particles. But this is not what physicists have expected: It is going to be one or the other. More recently, "super-symmetry" theories have extended Einstein's approach to the other forces, with the addition of ten extra dimensions of space to account for them. The mathematical, if not the observational, success of these theories have animated physics in the last several decades; but the strangeness of the whole business has come full circle with the conclusion of some that the extra dimensions are not "really" there but simply represent abstract mathematical dimensions that do not need to exist in the world. This would seem to leave things even more unexplained than before:  Einstein's geometry of reality becomes, once again, a calculating device.

Thus, indeed, a successful mathematical theory does not enable us to understand what is going on in reality. Multiple aspects of quantum mechanics reinforce this impression, since, as people say, no one understands quantum mechanics, but you get used to it. Why this could happen is explained by Karl Popper's view of scientific method:  Theories are simply logically sufficient, not necessary, to observations. That is because theories are only falsified, never verified. Thus, different theories, in principle, could explain the same phenomena; or, a theory could mathematically predict the phenomena, without otherwise making any sense. This seems to be the case with quantum mechanics.

Refusing to commit the "sin of Galileo" thus means the realization that scientific theories have both mathematical and conceptual sides to them, where the mathematics never represents more than an abstract fragment, the quantitative, of phenomena. A theory may be mathematically strong but conceptually weak, or vice versa. It is widely acknowledged that Einstein's Relativity is conceptually lucid and compelling, while quantum mechanics is mathematically exemplary while conceptually incoherent. It is revealing that, despite this, quantum mechanics for a long time was expected to replace Einstein. Great physicists like Richard Feynman positively reveled in the conceptual incomprehensibility of quantum mechanics. Now the shoe seems to be on the other foot, as super-symmetry extensions of Einstein's theory apparently embrace the other forces of nature as quantum mechanics never could gravity -- despite the people who don't seem to think it is important to posit the real dimensions required by these extensions.

Recognizing the "sin of Galileo" must put scientists, and their sympathizers in philosophy, in the uncomfortable position of admitting that "philosophical objections" are not always absurd vapors to be dismissed but can often be significant warnings about the deficiencies of a scientific theory. The objections themselves, indeed, will not always be cogent and will rarely produce the answer; but, like the canary in the mine, they are a significant warning that something is not quite right. Nor is it always philosophers voicing the objections. Albert Einstein himself was intensely unhappy with the direction quantum mechanics took with Werner Heisenberg and Niels Bohr. Either Einstein himself must be dismissed as an old fool, which he was for many years, or it must be recognized that philosophical objections are often germane. Indeed, a compelling vindication of Einstein's doubts is still not complete, though thankfully Roger Penrose's The Emperor's New Mind goes a very long way to completing it.

It is perhaps too much to expect that mathematicians and philosophers will ever collaborate in the way that composers and librettists do; but this is what, over time, will and must in effect happen. The major difficulty is with the philosophers: They tend either to be so enamored of their own theories, like the Hegelians, that they don't even notice real science, or they are so awe struck and humbled by science, like the Logical Positivists, that they cannot summon up the audacity to actually criticize it. The middle ground of philosophers like Kant, Nelson, and Popper, who mostly understand science rather well but retain the faculty of criticism, is quite rare. Schopenhauer demonstrates the difficulty of hitting the mean, since he has rather interesting things to say about the laws of nature but then makes extremely foolish statements about the wave theory of light. Preferring Goethe's theory of light to Newton's, Schopenhauer misses the point of the new physics in his day of Thomas Young and Michael Faraday. Loving the pure philosophical theory better than the good science, Schopenhauer could not have anticipated that science would ultimately return to his beloved qualitates occultae in the form of the strangeness, charm, top, bottom, leptonic charge, baryon number, etc. now found in particle physics. The Democritean Atomism that he thought he discerned in Young and Faraday is long discredited.

Thus, recognizing the sin of Galileo does not provide us with a method for distinguishing the true from the false, but only with a caution, like Popperian philosophy of science in general, for how we regard the results of science or its relation to philosophy. This is a real enough caution, however, which must rule out many commonly expressed attitudes, especially those that disparage the independence or usefulness of philosophical knowledge, or those which are eager to dismiss science as damned with some kind of political bias -- but that is another story.

The Strangest Man,
The Hidden Life of Paul Dirac,
Mystic of the Atom

by Graham Farmelo

Basic Books, 2009

Out of the blue, it occurred to Dirac that he had come across a special mathematical construction, known as a Poisson bracket, that looked vaguely like AB - BA. He had only a faint visual recollection of the construction, but he knew it was somehow related to the Hamiltonian method of describing motion...

Sure enough, as Dirac had surmised, the Poisson bracket, which first appeared over a century before in the writings of French mathematician Siméon-Denis Poisson, had the form of two mathematical quantities multiplied together minus two related quantities multiplied together, the multiplication and minus signs making it appear similar to the expression AB - BA. In one of his great insights, Dirac saw that he could weave an entire carpet from this thread -- within a few weeks of uninterupted work he had set out the mathematical basis of quantum theory in analogy to classical theory. Like Heisenberg, he believed that mental pictures of the tiniest particles of matter were bound to be misleading. Such particles cannot be visualized, nor is it possible to describe them using quantities that behave like ordinary numbers, such as position, speed and momentum. The solution is to use abtract mathematical quantities that correspond to the familiar classical quantities: it was these relationships that Dirac pictured, not the particles that they described. Using the analogy with the Poisson bracket, together with the correspondence principle, Dirac found connections between the abstract mathematical quantities in his theory, including the crucial equation connecting the symbols associated with the position and momentum of a particle of matter:

position symbol x momentum symbol - momentum symbol x position
symbol = h x (square root of -1)/(2 x π)

where h is Planck's constant and π is the ratio of the circumference to the diameter of every circle (its value is about 3.142)... The most mysterious part of the equation was on the right-hand side, especially for those unwise enough to think of the position and momentum symbols as anything other than abstractions: they are not numbers of measurable quantities but symbols, purely mathematical objects.

Graham Farmelo, The Strangest Man, The Hidden Life of Paul Dirac, Mystic of the Atom, pp.86-87, color added; do Farmelo's readers really need pi explained to them?

The book by Graham Farmelo on great physicist Paul Dirac gives us some excellent examples of the Sin of Galileo, such as we see in the passage just quoted, and on some other points in the philosophy of science. The most striking thing here is that Dirac himself seems to like the idea that the "symbols" in his equations don't even refer to anything. But, in fact, as "purely mathematical objects" they are already more than just "symbols." If they have no meaning or reference at all, then we have the kind of mathematics described in the formalistic project of David Hilbert, where the symbols could mean "beer steins" rather than numbers, and all that matters are the rules that manipulate them. This formalistic view of mathematics was refuted by Kurt Gödel, who proved that any formal system of mathematics must contain propositions that must be true but cannot be proven within the system. Thus, they can only be true because they refer to some external ground, and they must have meaning and reference for that to happen. So, in mathematics the symbols in equations cannot be just "symbols."

The idea that in physics the terms are only "symbols" and do not mean "measurable quantities" is even more remarkable, especially when we actually identify those quantities as things like "position" and "momentum." We could use the notion that our quantities aren't really these things in any way that we can understand as part of a modest philosophy, consistent with a rejection of the Sin of Galileo, that we don't know what is really going on. This is a natural reflection given the curious and mysterious characteristics of quantum mechanics. But it also leads us to a crossroads. Either we can endeavor to make sense of the reality of the objects we are studying, or we can adopt a Positivist approach that we don't care what the reality is and will make no effort to figure it out. Thus, all that matters to the Positivist is that the "black box" of a physical theory produces predictions that can be confirmed. As Roger Penrose himself has commented, it hardly even makes sense to study or do science if you don't what to know what is going on. To the extent that Dirac himself thought that the pure mathematics was the only object of his study, he was either a Positivist or embraced the Sin of Galileo itself. At least in the Sin of Galileo you think that something -- pure mathematics -- provides meaning for your theory. In Positivism, you have no ambition to actually understand anything. But the expressions used by Farmelo, like "momentum symbol," in which we have somehow simply named a meaningless hieroglyph, are deeply meaningless and pointless, or evasive.

Farmelo says:

The philosophers who least offended Dirac and other theoretical physicists were the logical positivists, who held that a statement had meaning only if it could be verified by observation. There are traces of this philosophy in three pages of notes Dirac wrote out by hand in mid-January 1933... [p.220]

Since Dirac was not otherwise interested in philosophy, he never had the perspective necessary to see the problems with Logical Positivism. Briefly, (1) to know whether or not a theory could be verified, one would need to understand it, which means that even theories that could not be empirically verified must have meaning, or their character could not be understood; and (2) the Positivists venerated Hume, and they were aware of Hume's treatment of the fatal Problem of Induction, but they never seemed to understand that this meant that scientific theories could never be "verified," which means they would necessarily all be meaningless. In these terms, and more, Logical Positivism was incoherent and self-refuting.

But Dirac did not have the sensibility of a Positivist. It was more important to Dirac that a theory be mathematically elegant than that it actually passed empirical tests. The mathematical "renormalization" technique of Richard Feynman, which defused the infinities that had plagued quantum electrodynamics, were offensive to Dirac; and he hated them the rest of his life, regardless of the success of the practice.

Although [Pascual] Jordan and [Eugene] Wigner's mathematics [of a field theory of the electron] was similar to Dirac's, their theory did not appeal to Dirac, who could not see how their symbols corresponded to things going on in nature. Their work looked to him like an exercise in algebra, though later he realised he was wrong; his mistake stemmed from his approach to theoretical physics, which was 'essentially a geometrical one and not an algebraic one' -- if he could not visualize a theory, he tended to ignore it. [p.139, color added]

This remarkable statement seems to contradict what Farmelo says, as we have seen, elsewhere. If Dirac was worried about the symbols corresponding to "things going on in nature," then obviously they are not just self-referential symbols. At the same time, we get the view, expressed elsewhere in the book, that Dirac's sensiblity was, not just for mathematical elegance and beauty, but for a particularly geometrical version of this. If he realized that it was a "mistake" to use geometry to dismiss algebra, this still leaves his sensibility as essentially mathematical, with an open question how any of it is going to correspond to things going on in nature. So there is a bit of a muddle here that needs clarifying, something that I'm not sure ever happened.

With astonishing boldness, Heisenberg had abandoned the assumption that electrons can be visualized in orbit around a nucleus -- an assumption no one had previously thought to question [!] -- and replaced it by an purely mathematical description of the electron. [p.84]

First of all, it does not seem like "astonishing boldness" for Heisenberg to abandon the "assumption" that electrons must be conceived, or "visualized," as in orbit around a nucleus. I think it was already obvious that they could not be in orbits, like planets around the sun, since they would be accelerated in such orbits and would, as accelerated charges, radiate away their energy. This was a profound problem for all theories of Rutherford's atom, where the nucleus is small, dense, and positively charged, while electrons somehow fill the rest of the volume of the atom. The model of orbiting electrons is so irresistable that we still constantly see images of it (as on the popular The Big Bang Theory television show); but it never could have been correct. If "no one had previously thought to question" this, then every living physicist was out to lunch -- which strikes me as unlikely. There may have been nothing to do about it -- a situation that often happens in science -- but that did not mean it could be accepted as correct.

Heisenberg wrote that some of the quantities in the theory have a peculiar property:  if one quantity is multiplied by another, the result is sometimes different from the one obtained if the sequence of multiplication is reserved. This was exemplified by the quantities he used to represent position and momentum of a piece of matter (its mass multiplied by its velocity):  position multiplied by momentum was, strangely, not the same as momentum multiplied by position...

Unlike Heisenberg, who had never come across non-commuting quantities before, Dirac was well acquainted with them -- from his studies of quaternions, from the Grassman algebra he had heard about at [Henry] Baker's tea parties, and from his extensive studies of projective geometry, which also features such relationships. [pp.84-85]

What Heisenberg was developing was his "Matrix Mechanics" version of quantum mechanics. As Farmelo mentions elsewhere, Heisenberg actually wasn't even familiar with mathematical matrices, which is why he was surprised that his own mathematics violated the algebraic principle of commutation (i.e. xy = yx). In effect, he was rediscovering a property of matrices. This is actually a nice example of a truth that Farmelo says was appreciated by Dirac, that unusual forms of mathematics may turn out to be the key to something in physics. At the same time, when this happens, it vindicates the Pythagorean and Platonic understanding of the world.

Dirac probably knew about matrices, but he also already knew about William Hamilton and quaternions, where Hamilton was forced to accept that commutation was violated with his strange multiples of imaginary numbers (i, j, and k; with ijk = -1), so that ij = -ji. Heisenberg, of course, was dealing with the sort of autistic mathematical abstraction whose physical meaning was entirely concealed -- and, with "boldness," left unexplained by Heisenberg. Before long, Heisenberg's Matrix Mechanics was matched by Schrödinger's Wave Mechanics, which turned out to be mathematically equivalent, but which also had a physical meaning. Nevertheless, Farmelo, like many other scientists, philosophers of science, and historians of science, has difficulty understanding or expressing the wave nature of matter, despite Niels Bohr's Principle of Complementarity, which is that the "wave/particle duality" means that in each physical event we can understand the operation of a wave, or of a particle, but not both at the same time. I don't think that Farmelo does a good job of explaining this principle, or even of showing that he understands, or accepts, it very well.

I must now display my own ignorance. My understanding is that the introduction of a negation as the result of commutation, as with Hamilton's quaternions, is the key to the Pauli Exclusion Principle of Wolfgang Pauli. I saw this clearly explained just once, when I was watching physics documentaries shown on "Thursday Night at the Physics Movies," put on by the Physics Department at the University of Texas in the late 1970's. Unfortunately, I wasn't taking notes, supposing that the explanation was something otherwise readily available. It wasn't; and in fact I have never seen anything of the sort since in popular expositions of science.

Indeed, we get a similar oversight in Farmelo's book. Thus, even though Dirac is immortal precisely for his equation of the electron, this equation is never given in its full form in the book and its features are never explained in the least way. All we are given is a "succinct version of the Dirac Equation," iγ.∂ψ = mψ, such as is carved on the monument to Dirac in Westminster Abbey [p.142]. Not even the elements of this "succinct version" are identified or explained in the book. "When set out in full, in the form he originally used, the equation looked intimidating even to many theoreticians simply because it was so unusual..." [pp.142-143]; but we are not even vouchsafed a peek at so "intimidating" and "unusual" an equation, let alone have any of it explained.

Perhaps this is the Sin of Galileo in action. Farmelo does not explain the equation because it cannot be explained except in its own esoteric mathematical terms, and those of us without the background will just have to leave it to the experts to know what is going on. This is rather different from saying that the mathematics is "just symbols" and a meaningless device for making Positivistic predictions. If it is really so meaningless, how can its formalism not be explained? Or at least shown? Indeed, the obscurity of these modern equations in physics mainly lies in their mathematical detail largely being concealed. Many of the symbols are mathematical operators that, far from being mere symbols, stand in for a great many other equations that unpack the function. These are just never given at the popular level, let alone explained to the ignorant.

Nevertheless, Farmelo gives some explanations that I have not seen elsewhere. The prediction of anti-matter from Dirac's Equation is because energy is expressed in the equation (although not in the forms shown here) as a square, which will have both positive and negative roots:

Dirac's problem was that his equation predicted that, in addition to perfectly sensible positive energy levels, a free electron has negative energy levels, too. This arose because his theory agreed with Einstein's special theory of relativity, which said that the most general equation for a particle's energy specifies the square of the energy, E2. So if one knows that E2 is, say, 25 (using some chosen unit of energy), then it follows that the energy E could be either +5 or -5 (each of them, multiplied by itself, equals 25). So, Dirac's formula for the energy of a free electron predicted that there were two sets of energy values -- one positive, the other negative. [pp.144-145]

This was a troubling feature of Dirac's theory. No one believed there could be negative energy. It could well mean that his equation was gravely flawed, or wrong. And it all depended on the basic rule of algebra that positive numbers have positive and negative roots -- it never occurred to physicists, apparently, that we could just change the rules of algebra (as suggested by Alberto Martínez).

Dirac and others developed different ways of dealing with the required negative energy electrons. Dirac's own strange idea was that space had become filled with negative energy electrons, the equivalent, in a way, of the ether, which generally would be undetectable but occasionally would have a "hole," a missing electron, which would then appear as a positive particle in interactions with other matter. Unfortunately, Dirac at first thought that these positive "holes" in space were actually protons. This bizarre theory, however, was soon vindicated, in part, by observation. Cosmic rays, first observed in cloud chambers, seemed to contain, or to create, particles with the mass of the electron but with a positive charge -- "positrons." If these were the "holes," they were not protons, which were much more massive. But they were something quite as bizarre as was required by Dirac's theory. It was anti-matter, like ordinary matter in mass (and energy) but opposite in charge and all other quantum numbers (the esoteric baryon number, lepton number, strangeness, charm, etc.). This is now a commonplace of science and science fiction. The engines of the starship Enterprise somehow operate by using anti-matter, which releases pure energy when brought into contact with ordinary matter.

Meanwhile, we might still wonder what it would mean if anti-matter is ordinary matter with negative energy -- let alone why the unverse seems to consist of matter rather than anti-matter (a cosmological problem). All we really need is a place to move the negative sign. As it happens, we can just move the negation down to time. Anti-matter is ordinary matter going back in time. Now, in its own way, this is as strange as negative energy. However, the fundamental equations of physics (like those of either Newton or Einstein) are symmetrical in time, so there is nothing particularly outrageous about particles going back in time -- the problem of physics in time is order and entropy, which a lot of physicists and philosophers would rather not worry about. And we don't need to worry about them here, especially when the particles going back in time are not easily and conventionally represented in Feynman Diagrams, which Feynman himself orginated and used to represent the intereactions of particles. In the diagram here we have an electron going forward in time, emitting a photon of energy, and then going back in time as a positron. The photon itself decays into a quark and an anti-quark, with the idea that the anti-quark is coming back in time and being scattered by the photon into a quark, which goes forward in time.

Farmelo does not mention or discuss this interpretation of anti-matter; but, if we don't worry too much about the metaphysics of time, it provides a simple and elegant representation of what "negative energy" could possibly mean. At the very least, it is a nice convention for use in the Feynman diagrams. At best, some like the idea because we could imagine that there is a dearth of anti-matter in the universe because all the anti-matter actually went back in time from the point of the Big Bang. This would make the Big Bang symmetrical in time, which has its own appeal.

In later life, Dirac liked to point out that quantum mechanics was the first physical theory to be discovered before anyone knew what it meant. He spent months on the problem of interpreting its symbols and came to see that the theory was mathematically less complicated than he had first thought. [Max] Born pointed out to Heisenberg that each array of numbers in his quantum theory was a matrix, which consists of numbers arranged in horizontal rows and vertical columns that behave according to simple rules spelt out in textbooks. Heisenberg had never heard of matrices when he discovered the theory, as Born often reminded his colleagues... [p.96]

However, one could easily say that Newton's theory of gravity was also a physical theory that was formulated before anyone knew what it meant. The "action at a distance" nature of Newtonian gravity was perplexing and objectionable to many, while that very feature of it has been serendipitously eliminated, as we have seen above, in both Einstein's Relativity and in quantum mechanics. At the same time, if Dirac decided that quantum mechanics was "mathematically less complicated than he first thought," because it can be arranged as mathematical matrices, as Born pointed out to Heisenberg, it is not clear how this gives us any notion of what quantum mechancis, as a "physical theory," is supposed to mean. Heisenberg's matrices were precisely how he could avoid explaining what the theory meant.

I listened to their arguments, but I did not join in them, essentially because I was not very much interested [...] It seemed to me that the foundation of the work of a mathematical physicist is to get the correct equations, that the interpretation of those equations was only of secondary importance...

Yet, whereas Einstein remained interested in philosophy, for Dirac it was a waste of time. What Dirac had retained from his reading of Mill, bolstered by his studies of engineering, was a ultilitary approach to science: the salient question to ask about a theory is not 'Does it appeal to my beliefs about how the world behaves?' but "Does it work?' [p.137]

So here, beginning with Dirac's response to Bohr and Einstein arguing about the meaning of quantum mechanics in 1927, we are back to a metaphysical, or even as physical, Know-Nothing-ism. The equations seem to exist for Dirac in their own reality; and if physics is a study of the world, the actual explanation that we end up with in those terms is actually "of secondary importance." Now, if Dirac were a Platonist, like Gödel, this would make a little more sense. The mathematics of the world would be the nature of the world. But it doesn't look like he was. So the result is a Skeptical ἐποχή (epoché), a suspension of judgment.

However, Dirac could not even be consistent about this:

According to Heisenberg, Dirac thought religion was just 'a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination.' For Dirac, 'the postulate of an Almighty God' is unhelpful and unnecessary, taught only 'because some of us want to keep the lower classes quiet'. Heisenberg wrote that he objected to Dirac's judgement of religion because 'most things in this world can be abused -- even the Communist ideology which you recently propounded'. Dirac was not to be deflected. He disliked 'religious myths on principle' and believed that the way to decide what was right was 'to deduce it by reason alone from the situation in which I find myself:  I live in a society with others, to whom, on principle, I must grant the same rights I claim for myself. I must simply try to strike a fair balance.' [John Stuart] Mill would have approved. [p.138]

Now, having decided that the meaning of a physical theory is of "secondary importance," Dirac passes judgment on religion as having "no basis in reality." Didn't we just see that he wasn't interested in reality? But if he was not interested in reality, then his inability to see anything in religion would make sense. So Dirac needed to make up his mind. At the same time, while his desire to deduce "what was right" by "reason alone" would be agreeable to John Locke, the "Communist ideology" that seemed to always be tempting Dirac would not have been commensurable with Locke's liberalism. In the same way, Mill's principle of liberty that for others "I must grant the same rights I claim for myself," would not be consistent with "Communist ideology" either. In any case, it is not clear how the principles of either Locke or Mill could be "deduced" from "the situation in which I find myself," which, as a matter of fact, could not logically imply, according to Hume, any moral principle at all, let alone one that established maxims of bourgeois morality rejected by Marx.

But then, as we see also in another physicist who expressed no interest in philosophy whatsoever, namely Richard Feynman, we cannot expect Dirac to have very sophisticated ideas about either metaphysics or ethics. Like Feynman, he has trouble even maintaining consistency when venturing into the perilous territory of those matters. Yet he occasionally had some things to say about them anyway, unlike Feynman. But those forays were indeed rare; and even his trips to the Soviet Union and friendship with Russian physicists seemed to be largely non-political. So, as we might expect, we never do get a systematic account from Dirac about science, reality, or morals.


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