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.
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
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.
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 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 Elements and Variety of the Sciences
Philosophy of Science