Saturday, January 07, 2012

Can the maths in physics be simpler than it is?


That was the short version of this blog entry addressed to the impatient readers. But the more patient readers may continue. ;-)
Motto: A scientific theory should be as simple as possible, but no simpler. – A former South German colleague of mine
In a comment, James G. summarized the ultimate driving force that makes so many people reject string theory, quantum field theory, quantum mechanics, or modern physics in general:
I undestand your defence of the depths of the mathematical techniques required for full assimilation of the modern theoretical physics papers – but I for one believe it is not really required i.e. Nature is not so [copulating] difficult.
Yes, the ultimate driver is math phobia. But as a wise web page explains, math doesn't suck: you do.

Allowing maths to play a role

The people who don't understand that the world is fundamentally controlled by maths are a mystery for me, in the same sense as women are mystery for Stephen Hawking. After all, this is not just an analogy, it could be a nearly equivalent statement because most people who deny that maths fundamentally governs the reality are women.

(Hawking has reiterated that the humans have to colonize Mars and outer space to escape from the looming nuclear Armageddon: yes, he echoed Fidel Castro. Because Hawking's IQ tops that of the climate alarmists by 40 points or so, BBC published a criticism of Hawking's summary for policymakers.)

I have taken the key role of maths in the world for granted from the first moment I began to think – which is really why the math deniers look like a different biological species to me. When I was 3 and learned how to write and read, finding the right theory matching the perceptions became a priority. The first framework I had when I was 4 or so was based on matter filling a three-dimensional Euclidean space. Using a modern terminology, it was a classical theory whose configuration space was made out of the maps
\[ f:\RR^3 \to \{0,1\} \] In other words, at each point of the world, there either "is" something, or there is nothing. I bet that for many of you, this was the template of the first theory you believed to match the natural phenomena around you. I was convinced that it had to be true, that properties of the materials are encoded in some microscopic patterns and shapes of the regions where the map is equal to one (regions with boundaries that I implicitly assumed to be infinitely smooth), and the only remaining task was to find out how the map \(f\) evolves in time.

My belief in this basic framework of physics (which may have been on par with the ideas of some of the ancient Greek philosophers) continued for nearly a decade – up to the age of 12 or so when the bulk of my actual knowledge of physics was already dominated by classical mechanics. In some sense, mechanics is compatible with the picture of the world given by the binary maps above.

Because of this first picture of the world, I had quite some problems to understand that there can be electromagnetic waves as well. It's not hard to see that the binary picture of matter above isn't quite naturally compatible with the concept of classical fields. The apparent fact that the radio is able to catch dozens of radio stations – something I could observe by playing with the radio – looked kind of incredible to me. How does the apparently empty volume of the living room (except for air that I considered clearly irrelevant for the phenomenon: I've never believed any kind of an aether) manage to contain so much information? Information that is needed to give me dozens of radio stations with a great sound when I tune them?

My naivite concerning the Fourier expansions was so high that when I was 8 or so, despite the fact that I played the piano (rather well) and "heard" the music, I still believed that when you press "C" and "E" on the piano, the resulting sound has to be kind of equivalent to "D". This was really stupid, wasn't it? Tones don't get averaged out in this way. Every frequency is independent of others. This was true for the piano; it was true for the radio stations.

As you might expect, I ultimately decided that my previous opinions had to be wrong: it directly followed from the observations. Because the "binary map" picture of the world was too discrete to account for continuous waves representing different radio stations that are added and that don't interfere with each other, I eventually abandoned the idea that the "binary maps" are the right description of Nature.

Of course, this conclusion wasn't important just because it ruled out a particularly stupid theory of Nature. The more important lesson was that the very conceptual beliefs that you treat as dogmas may turn out to be wrong. Someone has said a wise thing: if you haven't believed a general thesis about Nature for years, a thesis that you ultimately falsified, found incorrect, and abandoned, then you haven't started to think about Nature in a scientific way yet.

It's extremely important for a potential physicist to go through this experience. What one learns is that dogmas and preconceptions may be wrong. Most deluded people – those who oppose quantum mechanics or string theory or any other valid theory of Nature – have failed to be enlightened about this fundamental principle of science. I could have thought that the uncertainty about the dynamical rules governing the evolution of the "binary maps" was large enough. In other words, Nature had a lot of freedom how it could behave, while respecting my basic framework. You may live in a studio as well so why Nature shouldn't be satisfied with the space you reserved for Her? But the message I learned was that it wasn't up to me to decide whether Nature had enough maneuvering space: Nature may always decide to disagree. She may find the room I reserved for Her to be too constraining. The class of theories you are willing to even consider may look large to you but Nature may still find it suffocating and refuse to live there! General (and huge) classes of theories, and not only individual, fully specified models, may be and are often ruled out in science.

If I get back to the story, I obviously adopted the classical fields as an independent part of the reality. By the age of 15, I thought that everything in the real world had to be given by some kind of a classical field theory. It was obvious that the old "particles" I believed in the age of classical mechanics were unnecessary: they could be created out of fields, as solitons, if I use a modern terminology again (which I surely wasn't using when I was 15). During attempts to realize "Einstein's program", I found the skyrmions (modern terminology) as a possible explanation of the charge quantization, and I proposed that wormholes connecting pairs of solid holes of genera \(h=0,1,2\) corresponded to the electron, neutrino, and proton. For a month, I was sure that I had to instantly get a Nobel prize for that (which, I admit, wasn't the first time). The \(h=2\) surface may have a \(\ZZ_3\) symmetry which I thought was the final explanation why the stupid physicists had previously thought that there were 3 quarks inside a proton. Now, everything seemed clear: The fundamental topological theory of all the matter had been finally found.

Or was it? ;-) Of course, I figured out that the theory was incompatible with the electron-positron annihilation and other things; it had really nothing to say about the data that a theory of similar phenomena has to explain. (And I realized that the reason why physicists believed in quarks was very different from a \(\ZZ_3\) symmetry of the proton.) Most crackpots who insist on their similar theories never want to realize such inconvenient truths.

Then, about a year later, when I was 16, I began to study the atoms and spectra. In the attempts to understand all the known facts about the behavior of the Hydrogen atom, I was stealing ideas from the engineering textbooks and encyclopedias on quantum mechanics. At the end, I realized that I had to steal everything for my theory to work well. ;-) A story I mentioned was that I dedicated about 2 months to struggles to preserve determinism and/or realism (using modern terminology), and because my attempts to describe the phenomena within the classical framework made no sense and some conclusions or postulates of quantum mechanics began to look like "directly extracted from the observations", I stopped that activity and have been convinced for more than 20 years that quantum mechanics including its right Copenhagen-like "interpretation" is absolutely inevitable for a viable theory.

While I spent some time by trying to "preserve" some of the principles of classical physics, it could have been a much shorter time than the time I spent with beliefs in the "binary maps" or in the "non-existence of waves". What may have mattered was the general lesson that preconceptions may be wrong and conclusions systematically extracted by a mathematical analysis of the observations have the power to beat any preconceptions. In the case of quantum mechanics, I just applied the same principle – the main principle of science – to the questions about the behavior of atoms and elementary particles.

From this conceptual viewpoint, my learning or rediscovering of quantum field theory and string theory were already parts of a "routine". One could say that the number of "truly conceptual changes" one must adopt is lower than it is when you start to deal with quantum mechanics. But even if quantum mechanics is the "largest single step", it surely doesn't mean that it has to be the "final step". I've always understood that when you freeze your conceptual opinions, being convinced that a step beyond a conceptual framework is a "taboo", you are implicitly deciding to stop learning new physics.

In some sense, I think that all the quantum haters and string theory haters etc. must realize the same thing. They just decide to stop learning physics above a certain level. They must know extremely well that it's them who suck. James G. must know that he was better in physics while an undergrad, so he had a clear edge. But he just lost it when he began to deal with more advanced physics, the graduate school level of physics (which may be equivalent to the last 70 years in physics research, although the number is higher or lower for various subdisciplines of physics). The situations are completely analogous; in James' case, the only difference is that in the graduate case, he's already been on the losing edge.

Mathematics is pretty much by definition what is needed to understand the "precise truth" about Nature. That's because mathematics is the only precise language we may use. I am sure that whenever practical questions are concerned, people understand it's the case. They are careful whether they pay 340 dollars or 430 dollars even though the difference is just some silly permutation of digits, the kind of thing that nitpicking mathematicians could care about.

But the same people believe that when it comes to the "impractical" questions about the fundamental laws governing Nature, mathematics and the precision associated with it shouldn't matter. Of course, they're wrong. I think that the reason why they are ready to believe this silly conclusion – that maths applies in the supermarket but not in the "internal guts" of atoms and elementary particles – is that they ultimately don't give a damn about the fundamental laws of Nature, so they're not ready pay the price (the hard work needed to think mathematically) in the theoretical context.

Extensive vs intensive expansion of mathematics

When it comes to the advances that force us to learn "more maths" or "more difficult maths", there are two basic kinds of progress:
  • extensive progress: using the same rules and concepts but adding much more data, something that requires more brute power to evaluate
  • intensive progress: one has to learn completely new concepts and meta-concepts, new relationships between them, new mathematical structures to describe the same perceptions; the connection between observations and the theory is getting more indirect and more dependent on maths
The extensive progress happens all the time and people usually agree it's needed. When you replace 2 celestial bodies by 3 celestial bodies, well, you have to solve a 3-body problem instead of the 2-body problems. It's not hard to see that it's more difficult. But people may agree that it may be put on a computer. The situation may be simulated, and so on. A similar comment may apply to trillions of trillions of atoms in a piece of insect that we want to describe. There are clearly more data; the particles or other objects carry more information; longer calculations are needed to evaluate what's happening.

However, I think that math phobia isn't really an opposition to this "extensive progress". Math phobia is almost universally a fear of the "intensive progress". People just hate to learn completely new methods, new mathematical structures, or the representation of physical objects and phenomena by completely different structures than those that were used previously. That's really what leads various anti-quantum zealots, string theory critics, and other breathtaking idiots to invent excuses and redirect the criticism – which should be exclusively directed on themselves – onto someone or something else.

Lots of "intensive progress" or "conceptual breakthroughs" or "incorporation of new abstract concepts" have occurred in the history of physics. From some viewpoint, these advances are really the most important advances that have taken place. I am sure that most good theorists are ultimately measuring the importance of a paper in this way – how much it changes the "qualitative character" of our ways to think about Nature.

One should notice that the need for the "complicated concepts" or "complicated equations" may often be almost directly extracted from observations of Nature. Einstein's equations of general relativity may look difficult (look at all the Christoffel's symbols that are inside!) but we may pretty much directly extract the individual terms from observations in a one-by-one fashion. So the right theory has to include Einstein's equations or something equivalent to them (or more accurate to them). People who find the equations "too complicated" never have an alternative. However, a much stronger statement holds: it can often be shown that there can't really be any alternative because the individual pieces of the equations or new conceptual theories are "almost directly" observed. So any opposition is utterly irrational. A rational person may only try to "internalize" the insights, reformulate them in an equivalent way, or derive them as consequences of a deeper theory (which however requires a higher level of abstraction).

Another comment related to the previous sentence that I want to offer is that some of these breakthroughs "simplify things". But it is almost never a kind of a simplification that would make the science "less extensive" as well as "less profound or less abstract". If such a simplification occurred, it would really prove that the previous theory was strictly wrong. Instead, what may happen is that concepts get unified or new methods are developed to solve seemingly intractable problems. In those cases, the "extensive volume" of science may drop; however, the required ability of the physicists to think in "new abstract ways" is almost certainly going up.

A large number of molecules of gas may be described by continuous fields and their evolution is governed by partial differential equations. That's a different system than a large set of ordinary differential equations that governed the positions of molecules in mechanics. And the field description of gases may even be "less accurate" because it only remembers the statistical properties of the molecules. However, the field description may be substantially closer to our understanding of some properties of gas. It may be much simpler mathematically, assuming that we learn the basics of the relevant maths. And quantum field theory shows that the field description may actually be exact – in the quantum version of field theory, fields automatically do imply that matter is composed of particles, too.

The jump from classical physics to quantum mechanics has been discussed hundreds of times on this blog. It is a substantial step that qualitatively changes the way how we connect our observations with the mathematical objects in the theory we use.

Some people are afraid to even think about such a step simply because they are afraid of any big steps. They prefer to live in a world (seemingly) governed by classical physics; they are trying to convince (to fool) themselves (and others) by various would-be philosophical verbal tricks that no other framework may possibly work, without actually trying whether it works. But it's not possible to rule out new theories without trying how they work. They have surely the right to avoid the conceptual jump; but if that's so, they can't be surprised that their thinking about Nature will remain frozen in the 19th century. Quantum mechanics does work. Its heart is inequivalent to any classical theory but it is much more valid when describing the microscopic world. The fact that classical physics was found earlier (which is mostly a historical coincidence, not a part of hard sciences) plays no role if you want to rationally compare the validity of an old theory and its newer replacement.

(The same important comment applies to the comparison of string theory and quantum field theory and in this context, this obvious truth is being neglected by many prejudiced people, too.)

Science has made a huge progress in the 20th century and they're not being a part of it. If these people were consistent in their rejection of the 20th century advances, they should also abandon computers, cars (which were evolved in the context of constant interactions with the progress in theory) and return to horses and eating bananas on the palm trees. But of course, people love to be inconsistent, so they're ready to use the results of the advances in inventions that are practically useful for them; and deny the role of mathematics in questions where mathematics is clearly even much more important than it is for cars and computers. They deny the importance of maths only because maths is tough and in their cost-benefit analysis, their personal evaluation of the benefits of the pure truth about Nature is near zero. These people are still fundamentally animals so one shouldn't expect them to learn quantum mechanics or string theory.


At this moment, you may expect me to analyze ways how to improve the education system in order to get rid of math phobia that is controlling the bulk of the current population. Well, I am skeptical: I don't think it's really possible. And chances are that math phobia in the early 21st century is really nothing new, either. Math phobia is bound to remain a constant associated with the bulk of the laymen. It makes no sense to fight these particular wind turbines.

What I find much more worrisome is that math phobia is gradually penetrating even the sectors of the human society whose very goal – their raison d'être - is mathematics or its application. Math phobia has become a unifying theme of many science journalists, commenters in the blogosphere, science bloggers themselves, pseudoscientists on the fringes of the scientific community (e.g. Lee Smolin), and even seemingly legitimate scientists who are however very far from the top of their field.

I've been thinking about the most efficient ways to nuke this degenerative tendency and neutralize much of math phobia and its pathological carriers but you may have noticed that I haven't had the right devastating idea yet. Stay tuned. ;-)


  1. Hey Lumo:

    I enjoy reading your blog. Just started.I am not a physist (a chemistry PhD instead), but I find that so far I am almost in 100% agreement with both your views on physics and politics (scary, isn't it...;-). Regarding your post on the role of mathematics in physics - I also agree. Without math, physics becomes a philosophy, a mere opinion. But my fear (or gut feeling, if you will -gut feeling is all I have; I am a neophyte in math and physics), is that math, while an absolutely necessary tool, may not be enough to unlock the ultimate secret of what Nature is. I mean, at the end the math will just get too complicated, equations unsolvable. Same goes for the energies that would be needed to experimentally verify the Nature at its most basic level - never gonna happen. So, do you think that humans will EVER solve all the equations that fully desribe Nature? I personally don't think so; I think a monkey has a better chance of typing an Iliad on a typewriter than us developing an air-tight Theory of Everything. It does not mean that we should stop trying - far from it. But in the end we may have to admit that the best Theory of Everything we have is just a philosophy, an opinion. I think such an ultimate outcome is likely.

    Respectfully (and ducking) - Wojtek

  2. Thanks a lot, Wojtek, for the synergy etc. And I would love to know the answer to your question, especially if it were Yes and if I also had some proof of it. :-)

  3. Wojtek,

    My twin brother is a PhD chemist who holds the same opinion as yours regarding a theory of everything but pardon me if I differ. If a TOE is discovered, and it is likely to be, the equations will neither be either too complicated or unsolvable. Very compact mathematical notation will be developed, keeping them simple. What is almost inevitable is that the concepts will get ever more abstract and difficult to visualize. It is also probable that the equations will require numerical solution and more computing power than we have today but, in principle, they will not be unsolvable. Of course I am guessing here.

  4. Unlike the average layman today, the Greek and Medieval scholars had no aversion to mathematics.

    E.g., Aristotle wrote in Metaphysics Chapter 3: 994b 32-995a 20, regarding "The Method to Be Followed in the Search for Truth," that:

    173. Now some men will not accept what a speaker says unless he speaks in mathematical terms; and others, unless he gives examples; while others expect him to quote a poet as an authority. Again, some want everything stated with certitude, while others find certitude annoying, either because they are incapable of comprehending anything, or because they consider exact inquiry to be quibbling; for there is some similarity. Hence it seems to some men that, just as liberality is lacking in the matter of a fee for a banquet, so also is it lacking in arguments.

    174. For this reason one must be trained how to meet every kind of argument; and it is absurd to search simultaneously for knowledge and for the method of acquiring it; for neither of these is easily attained.

    175. But the exactness of mathematics is not to be expected in all cases, but only in those which have no matter. This is why its method is not that of natural philosophy; for perhaps the whole of nature contains matter. Hence we must first investigate what nature is; for in this way it will become evident what the things are with which natural philosophy deals, and whether it belongs to one science or to several to consider the causes and principles of things.

    Aristotle would agree—e.g., with Max Tegmark—that mathematics is most connatural to the human intellect, but disagree with his "Mathematical Universe Hypothesis" that it is everything. Some people's aversion to mathematics is because they do not know "metascience," which tells how mathematics relates to the real world.

    Elucidating this last part (175.), Thomas Aquinas comments:

    He [Aristotle] shows that the method which is absolutely the best should not be demanded in all the sciences. He says that the “exactness,” i.e., the careful and certain demonstrations, found in mathematics should not be demanded in the case of all things of which we have science, but only in the case of those things which have no matter; for things that have matter are subject to motion and change, and therefore in their case complete certitude cannot be had. For in the case of these things we do not look for what exists always and of necessity, but only for what exists in the majority of cases. [The jump from Newtonian to modern, quantum physics definitely corroborates this.]

    Now immaterial things are most certain by their very nature because they are unchangeable, although they are not certain to us because our intellectual power is weak [...]. The separate substances are things of this kind. But while the things with which mathematics deals are abstracted from matter, they do not surpass our understanding; and therefore in their case most certain reasoning is demanded.

    He also said in Sententia Metaphysicae, lib. 5 l. 15 n. 7 that “of all the accidents [properties of a substance], quantity [which mathematics treats] is closest to substance.”

    So, indeed, the Greek and Medieval scholars knew "the world is [almost] fundamentally controlled by maths."

  5. Nor did the Medievals have an aversion to applying mathematics to physics, which they called a "middle science," which, because of their "conclusions about physical matter from mathematical principles, are reckoned rather among the mathematical sciences, though, as to their matter they have more in common with physical [i.e., natural-philosophical] sciences." (II-II, q. 9, a. 2 ad 3).

    Here are some physicists we rarely hear about because of the myth that the Middle Ages were "dark ages":
    Nor did the Medievals have an aversion to applying mathematics to physics, which they called a "middle science," which, because of their "conclusions about physical matter from mathematical principles, are reckoned rather among the mathematical sciences, though, as to their matter they have more in common with physical [i.e., natural-philosophical] sciences." (II-II, q. 9, a. 2 ad 3).

    Here are some physicists we rarely hear about because of the myth that the Middle Ages were "dark ages":
    • The medieval scientist Thomas Bradwardine determined in 1300 that for uniformly accelerated objects, d = ½ a t², which Fr. de Soto, O.P., (b. ca. 1494) applied to free-falling objects. (Before Galileo!)
    Jean Buridan (d. ca. 1359) invented the momentum equation: p = m v. Some have proposed naming the unit of momentum after him, where 1 B = 1 kg m/s.
    • The French Bishop Nicole Oresme (d. 1382) determined mean speed theorem of uniformly accelerated body: v_avg = v_f / 2.
    • Bishop Oresme posed the famous Gedankenexperiment: “I posit that the Earth is pierced clear through and that we can see through a great hole farther and farther right up to the other end where the antipodes [poles] would be if the whole of this Earth were inhabited; I say, first of all, that if we dropped a stone through this hole, it would fall and pass beyond the center of the earth, going straight on toward the other side for a certain limited distance, and that then it would turn back going beyond the center on this side of the Earth; afterward, it would fall back again, going beyond the center but not so far as before; it would go and come this way several times with a reduction of its reflex motions until finally it would come to rest as the center of the Earth....” Quoted by K. V. Magruder from Le Livre du Ciel et due Monde (Madison: University of Wisconsin Press, 1968), translated by D. Menut, pg. 573.
    • Bishop Oresme wrote (before Galilean relativity): “If air were enclosed in a moving ship, it would seem to the person situated in this air that it was not moved.” Book of the Heavens, Book II chapter 25, from Grant, A Source Book of Medieval Science, pg. 505, Harvard, 1974

    I copy and paste these things almost ad nauseum to all my physics friends. Let it illustrate that there is a rich intellectual tradition to draw upon in resolving our current problems.

  6. My life consists of estimating probabilities of intersections of world lines in R^3xR^1 and juggling these probabilities in an often irrational way in my head.

    Seriously, a text book on nonlinear PDE for senior or first year graduate student physics majors is needed - as a follow on to the classical theory of orthogonal functions text books used for the first "mathematical physics" course.

    This has been keeping me busy.

    Suggestions on content to be included in such a text book on nonlinear PDE are welcome, right here, thanks.

  7. Lubos, Do you think quantum computers could be used to advance String Theory? I've translated the question to my native, lower brow (chimp) language:

    Of Bananas and String Theorists