Monday, June 14

This Will Either Fascinate You or Drive You to Drink with Boredom

skepticLet's find out which!

Once in a while, for better or for worse, the past comes back to haunt you. An instance of the “better” part of this assertion occurred with me recently when I saw a public TV documentary on mathematics. Much of the documentary revolved around what the physicist and mathematician Eugene Wigner described as the “unreasonable effectiveness” of mathematics in the natural sciences in an essay of that title. Wigner’s famous essay was written around 1960. I first encountered it as an undergraduate math and physics – and, significantly, philosophy – major at Wichita State University in Wichita, KS, during the late 1960s. It stuck around in the back of my mind to haunt me at graduate school in physics about ten years after it was written. But, finding little or no sympathy for my philosophical perplexity in the physics department – cite a philosophical issue and most physicists respond with a deer-in-the-headlights stare – I did not so much become indifferent as preoccupied with other pursuits. Until last week, when I encountered it again in that documentary. Remembering how impressed I was with Wigner’s text 40-plus years ago, I Bing’ed it up, printed it off, read it … and found that it had lost none of its power to perplex and to provoke. In particular, I found that it had lost none of its power – not so much to challenge – as to chasten what has by now become my habitual attitude of skepticism.

Eugene Wigner (1902-1995), Hungarian physicist. Eugene Paul Wigner was most famous for his deep exploration into quantum physics, especially in establishing a core algebraic theory on the symmetries in quantum mechanics, known as the Wigner D-matrix. He came up with several other theorems while researching into atomic nuclei and also co-wrote the classic book The Physical Theory of Neutron Chain Reactors with Alvin Weinburg in 1958, which he holds in this portrait. Five years later he was awarded half of the 1963 Nobel Prize in Physics for his symmetry principles, the other half being awarded to J. Hans D. Jensen and Maria Goeppert-Mayer for their discovery of nuclear shell structure.
Eugene Wigner

Wigner’s essay opens with a (possibly apocryphal, though that does not matter) story of a statistician explaining to a non-statistician friend the meaning of a bell-shaped-curve graph about population, and the graph’s associated terminology and symbology. The conversation proceeds: “And what is this symbol here?” “Oh,” said the statistician, “that is pi.” “What is that?” “The ratio of the circumference of a circle to its diameter.” “Well, now you’re pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of a circle”. Wigner’s point, which he illustrates with several other more abstruse and technical examples I will skip over, is that time and time and time again – in fact, with a frequency that by now verges on the customary and to-be-expected -- mathematical entities like pi, long believed to be relevant only to abstractions confined within the skulls of mathematicians, turn out, it may be generations or even centuries later, to have critical, fundamental -- even world-historical -- relevance to understanding aspects of the Universe outside those skulls. Like population. In fact, such episodes of “unreasonable effectiveness” tantalizingly call into question that very dichotomy of “inside” vs. “outside”. Rather, it seems that, at least sometimes, “deep calls to deep” (Psalm 42:7), Decartes’ dualism is transcended, and res cogitans (“thinking substance”, the “inside-the-skull” world) and res extensa (“extended substance”, the “outside-the-skull” world) meet and embrace.

There are other examples that Wigner does not mention, that in some cases did not exist circa 1960, when Wigner wrote his essay. Wigner tends to concentrate on contemporary instances of “unreasonable effectiveness” in relativity and quantum theory. But there are many others.

o Non-Euclidean geometry

In the middle 1800s, mathematicians like Bolyai, Lobachevsky, and Riemann developed consistent systems of geometries that were founded on variations of Euclid’s famous Fifth (or Parallel) Postulate: given a line and a point not on the line, one and only one line can be drawn through the point parallel to the given line.

Bernhard Riemann
Bernhard Riemann

These systems of geometry were surprising to many because they turned out to be consistent, but remained playthings for mathematicians and geometers … until a century or so later, when Albert Einstein formulated his Theory of General Relativity, at which point it became necessary to conceive of space, not in Newtonian terms as a static, flat, unalterable Euclidean plane, but as a vast sheet of something like rubber – albeit a four-dimensional sheet – that gravity could warp and deform in ways that could not be understood apart from the application of non-Euclidean geometries. (This explains the ubiquity of "rubber-sheet geometry" in popular expositions of general relativity -- though, technically, gravity is a tensor field, so the "rubber" is warped in ways not possible with a physical rubber sheet.) Non-Euclidean geometry makes General Relativity possible ... and along with it, virtually all of 20th- / 21st-century cosmology and astronomy.

o “Chaos” theory

When I was taking graduate-level courses in applied math, it was admitted, in certain cases, that while the coefficients of the partial differential equations governing certain physical processes could be non-linear, that we would only concentrate on the cases where the coefficients were linear because (a) only linear coefficients were tractable by the methods we were using because the non-linear cases pertained to “chaotic” systems, and besides, (b) only those partial differential equations with linear coefficients had physical significance, anyway.

rf_detail_261_0Or so we believed at the time. (Like I said, these were courses in applied math: no point in trying to apply math to cases with no physical relevance.) To cut to the Reader’s-Digest-condensed version of the story, it later turned out that, contrary to being physically irrelevant, the non-linear cases previously believed to be only bloodless abstractions were the very cases most relevant to the physical Universe in terms of population growth, certain resonances in the physics of elementary particles – and even the behavior of markets and political systems. (Another crucial factor was the development of computers powerful enough to process numerical-analysis / finite-element models of non-linear processes.) Because of its “unreasonable effectiveness”, our math told us far more than we realized, e.g., the long-term behavior of global weather systems.

o Abstract algebra

 Until well into the 20th century, it was believed that the most hard-core “useless” mathematical discipline was the area of abstract algebra. Surely nothing as rarefied as non-commutative groups and Lie algebras could have any practical significance whatsoever, outside of providing job security to professors of mathematics. But it turned out that the relationships between and symmetries prevailing among elementary particles – and whole families of particles – were understandable only with reference to some cognate of something called “Lie algebras” and “non-Abelian gauge theories”. Don’t worry about the “tech talk”. The point is that even such abstract algebraic structures turned out to have deep physical relevance.


This just scratches the surface, and even then only barely so. I could also mention “useless” things like Hilbert spaces in quantum theory, Calabi-Yau manifolds and Klein bottles (and other differential-geometry entities) in superstrings, etc., etc., etc., etc. … the list is almost endless. (Brian Greene's The Elegant Universe contains a challenging but relatively accessible discussion of such entities vis a vis string theory.) In all such cases, the exotic critter eventually escapes from the space inside mathematicians’ skulls and becomes "unreasonab[ly] effective[]" in enabling physicists – and sociologists and market analysts – to make sense of the world.

What’s going on?

It is near-miraculous enough that the world is understandable through mathematics.  That it may so often be understood through the mediation of mathematical structures never designed to enable us to understand it pushes the envelope from near-miraculous into the territory of the ... well ... the numinous ... often downright "spooky". (As Wigner notes in his by-now-classic essay, most pure mathematical abstractions, by definition of the word "pure", were initially developed with the intent of proving even more 'way-cool theorems and developing even more 'way-cool abstractions and making math even more 'way cool, not with the intent of explaining anything going on "'way out there".) That is what tantalizes me about Wigner's essay to this day:  the hint that -- whatever is going on -- it involves a potential way of transcending the dualism of Mind (in the sense of Descartes' res cogitans) and World (Decartes' res extensa). There are other hints, some buttressed by the "hard" science of data and experiment (the "measurement problem" in quantum theory, Feynman's classic "two-slit" experiment, Bell's Theorem) and others much more problematical (Jungian synchronicity). (I say "much more problematical" because it is far from clear how or whether anything like synchronicity could ever be scientifically investigated, even in principle, because of its non-causal nature, inasmuch as science deals in terms of cause and effect.) Even such disparate fields as deconstructionist literary critical theory, with its subversion of the distinction between text and author and its emphasis on the self-referential nature of the former, falls into the same pattern of calling into question the distinction between subject and object, of inner world and outer reality, of the observer and the observed -- of what we think in our heads and what happens in the world.

Maybe we are not, after all, strangers in a strange land.  Maybe we are not Sisyphus eternally rolling his rock uphill.  Maybe we are home.

James R. Cowles






  • Mr James Cowles, I didn’t understand a word of what you said – well, maybe I understood a couple of words. You always write with passion and authority (and references) and I would love a “cliff notes edition” of this post. (I’m serious)

    P.S. You didn’t drive me to drink but I did get up at the end of each paragraph looking for sugar & refined carbs.

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