(The following guest post was submitted by Ms. K.M., who initially shared the article referenced herein with me. We exchanged some short emails about this, both of us concluding that there must be some [unanalyzed] principle of analogy at work in Dr. Kim's work. This is a thought-provoking look at a recent revolution in mathematics from the standpoint of a wide historical context dating back to observations of Newton, Descartes, and Leibniz. Enjoy!)
Mathematicians and physicists, though they use many of the same tools, exist in a kind of cultural cold war. Mathematicians find themselves unable to refer to mathematical physics in their papers on number theory, while physicists hungrily borrow from the tool set of math while remaining uninterested in the puzzles which occupy the brightest minds in the math field. When I was in college, there was something of an "interservice rivalry" (Army v. Navy) between the two groups. The mathematicians would point out that some physicist incorrectly applied math in some theory or other. And the physicists distrusted mathematicians' lack of concern with the applicability of their work to problems "in the real world."
But many of us, this author included, bathed in the deep intuition that there was some core idea (or something like that) that would make it clear that math and physics were, at some very great level, both part of a greater holistic order, invisible to the methods in use even up to today. Physicists were the most likely to believe that math and physics were deeply related. But to be fair, the mathematicians' aversion to physics grew out of a real concern that the messiness of physical phenomena and the imperfection of experimental method would distract them from their attentions to pure reason.
Well, in a recent issue of Quanta Magazine, it was revealed that one of the world's top number theorists, Minhyon Kim, of the University of Oxford, has brought forth a unique and powerful theory that portends to tie some elements in physics directly to the theory of real numbers. Here is the link to this rather long but very well written piece in Quanta Magazine.
The author states:
"Over the past decade Kim has described a very new way of looking for patterns in the seemingly patternless world of rational numbers. He’s described this method in papers and conference talks and passed it along to students who now carry on the work themselves. Yet he has always held something back. He has a vision that animates his ideas, one based not in the pure world of numbers, but in concepts borrowed from physics. To Kim, rational solutions are somehow like the trajectory of light."
The new theory connects a type of field theory (which means objects affecting each other in action in a space) called gauge theory and the geometric representation of the real numbers. For example, if you have an equation, x squared plus y squared = 1, the solutions can be mapped as dots on a circle, but the distribution of those points on a circle will be all over the place, which means that symmetry suffers. Reduced symmetry makes comprehensive theories difficult. Can you imagine Einstein's relativity without a clear symmetry between the effects of mass and energy on space and vice versa not being symmetric? It would not work.
In order to increase symmetry, Kim created a special "space of spaces", which is the tie to gauge theory, that creates many more opportunities for real numbers to be analyzed with an eye to exposing symmetries. Interestingly, Dr. Kim describes in his work that a solution to his problem is analogous to the trajectory of light, which relies on what physicists call "the principle of least action", meaning the photon will take whatever route is the most convenient in terms of time, like a commuter seeking out all the shortcuts on the way home. He firmly believes that an analogous principle lies at the heart of real numbers. It's hard to explain why this is so important, but I'll try. Rene Decartes once quipped that when he deconstructed Euclid's elements, he noticed that the ancient geek (sic) seemed to rely upon a method of analysis that has since been lost. Descartes said he detected a "certain wicked slyness" in Euclid's work. Work like Dr. Kim's takes us back closer to the intellectual level of High Antiquity, as Jefferson put it. We need to recover that lost analysis.
The deeper picture here, which members of the Giza Community may observe, is that Dr. Kim has noticed the analogy between light and real number representations. And this clue opens the door to consider Dr. Kim's work in relation to Leibniz's Characteristica Universalis, and Goedel's incompleteness theorem.
With respect to Leibniz, Dr. Kim has taken us a few steps closer to revealing (at least publicly) the set of universal principles of systems that can be comprehended by thought. When implicate connections are revealed between two very disparate disciplines, that's a good clue that we are getting closer to Leibnitz's vision. A 17th century natural philosopher and mathematician and the inventor of integral calculus, Leibniz thought that if the universal principles, forms, and methods, of thought itself could be uncovered, that a form of analysis could be generated that would provide a universal analysis of situation that could be applied anywhere that thought could be applied. If it sounds a lot like artificial intelligence, that's because it is analogous to it. But Leibnitz himself was publicly aiming for a universal language of proof.
With respect to Kurt Goedel, Einstein's "best pal" at Princeton, the application of "space of spaces" sounds eerily like "the set of all sets" ensconced in his famous theory of incompleteness. The emergence of correlations between physics and real number theory, both the logic of encapsulating sets, and the difficulty of finding perfect symmetry, may point to the potential revelation of incompleteness in real number theory. To put it farm simple, "the closer you get, the further away you become."
Perhaps in fifty years, or two centuries Dr. Kim's intellectual descendants will discover that the real numbers are a kind of physics. But for now, the exciting possibility that mathematics and physics might someday reconverge is a remote, but actually real, possibility.