In this day one year ago, Freeman Dyson died at age 96. A brilliant mathematician and theoretical physicist, he gave a major contribution to quantum electrodynamics while being still a young research fellow. He was tutored by Hans Bethe and Richard Feynman, but he never completed his PhD, something he was kind of proud of. This is just one sign of him being a kind of anarchic scientist, despite being tenured at Princeton, a temple of academic studies where he worked for 60 years. His amazing mind could apply with the same depth at advanced technologies, such as the design of the Triga-Mark I nuclear reactor, as well as to crystalline mathematical problems, such as the problem of the zeroes of Riemann’s zeta function. Always keen of his role of iconoclast, although shy and self-effacing in public, he could confound the scientific establishment by dismissing the consensus about the perils of a man-made climate change as “tribal group thinking”. As much he distrusted climate models, exacerbating experts with his visionary and minoritarian predictions, as he doubted that superstrings or anything else would lead to a “Theory of Everything”, unifying all of physics with a succinct formulation that a kid could wear on a T-shirt. But his sharp criticism was honest and never ill-natured: for example, he gave a fundamental contribution to QED, but he did not begrudge when Feynman, Schwinger and Tomonaga were awarded the Nobel in 1965. “If you aim at a Nobel Prize, you should have a long attention span, get hold of some deep and important problem and stay with it for 10 years. That wasn’t my style.” he told The Times Magazine in 2009.
But Dyson was also known for inspiring futuristic visions, and for us kids of the ‘60s, grown in the excitement of the first space travels and dreaming of a man landing on the Moon, he was a leading figure, next to Werner Von Braun, Isaac Asimov, or Carl Sagan. He imagined exploring the solar system with spaceships propelled by nuclear explosions, and establishing distant colonies nourished by genetically engineered plants, the “Dyson spheres” that inspired not less than a dozen science fiction novels, one Star Trek TV episode, and planet Nidavellir in the Avengers movie. For all these reasons, Dyson had become one of my favorite role-models in science, since the times I was studying his “Dyson series” in perturbative quantum mechanics. He was a great and often poetic writer, with a distinctively literary style, not just a good science communicator. I found great pleasure in reading his popular books, Infinite in all directions and Disturbing the universe, and I love to use often quotes by him in my own writings.
A couple of years ago I had the idea of writing him with a question about the foundations of mathematics. This is the mail I sent him [excerpts, the letter was a bit longer]:
“Dear Freeman, I have always been a great admirer of your work in physics since I was a student. At that time, I read with the greatest interest your essay “Time without End”, for the Review of Modern Physics, and those ideas you explained so clearly, positively struck me. […] That’s why I would like to ask you a question that has been keeping my mind busy over all these years […]: do you think that mathematics exists outside us? Or, it is just one language of our mind, just like words, poetry, painting, music? Maybe it is just the one code that is the best adapted to describe how our brain regulates the information coming from the outside world? Does mathematics come from the universe around, and we must learn it? Or, instead, it is a product of our brain that fits at its best what we see around us? With my best regards and my highest admiration, Fabrizio Cleri.”
This is what he replied, a couple of days later:
“Dear Fabrizio. To answer your question, I would like to tell a little story. I have a friend, Jessica Park, who is autistic. I have known her for fifty years, since she was born until today. Her universe is radically different from ours. As a child she used the words I and you interchangeably. She had no conception of her own identity. She was a truly alien intelligence, with no understanding of human society as we understand it. One day, when she was twelve years old, she received a letter from a friend, a boy who is also autistic. The letter contained no words, only a long list of numbers. I could see that the numbers were the prime numbers between zero and a thousand. But Jessica could see more than me. Immediately after she saw the numbers, she became excited and said, “Mistake! Mistake!” She pointed to the number 703 which was on the list. She knew that it was a mistake because it is 19 times 37. The point of this story is that Jessica is a truly alien intelligence but her prime numbers are the same as ours. She demonstrates that prime numbers exist outside of our little universe. Prime numbers exist in a universe of ideas, as Plato said long ago. The universe of ideas is independent of the various kinds of creatures that may be observing it. The whole of mathematics belongs to the universe of ideas. Yours sincerely, Freeman Dyson.”
I knew already too well what his answer would have been: a straight Platonist declaration. Come on, he was the man who discovered a relation between Riemann’s function and the distribution of prime numbers while chatting over a cup of tea! Therefore, I expected he would be partisan of the universality of mathematics. But, all my admiration and respect for The Man notwithstanding, I maintain my point of view. And, if any, I think his little cute story proves just the opposite of what Freeman wished to demonstrate. Despite the beauty and fascination of the prime numbers, the fantastic figures and geometries of their distribution in the complex plane (please take some time to watch this fabulous video), they remain a purely mathematical entity: there are no known natural phenomena that should rely on prime numbers for an explanation. To me, the fact that “even” autistic children are aware of the existence of prime numbers proves exactly the opposite, instead: that prime numbers, like all the rest of mathematics, are a creation of the human brain, the way our brain is wired to the reality that surrounds us.
Then, the obvious question arises: if mathematics is created by human brains, how come it can be so effective in describing, explaining, and quite often predicting natural phenomena? A little more than 60 years ago, Eugene Wigner wrote a famous essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences (where, in fact, he rather meant “in the physical sciences”), in which the apparent success of mathematics in adhering to the physical reality was described in terms nearing on the verge of miraculous. This position spurred a rather intense discussion among scientists, philosophers and historians of science, and what I am going to write here will certainly not add much to the debate. I will just try to put down some personal considerations directed at what I see as excess of Platonism in science, for all the Tegmarks out there.
Despite the rich multidimensional depth of the current mathematical bestiary, it is not much the complexity of mathematical weapons that besets us, but – at the opposite – the evidence that just a few extremely simple mathematical tools (trigonometric functions, quadratic curves and conical surfaces, exponential functions, linear matrix algebra, Fourier series…) can explain so much of physics and chemistry, also including a few, occasional incursions in more distant domains as, e.g., biology, sociology or economics. To me, this is an especially important point: not only most of the beautiful (or weird) mathematics we invent is not (yet) used, but we often seem to invent the necessary piece of maths for the purpose at hand. For example, when scalars proved not useful for understanding forces, we invented vectors and tensors. And because of this “utilitarian” attitude (which in no way is meant to diminish the value and beauty of research in pure mathematics, much alike the development of arts or philosophical studies), it is impossible not to observe how much of our human experience lies beyond mathematical principles, requiring analysis tools from the philosophy of values, ethics, aesthetics, historical and political philosophy, psychology.
Mathematics can be seen as a very sophisticated evolutionary device for survival: we have no sharp teeth, or venom glands, nor a thick exoskeleton, but we can calculate and predict things. Other animals indeed have some computational capability, you can see it when your cat measures the distance to a wall and decides if it’s safe to make a jump; however, your cat cannot predict then next Moon eclipse. This makes me think that our Sapiens brain has grown by pushing to extremes the numerical and computational capabilities that at some point had started to emerge in mammalian brains. But our ability to find patterns everywhere should not be mistaken for the idea that patterns are always there: we see constellations by connecting stars that are millions of light-years away from each other. We call the regularities of our thoughts “mathematics”, and when some of these manifest as regularities of perceived natural phenomena we call them “physics”. However, most of what we think every day lacks the precision of mathematics, and regularities are rather the exception than the rule. (Note that such definitions are not limited to human-scale senses: our brain is largely capable of abstraction beyond the immediate sensory perception.)
But finding patterns is a true survival skill, for example comforting our impression of living in a three-dimensional world (although we occasionally glimpse that there should be more out there, for example when electrons are accelerated at near the speed of light). The flip side of the coin is thathumans only see what they look for, a kind of “evolutionary priming” in filtering out what can be understood with a long chain of reasoning. Also, the belief that our science is experimentally grounded is only partially true; rather, our intellectual apparatus is such that much of what we see comes from the glasses we put on. For example, the 1/r2 law of force that seems so universal to us, as applying to both distant planets and electrical charges, is the human translation of the principle of energy conservation in a space assumed three-dimensional (this can be easily shown by the Gauss’ theorem, but the key idea goes back to Kant, who had no theorems at hand): when we verify Newton’s law of gravitation between planets and galaxies, we are in fact verifying that the space between such objects is locally three-dimensional Euclidean, down to the limits at which another man-made concept (general relativity) starts kicking in.
Another example: there are physical limits to computation [see, e.g., S. Lloyd, Nature 406, 1047 (2000)], as determined by the speed of light c, the quantum scale ℏ and the gravitational constant G. Yet, we seamlessly and profitably use the notion of infinite and infinitesimal numbers in mathematical physics, although in nature there are no infinitely large, nor infinitely small quantities. If we were to take only experiments as the ultimate judge to attribute patents of reality to mathematical concepts, then the notion of integer should be revised: no closure relation can exist in counting apples, since at some point the experimental count of apples will be so large that they collapse in a black hole.
Superstrings, OrchOR, multiverses, wormholes and the like, appear as highly philosophical mathematical elaborations but devoid of a physical equivalent (and Freeman Dyson was adamant in his criticism of such untestable hypotheses), while we candidly admit that only 5% of the universe we perceive around us appears today as “explainable”. This is nothing more than a new form of anthropocentrism: trying to discover the mechanism of nature by assuming that everything is understandable through the human experience. Starting from Bertrand Russell’s quote, “Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.”, I would slightly rephrase it as: …it is only its human-brain understandable properties that we can discover.
Suggested readings: Where the mathematics comes from, by G. Lakoff and R. E. Nunez; Trick or Truth? The mysterious connection between mathematics and physics, collected essays by the FQXi foundation.