Writing Papers by Night
"The fact that crystal symmetry simplifies the Hamiltonian is based on people’s a priori conceptions of spacetime—which shows that science is ideational; the scientific world is nominalist, a world of ideas."
It is past midnight and I am working on a condensed matter physics paper. The section I’m writing concerns how crystal symmetry constrains the form of the Hamiltonian—a powerful technique that cuts through enormous complexity.
This started me thinking. The reason crystal symmetry simplifies the Hamiltonian is that we already assume spacetime has certain a priori properties—translational invariance, rotational invariance, and so on. These are Kantian: they are categories of our intuition, not facts we read off from nature. The scientific world, seen from this angle, is ideational. The scientific world is nominalist—a world of ideas, models, frameworks that we impose on the black box of nature and then test.
Kuhn’s paradigm shifts are now old news—every educated person knows the basic story. What’s underappreciated is that Aquinas was already saying something similar in the 13th century: our knowledge of the world is always mediated by our cognitive frameworks, and those frameworks can be superseded. The novelty of Kuhn was not the idea but the application to natural science, which had previously enjoyed a reputation for accumulative certainty.
Game physics engines are interesting in this context. They simulate mechanics, optics, and thermodynamics tolerably well. None of them simulates electromagnetism. The reason is not computational—EM is not more expensive than fluid dynamics in all cases. The reason is historical: EM is a purely modern discovery, emerging from Faraday and Maxwell in the 19th century. Ancient intuitions of the world did not include it. Our a priori conception of the physical world still doesn’t really include it, which is why EM feels counterintuitive in a way that mechanics doesn’t.
Nine years of Chinese science education makes the following error systematically: it presents science as a process that always produces revolutionary improvements, always supersedes superstition, always marches toward truth. This is Hegelian teleology applied to epistemology. The Popperian correction: science is falsifiable, not guaranteed correct; it is the best available approximation, not the final word. The natural world is a black box; our theories are interfaces.
The historical path of mechanics is instructive: Copernicus had the heliocentric intuition before anyone had the mathematics; Galileo and Newton formalized inertia before anyone could explain why; Brahe spent years making meticulous observations without a theory; Kepler fit the observations to ellipses; Newton synthesized everything into a single framework. None of these steps was logically necessary from the previous one. Each required a creative leap that was, at the time, unjustified.
Chinese classical thought maintained a double standard: astronomers knew the earth was spherical and used that knowledge for calendar calculation; the official cosmology remained the round-heaven-square-earth model, because it grounded the emperor’s ritual position. Knowing and believing are different; institutional truth and scientific truth have always been separable.
A postscript on comedy: our university physics textbook refers to Torricelli as 「托里拆利，托斯卡纳公国的数学大公」—“Torricelli, Grand Duke of Mathematics of Tuscany.” The correct Italian phrase is matematico del Granducato di Toscana—“mathematician of the Grand Duchy of Tuscany.”「数学大公」 is a word-by-word translation error from an old Chinese Wikipedia article, evidently not checked against any source. It has apparently survived in some textbook editions. I find this deeply endearing.