Mathematical Reality

Steve Landsburg (OK, I can’t help myself - I can’t quit you Steve. He’s frequently wrong but often writes about interesting stuff) is arguing that the universe, and its contents, are mathematical objects.

1. A “mathematical object” consists of abstract entities (that is, “things” with no intrinsic properties) together with some relations among them. For example, the euclidean plane that you studied in high school geometry consists of points, together with certain relations among them (such as “points A, B and C are collinear”). Mathematical objects can be very complicated. Mathematical objects can have “substructures”, which is a fancy name for “parts”. A line in the plane, for example, is a substructure of the plane.

2. Every modern theory of physics says that our universe is a mathematical object, and that we are substructures of that object. Theories differ only with regard to which mathematical object we happen to be a part of. Particles, forces and energy are not just described by equations; they are the equations (together with abstract, purely mathematical relations among those equations).

It’s not a new argument. It goes back at least to Plato and probably to Pythagoras. Lots of physicists continue to make the same argument. Given the central role mathematics in our theories of physics, can we believe otherwise? A weaker position is to argue that mathematics is effective in “describing the universe.” We also use language to describe the Universe, especially in sciences where mathematics has shown limited explanatory power, like biology. Should we also say then that the Universe is a “linguistic object.” It sounds odd, and that’s because it is.

There is an inherent danger, I think, in confusing our descriptions with the presumed reality that underlies them. One important reason is that we need to remember that our descriptions are tentative and subject to revision in the light of further information. Plato’s Universe consisted of fundamental substances made of regular polyhedrons – tetrahedrons, octahedrons, cubes, dodecahedrons, and icosahedrons – very pretty, but it didn’t work out. Kepler liked polyhedra too, but discovered that the orbits of the planets were ellipses – which allowed Newton discover that they weren’t quite ellipses, and Einstein had a further revision. There is every reason to think that our current mathematical description of nature – a world of fields and particles identified with irreducible representations of some groups – is also tentative. If we are lucky, we will find an even more powerful mathematical description in which to embed our next theory.

I like to think that mathematics is an offshoot of language, one that enforces some linguistic rules quite rigidly and disregards some other parts of language. It is a fact that this kind of language has special power to describe physics, a power that may suggest that God is a mathematician, or, more or less equivalently, that the Universe *is* a mathematical object. I prefer to suspend judgment.

Landsburg takes the other road and reaches a number of conclusions which I’m inclined to reject, but I won’t discuss most of them here. He ends with:

Finally: I never cease to be amazed by people who uncritically accept the reality of rocks, geese and butterflies but want to deny the reality of mathematical objects. Science tells us that rocks, geese and butterflies are mathematical objects. What else could they be?

I don’t buy it. They can’t be mathematical objects to my satisfaction unless you can tell me ***what*** mathematical objects they are, in sufficient detail that their properties follow from that mathematical description. Linguistic descriptions, with their inherently vague and incomplete character, capture rocks and geese much better than any specific mathematical description can.

The Universe presumably is some interacting whole, but we can only describes its parts (mathematically or otherwise) when we can sufficiently isolate that part from the rest. For a planetary orbit or an elementary particle interaction we can often do that well enough to come up with a mathematical description, but for geese and butterflies, not so much.