Mathematics is a powerful tool in physics, describing everything from the shapes of planetary
orbits to the properties of atoms. Math was used to predict the existence of the planet
Neptune, radio waves and the Higgs Boson. Some think it means nothing profound, or
that math is simply something we've made up to be useful. Others think it means that
there's something fundamentally mathematical about nature. The most extreme possibility
is that our universe is completely mathematical, in the sense that it has no properties except
mathematical properties.
But at first glance, our universe doesn't seem very mathematical at all. A sheep has
properties such as cuteness and fluffiness — not mathematical properties. Yet we know
that this sheep - and everything else in our universe — is ultimately made of elementary
particles such as electrons and quarks. And what properties does an electron have? Smell? Color?
Texture? No, only properties like -1, ½ and 1! We physicists call these properties electric charge,
spin and lepton number, but the electron doesn't care what we call them: the fundamental properties
that an electron has are just numbers, mathematical properties. As far as we know, all elementary
particles, the building blocks of everything around, are purely mathematical objects in the sense
that they don't have any properties except for mathematical properties. The same goes
for the space that these particles are in - for example, it has the property 3, the number
of dimensions. If space is mathematical and everything in space is also mathematical,
then the idea that everything is mathematical doesn't sound as crazy anymore.
Quantum mechanics and string theory introduce even more mathematical structures, with nerdy
names like Hilbert space, linear operators and Calabi-Yau manifolds. We physicists
still haven't found any properties of nature that we can prove are non-mathematical, which
is why some of us - including me - think that perhaps there just isn't anything else out
there, other than math. And that thought makes me one positively awestruck
mathematical object.