# Intuition arises out of Mathematical Intractability

Reading yet another profound post from The Curious Wavefunction, I got to thinking about the notion of intuition in science. In that particular essay, Curious explored the nature of intuition in chemistry, which started off from a remark that "in chemistry, intuition is much more important than in physics." It's a common position to take as mathematics is so integral to physics, such that even Riemmanian geometry can be co-opted into a description of gravity.

However, I want to hone a little bit more into exactly where intuition ends and mathematics takes on a life of its own. I think the particular quality we are looking is not simply the use of mathematics – the best quantum-mechanical chemists are as formidably mathematical as the best physicists. The key to mathematizing a scientific discipline (especially in physics) is the happy accident of finding tractable mathematical equations that admits of reliable first order approximations. Newtonian mechanics comes to mind where the first order solution of ellipsoidal orbits of the planets is rather straightforward, and later refinements comes at increasingly higher costs in the calculation.

It is somewhat accidental when this occurs. Quantum mechanics would not have had the explosive impact it had in the 1920's were it not for the fact that the solution of the electron spectra in the hydrogen atom comes out so easily from the first order approximation of Schrödinger's equation using the very familiar inverse square potential. However, things get messy very quickly as we add the number of atoms to the system. It took decades to find a good solution to higher number of atoms, which resulted in the wonderfully baroque but powerful methods of density functional theory.

One of the reasons that physics seems to be dominated by beautiful mathematical models that are reasonably easy to calculate, is simply that these are exactly the models that physics professors the world over has chosen to teach in the first few years of a physics course – statistical mechanics of simple gases, quantum mechanics of single atoms, and even the first order solutions of quantum electrodynamics (the relativistic generalization of quantum mechanics). Compared to the first order solution, the 8th order solution of quantum electrodynamics for the value of the fine-structure constant requires a dizzying calculation that involved summing 891 terms as represented by the feynmann diagram expansion of the equation. It's not even verified by hand, but by a computerized algorithm.

What physics professors don't teach you until later are the systems that do not have easy to calculate solutions, nor approximate solutions that match reality. And we are not talking about trivial or esoteric system. I didn't notice until much later in my physics education that you first learn about electrons and protons in atoms, but then skip right over the energies of the nucleus, and go straight to sub-atomic particles. There's a very good reason to that. We have a really crap model of the nucleus. In fact we have two, and both are kind of shit – the liquid drop model and the nuclear shell model. No one has been able to find a simple mathematical model of these models, or the ones that are calculable give some good answers and many bad ones. People who specifically want to work with nuclei energies rely on their intuition as much as chemists because no beautifully tractable or reliable mathematical model was ever found.

Similarly, we skip over fluid dynamics where the equations are simply unsolvable, and the Ising model of magnetic materials, where the great Onsager found a solution in 2D but the proof is some hair-raising mathematics that is beyond almost all undergraduates.

In conclusion, all I want to add to Curious's post is that chemist rely
on their intuition specifically because the mathematics in quantum
chemistry does not admit easily solvable equations, or reliable
low-order approximations. The intuition is needed simply because a
*useful* mathematical solution is not tractable. Similar problems plague
many sub-disciplines in physics but the physicists have a much better PR
machine.