Is mathematics real?

I didn’t think to think about the realism of mathematics until I got to high school, and encountered quantum mechanics.

Mathematics was at first just another subject, before becoming a tool with which to think intelligently about money and, later, with advanced statistical concepts in the picture, to understand the properties of groups of objects that couldn’t be deduced from those of individual ones. But by this time, mathematics – taken here to mean the systematic manipulation of numbers according to a fixed and rigid system of rules – seemed to be a world unto its own, separated cleanly from our physical reality akin to the way “a map is not the territory”.

Put another and limited way, mathematics seemed to me to be a post facto system of rationalisation that people used to understand forces and outcomes whose physical forms weren’t available for direct observation (through one, some or all of the human senses). For example, (a + b)2 = a2 + b2 + 2ab. To what does this translate in the real world? Perhaps I had 10 rupees in one pocket and 20 rupees in the other, and 29 other people turn up with the same combination of funds in their pockets. We could use this formula to quickly calculate the total amount of money there is in all of our pockets. But other than finding application of this sort, I didn’t think the formulae could have any other purpose – and that, certainly, knowing the formula wouldn’t allow us to predict anything new about the world (ergo post facto).

I was constantly on the cusp of concluding mathematics was made up, a contrivance fashioned to fit our observations, and not real. But in high school, I came upon a form of mathematics-based reasoning that suggested I should think about it differently, if only for the sake of my own productivity. In class XI, my physics teacher at school introduced Wolfgang Pauli’s exclusion principle.

The principle itself is simple, at least at the outset. Every particle has a fixed set of quantum numbers. An electron in an atom, for example, has four quantum numbers. Each quantum number can take a range of discrete values. A particular combination of the numbers is called a quantum state (i.e. the combination confers the particle with some possibilities and impossibilities). The principle is that no two particles in the same system can occupy the same quantum state.

Now, it is Pauli’s principle – a logical relationship between various facts – that animates the idea, and not any mathematical rule or prescription. At the same time, the principle itself is arrived at by solving mathematical problems. Why do electrons in atoms have four quantum numbers? Because historically we started off with one, because we perceived the need for one, and over time we added a second, then a third and finally a fourth – all based on experiments in which the electrons behaved in a certain way, but because direct physical observation was out of the question we invented mathematical relationships between the particles’ parameters in different contexts and ascribed meaning to them.

It was still ‘only’ empirical: scientists tried different things and those that worked stuck. There may be another way to make sense of the particles’ behaviour with, say, five dim sum (🥟) numbers, and reorganise the rest of quantum mechanics to fit in this paradigm. Even then, only the mathematical features of the topic will have changed – the physical features, or more broadly the specific ways in which particles are real, will have not. But this view of mine changed when I read about experiments that proved Pauli’s principle was real. A mathematical system we set up eventually led to the creation of a fixed set (not more, not less) of quantum numbers, and which Wolfgang Pauli eventually combined into a common principle. If scientists had proved that the principle was true and therefore real, could the mathematics undergirding the principle be true and real as well?

Not all fundamental particles obey Pauli’s exclusion principle. The four quantum numbers of an electron in an atom are: principal (n), azimuthal (l), magnetic (ml) and spin (s). Of these, the spin quantum number can take two kinds of values: half-integer (1/2, 3/2, …) and integer (0, 1, 2, …). Particles with half-integer spin are called fermions, and the rules describing their behaviour are defined by Fermi-Dirac statistics. They obey Pauli’s exclusion principle. Particles with integer spin are called bosons, and the rules describing their behaviour are defined by Bose-Einstein statistics. They don’t obey Pauli’s exclusion principle.

When some kinds of heavy stars can no longer continue fusion reactions outside their core, they collapse into a neutron star – an ultra-dense ball of neutrons. Neutrons are fermionic particles – they have half-integer spin – which means they obey Pauli’s exclusion principle, and can’t occupy common quantum states. So the neutrons in a neutron star are tightly packed against each other. Their combined mass generates gravity that tries to pull them even closer together – but at the same time Pauli’s exclusion principle forces them to stay apart and remain stuck in their existing quantum states, creating a counter-force called neutron degeneracy pressure.

We wouldn’t have neutron stars, or electronic goods or even heavy elements in the periodic table, if Pauli’s exclusion principle didn’t exist.

Most recently, three separate groups of scientists described a new physical manifestation of the principle, called Pauli blocking. Most atoms are fermions (as a whole); each group first created a gas of such atoms and cooled them to a very low temperature – to ensure that in each gaseous system, all of the lowest available quantum states were occupied. (The higher a particle’s quantum state, the more energy it has.)

A group at JILA, in Colorado, used strontium-87 atoms. A group from the University of Otago, New Zealand, used potassium-40 atoms. And a group from MIT used lithium-6 atoms. (The last one includes Wolfgang Ketterle, whose work I have discussed before).

Usually, when a photon and an electron collide, the photon is scattered off into a different direction while the atom absorbs some of the photon’s energy and recoils. The absorbed energy forces the atom into a higher quantum state, with a different combination of the quantum numbers than the one it had before the collision. In an ultra-cold fermionic gas in which the particles have occupied the lowest available quantum states, and are packed tightly together as if in a solid, there is no room for any atom to absorb a small amount of energy imparted by a photon because all of the ‘nearby’ quantum states are taken. So the atoms allow the photons to sail right through, and the gas appears to be transparent.

This barrier, in the form of the atoms being ‘blocked’ from scattering the photons, is called Pauli blocking. And in the three experiments, its effects were directly observable, without their validity having to be mediated through the use of mathematics.

My views in high school and through college being what they were, I don’t have any serious position on the matter of whether mathematics is real. In fact, my reasoning could have been flawed in ways that I’m yet to realise but which a philosopher who has seriously studied this question may consider trivial. But this said, having to work my way through different concepts in high-energy, astroparticle and condensed-matter physics (as a science communicator) has forced me to accept not anything about mathematics as much as the importance we place on the distinction between something being real versus non-real, and the consequences of that on what mathematics is and isn’t allowed to tell us about the real world.

Ultimately, dwelling on the distinction and its consequences distracted from what I found to be the most worthwhile part of discovery: the discovery itself. Even this post was motivated by an article in Physics World about the three experiments, whose second paragraph (and in fact most of whose second paragraphs) focused on potential, far-in-the-future applications of cold fermionic gases displaying Pauli blocking. I don’t care, and I think that from time to time, no one should.