Ode (Wonder)

I don't have anything particularly well-researched or super put together to say in this post; but I've had a very strong feeling of amazement and joy kind of brewing inside me for a long time. And I want to express that feeling, write down some general thoughts I've had and give some examples of things which gave me a feeling of warm excitement. I don't know how else to describe it, working on some problems and discovering a few patterns made me feel extremely free.

There are multiple sources of this sensation, but by far the main one has been mathematics and physics. It feels like the last year or so has opened my eyes to a completely new, more unified perspective on what these subjects really are, and it feels like I have begun seeing them in a completely different way, despite spending the majority of the last four years or so (and a decent portion of my time as a teenager) thinking about them. I can't imagine what the landscape looks like after dedicating your entire life to this subject. It must be incredibly beautiful.

The more time I spend studying the more I'm convinced of the fact that this is an incredibly mysterious and strange subject, and often feels very non-human (at least to me)[1]. Even if we assume that it's 'real', and exists independently of humans, the mathematics we know today exists only because of the efforts of curious humans, compounded over millennia. We can't touch it, and can't really see it or feel it in any way; and yet it's powerful enough to evoke very strong emotional responses in a wide range of different people. How is this possible?

Why is mathematics beautiful?

This is a very natural question to ask if you have never really come into contact with mathematical elegance. The math we all learnt at school definitely never had any glimmers of aesthetic sense; at best, we would get some nice Taylor series. To understand how mathematics can be considered in the same aesthetic vein as something like poetry, it's useful to first consider what exactly mathematics is, or at least can be - once we see an object and understand what it is, understanding its properties (e.g. big or small, ugly or beautiful) becomes much easier. In my view, different branches of math can be considered almost entirely different subjects, loosely held together by the fact that they both obey the rules of logic and inference: so kind of like jazz and rock are both music in the sense that they obviously are genres of art manifested through sound, yet often have very little in common with each other (although, of course, learning more math or physics leads one to inevitably begin realising how the whole subject is a single field manifesting in different ways). I present a few different views (perhaps they are all isomorphic to each other; shadows of a greater Platonic Form?) on what math is.

Mathematics is the study of time

The most basic mathematical notion that all humans intuitively understand is commutativity: shirts are put on before jackets, and jackets are taken off before shirts. It is obvious that the order in which we do these operations matters; so the time in which we perform these operations makes a big difference. The most general notion of commutativity everyone feels is that of time passing: for some reason, we seem to move forward from the past to the future, but never from the future to the past. This is a pretty remarkable fact, since we still have absolutely no working theory of why time doesn't commute; although we have the theories and tools to describe nature at a subatomic scale, we don't know why tomorrow happens tomorrow, and not yesterday.

A significant part of mathematics (especially algebra) studies a generalisation of this principle. What happens when we restrict the order in which things must be done? As it turns out this introduces a new set of logical constraints and surprisingly beautiful relationships.

I don't want to give a lot of examples here - if you are interested see this - but a striking one to consider is the mathematical formulation of quantum mechanics.

Yuri Manin considered quantum mechanics as a form of noncommutative geometry. This means that if we consider some set of physical space with phenomena happening inside it, the order in which we observe these phenomena (and the order in which they interact with each other) will result in completely different outcomes.

Mathematics is a 'generator'

This idea is a bit more wooey and admittedly not very rigorous, but it is a thought I've had for a while.

For some reason, our universe follows physical laws, and these laws are mysteriously well-suited to mathematical expression. It is completely unclear as to why this is the case.

One of the first 'easy' (although in my opinion still quite weak) explanations is that mathematics appears to be Platonic, and perhaps there exists some kind of 'mathematical universe' of Forms, which contains the set of all possible mathematical truths. These Forms include all the math we have ever invented (or, in this case, discovered). Many mathematicians and physicists subconsciously assume this to be the case, and regardless of your stance, finding new math often feels much more akin to discovery than invention; Descartes invented and introduced the notion of a complex number, but he did not invent the Julia set; and yet, just by using the idea of an imaginary number, we somehow magically uncover much deeper structures which came for free. They were already encoded into the ideas Descartes proposed, even though he had no intention of doing so, and clearly wasn't aware of the full power of complex numbers.

The Julia set, a wonderful fractal structure

This is quite different from regular 'invention'; although it's possible to find new, creative ways of using technology, discovering the Julia set would be like suddenly realising that car engines, for some reason, actually have more bolts in them than any engineer ever predicted. It doesn't make any sense!

The unusual thing about this Platonic idea, though, is that most math doesn't seem to have any bearing (or relationship) towards physical reality. Quite a few specific things - differential equations, groups, integrals, topology - seem to be intimately tied to physics, and they are some of the best tools we have for understanding reality. And yet the vast majority of math, particularly more abstract things (algebraic geometry, number theory, most algebra, Euclidean geometry, many pathological cases of analysis) have absolutely nothing to do with reality. Many pure mathematicians delight in this fact, and their work is primarily motivated by aesthetic principles rather than any kind of connection to reality. But this implies that the vast set of Platonic 'math' is only partially represented in our universe; in that sense, the Platonic world acts as a 'generator', from which our physical reality emerges as a kind of shadow. Perhaps some other universe exists which is a manifestation of another subset of the Platonic world.

I like this argument and it makes sense, but at the same time I usually find these kinds of directions frustrating, because it's a claim about some abstract metaphysical world which we can't see, feel, or even test the existence of - and therefore statements about Platonic worlds existing are completely meaningless. But I can't shake the feeling that math is an infinitely deep tapestry, much deeper and larger than any sum of human minds; which means that it's either an emergent property of human invention, or simply exists independently of humans. Max Tegmark wrote a whole book about this, aptly called Our Mathematical Universe.

Mathematics is a recursive structure - i.e. mathematics is mathematics

Bertrand Russell famously speculated that all of mathematics is just a tautology. There is a lot of logic and reason to that statement.

But that sounds boring and defeatist; I think that a better way of looking at it is to consider math as a loop.

The statements '2+2=4' and '2+2=4 is a valid statement' appear to not mean the same thing. One statement refers to the property of numbers; another statement refers to the statement on numbers as being valid. But isn't putting an equals sign just the exact same thing as saying 'this is valid and true'? Every single mathematical statement or proof is effectively structured as:

Theorem: Statement A is true
Lemma: Statement A is true if statement B, C, …, N are true
Proof: Statements B, C, …, N are true because statements B1, C1, …, N1 are true
Therefore Lemma is true
Therefore statement A is true, so theorem that statement A is true is also true, QED

You can see how this quickly becomes a problem of semantics; we use previously proven statements to prove new statements. But you eventually 'bottom out'; there have to be axioms which we accept as true, otherwise we get stuck in a loop of infinitely referring to more fundamental theorems. This means that every mathematical statement effectively becomes:

Theorem: Statement A is true
Lemma: …
Proof: …
Lemma: B1, C1, …, N1 are true, because B2, C2, …, N2 are true…
….
Axiom: We assume the truth of Axiom 1, Axiom 2, Axiom 3, …
…
Proof: B3, C3, …, N3 are true because our axioms are true.
…
Proof: Theorem is true because axioms are true. So, cutting out the middle man, the theorem is true because we defined it as true. QED

Mathematical and physical beauty are supreme simplicity; they are games, puzzles, total liberty, art, joy

This one is quite difficult to contextualize. How could the subject every single human finds difficult be simple?

Paradoxically, I've found that simple things are often much harder to think of than complex things. Simple doesn't mean trivial or easy. For example, going to the gym is very simple; you just go and lift weights and eat enough food to get strong. But that is not easy.

This is exactly why math and physics is so hard: because they require extremely deep insight past formalism, past accepted intuitions and psychological blocks; a practitioner of math or physics must see things as they truly are, not as we would like them to be. Of course, the same mathematical object can be interpreted in a huge variety of different ways. But at the end of the day the most beautiful theories are the ones which showed how complicated phenomena arise out of simple ideas. It's much easier to think up super elaborate reasoning than it is to get to the core of something, and to cut away all unnecessary details. Mathematical beauty arises exactly when we are able to cut past everything to understand deep, fundamental ideas - capturing everything in one fell swoop, instead of a hundred boring calculations.

The best way to explain this is probably through an example.

Everybody knows that, for some reason, quantum mechanics says that 'matter is both a particle and a wave'. This statement is, to the dismay of pop sci books, totally meaningless without a physical understanding. The double slit experiment is an excellent way of seeing where exactly this duality comes from, and also ties in to the next point (math, and particularly physics, being a game and an art).

Suppose we put a screen with one hole into a bathtub full of water, and put another screen with two slits in front. Playing with the water behind the screen will generate waves, which then pass through the first hole, and dissipate through the two next holes. This creates a familiar wave-interference pattern, where the crests and troughs of waves either build each other up or cancel each other out, depending on whether or not the waves are in phase with each other. We end up seeing something like this:

The final peaks around the detector alternate with troughs; this makes sense, since we expected interference.

Now suppose we do the exact same experiment, but with bullets being fired from a faulty gun, instead of water. Suppose the gun randomly sprays across the first screen with two slits. We then expect to see only two crests; the probability of a bullet being at any one spot just comes from adding the probability of it going through hole 1 or hole 2.

Now we do this experiment with electrons, randomly firing them at the slits. Intuitively, electrons are tiny balls which carry a negative charge. So we should expect them to either go through the first hole or the second hole, just like the bullets. But what we see instead is quite surprising:

It's the same pattern we saw with water! How is that possible?

Well, the first solution is either that our intuition around what an electron is is wrong (i.e. it's actually a wave, not a particle), or that the electron somehow 'interferes' with itself, or that it doesn't just go through hole 1 or hole 2 - does it go through both at the same time? Go through one and then loop back through the other? Regardless, the probability of an electron arriving in a given spot doesn't come from just adding the probabilities of coming through hole 1 or hole 2, so something in our model is not right. The answer is not clear, but we can try to check. It shouldn't be too hard to figure out what's going on: let's set up a light beam behind the double-slit wall. Electrons deflect light, so every time an electron goes through the bottom hole, the light will deflect downwards. Every time it goes through the top hole, the light will deflect upwards. If it goes through both at the same time (somehow), then the light will go both ways; if it's a wave, then we'll see our interference pattern change a bit.

Surprisingly, we now get back the 'bullet' pattern, with each electron now going through either hole 1 or hole 2, and the probabilities of arrival once again becoming P1 + P2. That doesn't make any sense, again! We just observed our experiment, and somehow completely changed the outcome.

Alright, it's not entirely clear why that happens, but we now at least have some intuition behind the particle-wave duality idea; depending on whether or not we watch what happens, the probability of the electron arriving in a given location seems to change. Very strange. Suppose that when we don't observe the experiment, the probability of an electron going through hole 1 or hole 2 is given by its amplitude; we introduce this term to differentiate between the usual probability we're used to with the bullets (P1 + P2) and the seemingly new probability calculations that arise when we don't use any measuring tools. So there is an amplitude associated with each path the electron could take, and adding the amplitudes together in some strange way will give us the final probability distribution we're looking for.

So far this is pretty cool, but kind of random, although it does demonstrate an unusual character of physical law: we can interfere with results simply by looking at them, and therefore need to revise our understanding of probability when considering particles at the smallest scale. But so far this has only held in a very particular, weird experimental scenario; how does this relate to our everyday experience?

This next step is where the magic happens, and I hope this illustrates what I mean when I say that simple is beautiful, and why physics or math are vehicles for creative expression and freedom.

Suppose we now repeat this experiment with more than two walls, and drill a bunch of holes in each wall. Then each electron will go along some path in this space; several alternatives are possible:

As we have seen before, the electron appears to only 'choose' a single hole once we observe it. If we leave it alone, then each path has an amplitude associated with it (like a wave); so the total likelihood of arriving at some location in B comes from summing together all the amplitudes from all the possible paths.

Now suppose we keep adding more and more walls and keep drilling more and more holes in each until there's nothing left; our electron is just moving through empty space.

But we already know that we need to sum over all possible paths; except now that we've taken the limit, our sum turns into an integral over all the possibilities, each one having some amplitude associated with it. So we understand that the motion of an electron in space can be modelled by integrating over all conceivable paths - i.e. we've just come up with our first theory of quantum mechanics, which strongly diverges from classical intuition. Notice that we proved a very deep result (a particle's position is determined by considering all of its possible paths, instead of just one!) simply by drawing pictures and thinking a bit; no advanced math or ornate structures needed. This model is called the path integral, and it's one of my favorite ideas.

This kind of reasoning and visualisation can be carried out by a middle schooler. But thinking carefully and deeply enough about what we're doing leads to a new model of reality, and explains why it makes sense to consider certain phenomena as probabilistic. This is the kind of thing I have in mind when I say that physics is a game full of freedom: the goal is to solve hard puzzles with as much elegance as possible.

Although it is impossible to predict who will be the next person to do great work, it is possible to see a unifying theme across all great work: the person doing it didn't do it because they had to, or because they were disciplining themselves into it, or because someone told them to do it: the only reason great work happens is because the person doing it is totally in love with their work.

So great work is simply a manifestation of play in its highest form; it is just fun.

Mathematics and physics are some of the few (only?) fields which are completely unconstrained by human conventions. The only constraints are logic and physical intuition. This provides an endlessly entertaining playground: you can literally do anything within the frames of reason, and the only limit or rule is your own creativity. In this sense, these fields are just like music. The only constraint on the kind of music you can (or, rather, should) write are the laws of musical harmony. Otherwise, you can do anything, in a way which doesn't depend on other people, or money, or much equipment; this is extremely freeing, and it provides an infinite platform for play.

This kind of play inevitably leads to a deep sense of personal liberty, since it shows the 'players' that they are limited only by their intelligence, creativity, interest and desire. It is also very honest and fair, since the only marker of good math or physics is that it makes sense. I find this aspect extremely appealing; it's impossible to bullshit anyone, including yourself. The same is not true in almost anything else.

Mathematics also provides humans with a unique opportunity to satisfy their curiosity: it provides a framework for anyone to experiment, explore, and discover the world around them, as well as the world inside themselves. It's like a game, since it usually doesn't have much to do with reality, and even when it does, you can treat it as a game: it's a puzzle-solving exercise, which, to me, is the most fun thing in the world.

Any game or art you can think of is also a puzzle-solving exercise. Math and physics elevate the process of problem-solving to a virtue.

Consider the complete fun to imagine: a quantum field, an electric field, the very particles which me and you are made of all become symbols and images you play with inside your head. The very reality we are part of is understandable and legible through a process of analogy, visualisation, and conceptualisation in the language of math. Isn't that incredible?

The best example I can give of mathematics being an art is through Escher's prints, or Islamic architecture:

The waterfall here is an unusual topological structure, which never appears in reality; however, the tools of mathematics let us construct these unusual, beautiful weaving patterns.

This Escher print is an example of hyperbolic geometry: what would the world look like if it was on the surface of a sphere?

Note that Lewis Carroll was himself a mathematician for his whole life: the ornate structure of Alice's Wonderland, as well as its characters, have a mathematical life to them. Everyone likes his books, but I think they are particularly appreciated by mathematicians and scientists, since Alice enters a world of swirling patterns. This is exactly the same kind of joy and depth mathematics provides to its practitioners. Martin Gardner, known for his Mathematical Games, wrote the most detailed annotation of Alice to date; the world of puzzles is a world of endless fascination.

Mathematics is the study and creation of metaphors

This perspective is due to Yuri Manin.

The goal of a mathematician or physicist is to reduce a currently intractable problem into something solvable, or maybe consider the intractable problem from an unusual, creative angle - since all the previous angles lead to failure. This means that a mathematician's main weapon is analogy; seeing how two ideas bridge together in an unexpected way is often the key step in cracking open valuable insights. In this sense, every single mathematical idea serves as a metaphor for some other idea. The machinery of abstract algebra - groups, categories - seems needlessly abstract and boring at first glance. But digging deeper and understanding why anyone would care about these abstract structures leads one to recognise that these abstract objects allow mathematicians to turn specific cases into general examples. The clearest example of this is in geometry, where shapes are studied through their symmetries, rather than just as pictures. This means that a group is a metaphor for any kind of object with symmetry; this lets us study any kind of geometrical figure we like, and quickly understand whether or not this figure is symmetric. This is much easier than counting vertices, and opens up a totally new avenue of creativity: the shapes we think of don't have to be limited to those we are used to.

In the realm of physics, mathematics serves as a metaphor of reality. Finding the right metaphor means understanding how different pieces of the universe are connected, in a single wonderful puzzle. In this sense the fields are not too different from poetry, since the mark of a good poet is the ability to find the right metaphor for describing a unique vision.