(Rough working notes, Recurse Center, April 2016. If you're hoping for polished work, you're in the wrong place.)
Let me show you a wonderful visualization of the Fourier transform, done by Steven Wittens. Turn on your system sound, and take a moment to page through the visualization.
A natural response to this visualization is to admire its beauty and the technical virtuosity involved (as of 2016) in implementing it in a web browser. But what's interesting about the visualization goes well beyond beauty and technical virtuosity.
When I saw Wittens present this demo, my jaw dropped. I glanced across the room to my friend Chris Olah, and he was looking similarly awed. Talking later to Chris, we agreed that much of the impact of Wittens' demo was that he was vividly showing us aspects of the Fourier transform that we'd both long understood, but had never before seen made explicitly visible.
When one begins learning about a mathematical object – say, the Fourier transform – one begins by reading the definition, seeing some examples, understanding some theorems about the object, and so on. Those are all external representations related to the object. After a lot of practice working with those external representations, one internalizes and becomes fluent with them. They're no longer awkward external properties, but rather fluid and natural parts of your thought.
Most people – even those who think of themselves as “good at mathematics” never get beyond this stage. But if you continue to work hard with the mathematical object, if you solve more and more difficult problems with it, your understanding deepens further still. You begin to develop a sense that goes even beyond extant external representations. The best description I know of this comes from the mathematician William Thurston, who framed it as a question to himself:
How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? … We have complex minds evolved over many millions of years, with many modules always at work. A lot we don't habitually verbalize, and some of it is very challenging to verbalize or to communicate in any medium. Whether for this or other reasons, I'm under the impression that mathematicians often have unspoken thought processes guiding their work which may be difficult to explain, or they feel too inhibited to try. One prototypical situation is this: there's a mathematical object that's obviously (to you) invariant under a certain transformation. For instant, a linear map might conserve volume for an ‘obvious’ reason. But you don't have good language to explain your reason — so instead of explaining, or perhaps after trying to explain and failing, you fall back on computation. You turn the crank and without undue effort, demonstrate that the object is indeed invariant. … Once I mentioned this phenomenon to Andy Gleason; he immediately responded that when he taught algebra courses, if he was discussing cyclic subgroups of a group, he had a mental image of group elements breaking into a formation organized into circular groups. He said that ‘we’ never would say anything like that to the students. His words made a vivid picture in my head, because it fit with how I thought about groups. I was reminded of my long struggle as a student, trying to attach meaning to ‘group’, rather than just a collection of symbols, words, definitions, theorems and proofs that I read in a textbook.
In other words, when you undersand mathematics deeply, you begin to develop internal representations that are not well captured or easy to explain using existing external representations, but which nonetheless provide powerful ways of thinking.
What makes Wittens' demo amazing is that it captured something of my internal representation of the Fourier transform, and made it externally visible. It was invisible understanding made visible.
This is not to say that I previously thought of the Fourier Transform in the way Wittens shows. I did not. Nonetheless, there was still a shock of recognition, a sense that Wittens' representation captured aspects of the Fourier transform that I felt, but which were not present in conventional representations.
Of course, this visualization is intended as a demo. It's not a working representation; it doesn't have a fully developed interface which one can use to work with and reason about the Fourier tranform. Still, it's tempting to muse: might it be possible to build out an interface based on this representation? Might such an interface reify important facts about the Fourier transform, facts which are not obvious in traditional representations?
I'm not sure how that would work** The ideas that follow in this paragraph arose in conversation with Chris Olah.. One nice property of the Fourier transform is that the Fourier transform of a convolution of functions is the product of the Fourier transforms of the individual functions. (This has many nice consequences, including making it almost trivial to prove the central limit theorem.) Might this beautiful theorem have an insight-giving representation in this approach? Might it, in consequence, then be possible to more deeply understand results like the central limit theorem?
Marshall McLuhan wrote “we shape our tools and afterwards our tools shape us”. When we invent powerful new external representations of concepts like the Fourier transform, that opens up a path to deeper understanding, and thereafter to the invention of still more powerful representations, in a kind of virtuous circle.
So what, if any, conclusion can we draw from this? I think it's something like this. Pick any phenomenon in the world. Ask if existing representations reveal the deepest existing understanding of that phenomenon. Not just your own understanding, but the understanding held by the world's most thoughtful experts. If not, there's an opportunity. How can we develop more powerful representations that let us see and act more deeply?
The notion of mathematical beauty: Mathematicians often talk of finding beauty in mathematics; equally often, non-mathematicians are bewildered by this. They look at a page of mathematics and see what appears to be a bunch of largely disconnected and incomprehensible scribbles. The page itself may not be much more directly meaningful to a mathematician. But they can use it as a bridge to connect them to their deeper internal representations. And that's the level at which they find beauty: in sudden, unexpected connections, revealing surprising order.
The non-mathematician's experience of mathematics may thus be compared to the experience of a severely short-sighted person visiting an art museum. It is difficult to appreciate a Rembrandt when all you see is a blur. You are not seeing Rembrandt; indeed, you have no direct access to what Rembrandt was doing at all.
Connection to Plato's caves: One might wonder if there's any connection to Platonic forms, and questions like whether mathematics is invented or discovered. I don't have a strong opinion. I will say that, to the extent one believes Plato's story, the purpose of building external representations is to better reveal the true underlying forms. And so this is perhaps a useful framing.
Tacit knowledge: There's a connection here to the concept of tacit knowledge developed by Michael Polanyi. One might suppose that I am proposing that we can use interfaces to make explicit what was previously tacit knowledge. I don't entirely disagree with this, but I don't entirely agree with it, either. The issue is that I'm not sure all of what Polanyi called tacit knowledge is necessarily reifiable in an interface. Conversely, I'm not sure we only want to reify tacit knowledge in an interface. There's all sorts of emotional content and affordances that we want to reify as well (at a minimum). So I think it would be a mistaken oversimplification to think merely of reifying tacit knowledge.
Acknowledgements: Thanks to Chris Olah and Toph Tucker for the conversations that led to these notes. And thanks to Steven Wittens' for making his wonderful visualization.