A New Type of Mathematics: David Dalrymple at TEDxMontreal (Full Transcript)

Great to be back on the TEDx stage. It’s been a little while.

No, I think a good theme for, at least, the egghead crowd at this event, seems to be life in a complex world. And yet, all of us, speakers, are really embedded in a very simple world. It’s just a big red circle. I think my theory is they just put the circle here to keep us from checking to see if the letters are real. They’re probably holograms or something.

But a lot has happened since I’ve been at TEDx. For one thing, I’ve gone through puberty, gotten some degrees and certifications, I’ve learned a lot, and Singularity University was really one of the defining moments of my life so far. I switched from computer science to neuroscience. I sort of got this strong sense that the next big thing is going to come from brains somewhere, and what I want to share with you is kind of how that sense has evolved over the last year. I was at this big conference, one of the biggest academic conferences around is called Society for Neuroscience.

Tens of thousands of researchers pack into one of the five convention centers in the country that can handle that kind of crowd. There’s a huge, huge array of panels and events, but at the evenings, groups of a dozen or so get together around the table and you get to have real discussions. One of those discussions was about modeling the brain large-scale, ambitious things were saying, “Let’s go after an entire organism, understand how it works.” That’s what I’m planning to do. And one of the people at the table was this old-school neuroscientist and electrophysiologist – was what he called himself – He had this sort of historical note, he wanted to put out a caution for those of us who would seek to understand the brain with mathematics.

He said, “You know, John Von Neumann the great mathematician, arguably the greatest of the 20th century, once said, “The brain does not use mathematics. There is something more mysterious at work there. And so you should be humble, young children, and do not attempt this sort of research.” And so I decided. You know, John Von Neumann is actually a personal hero of mine. Back when I was a computer scientist, I did a lot of work that followed up from what he was doing in cellular automata theory, and so, I decided to go to the MIT libraries and check out the source for this information. And it’s this adorable book, it’s called “The computer and the brain,” and it was written in 1957 when knowledge about both of those was sparse enough that to claim some connection was fairly radical. I think it’s really impressive that he even thought of that.

And the relevant quote, amazingly enough – here’s what he actually said, “The outward forms of our mathematics are not absolutely relevant from the point of view of evaluating what the mathematical language truly used by the central nervous system is. The above remarks about reliability and so forth prove that whatever that system is, it cannot fail to differ considerably from what we consciously and explicitly considered today as mathematics,” and there, that is the end. There’s nothing after this, except the due date. John Von Neumann actually got bone cancer while he was writing the manuscript, and it is unfinished. That is where he left it, and that trail, I think, is ripe to be taken back up again.

So, I want to explain why the sort of mathematics that we do seems to be not quite up to the task. One of those mottoes that we have in the AI world these days is “the hard stuff is easy, but the easy stuff is hard.” You know, if you want to play a good game of chess, that’s been solved, but if you want to do something like take a drink, for a robot to do that, that’s a pretty hard problem. We managed to do it pretty easily. What’s going on is there’s you looking at the cup and tracking it as you’re moving it towards your mouth, but also, inside your brain there is a model of you holding the cup and some kind of concept of what drinking is supposed to be, and that has an effect on what it is that you actually do, but your thinking ahead appear.

And so, it’s almost like reverse causality, it’s almost like what’s happening in the future is affecting what you do because your’re thinking ahead, you think, “Well, what’s about to happen is it about to splash in my face. Oh, I’d better stop now.” This something from the future is sort of affecting the present so it’s almost like your brain is traveling back in time, saying, “No, don’t do that. Stop! Stop there.” And this is really hard to model, because in modeling physical systems, we always take to what Richard [March] say, “Well, time instant one, two, three, four- each one depending rigidly on the last,” and you can sort of imagine it in this continuous way, but it’s we can do these things in simple cases, like for instance, if you have a thermostat.

You want to anticipate what’s going on when you turn on the heater or when you turn off the heater, that the temperature is going to dip down. And there’s some wind outside, you don’t know how windy it is, how fast temperature is going to dip down, but it’s pretty easy; you just have some wine, you call your set point when you get below that, you turn the heater on, and when you go sufficiently far above it, you turn the heater off, and you can analyze this in a purely rigid sense.

It’s trying to keep it; you can just look at this line and say, “Well, it’s staying near the dashed line. I can understand what’s going on.” But in a biological network, it’s not that you’re trying to keep something near a line, you’re trying to keep it near some multi-dimensional, abstract, manifold surface. And there are spike trains, and there are chemical gradients, and everything is connected in weird patterns. You don’t have these nice labels for what any of the signals are; you just get lots of data, and nobody knows what to do with it.

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It sort of seems like there’s an analogy to this other, very complicated many-body system that it took a while to figure out the mathematics of, namely the motions of celestial bodies. For a while, they were just the Sun, and the Moon, and the planets and you could get by – click – with what I call the spirograph model of astronomy. We have circles within circles, and circles within those circles, and you spin them out enough, and, of course, the Earth is in the center. And you see where the things move: here’s the planets that follow this trajectory for some reason. And after a while, we got the idea that maybe planets should just go in regular circles, and maybe they go around the Sun, but that still wasn’t really enough.

I mean, then, you still. It’s a little bit simpler, but you still use some corrections – they are called epicycles. And whenever you start to see this kind of thing in another scientific field as a common thing, you say, “Uh-oh! lt looks like you’re adding epicycles,” and they say, “Well, OK. Maybe this isn’t really what’s going on after all.” We’ve definitely gotten to the point in neuroscience for there’s a lots of epicycles. Try to take a physical neural network with hundreds of thousands of neurons and sample ten of them and make a guess as to what’s going on in the whole thing. It’s basically like looking at the Solar System without seeing most of the planets, or, in particular, the closer analogy is we don’t see Jupiter’s moons just yet. And that’s what really gave Galileo the clue of what’s actually going on.

It’s not just that things are moving in circles. It’s not just geometry, there’s more to it than geometry. There’s calculus. And it really spawned a whole branch of mathematics; there is numbers, and numbers get you algebra, and there’s geometry which gets you trigonometry, and then there’s calculus which gets you differential equations, and it gets you computing volumes of arbitrary shapes, and basically, all of the things that engineers do rely on that. What I am anticipating, without any real evidence to back it up – but it’s a big stage, I’m going to make a provocative claim – when we study the brain and really understand in detail what’s going on, we’re going to find out in retrospect that we’ve invented a new kind of math, something that is as powerful as calculus and as inconceivable; if you would ask John Napier, who invented the logarithm 50 years before calculus was known, if there would be a completely new field of mathematics that would dominate everything that was known in another 50 years, he probably would have said no.

So it seems a little silly in advance, but I think the signs are there, and historically, we’ve seen – in this case, at least- when things start getting really complicated and at the same time, you’re getting more and more accurate ways of looking at what’s going on, there is something very profound that you might not be able to discover. And what’s going on in neuroscience really is along those lines: we’re starting to get tools that can look into all of the neurons in a region of brain while it’s living, and it’s kind of a standard thing to do – going by the name of connectomics – to take a brain, and the first thing you do is kill it, then you slice it up, and then you image the remains, and you try and figure something out from that. There’s definitely things that can be learned there, but you’re not going to understand the dynamics which are what power the computation. You can’t probe a dead brain and expected it to respond and test your theory about how it would respond. But we are now getting the technology where you don’t have to do things in dead brains.

You can actually probe living brains with light waves of all things. You know, I used to think, when I was very young, and I first learned about lasers, that something special what happened, if you intersect two lasers, that the air there would heat up a lot, or something like that; and then I learned more about lasers, “OK, I guess lasers just go past away each other,” but then, when you learn even more about lasers, you realize that something special can happen there if two photons arrive at exactly the same instant on molecular excitation timescales.

And so, if you have fancy-enough lasers, you could actually do that, and you can do it inside of a brain and you can put molecules into that brain that will cause the neuron to be active when you fire two lasers at it simultaneously. We are living in the future, my friends! And you can also read what’s going on because you can put fluorescent reporters into the brain and so when you fire two lasers at it, it will activate if the neuron is already active, and it will light up, and you could look at it. It’s like Richard Feynman said in “There’s plenty of room at the bottom.” All of these fundamental biological questions can eventually be addressed by just looking at the damn things. And we’re getting to that point, it takes a lot of technology to just look at things.

It took a very powerful telescope to look at Jupiter’s moons and understand that there were some orbits happening that it wasn’t just about circles, but it was about two things orbiting each other. And I think we’re just starting to get to the level of optics if you will, on inner space as opposed to outer space that we can look in and see the profound principles that are at work there. So this is that book, and I took this photo this morning. The last page is still unwritten, but there is a pen there, ready to write it, so it’s exciting times and I hope you will follow it as it progresses. Thank you.