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.

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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.

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