Rodney Brooks recently penned an essay on the need for a new mathematics "that is so revolutionary and elegantly simple that it will appear in high-school curricula." He claims that understanding biological systems "demands it."
Brooks suggests that current tools (mathematics) are brittle when it comes to the challenge of describing "systems" (emphasis added):
Currently, many different forms of mathematics are used to model and understand complicated systems. Algebras can tell you how many solutions there might be to an equation. The algebra of group theory is crucial in understanding the complex crystal structures of matter. The calculus of derivatives and integrals lets you understand the relationships between continuous quantities and their rates of change... Boolean algebra is the core tool for analyzing digital circuits; statistics provides insight into the overall behavior of large groups that have local unpredictability; geometry helps explain abstract problems that can be mapped into spatial terms; lambda calculus and pi-calculus enable an understanding of formal computational systems...
...all these tools have provided only limited help when it comes to understanding complex biological systems ... inadequate to explaining how networks of hundreds of millions of computers work, or how and when artificial evolutionary techniques -- applied to fields like software development -- will succeed.
Virtual worlds - insofar as they represent large dynamic systems that comingle art, technology, sociology, engineering, economics, and yes, A Theory of Fun - might seem to present a messier take. Given the size and interconnectedness of this realm how is it possible to usefully talk about any substantial piece of it without lashing together many stove-pipe intuitions?
From time to time some do scratch out notations to help quantify a little of these surfaces. Perhaps coming one day to a high-school near you will be new and better tools for so many blind people to converse about this very large elephant.
Comments on New Math:
The key statement:
"We try to impose a computer metaphor on a system that was not intelligently designed in that way but evolved from simpler systems."
Posted Mar 26, 2006 3:29:19 PM | link
Retort to Key Statement: Unless it was...
As I see it, the only way to implement such a system within a reasonable timescale would involve a sizeable donation to a private school under the condition that they begin teaching group theory, Boolean algebra, lambda calculus and pi-calculus starting at the 8th or 9th grade. We are already moving towards this, I think, as my generation saw calculus strictly as a college class whereas we now are seeing juniors in highschool taking calculus as if it was no big deal. The question might be "is it fast enough?" Are school curriculums advancing quickly enough to provide young thinkers the tools they need to describe new events? Personally I believe that mathematics deserve more emphisis in the States than they currently recieve but with out really banging up the current system it will still be another 20-30 years.
Posted Mar 26, 2006 4:07:48 PM | link
I didn't interpret it as a "teach everything faster, sooner", but rather as something new that high schoolers could understand without having covered all of the above material. It would be elaborated by these other higher maths, but it wouldn't be dependent on them.
Don't know if that's possible, but I'd definitely like to see it happen.
Posted Mar 26, 2006 6:10:57 PM | link
Queueing theory is good for some of Brooks' goals, and control theory for others, but both are usually taught as dependent on a huge array of earlier mathematics. Queues could be taught as a toolbox approach rather than intense probability and statistics, but then I'm not sure how useful they'd really be. (Kind of like differential equations are today, or were when I was an undergrad.) Basic control theory tends to be taught as a toolbox, but it still requires comfort with imaginary numbers and Fourier/Laplace/Z transforms, which are usually backed up with some calculus.
Isn't a "new kind of math" what Wolfram was trying to do with Cellular Automata? One of my favorite sci fi authors has CA replacing calculus as the end of the math sequence, but I'm not convinced we're anywhere near that yet.
Posted Mar 27, 2006 9:57:49 AM | link
Didn't Wolframs "A New Kind of Science" already propose exactltly the system he's mentioning?
Posted Mar 27, 2006 12:40:40 PM | link
An article titled "The Biology of Cognition" by Humberto Maturana is probably one of the most important writings on the epistemology of biological systems. Any mathematical theory of biological behaviour must first be premised upon a root metaphor that is *not* computational, but biological. I strongly suggest sludging through this (rather cryptic) article for anyone that takes biological systems theory seriously.
Posted Mar 27, 2006 2:26:23 PM | link
woops. forgot a link to the article: http://www.enolagaia.com/M70-80BoC.html
Posted Mar 27, 2006 2:26:55 PM | link
There do seem to be echos of Wolfram's proposals in the essay. And wasn't Wolfram also blasted pretty hard at the time for lifting his work without attributing Schmidhuber and others?
Regardless, I question whether the same goals can be reached by teaching theory and application of Bayesian Networks. This quote came out of a discussion we were having on my blog regarding Genetic Algorithms, Neural Networks and Bayesian Networks (for a different purpose than the subject here):
Way to low level. PNN's are likely good for modelling the low level dendrite-axon firing of neurons.
But what are the neurons doing? They are making predictions. What is the best statistical framework for making predictions? BN's are. It is very likely that collections neurons are implementing BN's (conveys a survival advantage).
I'm not convinced teaching CA should be a high priority given some very serious deficiencies in first-principals mathematics and sciences (not to mention language-arts). I definitely resist the notion that higher maths can taught in any useful fashion without first mastering dependent or related basics. We should endeavor to create mathematical *thinking*, not just mechanised doing.
Posted Mar 27, 2006 3:57:38 PM | link
"Any mathematical theory of biological behaviour must first be premised upon a root metaphor that is *not* computational, but biological."
Computational: The act or process of computation
Computation: To determine by mathematics
So what you're saying, is that we must create a paradox, by constructing a tautology? :)
I'm not entirely certain, but to me this looks like a view that ignores the ability of forms of mathematics and computation to deal with non-determinism. Either that, or it sounds a bit like saying that the base metaphor for the math of architecture lies in buildings. But then, I'm a mathematician, to me, mathematics is about taking any problem and then abstracting it - the math of abstract algebra isn't obviously connected to codes ("how can algebra deal with rotating keys?") when you first start, but it is when you understand them both and get further in.
Posted Mar 30, 2006 7:15:50 AM | link