Screaming Penguin - Source Does Matter

Rodney Brooks @ MIT TR says we need a whole new mathematical lexicon for complex systems:

It seems math is just basic stuff that's true; there won't be anything new discovered that's simple enough to teach to us mortals.

But just maybe, this conventional wisdom is wrong. Perhaps sometime soon, a new mathematics will be developed that is so revolutionary and elegantly simple that it will appear in high-school curricula. Let's hope so, because the future of technology -- and of understanding how the brain works -- demands it.

My guess is that this new mathematics will be about the organization of systems. To be sure, over the last 50 years we've seen lots of attempts at "systems science" and "mathematics of systems." They all turned out to be rather more descriptive than predictive. I'm talking about a useful mathematics of systems.

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. Such a calculus is essential to predicting, for example, how long a tank of water would take to drain when the rate of flow fluctuates with the amount of water still in the tank.

The list goes on: 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.

Still, all these tools have provided only limited help when it comes to understanding complex biological systems such as the brain or even a single living cell. They are also 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.

While I am not imminently qualified to pontificate on this matter, I am going to anyway.

"It seems math is just basic stuff that's true" I think is not bad.

I have actually found myself thinking about this a lot lately. Reading God Created the Integers one thing I have found is that for me to begin to understand the math in these books -- be it Euclid or Newton -- I have to first stop and translate it into the notations I was taught in various classes, from 8th grade on, to understand it.

The point here is that mathematics is a language as much as it is a tool. There isn't a unified way of expressing everything, rather each subset of "math" is like a different language geared to specific tasks. Much of learning "math" isn't necessarily even learning the theoretical constructs of math, but learning these languages and how to manipulate them to accomplish your goal. This is not unlike developing computer software: you have to know Java or C-something to develop your general purpose components, SQL to talk to the database (though less and less these days), HTML/XML/whatever to work with other systems. Finally you need to know the "glue" that binds these together, and when doing something in Java is easier than doing it in SQL or vice versa. While you *can* do everything in one language if you want, it doesn't serve you well. Like Newton, wanting not to reveal his developments w/r/t calculus presenting Principia done entirely in insanely complex geometry the first time around, just because you can do it with one system, doesn't mean that is the best way to go. It took a long time for experts to finally work through all of Principia before it was widely renowned for the great work it was. Compare to Einsteins miracle year publications, which were almost immediately seized upon.

I also find this interesting when viewed in terms of music. It has long been established that kids who take music lessons as a kid tend to do significantly better in mathematics. Articles in the media tend to describe this as the fact that music is expressed in a lot of mathematical ways: harmonies, fractions, linear progressions. There may be truth to this, but what stands out in my mind isn't the common concepts between music and math, but the fact that learning to read sheet music, like math or computer languages, is learning a non-spoken language special-purpose notation. It is teaching kids to understand how to build interpreters on top of the Turing machine in their head, which is exactly what mathematics or physics or chemistry mostly demands of the student.

I am also taken by the relationship of graphic representations to these various notations. UML and blueprints and geometric (vs algebraic) proofs. Again, these graphic systems help us see things that the notations on their own do not. While they are still representative of the same information, seeing a picture of an ellipse communicates different information than x^2/a^2 - y^2/b^2 = 1, while the understanding that planets on elliptical orbits move at a speeds thus that arc area/time rather than distance along the perimeter over time gives you a completely different meaning graphically than algebraically. And z(t) + 1 = z(t)^k + C belies the image of the Mandelbrot set.

Much like finding a universal programming language is kind of a waste of time, I think trying to find a "Grand unified math" isn't necessarily the best idea. Remember the lessons of Godel and Turing: A singular system can never be self-consistent, and you can't derive the solubility of some problems within a single system. In order for Arabic algebra to develop, it took the Greeks going back and changing basic arithmetic theory by proving things geometrically that couldn't be worked out arithmetically. Each of these systems reinforce each other and allow us to move past the "Godel problem". We have multiple systems, because no single system can ever be expressive of a complete truth.