By Dr. Rodney Brooks, Dept Head of the MIT Computer Science and Artificial Intelligence Lab

Quote: Newer Math? A new high-school mathematics might someday model complex adaptive systems. By Rodney Brooks While prognostications about "the end of science" might be premature, I think most of us expect that high-school mathematics, and even undergraduate math, will remain pretty much the same for all time. 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. These are just a few examples of what are sometimes referred to as complex adaptive systems. They have many interacting parts that change in response to local inputs and as a result change the global behavior of the complete system. The relatively smooth operation of biological systems -- and even our human-constructed Internet -- is in some ways mysterious. Individual parts clearly do not have an understanding of how other individual parts are going to change their behavior. Nevertheless, the ensemble ends up working. We need a new mathematics to help us explain and predict the behavior of these sorts of systems. In my own field, we want to understand the brain so we can build more intelligent robots. We have primitive models of what individual neurons do, but we get stuck using the tools of information theory in trying to understand the "information content" that is passed between neurons in the timing of voltage spikes. We try to impose a computer metaphor on a system that was not intelligently designed in that way but evolved from simpler systems. My guess is that a new mathematics for complex adaptive systems will emerge, one that is perhaps no more difficult to understand than topology or group theory or differential calculus and that will let us answer essential questions about living cells, brains, and computer networks. We haven't had any new household names in mathematics for a while, but whoever figures out the structure of this new mathematics will become an intellectual darling -- and may actually succeed in designing a computer that comes close to mimicking the brain. Rodney Brooks directs MIT's Computer Science and Artificial Intelligence Laboratory.

I've been thinking about this for sometime. I suppose one would start with a simple graph and derive a calculus to model the evolution of the graph? I'm still tinkering with the idea, and I'll write up what I figure out tonight. Any thoughts on his proposal?

Wolfram has been trying to develop his version of this new kind of mathematics for 20 years, with the recent release of his volumous tome "A new kind of science" (iirc). Basically it's his take on automata theory. (one sentence cannot do it justice though... read the book if you have a year to spare and want to form your own opinion of his ideas/results).

I agree with Brooks though; we need a fundamental mathematical language for describing complex adaptive systems and their dynamic bahviour. In addition to this though, we also need to understand how to analyse this behaviour, which is really what science and mathematics has been working on for many hundreds of years (and especially since the 30s and 40s). The past 70 years has seen huge advancements in our abilities to analyse these systems... but we're not much closer at describing why they work the way they do... the best answer I've heard so far is "because they do"! (Think about that for a moment before dismissing it as a witty one-liner).

Cheers,

Timkin

I draw a paralell to AI here.

If you replace 'complex adaptive systems' with AI, the text makes sence to me, as an master student in the field of AI.

The prolem is that nobody realy know what intelligence is, and as the text sais, most AI is simplifications of the real world, simplified to a point where we _can_ understand it (Our math tools are limiting the systems complexity).

There are different opinions on what is the right way to go, and there are no standard notations / rules that apply to AI over the algorithem level. (or in wery spesific cases like ANN)

So are these the same? Is this what theoretical AI people have been working on since the 70es? If i could express my little system in a standarised notation, and the apply all time valid mathematical tools to this, to understand it better, then scaleability would explode, and incredible complex systems could be created, just as increadible complex simulations or matrix operations or analysis is possible because of the definitions, the rulse, and their proofs...

now, what is your take on this? :-)

Quote: Original post by Timkin
Wolfram has been trying to develop his version of this new kind of mathematics for 20 years, with the recent release of his volumous tome "A new kind of science" (iirc). Basically it's his take on automata theory. (one sentence cannot do it justice though... read the book if you have a year to spare and want to form your own opinion of his ideas/results).

I agree with Brooks though; we need a fundamental mathematical language for describing complex adaptive systems and their dynamic bahviour. In addition to this though, we also need to understand how to analyse this behaviour, which is really what science and mathematics has been working on for many hundreds of years (and especially since the 30s and 40s). The past 70 years has seen huge advancements in our abilities to analyse these systems... but we're not much closer at describing why they work the way they do... the best answer I've heard so far is "because they do"! (Think about that for a moment before dismissing it as a witty one-liner).

Cheers,

Timkin

I was going to make this my secondary research project for my Research Computer Science class. My approach is to study how data flows recursively through the network, and try determine a calculus to study the instantaneous change of the network. I'm creating a nice proposal right now on what exactly I want to do. I'll put it up sometime tomorrow.

I don't know much about advanced Math, all I have is high-school in my bag of tricks...

...but how can this be done? If each individual node in a system has an internal set of variables, and is itself a variable, the correct formula to extract information about that system is another function with the same number of variables, so in other words, if you want to know what Adam's brain will do next, you need a second atomicly-cloned Adam's Brain.

Any other solutions will present deviations that after "n" iterations will have deviated too far off. Now that I think of it it's a bit like weather-simulation.

We can extract generalistic data from the system and be somewhat sure that it holds true for the next "n" time units, the same way we can say a hurricane is going to hit an island in the next 3 days, but it will never cease to be a "statistical" aproach to it, you'll allways have to say "there is a 94% chance of that happening", but I don't think we'll ever develop the math to write something like:
sim(AdamBrain) == Hunger?

...unless I'm not reading this correctly. Interesting subject none the less.

I'll also point out that "complex adaptive systems" theory actually boils down to the ability to "solve" and analyze nonlinear equations. Humans have been trying to do this for a long, long time and our current methods are only qualitative or numerical, yielding insight but no real ability to design new systems with desirable properties. These nonlinear systems come up in every field -- from a simple pendulum swinging (one of the very few solvable ones) to the analysis of neural networks. The fact is that our world is not linear, particularly not the brain, yet humans have only been able to survive thus far with mathematics by approximating nonlinear systems with linear models. So yes, the current math is very difficult to understand and to perform, but it may be a very long time before the "right" patterns are found to simplify it all.

The problem is that math tools we have right now can solve the problems in numerically, but not analytically. So we have a lot of variables, and solve those in simulation takes just extrimely time.

Another problem is that we know so little bit. So we have to use to general level formula.

How do you guys feel about the whole "so simple it can be taught in a HS curriculem." Now maybe its just me, but I don't think Group Theory, Topology, or differential calculus is taught to the average HS'er(ok... the latter is coming into play for sure).

I have a nice ppt I'll put up later tonight explaining my approach to this whole problem.

Let me first say that I don't fully understand what analysis Brooks wants to perform. Despite this fact, I'll give my reaction to the topic.

I think the idea is ill-founded. While some may view this as closed-minded, I don't forsee the conception of a new type of mathematics that will suddenly make analysis of such complex systems a simple task. The fact of the matter is that complex systems are just that. Trying to analyze or interpret the state of the entire internet or all of the cells in a creatures body, to me seems like an misguided exercise. It is not the entire state that matters, but each entities dynamic role in a system's operation that is key. I disagree with his statement that the operation of "our human-constructed Internet -- is in some ways mysterious". Such a system, despite its amazing and beautiful complexity, is the result of a huge number of small, simple entities that have come to a concensus about a standard means of operating and communicating. The cells in a creatures body are fundamentally no different: each performs a simple task and due to extreme levels of standardization, extremely complex system behavior can result. Like I said, I don't exactly understand what type of analysis Brooks would like to perform, but I seriously doubt that a new branch of mathematics will suddendly simplify a human's ability to understand the interactions that take place in a system with vast numbers of entities. The key is understanding the standards that make the internet, and more impressively a creatures body, work the way it does.

I am not a student of advanced mathematics so maybe none of this makes any sense and I sound like an idiot...

Quote: Original post by janoside
...It is not the entire state that matters, but each entities dynamic role in a system's operation that is key. I disagree with his statement that the operation of "our human-constructed Internet -- is in some ways mysterious". Such a system, despite its amazing and beautiful complexity, is the result of a huge number of small, simple entities that have come to a concensus about a standard means of operating and communicating. The cells in a creatures body are fundamentally no different: each performs a simple task and due to extreme levels of standardization, extremely complex system behavior can result....

Yeah, Brooks is hoping for some way to go from each individual's simple rule set to the resulting aggregate behavior. Even more ambitious, to go from some desired aggregate behavior to a simple rule set that would produce that behavior given enough individuals.

Personally, I believe Brooks won't get his wish-- there's little (no?) evidence of natural systems that can *easily* design simple rules for some desired complex aggregate behavior. Most of the natural systems typically used as examples (epigenetic development, neural networks, ecosystems, etc.) took millions of years of incremental search (evolution) to get there. *Maybe* we can design some representation that could quickly do the same kind of work, but I doubt it.