[urll=“http://abcnews.go.com/sections/business/SiliconInsider/SiliconInsider_020528.html”]Thoughts?
Ugggh. Mods? A little help.
Beats me. There’s not nearly enough detail in that article for me to form an opinion. I’d probably have to read his book to even have a chance of forming an opinion.
I will say that my gut reaction is skeptical but I’m not certain I could say why. I will admit that since it seems his theory isn’t being cast down outright that there is probably something interesting in there.
Till I read it or hear a bunch from those who have I won’t know but I look forward to seeing this develop.
I tried to find a copy of his book this weekend so I could flip through it, but it was sold out at both bookstores I checked. I’m only marginally interested, so I haven’t pursued it.
His thesis as I understand it (you can reproduce natural patterns with simple cellular automata algorithms) sounds like it’s great for sound bites, but not useful for much else. Simulation is a powerful tool, but it’s not the same as true predictive theory. I’ve heard nothing that makes me believe he has discovered anything new, apart from how to write a successful popular science book. People who really want to change science go through peer-reviewed journals, not Borders and ABC news.
I am interested to hear more details about the book, though, just to see what his thesis actually is.
His thesis is not that cellular automata algorithms are at the root of all things; it’s that computation, not mathematics, is at the root of all things. Cellular automata merely provide the best examples of what he’s talking about: complex behaviour from simple, iterative, rule-based programs. In other words, scientists shouldn’t be looking for equations, they should be looking for algorithms, and short ones at that.
It’s not a particularly breathtaking thesis, until you realize that mathematics is a shortcut to computation, obtaining results without all the work. If Wolfram is right (and it’s a big if), then for any phenomena more complex than the simplest class, there is no accurate means of prediction and control. Only a full-length simulation will do.
Mathematics is a shortcut to computation? Umm… No. Very small subsections of mathematics are a shortcut to computation. The large majority of maths is too ‘big’ for computation to touch: There are only countably many possible computer programs, and yet there are uncountably many functions from the natural numbers to themselves to take the simplest example. You can’t do computation on the real numbers either at there uncountably many.
My take on it is that computation simply isn’t sufficiently interesting to provide a theory of everything. You may be able to use them to get arbitrarily good approximations to one - A countable set can be dense in an uncountable one - but no better than that.
Of course, this is all opinion. Also, I’m a mathematician so I may be biased. 
For the record, I do think computation is an interesting subject. It’s just not interesting enough to solve everything.
kitarak,
Help me out here. Aren’t the fields of nonlinear dynamics (including chaos theory, complexity theory, neural networks, etc) all part of mathematics? Isn’t the point really only that a more accurate model of reality must not ignore that the universe is a nonlinear beast?
Cool. This is getting interesting quick (even if it’s out of my league - I flunked every math course after Algebra 2 - my guidance counselor said it was due more to laziness than ability, but the result is the same). I’m going to have to give this book a read. Sounds like he makes it approachable to the layman.
DaLovin’ Dj
There’s already a thread about this topic here in Great Debates:
http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=116787
I’m only a few hundred pages into the book, and this is Wolfram’s opinion, not mine, so be a little patient with my attempt to argue from his perspective.
Mathematics is a shortcut to computation insofar as science depends upon mathematical equations to provide shortcuts to prediction and control of scientifically tractable systems. Scientists look for neat little equations that describe relationships, and use those equations to extrapolate from data to predictions (which permits control).
If Wolfram is correct that computation is a superior means of modelling systems to mathematics, it means that mathematics no longer provides a shortcut–ultimately, full length simulation is the only means of modelling a complex system, and an approximate means, at that. Complex systems are “computationally irreducible”, meaning that they can’t be simplified to mathematical (or algorithmic) abstractions that accurately predict arbitrary parts of the system.
Wolfram is not suggesting that all mathematics is a shortcut to computation. He’s concerned with the role of mathematics in the formulation of scientific theories. Computation is proposed as a generalization of mathematics in that area.
tracer is right. If the Mods would like to close this one, I’m glad to defer to the already established thread.
DaLovin’ Dj
The OP cited the following ABCNews.com article:
http://abcnews.go.com/sections/business/SiliconInsider/SiliconInsider_020528.html
… which I must take with a huge grain of salt, because the reporter made the outlandish claim that Stephen Jay Gould was a fan of the Boston Red Sox. (Gould has always been a fan of the NY Yankees.)
hansel, does he give any specific examples of how computation can make predictions which go beyond a mathematical description?
In my experience, simulation is primarily useful for numerically solving equations that are too difficult to solve analytically. For example, I can describe a box of electrons exactly using the Schrodinger equation. However, to calculate the many-body wave function for more than a few electrons would take forever, just using pencil and paper, unless you make approximations. Instead, we use computers to solve the problem numerically, allowing us to calculate properties of systems that are far too complicated for analytic theory. In this case, though, the computer is not coming up with any new description of the system – it is just a tool to solve the mathematical equations which already fully describe the system.
Cellular automata, as I understand it, is no different. You establish some criteria for a “good” system (e.g. low energy), and then you essentially sample many different configurations by letting the system evolve and interact. This allows you to find configurations that you would never guess from the mathematical equations which describe how the particles interact. That doesn’t mean that the computational method is somehow beyond the mathematical description – your answer is still based on how the system interacts and your goodness criterion (e.g. the total system energy). The fact that computers are powerful tools which can go far beyond pencil and paper math in solving equations and/or searching phase space, doesn’t make them fundamentally any different than, say, an integral in how we describe systems.
This is not to refute your point, by the way, but just to let you know where I’m coming from. Does Wolfram address how his algorithmic descriptions are somehow different than the underlying mathematics?
I could be wrong, but he seems to be claiming that no (accurate) predictions are possible for complex phenomena, just because the phenomena are computationally irreducible. You can run the simulations several times, and presumably find general rules or hard limits from them, but prediction will get no better.
The trade-off is that, by looking for computations, rather than equations, you can find better simulations in a stronger theoretical framework that better model complex systems.
I think you may have hit on his point right here: simulation is useful for solving equations that are too difficult to solve analytically. Wolfram is interested in the stronger thesis that computational simulation is closer to reality than analytical methods can get us, and this entails certain things about science.
Be careful, here. Cellular automata are merely the best example of what he’s talking about, and he’s careful to demonstrate that cellular automata are not unique–in fact, he spends about 60 pages generalizing automata to computational systems in a way that removes all properties specific to cellular automata (parallel evaluation, etc.), just to prove that cellular automata aren’t at the root.
I’m not sure he’s saying that it does go beyond–after all, his cellular automata and other systems were all done from inside Mathematica. He states that computation of this sort is a generalization of mathematics, not a replacement.
The key points seems to be one that you make: finding configurations the equations don’t suggest; moreover, simulating things that are not analytically tractable. To put it another way, complex systems are not mathematically reducible, so we are limited to computational simulation–but with computational simulation, we can reduce the phenomena to simple computational rules to describe the system as the result of an algorithm’s operations (where we can’t reduce it to an equation). This doesn’t allow better prediction, but it does make complex systems easier to handle, and to view as a class of similar phenomena.
But all simulations (that I know of) are based on equations. The equations define the system. The method you use to solve the equations, whether it is a computer program or an integral is irrelevant. If you have the correct equation to describe the system, you will get the right answer. If you don’t, you will not. Our physical understanding comes from being able to write down the equations correctly. Our ability to make predictions comes from being able to solve the equations. The latter is a practical problem, not a scientific problem, in my opinion.
The equations that we write down are based on experimentally-verified observations (e.g. how charged particles interact) which have a simple, consistent form (e.g. 1/r). Is he saying that there are systems whose rules can’t be expressed analytically, and hence by comparing the output of certain algorithms with experimental phenomenon, you can describe the rules of a system by comparing the two? Or is he saying that the solutions of some equations can’t be found analytically, and that describing the algorithms used to solve the equations is somehow meaningful?
**
Here it sounds like you think he is saying the latter. If he is, then I don’t see what is new or even interesting about his thesis.
**
Again, I don’t see how this last point is revolutionary, but then again, I may not understand what he is saying.
Thanks for the information, by the way. I appreciate your taking the time to describe the book. What is your take on his ideas?
kitarak said:
Is this true? Can’t one use a variation of Cantor’s diagonalization to count these functions?
Isn’t it Cantor’s diagonalization that proves the functions are uncountable?
hansel
Since you’re the only one here who seems to have read at least part of the book.
What seperates Wolfram’s new work from Chaos/complexity theory?
Everything seems textbook so far, except that he’s done some research on cellular automata that’s seems like a breakthrough.
I’ve started reading Kurzweil’s “challenge” to Wolfram. While it beats reading 1200 pages(!) on cellular automata it’s has annoying font formatting that makes it a pain to read.
Kurzweil is pretty underimpressed by the “new kind of science” part, although he thinks Wolfram is on to something.
Oh,BTW, is it a decent read?
I’m not as up on my math as I should be, but from Wolfram’s examples, he’s examining algorithmic simulations that aren’t based on equations (or, if you want to view them as equations, are so ludicrously simple as to not qualify as interesting math). His cellular automata are unidirectional iterations of simple comparison rules to determine whether or not each cell is black or white. At any rate, his automata are just as deterministic as equations. You should look at them yourself to decide.
An impression I’ve gotten so far is that the complexity Wolfram is demonstrating is orthogonal to the method generating it. This isn’t a case of obtaining a result–it’s a matter of observing complex patterns that arise as part of the representation of the automata. That’s not to say he’s finding pictures in clouds, but that the incidentals of the method are less important to the end result than the observable complexity of the result.
I would say the former. He’s pretty explicitly claiming that complex phenomena can’t be described analytically (i.e., as an equation yielding arbitrarily accurate results).
To be honest, I’m a bit underwhelmed at this point. He’s certainly demonstrated that surprisingly complex behaviour can result from very simple, rule-based systems–basically, very complex patterns that are apparently perfectly random, or with very complicated nesting or repetition, that are the representation of the output from an automata. He’s not hand-waving, either: the book is excellent for presenting graphical evidence of what Wolfram is pointing at, and you can see exactly what’s got him so excited. What I can’t decide yet is whether or not I should be excited. Wolfram is thorough to a fault (though in fairness, he has to be if he’s claiming to rewrite the book of science), and I’m waiting for the later chapters where he draws the connections between the complexity he’s found and the real world. Flipping through the book, there are pictures comparing the patterns found in things like leaves and nuclear explosions, and the patterns he can create with his automata, and there are definite similarities that are suggestive. What I haven’t gotten to yet is a connection between the two that isn’t incidental.
It’s plausible to me that he’s on to something. I have no conceptual problem with the idea that simple algorithms are responsible for the apparent complexity of the world, and that it’s impossible to capture that observed complexity analytically with any efficiency or effectiveness (though I haven’t accepted the idea yet). While Kitarik is suspicious because he’s a mathematician, I’m in computer science, so perhaps I’m predisposed to his ideas :). Basically, I’m waiting for the proof.
I can’t answer the first question, except to say from Wolfram’s introduction that both chaos and complexity theory are narrow cases of what he’s proposing. In the first chapter he lists a bunch of fields, from physics to philosophy, claiming his idea extends, subsumes, or solves traditionally intractable problems in all of them.
As for whether it’s a good read, yes, it’s a very good read. The prose is very clear, very accessible, and Wolfram is good expository writer. The text is extremely polished, and you won’t have a problem turning the pages. BTW, of the 1200 pages, the last 400 are notes, so it’s really not so long a book, and the font isn’t tiny, either. I’d recommend it.
I’m taking notes on it as I work through it, and posting them on my site. As soon as I have a significant amount, I’ll post a link here.