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The Universe in 4 Lines of Code?
serendigital writes "Stephen Wolfram, founder of Wolfram Research and creator of Mathematica has, after 10 years+ finished his book, "A New Kind of Science." In a "Wired" article entitled: The Man Who Cracked The Code to Everything ...," Steven Levy talks about how and why the book was written and more importantly, what it is about. The best part of the article is in this exchange: 'I've got to ask you,' I say. 'How long do you envision this rule of the universe to be?' ... 'I don't know. In Mathematica, for example, perhaps three, four lines of code.'" This book seems a little... nutty. But it's been submitted a bunch of times. If anyone wants to review it, go right ahead.
The book sounds superficially like David Deutsch's "The Fabric of Reality", which tries to try everything together using a computational theory of reality + the multiverse intrepretation of quantum mechanics.
Deutsch believes that the simulation of something at a deep enough level is entirely equivalent to the real thing -- which is another way of stating this authors belief that reality is just an algorithm. I personally think it's at least as good a metaphysics as anything else I've read...
Websurfing done Right! StumbleUpon
augment your senses: http://sensebridge.net/
-
"If anyone wants to review it, go right ahead."
Ouch... It'll be a while before any reviews get submitted, Michael -- it's HUGE! (Page 2 of the article: "At 1,280 pages, the book pushes the limit of what can be physically bound between two covers.") Levy talks about it dwarfing (!!!) a phone book... though it would depend on what phone book you're trying to dwarf.Wolfram's demands regarding publishing are interesting -- the book is going to cost $12 to actually produce (5x to 6x that of a "normal" book, though the extra size certainly has to be a factor!), and be priced at $45 -- it includes large quantities of high-rez graphics. Also, it went through alphas and betas, like software -- not versions or revisions as writers are familiar with.
Definitely something I'm going to read... although I doubt I'll achieve full comprehension. The "A New Kind of Science Explorer" software should be fun to play with -- but will I have to wait another 10 years for that?
"...America's great minds of today, teaching America's great minds of tomorrow. Poor bastards." -- A Beautiful Min
Ray Kurzweil, the inventor, AI theorist, and author of The Age of Spiritual Machines, has a long review of the book available here.
One of the key points of the review is that while Kurzweil agrees that certain levels of complexity can be achieved, higher levels of complexity are simply not derivable from cellular automaton, the generator of Wolfram's complexities.
To quote Kurzweil: There is a missing link here in how one gets from the interesting, but ultimately routine patterns of a cellular automaton to the complexity of persisting structures that demonstrate higher levels of intelligence. For example, these class 4 patterns are not capable of solving interesting problems, and no amount of iteration moves them closer to doing so.
Quantum Mechanics has already suggested that both space and time are discrete on small scales, and I believe there is quite a bit of indirect evidence to support this claim. The discreteness is based on Planck's constant, and the unit length and unit time are approx.10^-33 cm and 10^-43 sec respectively (which if you do the math are equivalent if you equate 1 year and one 1 light year). All lengths of space or time are either multiples or one over a multiple of length.
The claim that they are structured as a network of nodes is certainly speculation, but it is at least a logical speculation. The points would probably have to be connected somehow.
As a letter writer to Salon points out, it seems that Wolfram thinks that he's discovered Complexity Theory all by himself. The Salon article certainly gives that impression -- not having read the book, I can't make my own judgment.
The Salon writer writes as if cellular automata were some silly mathematical curiosity (or worse, the writer thinks that CA is recent to computing) that Wolfram "rediscovered" and took seriously for the first time. Of course that's absurd.
The Santa Fe Institute was founded jointly around 1984 by the eminent Nobel Laureate, physicist Murray Gell-Mann, and several others. Stuart Kauffman has researched and written on complexity for many years.
I myself have been following, as a layperson, complexity theory for about fifteen years. In 1991 I had the opportunity to be an undergraduate intern -- an opportunity I didn't follow up on because of my severe academic workload, but an opportunity I will always regret not taking advantage of. Undergraduate intern positions are much more competitive now. This eleven years has made the difference between "bleeding edge" and "cutting edge". Or perhaps complexity theory is even mainstream. I've noticed a burgeoning graduate school interest in complexity studies programs.
Complexity theory intersects many disciplines, and it involves several related ideas such as chaos theory, modeling, self-reference, artificial life, and others. It's evolved into a fairly rigorous discipline, and the more formalized idea of "complex adaptive systems" forms the core. For those who have read Douglas Hofstadter's book, Godel, Escher, Bach, (a very influential book for many of us) published around '82, many of these ideas will be familiar.
Wolfram's quip that seems so risible is really only an overstatement of the central idea of complexity theory: that a limited number of "rules" can give rise to extremely complex behavior. This was the surprise of cellular automata, exemplified by Conway's "Life", invented in 1970. But the underlying idea goes as far back as John von Neumann. Wolfram has done some interesting work in CA. But it sure as hell isn't his idea. For many in the Slashdot community, this is all as familiar as the back of their hands. But apparently there's still a lot of people that should be aware of this stuff that are not.
Finally, many people here would probably be interested to know that SimCity's designer, and Maxis, have had some association with SFI. This makes sense because the emergent behaviors of complex systems are not (as a practical matter) deductively predictable -- their behavior must be studied. The techniques of systems modeling are requisite. SimCity was the general public's first accessible insight into just how fascinating and educational systems modeling can be.
Well, we start with a differential equation, in particular, a partial different equation (pde): A pde is an equation that describes how a quantity changes with respect to several variables (which we will take to be time and space). Imagine when someone farts in a corner of a room. We want to describe how the concentration of farted gas (the quantity we are interested in) changes when time advances, as well as how the concentration of farted gas changes with space. Using molecular dynamics arguments, we can write down an equation
dc/dt = D (d^2c/dx^2 + d^2c/dy^2 + d^2c/dz^2)
where c is the concentration of farted gas, and t represents time and (x,y,z) represent three-dimensional space. The actual form of the equation is not important (but it is the diffusion equation in case you are interested). The point to note here is that we have written down a pde for c as a function of t and (x,y,z). We can then proceed to solve for c at any t and (x,y,z) that we are interested in, using techniques from calculus. This, in a nutshell, is the basis of many equations of physics -- Newton's, Maxwell's, Schrodinger's, and Einstein's equations are all pde's.
Now, imagine a discretized version of a pde, in which time t, space (x,y,z), and the quantity itself c, are all discretized. Discretized in the sense that they take discrete values, i.e., we measure time in "time steps" t=1,2,3,etc. and space in "space units" x=1,2,3,etc. and the quantity c in, for example, "smelly", "moderate", "not too smelly", etc. Then the discretized version of a pde is a cellular automaton.
By considering only two dimensions (one time and one space), and by explicitly enumerating all possible rules that one can get, Wolfram found that there are several automata that cen generate extermely complicated behavior.
Now, what his book seem to be proposing is that, by moving away from the calculus of a pde, and venture instead into discrete space, he seems to have uncovered a profund law governing all cellular automata. This in itself is a cool result! However, add that to his belief that everything (including the universe) is a cellular automaton, and people get less enthusiastic. Anways, hope this brief treatise on cellular automata helps!
The concept is deceptively simple. Every interaction in the universe can be reduced to a series of mathmatical equations of iteration that can be represented in two dimensional space. The clustering of solution follow extrememly simple rules, that even a child could learn in a few minutes. The reprocussions if this is proven to be true would be nothing short of revolutionary. Imagine the paradigm shift when the world finally realized that the earth revolved around the sun... this beats that by a factor of 100.
;-)
And he's got lots of hard data to back up his claim. Sampling from dozens of sciences, he shows the same patterns emerging over and over again. It's stunning to see some of the work because it becomes intuative after only a few examles and you can see the patterns in so many different places.
So either he's a complete nut, who has taken something that's absurdely simple and mis-applied it to all the major scientific endeavours, or he's a certifiable genius who has just opened the window to understand the universe in the most basic of ways.
I'll let you know after I read the book.
Wolfram's first CA book (the collection of his papers) is out of print but available for download at http://www.stephenwolfram.com/publications/books/c a-reprint/
A good place to get started with complexity theory is the book Computational Complexity by Papadimitriou, if you're interested. The definitions of a "complex system" are given in this book, and they have nothing to do with analogies of our experience of being human. Complexity is a mathematical object.
By the way, one of the open problems in complexity theory is the famous P=NP problem, and if you solve it, you will win $1 million.
The book
a ms/
http://www.wolframscience.com/
The downloadable code (4 lines, I suppose)
http://www.wolframscience.com/nks/progr
Stephen Wolfram
http://www.stephenwolfram.com/about-sw/
You seem to be unfamiliar with mathematical proofs. Grinding through many cases does not a valid proof make. In order to prove a theorem, you have to verify its validity for ALL cases, and in order to disprove a theorem, you only have to find one case where it is not valid. Just because you ran your theorem on a supercomputer for three months does not mean you have proved its validity for all cases. Example: You are trying to prove some theorem, and you use only positive integers. The supercomputer runs for a year and finds no holes in your theorem. Then your girlfriend comes over and enters -1, and your supercomputer barfs at you.
You seem to be unfamiliar the concept of proof by cases. A proof by cases is valid if and only if the cases are exhaustive. For example, if you prove something for all even numbers and all odd numbers, you have proven it for all integers. The proof of the Four Color Theorem broke the problem, or some lemma used in the problem, into around 1000 cases. The cases were exhaustive, or it would not have been a proof. Some curmudgeons didn't like the fact that the cases were checked by computer.
'What will other people think?' After a while I realized, 'Why am I really doing this? Is it really worth my while to spend 10 years of my life doing something to get other people to say positive things about it?' No, it's not. Absolutely not. And actually, from some very cynical point of view, my opinion of the world at large isn't high enough for me really to be interested in what they have to say."
Arrogant? Maybe. But you need the full quote to bring perspective to this issue. He spent 10 years of his life not to please people, but to do the right thing.
It doesn't matter if people think it's wrong or right. What it matters to him is being right (in the objective sense and not the subjective sense). So he DOES care. But his motivation is not "acceptance" biased. That is a good thing.
I have always found economics to be a stagnated field. By? Because you can only try to "extend" or complicate the "orthodoxial" core. Everything else will be filed to the trashcan without further analisis.
What this guy is doing is the way to go. If nobody believes in you, then you need selffinancing. That means you need to reach 3 achievements:
1) Be a genius (natural)
2) Make money (luck)
3) Think different and question mainstream if need be
That's hard. And it's really noteworthy that someone has met the requirements. I wish I could make some money so that I can begin to work in the way I believe (as oposed to the way "to please other people, so i get food").
Thanks!
unfinished: (adj.)
Anyway, it's quite clearly a romantic book. Romance novel, even.
-jon
Remember Amalek.
exactly. the reason why each of these cases required so much computer power, is because many graph theory proofs require exhaustive manipulation of abstracts such as vertices / edges / regions and in this case the coloring, X(G). the cases not only are exhaustive but most importantly categorize a graph into different types. Each of these types has a proof associated with it that makes the case valid. I am assuming that the cases are fairly similar because otherwise a computer could not prove them. Due to their similarity, the computer can grind away and prove all the somewhat similar but somewhat differing cases one by one.
QED
BSD is for people who love UNIX. Linux is for those who hate Microsoft.
To his suprise, it started doing completely different things than it did before. It turned out that the printout rounded the numbers. Only a few digits were missing, but that was enough.
What's the moral? Even if you know every detail about how a system works, you can't always predict it; the accuracy of the measurements matters. It's the same with the real weather: The biggest problem is knowing the situation we are in now; I once read that even if we had sensors in every square foot of the atmosphere, we would not be able to predict more than a few weeks. Not because of our model, but because even that isn't accurate enough.
By the way, even much simpler systems have this sort of behaviour. For example, take the function f(x)=3.8*x*(1-x). There's a value of x such that f(x)=f(f(x))=f(f(f(x))) and so on, meaning that if we iterate the function, this value of x is a fixed point. If we do it on a calculator, we find that it jumps away from this value after a few hundred iterations, just because of rounding error! Think about it... being within about 10^-10 (it depends on the particular calculator, of course) isn't enough!
"But really, I think life is just a game of Mao Nomic." -Purplebob
This is silly. The universe is far too simple to be explained by mathematics
Actually the book has more to do with cellular automatons than with mathematics,
although, arguably, you could describe cellular automatons using link theory (which is a theory of structure, logic and math, and Wolfram's automatons are specially well suited for it) and with more classic mathematical tools.
Here is my little biased review (biased because I have a take in that kind of stuff, only more mind related).
I wont reiterate the claims of the book because you can find all sorts of review that do that (oh wait, now that I reread this it appears I'm doing just that later, oh well, still not bad an intro, heh), suffice to say, this book could become the "Bible of Reductionism" for many generations of scientists to come. I do not use the word Bible trivially here, this book is about belief, and that is the biggest problem anybody will have with it. You can agree or disagree with Wolfram as to wether or not the boradness of his conclusions will hold up to scrutiny, but the transfer of those conclusions to to the real world is a completely different step. It is a matter of belief.
If you torture data sufficiently, it will confess to almost anything. (Fred Menger, Emory University Organic Chemist)
Nobody is immune to this mistake, a good part of the field of artificial intelligence research is faulty of the same (I myself do it often, but I don't publish), it is the reason why connectionism as a paradigm was so succesful among the community even if it still has to deliver on some of its most basic promises.
In a nutshell, Wolfram found a set of simple rules for cellular automatons that lead to complex behavior. The second part of his discovery is the principle of computational equivalence, again, summed up, it means that passed a 'threshold' (more or less), two computational processes can be regarded as equally complex. This is a BIG claim, one that will be investigated thoroughly by mathematicians. But the point is that if it holds, you have explained many things : randomess, free will, and you have put in terms that are all but vague what it means for connectionism to cross the threshold of self awareness (in a broad sense).
How, you aks, can he do that with cellular automatons ? Simple once you drop the concept of linear time. What he realized along with many other researchers (and I'll grab the opportunity to pat myself in the back and include myself in that group), is that time is a poorly defined concept today, until you dive into quantum physics when it starts to make sense. What is needed is to redefine causality. Again in a nutshell, classical causality says that an effect always follows a cause, but that is a definition that itself includes time, and since causality is supposed to define the arrow of time, this definition is not acceptable.
The new definition becomes "an effect always has a cause", now you can immediately see that the idea of causal directionality has been removed, but that doesnt mean that time flows backward, just like things didnt start falling up once Newton realized up and down were foolih concepts. Shortly put both future and past exert constraints on a local event (think about Marov states in the future and in the past). When equally balanced, those consraints map to classical quantum physics.
So Wolfram's cellular automatons integrate that concept, you can link events to cells that are in the same discrete time slice as your event. You can link to events in the past, or (like in classical physics), link to events in the future. That itself assumes that time is a discrete phenomenon, it is again a BIG assumption, it is a statement of Wolfram's belief (he uses that word) that time in the physical universe IS indeed discrete, and that thus, his discoveries about causal networks map directly to our world. Lets make it clear here that if he is wrong, then none of these claims map to the physical universe, and the book is just about having fun (a lot of it, tho) with computers and the concept of time (now of course that in itself could be very useful for quantum computing).
And then he goes on to describe how you can then use this stuff to make elemetary particles, or even space-time itself.
All in all this is genius stuff, if not completely revolutionary. I would describe it as the Game of Life meets Link Theory. It is a brilliant reformulation of Link Theory in terms of cellular automatons, and since Link Theory is a bit hard to work on, an easy way to use it with computers is extremely welcomed. For my part, I cannot wait for a version of Mathematica that integrates non-linear time processes. My own neural net models would become that much easy to write as I wouldn't have to deal with C++ journaling memory templates, and once quantum computers are out, I can just run the thing and not wait an arbitrary long time.
But again the flaw is one that we often make, if usually not that publicly: we start to believe in our stuff. Yes, it could work that way, but everything here is the result of a computer experiment, and that is the hard truth of it. It is a beautiful theory, easy to understand, even for the non scientist, but its predictions are minimal, distinguishing it from a physical model of reality in order to test it is going to be a hard task.
Arguably connectionism's biggest problem is that its promises are quite vague, and thus, it is hardly disprovable as a paradigm, and the same problem applies to Wolfram's work, it is very apealing, but things are explained in very tiny details or in broad strokes. There is no equation that will tell you the bigger picture because there is no bigger picture, the world is a soup of events, and as apealing as this might be, as natural as the patterns the simulation generated seems to be, this does not mean that the physical world is actually operating like this.
Even going further, it is worthless as a replacement for 'bigger laws', laws that supervene other laws, gaz propagation can be predicted by such laws, but Wolfram's laws are too tiny, their nature is to lead to chaos and non predictibility, to actually generate the supervenient laws, but again, predictive power is non existant or lower than current science.
But again, this will not prevent many from holding this book quasi religiously, even unknowingly (as many people do today with broad connectionism), because it is simple, elegant, and accounts for a lot, or so it seems (but again, some people think that the pyramids were built by aliens because they think it's simple, elegant, and explains a lot). This book will be about belief, in the next decades and centuries, it will be held as the Bible of Reductionism, because it provides the self consistent argument some philosophers like Dennett needed to explain away consciousness as a pure illusion.
This is my second problem with this book, Wolfram basically says he is presenting us with a theory of everything, but there is not much about perception, qualias, and more generally, the phenomenal aspects of consciousness. Wolfram, as the Priest of Reductionists I think he is going to become, simply leaves the matter out, talking about perception in terms of representational spaces (even if not in quite those terms), but the phenomenal aspect of those spaces is let out, as if we actually were Chalmers' zombies.
To conclude, this will be a delightful read for most slashdoters, at least, all of those with a scientific 'way of life' (no strong backround needed), they will see it as the crystalization of their materialistic views. Religious people might have a problem with this book as it depicts us as automatons, literally.
And then there are people like me, lost between the duality of phenomena and matter and the universe being-causally-closed-sad-state-of-affair. To us, sometimes known as naturalistic dualists (qualia as part of natural laws), the strong deterministic framework that Wolfram imposes seems to point to a strong epiphenomenalism for consciousness, where other theories based on quantum indeterminacy (and quantum theory has been throughly tested for 60+ years) do open possibilities of weak epiphenomenalism. In a few words, I'm not completely convinced by Wolfram's version of free will.
I'm a bit more than two third into the book, reading it quickly at first to grasp the feel of it, and then to read it slowly a second time, so it is possible that some of the things I have said may not be fair, and for this I apologize in advance.
I'm loving every part of it, and if you feel my remarks are too harsh, just assume that I'm jealous I didn't write it. If anything this will make mentioning reverse causation much easier in academia without being laughed at, and Link Theory is going to get a huge boost. Having made 4 computer languages already, I plan to have my fifth be able to run reverse causation in typical link theory problems or simulate my causal backpropagation neural network model. If I can use some of Wolfram's formalism to help this task and if he has cleared up the mess with causality, or helped people make the distinction between predictability and determinism for the rest of us too, then I'll be eternally grateful.
lone, dfx.
http://www.causaergsum.net/
WTF? What is your mathematical background to say this? (..) At worst, you have to solve sets of differential equations (..)
To say that we don't have an equation is either obtuse, naive, or a deliberate troll.
Both of you are imprecise. The first poster complained that there is no analytic solution. Which is true. The second poster counterargumented that it is easy to solve by some iterative procedure. Whic is true as well but misses the point of the first poster.
What we deal with here is symbolic integration. Derivation (finding the f' for a given function)is simple because there are easy rules that yield the formulas of derivatives, integration (finding a function f for a given f') is an art because we quickly end up with formulas that can't be simplified with the usual set of elementary functions and we are stuck with the integrals (which might be used to define functions, like erf). Look for Liouville's theorem to see how stuff like this is proved rigorously.
The more general problem is solving differential equations, systems of differential equations both in one or several changing variables.
Most physical laws tell you how to assemble the set of differential equations. Writing down the newtonian forces for the planets is exactly that.
Solving these systems of differential equation is again called integration.
What turns out is that you can't write down the solution to the three body problem in general as some simple combination of elementary formulas. It is not much different from the one dimensional integration case. No magic. Just that you can't write down the solution in a simple closed form. The one who proved that was Henri Poincare in his celestial mechanics treatise by the way.
It just means that the space of all solutions we can construct by assembling the usual cast of simple functions we employ is not large enough to hold every function which is singled out by the solution space of a differential equation.
The first was poster wrong in that he doesn't understand that the set of differential equations plus conditions is the precise description (if we neglegt general relativty and quantum effects :)
and that solutions are necessary of approximative nature if we don't want to extend our basic set of functions by lots of integral functions.
The second poster is wrong in labeling the first poster a troll, because he didn't understand his concern about closed solutions.
Regards, Marc