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In the article, Dr. Wolfram claims that computer programs, not mathematics, is the best way to model the universe. However, hasn't Lambda Calculus (and therefo

by derekp7 15y ago

In the article, Dr. Wolfram claims that computer programs, not mathematics, is the best way to model the universe. However, hasn't Lambda Calculus (and therefore functional programming) shown that the two are the same?

knowtheory 15y ago
Well, Wolfram's thesis is that analytical representation is not the correct frame for modeling the world, and algorithmic/procedural representation (e.g. simulation) is the correct method. This isn't particularly novel, but it is a key insight of computer science. I just don't think that Wolfram deserves credit for it. Edit for clarification: The key insight being that you can model things algorithmically. Whether it's "correct" or not is a different matter. But this type of modeling is different from analytical modeling.
- taliesinb 15y ago
  No, this is a mischaracterization of what Wolfram is saying. Wolfram says: why consider computer programs to be valuable as mere approximations to the underlying analytic models, when you can consider computer programs themselves to be an entirely new and unexplored land of complex and interesting models. Traditional computer science considers computer programs to be merely a means to an end. A quite delicious irony is given by the example of Naiver Stokes equations. They are themselves an idealization of the flow of fluids composed of discrete particles. Because of our obsession with continuous models, we mostly resort to laboriously solving them numerically -- but it turns out that simple lattice gas cellular automaton models are actually 1) fairly accurate 2) much more computationally efficient, and 3) more suggestive of the underlying microscale physics. To step back a bit, a helpful analogy would be that simple computations are the 21st century equivalent of differential equations, which were studied rigorously in their own right starting in the 18th century, often prior to their application to concrete problems. Whether this intuition will turn out to be prescient, or whether it will fizzle out, is another question.
  - knowtheory 15y ago
    That doesn't make sense in the context of my comment, and besides, i don't think that's true. If you take the Turing Test to it's logical conclusion, a program that is indistinguishable from a human intelligence is a human intelligence. There's no means to some other end. The program is the intelligence. This is also something that Turing came up with in the 40s & 50s. Not exactly novel.
    - taliesinb 15y ago
      Uh... well, I'm getting Wolfram's take right. So you don't think my analogy is true? It's just an analogy. But it's clear that it has methodological implications for how one does science: For example, if you think computers are just a way to simulate continuous systems, it would not occur to you to sample random programs and see what they do. It would not occur to you to enumerate simple programs. And you wouldn't think that it is very interesting that such and such a simple program can do such a such a computation. If you did, it would. And if you were ambitious enough, you would actually try hunting for the program that computes the universe, as Wolfram has been doing on a cluster in his basement (I love this tidbit) for some years now: http://www.ted.com/talks/stephen_wolfram_computing_a_theory_of_everything.html http://www.ted.com/talks/stephen_wolfram_computing_a_theory_...
      - knowtheory 15y ago
        No, you're making the assertion first, that everyone positively asserts that the universe is continuous, and second that computer scientists see algorithmic systems as being simulations of analytical systems. The first i am agnostic to (although i do like the feynman quote at the top of http://arxiv.org/pdf/quant-ph/0206089v2 http://arxiv.org/pdf/quant-ph/0206089v2 which was linked to above), and the second is most certainly false, as i have indicated above. The evidence in NKS to support wolfram's assertion that the universe is a simple program is circumstantial at best, and so his windmill tilting quests for the program that is the universe seems quixotic at best, and arrogantly foolhardy at worst. Analytical modeling, algorithmic modeling, or whatever other model someone wants to use to represent reality are models until you can prove them to actually be fundamentally connected with the manner in which reality functions.
pnathan 15y ago
Yes. There are multiple models of computation, and many have been demonstrated to be 'equal' (the technical term escapes me). Lambda calculus, turing machines, neural networks, certain cellular automata, and others I can not recall have been shown to have this equivalency. I am not impressed by Dr. Wolfram's arguments. The Amazon Review cited elsewhere in this thread has a very cogent discussion of his points.
primodemus 15y ago
Turing proved the existence of non-computable reals, so the two are not the same. Also Chaitin's constant is a non-computable real number with a precise mathematical definition. See: http://en.wikipedia.org/wiki/Chaitins_constant http://en.wikipedia.org/wiki/Chaitins_constant
- pygy_ 15y ago
  The Curry-Howard isomorphism states (and this has been proved) that for each formal proof in math, there is a corresponding computer program that halts, and reciprocally. This means that it is possible to express the proofs of your statements in code. On the other hand, non-halting programs are AFAIK beyond the reach of formal logic, and as consequence, the set of logical proofs (or more accurately, its isomorphic twin) is a subset of the set of all computer programs.
grinnbearit 15y ago
I've thought about this as well and IMO computer science makes sense as mathematics with a concept of side effects. In that way the universe is more a computer program than a mathematical equation.
maurycy 15y ago
The results you refer to are probably Curry-Howard correspondence, and Curry-Howard-Lambek correspondence.

by derekp7 15y ago

knowtheory 15y ago

Well, Wolfram's thesis is that analytical representation is not the correct frame for modeling the world, and algorithmic/procedural representation (e.g. simulation) is the correct method. This isn't particularly novel, but it is a key insight of computer science. I just don't think that Wolfram deserves credit for it. Edit for clarification: The key insight being that you can model things algorithmically. Whether it's "correct" or not is a different matter. But this type of modeling is different from analytical modeling.

taliesinb 15y ago
No, this is a mischaracterization of what Wolfram is saying. Wolfram says: why consider computer programs to be valuable as mere approximations to the underlying analytic models, when you can consider computer programs themselves to be an entirely new and unexplored land of complex and interesting models. Traditional computer science considers computer programs to be merely a means to an end. A quite delicious irony is given by the example of Naiver Stokes equations. They are themselves an idealization of the flow of fluids composed of discrete particles. Because of our obsession with continuous models, we mostly resort to laboriously solving them numerically -- but it turns out that simple lattice gas cellular automaton models are actually 1) fairly accurate 2) much more computationally efficient, and 3) more suggestive of the underlying microscale physics. To step back a bit, a helpful analogy would be that simple computations are the 21st century equivalent of differential equations, which were studied rigorously in their own right starting in the 18th century, often prior to their application to concrete problems. Whether this intuition will turn out to be prescient, or whether it will fizzle out, is another question.
- knowtheory 15y ago
  That doesn't make sense in the context of my comment, and besides, i don't think that's true. If you take the Turing Test to it's logical conclusion, a program that is indistinguishable from a human intelligence is a human intelligence. There's no means to some other end. The program is the intelligence. This is also something that Turing came up with in the 40s & 50s. Not exactly novel.
  - taliesinb 15y ago
    Uh... well, I'm getting Wolfram's take right. So you don't think my analogy is true? It's just an analogy. But it's clear that it has methodological implications for how one does science: For example, if you think computers are just a way to simulate continuous systems, it would not occur to you to sample random programs and see what they do. It would not occur to you to enumerate simple programs. And you wouldn't think that it is very interesting that such and such a simple program can do such a such a computation. If you did, it would. And if you were ambitious enough, you would actually try hunting for the program that computes the universe, as Wolfram has been doing on a cluster in his basement (I love this tidbit) for some years now: http://www.ted.com/talks/stephen_wolfram_computing_a_theory_of_everything.html http://www.ted.com/talks/stephen_wolfram_computing_a_theory_...
    - knowtheory 15y ago
      No, you're making the assertion first, that everyone positively asserts that the universe is continuous, and second that computer scientists see algorithmic systems as being simulations of analytical systems. The first i am agnostic to (although i do like the feynman quote at the top of http://arxiv.org/pdf/quant-ph/0206089v2 http://arxiv.org/pdf/quant-ph/0206089v2 which was linked to above), and the second is most certainly false, as i have indicated above. The evidence in NKS to support wolfram's assertion that the universe is a simple program is circumstantial at best, and so his windmill tilting quests for the program that is the universe seems quixotic at best, and arrogantly foolhardy at worst. Analytical modeling, algorithmic modeling, or whatever other model someone wants to use to represent reality are models until you can prove them to actually be fundamentally connected with the manner in which reality functions.

pnathan 15y ago

Yes. There are multiple models of computation, and many have been demonstrated to be 'equal' (the technical term escapes me). Lambda calculus, turing machines, neural networks, certain cellular automata, and others I can not recall have been shown to have this equivalency. I am not impressed by Dr. Wolfram's arguments. The Amazon Review cited elsewhere in this thread has a very cogent discussion of his points.

primodemus 15y ago

Turing proved the existence of non-computable reals, so the two are not the same. Also Chaitin's constant is a non-computable real number with a precise mathematical definition. See: http://en.wikipedia.org/wiki/Chaitins_constant http://en.wikipedia.org/wiki/Chaitins_constant

pygy_ 15y ago
The Curry-Howard isomorphism states (and this has been proved) that for each formal proof in math, there is a corresponding computer program that halts, and reciprocally. This means that it is possible to express the proofs of your statements in code. On the other hand, non-halting programs are AFAIK beyond the reach of formal logic, and as consequence, the set of logical proofs (or more accurately, its isomorphic twin) is a subset of the set of all computer programs.

grinnbearit 15y ago

I've thought about this as well and IMO computer science makes sense as mathematics with a concept of side effects. In that way the universe is more a computer program than a mathematical equation.

maurycy 15y ago

The results you refer to are probably Curry-Howard correspondence, and Curry-Howard-Lambek correspondence.