Rusting Car Bumper

Then you dont understand the idea or you are intentionally hoisting a straw man.

The idea is different from current thinking. The mental process for using mathematics, as we now do, (differential equations, statistical methods, etc) requires a different mindset than what he proposes.

That is why he made the analogy to the discovery/invention of calculus by Newton and Liebnitz. The point of that was that eventually it became apparent that one could derive new explanations or predictions from the math itself with amazing accuracy.

This is how new particles were predicted in high energy physics for example.

The way he views cellular automata (CA) is not like this. In fact the math tells us that it can't be like this. If his view of cellular automata modelling nature is correct, then one cannot derive relationships from the math. One has to go through and do a form of pattern matching for a lack of a better term.

Thus, the two methods are different. It's bold because, again, if he's correct then sets of simple rules might be shown to model divergent sets of phenomenon ranging all across the spectrum of studies: geology, biology, physics, and so on.

There are only a few people who do. Its not about computability theory. Its about the relationship of mathematics to nature and how one goes about determining meaningful relationships with these as a human enterprise. The only thing anyone ever did in the past was toy with the idea but through the lense of the older math/nature realtionships. That's why they found it dissapointing.... because they couldn't do what they did with differential equations.

DC

CardinalMan

I'm sorry if I came across this way. I'm not intentionally hoisting a straw man. I'd like to think that I have the background to understand his ideas. Precisely what do you think I am missing? Do you think that my original post on this thread missed a key aspect of Wolfram's book?

I can see how what he proposes may seem different to many natural scientists since they usually study "continuous" aspects of mathematics (like calculus, differential equations, statistical methods, etc.), but what about the huge number of mathematicians and computer scientists who study "discrete" aspects (like combinatorics, graph theory, automata theory, complexity theory, etc.)? How are the mental processes that mathematicians use to study these subjects differ from what Wolfram is proposing? As far as I can tell, they are identical.

Studies in mathematical logic and automata/computability theory allowed mathematicians to predict that many natural statements in mathematics could turn out to be unprovable from our current axioms. These predictions were fulfilled decades ago. Studies in complexity theory suggest that many natural computational problems have no efficient algorithms and that you can hardly do better than brute force. This remains an open mathematical question. Again, what exactly is new here? How does Wolfram's analysis and speculations differ from what many "discrete" mathematicians have done for decades? Do you believe that they are different because Wolfram is now applying them to natural world instead of just mathematics?

I'm not sure I fully understand what you're saying. If my comments do not relate to what you're thinking, please let me know.

As far as I can tell, his notions of "computational irreducibility" do not apply to every aspect of mathematics or nature. Our current framework for physics does allow us to derive relationships for a remarkable number of phenomena more quickly than going through each step. Wolram's claim, as I read it, seems to be that we shouldn't expect this to hold in general, and again this is an old idea (at least in certain mathematical circles). How does this qualitatively differ from the central belief in computer science that P does not equal NP, i.e. that there are many natural computational problems that basically require a brute-force calculation instead of using shortcuts?

As I alluded to above, perhaps our disagreement comes down to do the following. It seems to me that there are two central claims in Wolfram's book:

1) Wolfram has developed a new mathematical theory that leads to radical new ideas about the nature of computation and complexity.
2) These ideas can be applied to the natural world around us to gain insight into scientific questions in a completely new and unified way.

I reject the first claim without reservation as I've outlined above. I think that the second claim is open to debate. While I don't think that Wolram has many fundamentally new ideas in this realm either (although it may very well seem that way to many natural scientists), I will agree that he has presented a fairly nice and unified perspective in a manner that has not really appeared in the past. Do you agree with the first claim above, or are you just arguing for the second?

CardinalMan