Wolfram has done a lot of fantastic work on emergent mathematical phenomena. (TED talk given to a non-technical audience, but still worth watching.) One of the highly counter-intuitive things that he has worked on is computational irreducibility. Irreducible functions are ones where you have to physically run the function to find it's outcomes, and the emerging patterns. For this class of function, the emergent patterns cannot be predicted in advance.

There seems like the next step to build on older work on the halting problem, which states that some types of problems require unknowable amounts of computational power to solve. It's not possible to know whether such problems will be solvable in a finite time, or whether the program will run forever without finding an answer.

This is relevant to Yudkowsky's criticisms of emergent phenomena, because it demonstrates that emergent phenomena are not just fake explanations in these cases. The term is used to describe a specific class of mathematically defined problems, which are irreducible. These problems can't be broken down to any more simple explanations. In these instances, the phrase "emergent" isn't a semantic stopsign telling someone not to ask any more question, but rather a useful marker letting someone know that they physically can't break the problem down any simpler without breaking the rules of mathematics.

To be fair, Yudkowsky does go through great pains to specify that

It's the noun "emergence" that I protest, rather than the verb "emerges from".

But that's not quite the right distinction to make. Perhaps the noun form is more often used as a meaningless buzzword, and has been used to refer to larger and larger groups of things. Maybe you could argue that "emergence" should only be used to refer to this formal case where complexity arises from simplicity in an manner which may be impossible to predict in a finite amount of time. But, by definition, we can't actually know that any particular problem won't halt, because the test would take an infinite amount of time.

So do we just use this "emergence" word to refer to any problems which haven't yet been computed from more fundamental principles, or do we use it more generally to refer to problems where the math is just really complex? Personally, I'd lean toward the latter. The term is already in common use to describe complex systems which arise from simple ones. The term does still become meaningless if we use it to refer to slightly complex systems emerging from fairly simple ones, but it's still a useful and descriptive word for other cases. We shouldn't stop using terms like "toxins", "energy", "quantum", or "exponential" just because they have been re-purposed and watered down, so why should we do so with "emergent"?

## Comments (125)

OldWolfram has done a lot of fantastic work on emergent mathematical phenomena. (TED talk given to a non-technical audience, but still worth watching.) One of the highly counter-intuitive things that he has worked on is computational irreducibility. Irreducible functions are ones where you have to physically run the function to find it's outcomes, and the emerging patterns. For this class of function, the emergent patterns cannot be predicted in advance.

There seems like the next step to build on older work on the halting problem, which states that some types of problems require unknowable amounts of computational power to solve. It's not possible to know whether such problems will be solvable in a finite time, or whether the program will run forever without finding an answer.

This is relevant to Yudkowsky's criticisms of emergent phenomena, because it demonstrates that emergent phenomena are not just fake explanations in these cases. The term is used to describe a specific class of mathematically defined problems, which are irreducible. These problems can't be broken down to any more simple explanations. In these instances, the phrase "emergent" isn't a semantic stopsign telling someone

notto ask any more question, but rather a useful marker letting someone know that they physically can't break the problem down any simpler without breaking the rules of mathematics.To be fair, Yudkowsky does go through great pains to specify that

But that's not quite the right distinction to make. Perhaps the noun form is more often used as a meaningless buzzword, and has been used to refer to larger and larger groups of things. Maybe you could argue that "emergence" should only be used to refer to this formal case where complexity arises from simplicity in an manner which may be impossible to predict in a finite amount of time. But, by definition, we can't actually know that any particular problem won't halt, because the test would take an infinite amount of time.

So do we just use this "emergence" word to refer to any problems which haven't yet been computed from more fundamental principles, or do we use it more generally to refer to problems where the math is just really complex? Personally, I'd lean toward the latter. The term is already in common use to describe complex systems which arise from simple ones. The term does still become meaningless if we use it to refer to slightly complex systems emerging from fairly simple ones, but it's still a useful and descriptive word for other cases. We shouldn't stop using terms like "toxins", "energy", "quantum", or "exponential" just because they have been re-purposed and watered down, so why should we do so with "emergent"?