A field can be in a state such that unassisted human reasoning has reached a saturation point but that human reasoning assisted with sufficient additional computational power can still provide new insights.

It’s unclear to me whether the role of the scientist is actually different in such a scenario, or if computers simply play a role analogous to that of pen & paper, telescopes, and the rest of the paraphernalia of science.

Re: Re: 2) I’m being a bit pedantic, but I think there’s still a morself of historical bias hiding in your last sentence. You characterized experimental mathematics as trading off the use of powerful new tools for a decrease in foresight. Why? Just because one plays Advanced Chess, where all the moves one makes are vetted by a supercomputer, doesn’t mean one stops thinking as much. It’s just that you can now think about different things. You can ask different, higher-level questions that for their details require computations of mind boggling complexity that no human could ever perform.

Re: 2) Interesting. This suggests another way to look at saturated fields – as disciplines flirting with the edge of human analytical intelligence. That is, there may be plenty more ideas to be discovered, but that they are far more amenable to numerical exploration, data mining, or other tools that do not rely as much on foresight.

I’m tempted to think that Wolfram’s self-styled “NKS” presents one of these new fields: the universe of simple computational systems and their behavior. Two things make it ripe:

1) for a couple of reasons to do with undecidability, there isn’t very much that traditional analysis can say about these systems, and so people haven’t thought them worth studying. However, consumer computing is just now hitting the point where one can perform large-scale data mining on the space of possible simple computational systems

2) Heavily-architected systems like CPUs are all well and good at the micron scale, but get small enough and traditional engineering stops working. At some point we’re going to have to exploit emergence, self-assembly and all the other sexy buzzwords, and NKS-y type science is suited for exactly this.

Disclosure: I’m a Wolfram employee who works on Wolfram|Alpha.

One, I think it depends on what set of problems you’re interested in. Computational physics is just as broad a category as are experimental and theoretical physics. The trick is to apply those methods fruitfully to some unique area, be that of “physics” proper or another field (*cough* brains!).

Two, science in the age of massive data sets is still wide open. Check out that Fourth Paradigm e-book I sent to you a while back. We’re learning to acquire data faster than we’re learning how to deal with it and new things are becoming “datafied” and placed into the realm of analysis (i.e. the process of science itself via projects like CoLab).

Perhaps what I’m saying is that computational science is ripe with open problems.