Stephen Wolfram has posted here some interesting news about one of his hobbies - hunting for the fundamental laws of physics, where he outlines how he is both developing and using Mathematica to search for fundamental laws of physics that generate simulated universes which have properties that resemble our own real universe.
The power of his approach lies not only in his use of Mathematica, but more fundamentally in his use of very few axioms to define what the fundamental laws of physics are in the first place. His basic object is a network (i.e. nodes and links-between-nodes), and his basic operation is the mutation of a piece of the network (via the application of a set of rules), which thus allows the network structure to have a dynamical behaviour. In this approach the fundamental laws of physics are determined by the choice of the set of network update rules.
It turns out that various simple consistency criteria cause this approach to give rise to both special relativity and general relativity. That is impressive, starting from a rule-based approach!
One of the challenges is to determine the consequences of a particular choice of network update rules, and to ascertain if they correspond to the behaviour of our known real universe. In general, the network behaviour in response to its update rules can be extremely complicated, and working out what is going on can thus be very difficult and time consuming. The development of Mathematica itself is partly driven by the need to create tools for addressing problems such as this.
Wolfram says that he has not yet found a viable candidate for the fundamental laws of physics using this approach, but that he is hard at work both developing and using Mathematica to achieve this goal. As he says
I certainly think it'll be an interesting - almost metaphysical - moment if we finally have a simple rule which we can tell is our universe. And we'll be able to know that our particular universe is number such-and-such in the enumeration of all possible universes.
I wish him luck in this venture. It would be very impressive if he found that a 3-line Mathematica program was all that was needed to generate the behaviour of our known real universe. Even if he is destined not to discover the fundamental laws of physics using this approach, he will nevertheless have created along the way a very useful toolbox for doing lots of other things, i.e. Mathematica.