Steve Wolfram argues for a kind of technology mining instead of say mathematical modeling. I and Jacob Bronowski will point out that your not going to find the quadratic formula laying around on the ground cruncking out the patterns of nature. We know that mathematics is an idealization. But, Mr Wolfram wants to go much further.
Even if he's right, I feel that there's a bit of a problem with how far he's going with this. Assuming he's right, wouldn't the use of fire by Homo Erectus, the use of animals, windmills kind of like technological mining of stuff they didn't really understand? How about artificial selection of animals ten thousand years ago of both animals and planets to make agriculture civilization to happen? Well, maybe! But, it's mathematics that has made those things really fly; genetics and molecular biology period. Or, thermodynamics and steam engines. Shoot, there's natural nuclear reactions found by geologists; maybe if humans(or intelligence in general) had figured out how to mine enough uraneum and just put enough together, they could have made at least a dirty bomb back thousands of years ago! But, without the mathematics, nobody would have made the trinity test happen(or a pathway to the stars - nuclear powered spacecraft).
The above points to something I've been trying to think about more recently. That despite the fact that mathematicians can come up with much mathematics, the mathematics isn't finished until it's logically proved; and, the process of doing these proofs often leads to more mathematics(the irrationality of the square root of two is one immediate example; linear algebra coming from substitions of equations to solve the general third degree equation is another recent one I've found; the recent solutions to Fermat's Last theorem and the Poincare conjecture leading to much of the topology of the 20th century are other more recent examples). Yes, one can do technology without mathematics; but, mathematics gives it wings.
Another more science and mathematics topic I've been trying to think about recently to post(other than all this philosophy!) is chaos theory. The science of astronomy eventually led to great clocks from the 1300s to the Nuremberg egg as shown by James Burke connections videos(and his books; the books have much the same but some more details of course; the videos actually have details not shown in the books as well!) posted throughout this blog). What about the mathematics since? What is that leading to? Some people say number theory has no applications; well, computers for one disproves that. But, I'm starting to like chaos theory as another answer. I've seen chaos theory applications recently applied to nuclear fusion(instead of fission that's we've been using so far; in case my readers don't know; i don't know who most of my readers are!) and electron microscopes! Chaos theory in terms of, or related to Steve Wolfram above below,
I'd like to point out that the chaos theorists have this idea of going from a strange attractor state to a regular attractor state and then back to a chaotic state; and, they have shown they can go from chaotic to any of a given chaotic attractors possible stable states. Could one say they can go from irreducible complexity to reducible computational complexity - kind of like Mr Wolfram's technology mining he's talking about here?