(Apology: The question "How has the Internet changed the way you think?" is a difficult one for me to answer in an interesting way; the truth is, I use the Internet as an appliance, and it hasn't profoundly changed the way I think, at least not yet. So I've taken the liberty of interpreting the question more broadly, in the form "How should the Internet, or its descendants, affect how people like me think?")
If controversies were to arise, there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, to sit down to the slates, and to say to each other (with a friend as witness, if they liked): "Let us calculate." — Leibniz (1685)
Clearly Leibniz was wrong here, for without disputation philosophers would cease to be philosophers. And it is difficult to see how any amount of calculation could settle, for example, the question of free will. But if we replace, in Leibniz' visionary program, "sculptors of material reality" for "philosophers", then we arrive at an accurate description of an awesome opportunity — and an unanswered challenge — that faces us today. This opportunity began to take shape roughly eighty years ago, as the equations of quantum theory reached maturity.
The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. — P. A. M. Dirac (1929)
Much has happened in physics since Dirac's 1929 declaration. Physicists have found new equations that reach into the heart of atomic nuclei. High-energy accelerators have exposed new worlds of unexpected phenomena and tantalizing hints of Nature's ultimate beauty and symmetry. Thanks to that new fundamental understanding we understand how stars work, and how a profoundly simple but profoundly alien fireball evolved into universe we inhabit today. Yet Dirac's bold claim holds up; while the new developments provide reliable equations for smaller objects and more extreme conditions than we could handle before, they haven't changed the rules of the game for ordinary matter under ordinary conditions. On the contrary, the triumphant march of quantum theory far beyond its original borders strengthens our faith in its soundness.
What even Dirac probably did not foresee, and what transforms his philosophical reflection of 1929 into a call to arms today, is that the limitation of being "much too complicated to be soluble" could be challenged. With today's chips and architectures, we can start to solve the equations for chemistry and materials science. By orchestrating the power of billions of tomorrow's chips, linked through the Internet or its successors, we should be able to construct virtual laboratories of unprecedented flexibility and power.
Instead of mining for rare ingredients, refining, cooking, and trying various combinations scattershot, we will explore for useful materials more easily and systematically, by feeding multitudes of possibilities, each defined by a few lines of code, into a world-spanning grid of linked computers.
What might such a world-grid discover? Some not unrealistic possibilities: friendlier high-temperature superconductors, that would enable lossless power transmission, levitated supertrains, and computers that aren't limited by the heat they generate ; super-efficient photovoltaics and batteries, that would enable cheap capture and flexible use of solar energy, and wean us off carbon burning; super-strong materials, that could support elevators running directly from Earth to space.
The prospects we can presently foresee, exciting as they are, could be overmatched by discoveries not yet imagined. Beyond technological targets, we can aspire to a comprehensive survey of physical reality's potential. In 1964, Feynman posed this challenge:
Today, we cannot see whether Schrodinger's equation contains frogs, musical composers, or morality — or whether it does not. We cannot say whether something beyond it like God is needed, or not. And so we can all hold strong opinions either way. — R. P. Feynman (1964)
How far can we see today? Not all the way to frogs or to musical composers (at least not good ones), for sure. In fact only very recently did physicists succeed in solving the equations of quantum chromodynamics (QCD) to calculate a convincing proton, by using the fastest chips, big networks, and tricky algorithms. That might sound like a paltry beginning, but it's actually an encouraging show of strength, because the equations of QCD are much more complicated than the equations of quantum chemistry. And we've already been able to solve those more tractable equations well enough to guide several revolutions in the material foundations of microelectronics, laser technology, and magnetic imaging. But all these computational adventures, while impressive, are clearly warm-up exercises. To make a definitive leap into artificial reality, we'll need both more ingenuity and more computational power.
Fortunately, both could be at hand. The SETI@home  project has enabled people around the world to donate their idle computer time to sift radio waves from space, advancing the search for extraterrestrial intelligence. In connection with the Large Hadron Collider (LHC) project, CERN laboratory — where, earlier, the World Wide Web was born — is pioneering the GRID computer project, a sort of Internet on steroids, that will allow many thousands of remote computers and their users to share data and allocate tasks dynamically, functioning in essence as one giant brain. Only thus can we cope — barely! — with the gush of information that collisions at the LHC will generate. Projects like these are the shape of things to come.
Chess by pure calculation in 1958, and rapidly became more capable, beating masters (1978), grandmasters (1988), and world champions (1997). In the later steps, a transition to "massively" parallel computers played a crucial role. Those special-purpose creations are mini-Internets (actually mini-GRIDs), networking dozens or a few hundred ordinary computers. It would be an instructive project, today, to set up a SETI@home -style network, or a GRID client, that could beat the best standalones. Players of this kind, once created, would scale up smoothly to overwhelming strength, simply by tapping into ever larger resources.
In the more difficult game of calculating quantum reality we, with the help of our silicon friends, presently play like weak masters. We know the rules, and make some good moves, but we often substitute guesswork for calculation, we miss inspired possibilities, and we take too long doing it. To do much better we'll need to make the dream of a world-GRID into a working reality. We'll need to find better ways of parceling out subtasks in ways that don't require intense communication, better ways of exploiting the locality of the underlying equations, and better ways of building in physical insight, to prune the solution space. These issues have not received the attention they deserve, in my opinion. Many people with the requisite training and talent feel it's worthier to discover new equations, however esoteric, than to solve equations we already have, however important their application.
People respond to the rush of competition and the joy of the hunt. Some well-designed prizes for milestone achievements in the simulation of matter could have a big impact, by focusing attention and a bit of glamour toward this tough but potentially glorious endeavor. How about, for example, a prize for calculating virtual water that boils at the right temperature?