The Internet hasn't, so far, changed how we think. But it has radically altered the contexts in which we think and work.
The Internet offers a vast realm for distraction but then so does reading and television. The Internet is an improvement on television in the same way that Jane Jacob's bustling neighborhood sidewalk is an improvement on the dullness of suburbia. The Internet requires an active engagement and as a result it is full of surprises. You don't watch the Internet, you search and link. What is important for thought about the Internet is not the content, it is the new activity of being a searcher, with the world's store of knowledge and images at your fingertips.
The miracle of the browser is that it can show you any image or text from that storehouse. We used to cultivate thought, now we have become hunter gatherers of images and information. This speeds things up a lot but it doesn't replace the hard work in the laboratory or notebook which prepares the mind for a flash of insight. But it nonetheless changes the social situation of that mind. Scholars used to be more tied to the past through texts in libraries than to their contemporaries. The Internet reverses that by making each of our minds a node in a continually evolving network of other minds.
The Internet is also itself a metaphor for the emerging paradigm of thought in which systems are conceived as networks of relationships. To the extent that a Web page can be defined only by what links to it and what it links to, it is analogous to one of Leibniz's monads. But Web pages still have content, and so are not purely relational. Imagine a virtual world abstracted from the Internet by deleting all the content so that all that remained was the links. This is an image of the universe according to relational theories of space and time, it is also an image of the neural network in the brain. The content corresponds to what is missing in those model, it corresponds to what physicists and computer scientists have yet to understand about the difference between a mathematical model and an animated world or conscious mind.
Perhaps when the Internet has been soldered into our glasses or teeth, with the screen replaced by a laser making images directly on our retinas, there will be deeper changes. But even in its present form the Internet has transformed how we scientists work.
The Internet flattens communities of thought. Blogs, email and Internet data bases put everyone in the community on the same footing. There is a premium on articulateness. You don't need a secretary to maintain a large and varied correspondence.
Since 1992 research papers in physics are posted on an Internet archive, arxiv.org, which has a daily distribution of just posted papers and complete search and cross reference capabilities. It is moderated rather then refereed; and the refereed journals now play no role in spreading information. This gives a feeling of engagement and responsibility, once you are a registered member of the community you don't have to ask anyone's permission to publish your scientific results.
The Internet delocalizes your community. You participate from where-ever you are. You don't need to travel to see or give talks and there is less reason to go into the office. Travel is no reason not to stay current reading the latest papers and blog postings.
It used to be that physics preprints were distributed by bulk mail among major research institutes and there was a big advantage to being at a major university in the United States; every one else was working with a handicap of being weeks to months behind. The increasing numbers and influence of scientists working in Asia and Latin America and the dominance of European science in some fields is a consequence of the Internet.
The Internet synchronizes the thinking of global scientific communities. Everyone gets the news about the new papers at the same time every day. Gossip spreads just as fast on blogs. Announcements of new experimental results are video-cast through the Internet as they happen.
The Internet also broadens communities of thought. Obscure thinkers that you had to be introduced to, who published highly original work sporadically and in hard to find places, now have Web pages and post their papers along side everyone else's. An it creates communities of diverse thinkers who would not otherwise have met, like the one we celebrate every year at this time when we answer the Edge Annual Question.