Semantics and syntactics
According to Robert Rosen, the study of mathematics took several bad turns more than 600 years ago, when several people failed to appreciate the richness of real numbers (a world in which there are infinite, uncountable points between zero and one). Their mistake was to try to jam everything into syntax (or grammar), when in fact real numbers offer many semantic (language-like) opportunities, full of relational interest. (I am not a mathematician and I only barely understand this, but stay with me for a few more brief paragraphs.)
Rosen suggests that the history of math is filled with examples of people trying to work with bigger and more complex sets of data while using the same puny, simple, and syntactic tools they have always used in the past. His view is that we need a richer mathematics that looks at the connections between things — their semantics. But we don't have the language to do this yet. He points to something called “category theory” that may help. From what I can tell, category theory helps with clumping together kinds of relationships and allows people to work with them in the same kinds of ways we now add and subtract.
Why think about this? Well, the telephony/internet split, which has produced two entirely different mindsets looking at the same set of technologies, provides a simple example of the difference between syntactic and semantic thinking. Syntax, or grammar, says we're looking for hierarchical, traditional operations on the network to provide the outputs we're interested in and used to. This is the telephony view — the same mindset that tells wireless carriers that they don't want to allow phones to connect to their networks that allow users to download music from users' computers. Our network, our music.
The internet mindset, which is much more semantic in nature, says that we want everything to connect because we're not sure what the results will be. Something very lively and organized will unpredictably emerge from dense relational interactions, if we can only let ourselves let this happen. We need better semantic tools to describe what happens when we let these processes run.