Innovation Is a material basis for technology plus a knowledge topology equal to a model of technology? I suspect not - surely there are emergent effects. But there must be a relationship.
Related question: What is the shape of the vocabulary of communicating people? When do we denote new things? See semantics.
What is the shape of collected human knowledge?
This is vague. It would need to be made precise to be interesting. I'm thinking of hyperbolic and other non-euclidean geometries and wondering about how you can project the articles of an encyclopedia onto them, preserving some notice of similarity, dependency or priority.
What kind of attachment mechanisms are plausible? Could you mine patent networks or theorem networks to parameterise a stochastic process for this model which made it a plausible model for theorem growth? If not, what quality does knowledge posses which this could not encapsulate?
Can we represent this as a network (or a landscape?) that accretes around agent activity? Some kind of growth process? (keywords: “models of growth aggregation”, “rough interfaces”, “growth with surface diffusion”, “nucleation”, “morphogenesis”) Is this a constrained growth problem, like the one that governs coral drills?
Investigate configuration spaces of technologies. (see configuration space of the economy Maybe use genes as a model? genotype-phenotype interactions as a model of knowledge-economic systems? What is the most basic stochastic process that would serve as a statistically equivalent model of these?
How much area must a new thesis carve out from the unmade world?
Now, going out on a limb, consider a problem domain that looks evolutionary if you squint at it: creating mathematical theorems. Certainly Gödel and Turing invite looking at the things as symbol strings. I saw a presentation by Greg Leibon suggesting that there was a natural embedding of mathematical field onto hyperbolic geometry. Sure, his data set was Wikipedia mathematical article links, and the whole idea was tongue-in-cheek. But it feels like there is something in there, if not a whole-cloth topological theory of human knowledge. Is there some process driving mathematical innovation that means that the links between fields sit so naturally in hyperbolic space? Is it some characteristic of the subject matter itself? If either of these are true, would they be true of other fields? Science in general? Philosophy? Engineering? Design? Biological fitnesses?
To Investigate, possibly related
Topic modelling in text databases
Skip-grams provide a super trendy way of topic modelling free text. Is the growth of the inferred model with the size of the database in some way reflective of the growth of the knowledge that a database represents? If so, what geometry would that imply? Sounds like something like an inverse Johnson-Lindenstrauss lemma.
The class of things I think of as string dynamical - autocatalytics systems and the like. (c.f. Fontana's “Turing gas”)
Tagging systems and interfaces
A feature of technology that I would expect to find created by such feeble things as our human minds is somple interaces. Engineers in any project work hard to divide it up into nearly decomposable chunks. We work with technologies where as incremental as possible a change can be considered. APIs, industrial standards, conventions, norms. Reducers of search time and dimensionality of configuration space.
Theorems are like that, yes? Heaviside's vector calculus prevailed because it reduced Maxwell's equations from 24 to 4.
How could we identify that in an innovation data set?
An infinite number of papers trying to construct a netowrk topology for citation networks. Here's one: