[anonytrary]

Suppose you have a list of N things you know, and we define an invention as being any subset of those things. There would be 2^N possible combinations, or the size of the set of all subsets, or the size of the set of all inventions.

That's my toy-definition of an invention. Not very good, but it's a start. Let's take it further and say that each item in the list of knowledge is actually a basis vector, and that an invention is simply a vector in the space spanned by the basis set.

> We can find new knowledge, insight, and/or tool without increasing the "space of inventions", much less exponentially increasing it.

In my model, I will prove this is impossible. The dimension before finding the new vector is N. Suppose we find a new knowledge vector k'. If it is truly new, then it will be orthogonal to the other knowledge vectors, and the new basis will span N+1 dimensions, meaning the "space of inventions" increased. The only way for the dimension to remain the same is if k' could be written as a linear combination of k_i, which would imply that our assumption that k' is new was false.

ENOUGH METAPHYSICS!