The Algebra of Thought: a theory of human creativity

How is it possible to have an original thought?  This question seems to be burning in the minds of various corporate executives; who have apparently come up with the idea that a good solution would be to impose an additional layer of bureaucracy  in the form of a "Director of Innovation", for some current examples see here.  On the other side of the debate, we have the notion that innovation kinda just happens if you leave "creative" people alone to "do their thing" (for example, Google's 20% time, or Valve's unique approach to project management).

It seems to clear to me that more bureaucracy will not help people be more creative, but unfortunately we are left with a difficult and profound question: how does the creative process work?

To summarize my opinion on the matter, I argue that it's not possible to have a completely original idea (the key word being "completely"); but rather that every "creative" idea is really the result of a process of breaking down idea's into their component parts, and then reassembling those idea's in a way that produces something "new".

This raises several interesting and important questions:

If thoughts can be broken down into component pieces, is there such a concept as an "indivisible thought", or a "fundamental building block" of ideas?  And if so, what is the nature of these fundamental building blocks?  Furthermore, is it possible to construct an "algebra" so that we can have a notion of "adding" two thoughts together, or other relevant operations such as subtraction?

One possible answer to the question is to say: since our knowledge of the outside world is purely a result of our 5 senses, then the fundamental building blocks of thought should correspondingly be the sense input we receive from them.  

For example, I know what an elephant looks like, I know what the color pink is, therefore I can "add" these two ideas together to come up with the "new" idea of a pink elephant.

Similarly, I know what an elephant smells like and sounds like, I also know how a flower smells, what an alarm clock sounds like, and what it feels like to be stung by a bee.  So now I can come up with the original idea of a pink elephant that smells like a flower, sounds like my alarm clock, and stings like a bee!  I'm so creative aren't I!

Certainly the above idea is pretty dumb, but the process is actually the process by which humans are creative: we break down idea's we already have into smaller parts and then recombine them to come up with something "new".  There is an excellent series of video's called "Everything is a Remix" which argues the same point, and which I recommend you watch, if you have not done so already.  

A question which should be raised at this point is: if every thought can be broken down into building blocks which come from the 5 senses, then how is it possible that people can understand mathematics, or other abstract concepts?

I claim that the answer is that mathematics is a language, that the human mind has a natural capacity to understand language, and that the building blocks of thought are not sense experiences in and of themselves, but rather these building blocks are "abstractions" which are processed by the language centers of the brain.  Consider that language itself is perfectly suited for "adding" and "subtracting" ideas together, we simply add some words to a given sentence, or else remove some words from a given sentence.  It seems likely that the creative process works something like this: we abstract away from our sense data "ideas" which can be processed by the language centers of the brain, which has a coherent and consistent system for adding, subtracting  and recombining these "ideas", and then we can express those ideas and give them life in the external world in the form of some kind of object that we can experience with our senses.