Programs are made of ideas

My old supervisor, an acm fellow, admonished me: stop thinking about the theory! Look at the data!

Henri Poincaré said, C'est par la logique qu'on démontre, c'est par l'intuition qu'on invente. (It is by logic that we prove, but by intuition that we discover) http://www-history.mcs.st-and.ac.uk/Printonly/Poincare.html

Implicit in this idea is that there is some kind of search going on - some way of thinking about the problem - which is not logic, and which is beyond notational reasoning; and indeed, mathematicians create new mathematics and notation at the drop of a hat (even if, or sometimes it seems, especially if, someone else's notation already exists. They are like Lispers - or, perhaps, lispers are like them.)

Now, the truth is that our thoughts and ideas are also constrained by a system - our own minds, by which I mean both the specific development of a particular mind at a given time; and also the biological constraints of our thought.

Obviously, there are thoughts which a specific person cannot yet conceive, because they simply don't have the background: consider great geniuses of the past, born before crucial mathematics had been invented, or before crucial facts about the world were known.

But are there thoughts that are biologically impossible for humans to think? I think so. For example, it is very difficult for humans to understand complex ideas without hierarchy. All our systems are founded on hierarchies, or layers of abstractions, because our short-term memory - even of the best of us - is limited. I'm sure this has been discussed academically, but I'm only aware of a couple of science-fiction writers who have mentioned this: Niven/Pournelle's Motie Engineers, who effortlessly improvised simple solutions that participated in solutions to many problems at once; and Vinge's Rider design (clearly of Transcendental origin, apparently simple, even mechanical operational, but with extraordinary emergent properties.)

As the old joke goes: any problem can be solved by another layer of abstraction, except for the problem of too many layers of abstraction.