Invention and Discovery

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Continuing mythought from last week, a thought instigated by Ciceronianus: 
Do humansactively shape the world?  Do we inventreality? Or do we merely discover it?
Surely webuild skyscrapers and bridges, in much the same way that birds make nests. Weseek to shape our environment for the sake of our own survival.
Is there somesense that doesn’t immediately involve motor activity in which humans inventthe world? Something more constructivist?
A “yes”answer seems to make more sense to me than it does to Ciceronianus. In partthis is because of reflection on the history of mathematics A very shortstatement is this: mathematics is aseries of outrageous re-definitions of what it means to be a number. We learnto count when very young, and I suppose it has always been thus. The firstconception of number derives from the act of counting.
But through our lives, if wereceive any sort of education, we learn about ever more outlandish sorts ofnumber. The strangeness of zero, for example. Or irrational numbers, those wild things like pi that never repeat andnever end.  How uncanny!
We may also wrestle with negativenumbers. Then the idea of an "infinitesimal." I remember an oldSesame Street episode with the question whether a circle is “all one side” orwhether a circle has “a whole lot of very little sides.” Ernie was raising thequestion of infinitesimals.  Circles (orother curves) can be thought of as an infinite number of tangent straightlines, each line always receding in size, with the Euclidean point as a limit.
Beyond even that, there is thenotion of imaginary numbers. In the real number system, the basic rules ofmultiplication and division make it impossible that there should be such athing as the square root of a negative number. But forget about that and inventthe square root of -1 anyway! Call it i.
These increasingly absurd seemingsteps of human reason are also steps of human imagination.  They seem as sheerly inventive as anything elsewe as a species can do. The paradox, then, is that the inventions of theseoutlandish notions by clever humans working at a very high level ofabstraction, and often unconcerned with practical consequences, always turnsout to have enormous practical consequences.