Invention and Discovery

Continuing my thought from last week, a thought instigated by Ciceronianus: 

Do humans actively shape the world?  Do we invent reality? Or do we merely discover it?

Surely we build skyscrapers and bridges, in much the same way that birds make nests. We seek to shape our environment for the sake of our own survival.

Is there some sense that doesn’t immediately involve motor activity in which humans invent the world? Something more constructivist?

A “yes” answer seems to make more sense to me than it does to Ciceronianus. In part this is because of reflection on the history of mathematics A very short statement is this: mathematics is a series of outrageous re-definitions of what it means to be a number. We learn to count when very young, and I suppose it has always been thus. The first conception of number derives from the act of counting.

But through our lives, if we receive any sort of education, we learn about ever more outlandish sorts of number. The strangeness of zero, for example.  Or irrational numbers, those wild things like pi that never repeat and never end.  How uncanny!

We may also wrestle with negative numbers. Then the idea of an "infinitesimal." I remember an old Sesame Street episode with the question whether a circle is “all one side” or whether a circle has “a whole lot of very little sides.” Ernie was raising the question of infinitesimals.  Circles (or other curves) can be thought of as an infinite number of tangent straight lines, each line always receding in size, with the Euclidean point as a limit.

Beyond even that, there is the notion of imaginary numbers. In the real number system, the basic rules of multiplication and division make it impossible that there should be such a thing as the square root of a negative number. But forget about that and invent the square root of -1 anyway! Call it i.

These increasingly absurd seeming steps of human reason are also steps of human imagination.  They seem as sheerly inventive as anything else we as a species can do. The paradox, then, is that the inventions of these outlandish notions by clever humans working at a very high level of abstraction, and often unconcerned with practical consequences, always turns out to have enormous practical consequences.