My nomination for the most important invention of the past 2000 years is probability theory, which was mainly put together in a series of steps between 1654, when Blaise Pascal proposed a solution for splitting the pot in an unfinished game of chance, and 1843, when Antoine Cournot offered a definition of chance as the crossing of two independent streams of events. I don't nominate it simply because probability theory laid the foundation for statistical analysis, which provided us with a vocabulary without which most scientific discoveries made in the last century would have been (literally) unthinkable.

Nor do I nominate probability theory because it gave us for the first time a trustworthy tool for deciding how to apportion belief to multiple sources of evidence. Probability theory had even more fundamental epistemological implications whose importance is under-appreciated in our time because those implications are so seamlessly integrated into the foundations of our modern world view. Until the nineteenth century, the idea that there could exist deep regularities underlain by pure chance — regularities arising from distributions of events which were themselves the result of multifarious unmeasurable causes — was not only almost unknown (Aristotle had hinted at it, as he seems to have hinted at everything), but actually philosophically repugnant. It required the invention of probability theory to make this idea thinkable.

In making it possible to think about such abstract regularities, probability theory rescued us from two philosophical shackles which had held us back from the beginning of history: that of needing to postulate a centralized controller that made everything come out right, and that of assuming that "what you see is what you get" — i.e. that the proper objects of scientific study are roughly identical to the direct objects of the senses. Though perhaps they have still not been totally removed, those philosophical shackles needed to be at least loosened in order for science to get moving.

A whole new world of law-obeying objects to be studied was opened up by probability theory. Neither Darwin's theory of natural selection, nor Maxwell's theory of statistical mechanics (both published in the same year, only 140 years ago) would have been thinkable before probability theory was thinkable. Without probability theory, human kind would be (and was) unable to even conceive of the explanations for many — probably most — of the phenomena which we have ever explained.