computer scientist
Like others who have responded, I think the choice is obvious. The remarkable thing is that "the obvious choice" is different for everyone! To my mind, the most important inventions are those which have forced the largest changes in our world-view. On the basis of this criterion, I pick two (for reasons listed below): The telescope, and the theory of evolution by natural selection.
I pick two because it seems to me that there are two major categories of important inventions: a) complexity increasing, and b) complexity decreasing.
By complexity increasing, I mean those inventions that open up vast new realms of data, which can not be accounted for on the existing world view, making the universe less understandable, and therefore seemingly more complex.
By complexity decreasing, I mean those inventions that identify a pattern or algorithm in vast realms of data, ridding that data of a good deal of its apparent complication. These inventions force alterations to our world view to account for previously unaccountable data, or to account for it more directly and simply, making the universe more understandable, and therefore seemingly less complex.
The former tend to take the form of instruments or devices — physical constructs — while the latter tend to take the form of concepts, theories, or hypotheses — mental constructs. Both qualify as inventions.*
(*To be careful, the former also involves a mental construct — a device alone is useless without the mental construct that points it in the right direction.)
In the former category, nothing rivals the telescope.
No other device has initiated such a massive reconstruction of our world view. It forced us to accept the Earth, and ourselves, as "merely" a part of a larger cosmos. Of course, numerous theories besides the earth-centered universe existed before its invention, but the telescope opened the doors to the flood of data that would resolve what were previously largely philosophical disputes. The microscope — a relative of the telescope — also opened the door to a previously unimagined universe, and runs a close second to the telescope on the world-view shaking Richter scale.
In the latter category, there are many brilliant candidates, but I think that Darwin's invention of the theory of evolution by natural selection outshines them all. It is perhaps the only truly general theory in Biology, a field much more complex than physics. If we discover life elsewhere in the universe it is likely to be the only biological theory that will carry over from our terrestrial biology. Darwin's theory reduced tremendously the complication of zoological data. Critically, as with the telescope, it has put tremendous pressure on the previous world-view to accommodate man as "merely" a part of a much larger nature. This pressure is still largely being resisted, but the outcome is clear.
A close second would be the Second Law of Thermodynamics. Although the Second Law has not, perhaps, posed such a profound challenge to our collective world view, it has tremendously reduced the complexity of a great body of data (and it profoundly affects the world view of anyone who studies it in detail!)
I would have nominated the computer, but I think that, although it has profoundly affected our daily routines, it has not yet profoundly affected our world view. The computer is a kind of mathematical telescope, revealing to us a vast new realm of data about what kinds of dynamics follow from what sorts of rules — we are constantly discovering new galaxies of mathematical reality with computers. However, it will be a while before these empirical discoveries force a profound alteration of our world view.
