The most important invention, I believe, was a mathematical idea, which is the notion of representation: that one system of relationships, whether mathematical or physical, can be captured faithfully by another.
The first full use of the idea of a representation was the analytic geometry of Descartes, which is based on the discovery of a precise relationship between two different kinds of mathematical objects, in this case, numbers and geometry. This correspondence made it possible to formulate general questions about geometrical figures in terms of numbers and functions, and when people had learned to answer these questions they had invented the calculus. By now we have understood that it is nothing other than the existence of such relationships between systems of relations that gives mathematics its real power. Many of the most important mathematical developments of the present century, such as algebraic topology, differential geometry, representation theory and algebraic geometry come from the discovery of such relationships, of which Descartes analytic geometry was only the first example. The most profound developments in present mathematics and theoretical physics are all based on the notion of a representation, which is the general term we use for a way to code one set of mathematical relationships in terms of another. There is even a branch of mathematics whose subject is the study of correspondences between different mathematical systems, which is called category theory. According to some of its developers, mathematics is at its root nothing but the study of such relationships, and for many working mathematics, category theory has replaced set theory as the foundational language within which all mathematics is expressed.
Moreover, once it was understood that one mathematical system can represent another, the door was open to wondering if a mathematical system could represent a physical system, or vise versa. It was Kepler who first understood that the paths of the planets in the sky might form closed orbits, when looked at from the right reference point. This discovery of a correspondence between motion and geometry was far more profound than the Ptolemaic notion that the orbits were formed by the motion of circles on circles. Before Kepler, geometry may have played a role in the generation of motion, but only with Kepler do we have an attempt to represent the orbits themselves as geometrical figures. At the same time Galileo, by slowing motion down through the use of devices such as the pendulum and the inclined plane, realized that the motions of ordinary bodies could be represented by geometry. When combined with Descartes correspondence between geometry and number this made possible the spatialization of time, that is the representation of time and motion purely in terms of geometry. This not only made Newtonian physics possible, it is of course what we do every time we graph the motion of a body or the change of some quantity in time. It also made it possible, for the first time, to build clocks accurate enough to capture the motion of terrestrial, rather than celestial, bodies.
The next step in the discovery of correspondences between mathematical and physical systems of relations came with devices for representing logical operations in terms of physical motions. This idea was realized early in mechanical calculators and logic engines, but of course came into its own with the invention of the modern computer.
But the final step in the process began by Descartes analytic geometry was the discovery that if a physical system could represent a mathematical system, then one physical system might represent another. Thus, sequences of electrical pulses can represent sound waves, or pictures, and all of these can be represented by electromagnetic waves. Thus we have telecommunications, certainly among the most important inventions in its own right, which cannot even be conceived of without some notion of the representation of one system by another.
Telecommunications also gave rise to a question, which is what is it that remains the same when a signal is translated from sound waves to electrical impulses or electromagnetic waves. We have a name for the answer, it is information, but I do not have the impression that we really understand its implications. For example, using this concept some people are claiming that not only is it the case that some physical or mathematical systems can be represented in terms of another but that, there is some coding that would permit every sufficiently complicated physical or mathematical system to be represented in terms of any other. This of course, brings us back to Descartes, who wanted to understand the relationship between the mind and the brain. Certainly the concept of information is not the whole answer, but it does gives us a language in which to ask the question that was not available to Descartes. Nevertheless, without his first discovery of a correspondence between two systems of relations, we would not only lack the possibility of talking about information, we would not have most of mathematics, we would not have telecommunications and we would not have the computer. This the notion of a representation is not only the most important mathematical invention, it is the idea that made it possible to conceive of many of the other important inventions of the last few centuries.