After reading the very interesting Examples of mathematical discoveries which were kept as a secret I came to think of something: most math discoveries seem to have been made centuries ago, and with Pascal and Leibniz we already had machines that could add and multiply (some consider them as the first computers). Of course modern computers needed electricity and electric signals that can be converted to 1's and 0's, but those we have since the 19th century. First computers were also built with transistors, but we have those since the 1940's.
Modern computers are, at their core, not so different from the first computers that followed the Von Neumann architecture (with its arithmetic-logic unit). That is: they know how move data from one place to another, how to add and how to multiply, as well as perform logical operations (and, or, xor, not), and from that they get all the other operations (some can substract, divide and do some other stuff, but it's all very primitive in that sense). Of course, modern computers work at a much higher level than their predecessors: while in the beginning programming had to be done using commands composed of 0's and 1's or, if they were lucky, using assembly language, now we have high level programming languages that (to say it roughly) enclose a bunch of those single commands into one high level command.
And when I see pretty graphics in video-games, programs like photoshop, 3D rendering, CAD programs and such, I always think of all the calculus stuff I learned and how it must be applied to achieve those wonderful results.
And this is where my question arises: all of this mathematical knowledge has been available to us for way longer than computers existed. So, are there any modern mathematical discoveries that enabled the giant leap we took from the first computers we had in the 40's to what we have now? Or maybe old math started being applied in a different way at some point in time?