I have this vague notion that thinking about the Liar paradox has led to the development of important modern technologies. But I’m not sure whether this impression comes from something I read, or something I speculated a long time ago, or what.
The vague idea is that thinking about the Liar paradox inspired diagonalization and incompleteness arguments, which in turn led to results in the theory of computation, which in turn had an effect on our ability to design computing devices. Also, the theory of infinite cardinalities (enabled by diagonalization reasoning) led to advances in set theory which led to advances in math which helped out in Physics.
Is any of this true? Or am I misremembering something or making it all up?
Well, The Liar is a paradox of self reference (possibly the most basic one), and is thereby related to Russell’s paradox (“Who shaves the barber?”), which played an important role in stimulating the development of modern symbolic logic, in Principia Mathematica (and, later, Wittgenstein’s invention of truth tables, in the Tractatus). That is all pretty important for the development of computing.
I guess Gödel’s incompleteness theorem also has to do with the pitfalls of self reference.
I am not sure if that is what you are looking for.
I would have thought diagonalization, which (inasmuch as I understand it) led to the formalization of the notion of different types of infinities, owed more to Zeno’s paradoxes.
Obviously the study of paradoxical statements led to the development of formal logic. This sounds like the kind of story told on the show Connections. It is a set of links on the path from the ancient marketplace to modern science and technology. I’d think there would be a lot of other intermediate advances along the way that were as critical to the end result as those.
It might be more accurate to say that both Russell’s paradox and computers stem from the same drive, in the latter 19th Century, to regularize and formalize all of mathematics (and, ultimately, all of thought) by placing it on a sound foundation known as ‘set theory’. In both cases, the basic question is the same: How much can we mechanize thought? What kinds of things can machines think about? The existence of Russel’s paradox on one hand, and the insolubility of the Halting Problem on the other, means there are very real, very firm limits to what machines can do.
This is also connected to Alice in Wonderland: In his Charles Lutwidge Dodgson persona, Lewis Carroll was a mathematician who saw the then-new field of non-Euclidean geometry (which is, depending on your perspective, either doing all the usual geometric things by drawing on spheres and saddles instead of flat planes, or a huge revolution in our knowledge of how mathematics works) as a huge bunch of nonsense, like having to run at a tremendous rate simply to stay in the same place, or a cat disappearing and leaving its smile behind.
I don’t think that the development of computers had that much to do with mathematical logic (or the paradoxes that inspired it). Read the Wikipedia entry on it:
It suggests that the development went like this:
Tally stick
Abacus
Astrolabe
Slide Rule
Mechanical calculator
Jacquard’s loom
Hollerith cards for census
Cash registers
World War II computing equipment for codebreaking and other things
Commercial computers
It’s true that there were people who worked on the development of computers who also were experts on mathematics and logic, like Alan Turing. There were also people who worked on them that had no knowledge about or interest in such things, like Konrad Zuse. Even without the existence of mathematical logic as an academic discipline, someone like Zuse would have seen the usefulness of automating calculations and eventually have developed the computer along different lines. For that matter, Charles Babbage developed the idea of a general-purpose computer in the nineteenth century without having any particular interest in mathematical logic or logical paradoxes or whatever. The reason that he couldn’t actually build one was his inability to get past the hardware problems, not any problem with the design.