rationalskepticism.org

Bernoulli wrote:Regarding mathematical logic, I really don't know what came first - the philosophical chicken or the mathematical egg? Anyone have an opinion? My intestines are telling me that philosophy is responsible for mathematical logic.

I've got stories, but I advise that no-one takes them too seriously. The history of ideas is more complex than this.

George Boole is often credited with kicking off mathematical logic in Laws of Thought, but you could just as easily say that he kicked off abstract algebra. Boole could probably only achieve what he did because of the introduction of abstract symbolic notation, and because he lived at a time when mathematics was becoming much more inventive and free to explore arbitrary structures: in that climate, he was able to notice that addition, multiplication and negation look suspiciously like "or", "and" and "not", and then run with the idea.

In the same century, you have mathematicians trying to get back to the levels of rigour that the ancient Greeks enjoyed, and so you have a return to the axiomatic method. With the availability of new notation, and Boole's insight that logic might be algebraic, you then get the Italian school led by Peano who try to come up with a universal notation for all mathematical arguments, a sort of idealised mathematics. Critics at the time, such as the great Poincare, suggested that Peano should spend less time inventing needless notation and more time inventing space filling curves, and to be honest, I think he had a point. In any case, we still use most of the Italian school's notation.

Around the same time, there are some plausible attempts at mechanising logic, with Stanley Jevons inventing a "logic piano" that could check syllogisms, and it seems the idea was prevalent that, if you wrote your mathematical arguments out symbolically, you could build machines that would check or even figure the arguments out for themselves. This harking back to Leibniz' goals for all argumentation predates working computers by a few decades.

Then there's Frege, who probably invented the first ever formal system for maths, and who said that every mathematician is at least half a philosopher and every philosopher at least half a mathematician. However, his contribution with his formal system was primarily targeted at mathematicians. Contrast this with Aristotle, whose pathetic system of syllogisms was clearly not up to the task of handling the geometry of his time. And Frege wasn't just waffling. His ideas about how to encode induction schemes in higher-order logic are still used today by computer scientists and mathematical logicians.

Then there's some bloke called "Russell", who is watching all of this carefully, inspired by the work being done by the Italian school, and particularly impressed by Frege, until he showed that Frege's formal system is bust by some sort of paradox. Frege doesn't recover from this, and decides to blame the Jews for his failings. Russell has more fortitude, and puts in a bunch of fixes to Frege's system and then starts working through foundational constructions in maths. The result is the Principia Mathematica, a text of surprising fame considering that I doubt hardly anyone has read it, that Poincare said it was a waste of time, and that Russell ended up conceding the same once computers were invented that could generate all his proofs automatically. Nevertheless, the Principia was the first text to crystalise precisely what formalised mathematics would look like, and could thus become a definite target for metamathematicians such as Goedel.

However, metamathematics was actually first invented by the mathematician David Hilbert contemporaneously with the Principia, as the mathematical field studying axiomatic systems.

So mathematical logic grew out of mathematics, though the sort of people who pursue mathematical logic tend to be sympathetic to navel-gazing. There were critics of Cauchy and Bolzano in the 18th century who regarded their analysis of limits in terms of epsilon and delta as being "mere philosophy." Nowadays, the navel-gazers are the category theorists.

I sit in an IRC channel devoted to mathematical logic. We occasionally get philosophy types show up, but they soon find out that if they can't phrase their question in the form of a mathematical problem, no-one in the channel will be interested. Mathematical logic is just a branch of maths, and there's a story that it always was.