To borrow notions from computers, mathematical logic can be regarded as the “machine language” for mathematicians who usually use much higher languages and who do not worry about “compilation.”Of course there would be analogues to the fact that certain computer languages are higher-level than others as well. To take an example dear to me, the theory of generating functions might be at a higher level than the various ad hoc combinatorial arguments it's often introduced to students as a replacement of. I don't want to press this metaphor too hard because it'll break -- I don't think there are analogues to particular computer languages. But feel free to disagree!
03 December 2008
Logic as machine language
Gil Kalai mentions a metaphor I hadn't heard of before about the foundations of mathematics: