So I'm currently taking a history of math course and I need to write a 15 page paper in place of my final. It's a 400 level course (high undergrad) so the paper needs to have emphasis on the mathematics and not just the history.
I have been thinking about doing something with logic from the late 19th century to early 20th century, but am not quite sure how to really narrow it down.
I was thinking about doing something with the evolution of classical logic from in that time period Frege, Russell, Wittgenstein, Post, etc.. But that is too broad I think.
I was looking into Emil Post and his work on functional completeness and was thinking maybe something with boolean functions. Maybe incorporating that and show that a formal system can be done with just the sheffer stroke, but am I am not sure.
It seems there was a large push to really see how much could be done with very little (polish notation, Sheffer stroke, etc..) So maybe I could do something in regards to that?
I am fond of Wittgenstein's Tractatus, but I don't think there is too much solid logic there to really work with.
Russel and Whitehead's principia is a bit overwhelming.
I'm looking for advice for a topic that has to do with these things, but won't be too too difficult (I have many other things going on and this is my least highest priority..but important nonetheless)
(So I would not like to do anything with the incompleteness theorem, as I don't have the time to really devote myself to that)
Thanks
A few possibilities:
The development of the logic of relations: Schroeder's "relational algebra", through Frege, Principia, and Hilbert & Bernays. "Classical" logic (Aristotle's, Aquinas', (almost?) all logic before the advances of the later 19th century) dealt with monadic logic — one-place relations, suitable for syllogisms and the like but incapable of rendering even the definition of, say, continuity at a point or convergence of a sequence. As the example of set theory shows, stepping up to just a single two-placed relation ($\in$) confers an enormous increase in power.
Another possibility: history of choice principles and the meaning of "existence". In addition to the full Axiom of Choice (AC), various weaker principles have been used by mathematicians skeptical of AC. The French analysts Borel, Lebesgue, Baire used Countable Choice, and Dependent Choice, fairly freely (iirc this is true of all three of them), while at least one was (again, iirc) suspicious of full-blown AC.
The topic of choice principles intersects the broader topic of constructive vs nonconstructive proof (e.g. Intuitionism), which is certainly a debate about what it means to prove that a mathematical object exists. This broader topic in turn impinges on debates about predicativity, concerns about which caused Russell & Whitehead to gum up their work with cumbersome sub-hierarchies, only to blow them away with an axiom stating that the sub-hierarchies all collapse. Weyl wrote a book ("Das Kontinuum") in which he investigated how much of classical mathematics can be developed in a purely predicative formal system. Feferman's survey paper Predicativity is a good read in any case. The wikipedia article on the subject appears to be Impredicativity.