# Ideas for a history of math paper (with an emphasis on the mathematics), having to do with 19/20th century logic?

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

• Do you aim to present a general overview, or would you like to find and present one specific problem (and its context)? Do you have an opinion on Hilbert's Problem? – Roland Mar 7 '16 at 23:05
• Well, something fairly specific, but I would still talk about it in a general way incorporating the impact/history of it. The topic could be broad, I suppose, but as long as I go fairly in depth with the logic/math and explain what's going on. I'll look over those problems now, thanks – Boolean_functions Mar 7 '16 at 23:14
• What about the history of the development of non-Euclidean geometry? – Brian Tung Mar 7 '16 at 23:31
• Your idea about Post and functional completeness is nice. It lets you go back to the beginnings of propositional logic with Boole in the 19th century and forward to universal algebra in mathematics and digital hardware in computer science in the second half of the 20th century. (I hate to think how many NANDs or NORs have been calculated to enable me to type this comment $\ddot{\smile}$.) – Rob Arthan Mar 8 '16 at 0:05
• @Rob Arthan, that's what I was thinking. I could go into the criteria for a set of connectives to be functionally complete, show it holds for sheffer stroke, and even work my way up to the importance of NAND within logic gates....but what worries me is the accessibility of Post's work. There also doesn't seem to be too much secondary info on it. There is a good amount on the crtieria of montone, linear, dual, etc... but not so much on proving that these are the criteria – Boolean_functions Mar 8 '16 at 0:10

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.