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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

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  • $\begingroup$ 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? $\endgroup$ – Roland Mar 7 '16 at 23:05
  • $\begingroup$ 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 $\endgroup$ – Boolean_functions Mar 7 '16 at 23:14
  • $\begingroup$ What about the history of the development of non-Euclidean geometry? $\endgroup$ – Brian Tung Mar 7 '16 at 23:31
  • $\begingroup$ 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 $\ddot{\smile}$.) $\endgroup$ – Rob Arthan Mar 8 '16 at 0:05
  • $\begingroup$ @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 $\endgroup$ – Boolean_functions Mar 8 '16 at 0:10
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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.

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  • $\begingroup$ Yea, I was also thinking about axiom of choice/Zorn's lemma type of thing. I didn't think of doing something on constructive v nonconstructive proof though, that could be interesting. thanks $\endgroup$ – Boolean_functions Mar 7 '16 at 23:33
  • $\begingroup$ You're welcome. I'm curious what you finally settle on -- update us :) $\endgroup$ – BrianO Mar 7 '16 at 23:35
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Another possibility is to consider some essay from "Essays on the Theory of Numbers by Richard Dedekind" (available from http://www.gutenberg.org/ebooks/21016 ). The essays are about what it is nowadays called foundations of mathematics, but at that time it was pretty close to logic. Indeed, one essay is about using Peano axioms (wrongly assumed to be introduced by Peano) to characterize natural numbers; and another one is about how characterize the real numbers (the challenge here is continuity).

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