Do you have a vigor? A vigor for rigor?
August 8, 2016 11:34 AM   Subscribe

A colleague mentioned a college course he took on rigorously proving commonly handwaved aspects of math. Help me find books or a syllabus relating to this!

Much like Lyle Lanley's salesmanship, my mathematical education has a bunch of handwaving (e. g. how the fundamental theorem of calculus is usually presented). A colleague mentioned once that he took a very satisfying course (don't know if it was undergrad or not) that consisted entirely of taking foundational mathematical concepts that are presented with a bunch of flimflam and rigorously proving/defining them.

I want to reconstruct as much of that course, or something like it, as possible.

Do you know of any books dedicated to that or publicly available syllabi covering courses like that? Help me actually know things that I just pretend to know.

(Presented in the science & nature category rather than the more accurate philosophy & religion in the hopes of casting a wide net.)
posted by The Gaffer to Science & Nature (9 answers total) 18 users marked this as a favorite
I'd love to hear about a whole book or set of lessons like this, but your question reminds me of Paul Lockhart's A Mathematician's Lament (PDF), which does one lesson on the formula for area of a triangle and helps you fully understand it instead of just knowing a formula.
posted by rollcredits at 12:03 PM on August 8, 2016 [2 favorites]

The class you're looking for is Real Analysis - it's typically a higher-level undergrad math course which rigorously proves the fundamentals of calculus. It's also typically the first serous proof-based class math majors take.
posted by Jaclyn at 12:51 PM on August 8, 2016 [5 favorites]

Landau, Foundations of analysis . There seem to be lots of pdf's available. I seem to recall he was prompted to write it by his young daughter asking why 2+2=4. In any case, I absolutely loved it.
posted by Pre-Taped Call In Show at 12:51 PM on August 8, 2016 [1 favorite]

Yes, well, math is a big place with different foundations that you might find of interest. Googling "foundations of mathematics" brings up some texts. The foundations of number systems revolve around the Peano postulates on one hand and set theory on another hand. Problems with set theory led to a number of paradoxes by Russel and others.

For a long time Euclid's Elements was considered the foundation of plane geometry, but in the early 20th century, it was pointed out (by Hilbert?) That there were a tacit assumptions to account for.

Finally, the limitations of logical systems were discovered byvGodel and others.

I think the first thing is to pick which part of math you want to learn about.
posted by SemiSalt at 12:55 PM on August 8, 2016

You will find "Intro to Proofs" courses at most all universities in some form or another. For instance, here is UCLA's and MIT's. I suspect your friend took a class in number theory, which forms the basis of natural numbers starting at the Peano axioms. Many people find number theory to be quite enjoyable, partially because results can be proved very rapidly from a small number of base postulates.

This is quite distinct from the Fundamental Theorem of Calculus, which is usually taught in a "real analysis" course. This is a significantly more complex undertaking - in particular, it may take upwards of half a semester to an entire semester (depending on difficulty/rigor) to get to the Fundamental Theorem of Calculus.
posted by saeculorum at 12:57 PM on August 8, 2016

Response by poster: Thanks for the answers - to clarify, my friend's class was explicitly a survey across subfields of formally proving things that likely had been glossed over and assumed familiarity with formal proofs and a certain amount of education already.
posted by The Gaffer at 1:14 PM on August 8, 2016

It's funny that you bring up Lockhart, rollcredits, because he actually wrote a book that's pretty much just a series of lessons very similar to his explanation of how to find the area of a triangle in his famous Lament. It's called Measurement, and in addition to being the only mathematics text I've ever enjoyed reading, it's also one of my favorite nonfiction books, period. It covers pretty much all of traditional high school mathematics as well as calculus, but fills in all the gaps- why does a^2 + b^2 = c^2, why is the volume of a sphere equal to (4πr^3)/3?

Some caveats: Lockhart is very, very nontraditional in how he explains things, which I loved, but if you're looking for a "normal" mathematics textbook, look elsewhere. I'm nowhere near a mathematician so I can't vouch for how 'complete' a book it is in terms of the ground it covers. Also, the book is aimed towards a general audience- I can't gouge how experienced you are with math from your description, but if you have, like, a math PhD it's probably beneath you. I myself read it after having taken courses in pretty much all the material it covered, and I still got a hell of a lot out of it.

And a quick recommendation, if you do get it: it's fine to skip some (or all) of the problems, especially on a first reading.
posted by perplexion at 1:19 PM on August 8, 2016 [3 favorites]

Best answer: I took a class like this as a math undergrad and it was called "Foundations of Analysis." I second the suggestions above that you're looking for something that's called "Real Analysis" or or "Foundations of Analysis." I can't recall the book we used but a quick google of my alma mater's webpages show that they're using Abbott's Understanding Analysis.

Of all the classes I took in college, that one was uniquely memorable for the professor declaring--about halfway through a lecture a few weeks into the course--"As you can see, all of mathematics is based on a foundation of sand" (basically, you get down to a few certain assumptions such as parallel lines will never touch, and you have to assume those basic principles without proof).
posted by iminurmefi at 1:39 PM on August 8, 2016 [1 favorite]

You may want to look into Bertrand Russell's Principles of Mathematics, and the later Principia Mathematica which he wrote with Alfred Whitehead (and works/syllabi that cite them).

For some light? distraction reading, you may also be interested in Logicomix, which is a graphic novel about Russell, his philosphy, and the Principles.
posted by sparklemotion at 1:55 PM on August 8, 2016

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