SUNY Geneseo Department of Mathematics
Monday, February 1
Math 239 03
Spring 2021
Prof. Doug Baldwin
(No Previous Lecture)
Welcome to Proofs, Math 239.
I’m Doug Baldwin.
(No.)
Mathematicians have discovered lots of strange and beautiful things over the years, to appreciate which you need to be able to think like a mathematician.
Some of the strange and beautiful things you’ll be able to understand by the end of this course, in more or less the reverse of the order we’ll get to them, are…
Infinity. Intuitively, it’s a notion of “going on forever.” But do some things go on forever more than others do, i.e., are there bigger and smaller infinities? It turns out that there are, for example the infinite number of real numbers is bigger than the infinite number of integers. And yet at the same time, “bigger” and “smaller” infinities are not at all intuitive. For example, you would think that the number of points in a square (a “squared infinity” if you will) would be smaller than the number of points in a cube with that square as its base. And yet it isn’t — the two infinities are the same.
Sets and functions. The key to talking sensibly about “infinity,” and in particular doing so in a way that lets us to do things like comparing different infinities, is to formalize “infinity” as a generalization of the idea of the size of a set. Intuitively, a set is a collection of things, for example {a, b, c, d} is a set of letters, and its size is 4. You can also have sets whose size is intuitively infinite, for example the set of positive whole numbers {1, 2, 3, 4, ...}. So “size of a set” is an idea that seems to cover both finite numbers and infinity, thus bringing a certain unity to the two ideas. Functions, in particular functions that establish an equivalence or “mapping” between the elements of two sets in certain ways, are the key to defining when 2 sets have the same size.
But is the informal notion of a set as a collection of things a sufficient definition of “set”? It turns out the idea isn’t quite that simple.
(Interesting historical digression: Difficulties in notions of set, and for that matter logic underlying math more broadly, came to light when Bertrand Russell and Alfred Whitehead tried to give a rigorous logical foundation to all of mathematics in the early 20th century. This was a time when lots of math was being formalized, and so it seemed plausible that there should be some small but powerful foundation for the whole field. But as Russell and Whitehead tried to define that foundation, they found paradoxes lurking in their work, and the more they tried to resolve the paradoxes, the more new paradoxes kept popping up. Eventually, in the early 1930s, Kurt Gödel showed that it is in fact impossible to have a mathematical system that can deduce everything that’s true but not deduce anything false (another surprising result, but one we won’t get to in this course — take Math 301, the mathematical logic course, if this sounds interesting). So attempts to find a consistent foundation for all of mathematics as we know it can never succeed.)
In the case of sets, let’s think about slightly more complicated collections, in particular collections that contain other collections. There’s nothing a priori wrong with this, for example {2, 4, {3,5}, 8, {a,b,c} } is a set that contains 2 other sets. If we can talk about sets that contain other sets, surely we can also talk about sets that don’t contain certain other sets. For example, sets that don’t contain the empty set include {1,2,3}, {a,b,c,d}, {2, 4, {3,5}, 8, {a,b,c} }, etc., but not, for example, { 2, 4, {}, 3 }. But continuing this line of thinking leads to something called “Russell’s Paradox”: what about the set P (for “paradox”) of sets that don’t contain themselves? In particular, is P in P, or not in P (it has to be one or the other)? Well, if P is in P then P contains itself, which violates the definition of P, so clearly we were wrong, P is not in P. But then P would be a set that doesn’t contain itself, which meets the definition of P, so P would be in P. Either way, we have an impossibility, namely P is in P exactly when it isn’t.
The surprising result here is that even simple ideas like “set” can have unexpected pitfalls if not considered carefully. So the first things we’ll look at in this course are mathematicians’ tools for “considering carefully,” namely proofs, and the logic on which they rest.
We’ll start with basic logic.
But first we need to be clear on how the course will work. So…
When you enter this course from the Canvas dashboard, you’ll see a front page that displays the 3 most recent course announcements, and buttons that link to key pieces of content. You can also use Canvas’s own navigation menu, on the left side of the screen, to get to key content, and I actually find that more convenient than the buttons.
The most important course content is the Modules page, where everything you need for this course is laid out by topic, in chronological order. The 1-sentence take-away about Canvas and this course is therefore “go to the Modules page.”
The Syllabus page in Canvas is a summary of the complete syllabus, with links into it. Below this summary, there are some calendar links that I’ll try to use to give you short summaries of what to expect in upcoming classes and what you need to do to get ready for them.
I’ll use a lot of online discussion forums in Canvas to get discussions of course material started, and to ensure that you don’t miss out on discussions if you can’t come to in-person classes. There will be links to all of those discussions from the Modules page, or you can find them on the Discussions page.
Finally, I’ll use Canvas announcements as my main way to communicate with you about new materials being available, deadlines, etc.
Course policies and practices, i.e., the syllabus.
Please read the syllabus.
Also start identifying questions about it, important points, etc. in this Canvas discussion.