"Similarly Complicated", and other mathematical niceties, Part 2

But the first thing that made me want to write today happened much earlier, in one of my favorite classes – Algebra. I could talk about this subject all day. But I'm going to talk about one kind of proof in particular, of which we just happened to have an example today, and also about one of the funny comments made by my slightly crazy professor.

Math is all about proofs once you get done with the stuff they make you do in high school. Proofs are about, well, proving that something is true or not true, proving that a certain object has certain properties, proving that certain sets of objects all have something in common, or proving that various definitions or statements when combined lead to a much stronger statement. One fun type of theorem (that leads to a fun type of proof) has the following syntax:

Theorem (Name): Let X be a thing. Then the following statements are equivalent:

(i) X has property A.

(ii) X has property B.

(iii) X has property C.

(iv) X has property D.

“The statements are equivalent”– What does this mean? This means if you successfully prove something like this and you come across a beast (in the mathematical wilderness) with the quality of B, then you automatically know that it also certainly has the qualities A, C, and D. That’s the whole damn point. If I know that when bears (X) have blue fur (A), they also automatically are vegan (B), then if I see a blue bear I don’t have to ask it if it’s vegan or not. I already know, since A implies B.)

And how do you prove such a thing as this? Well, you start with whichever of the options A-D you are most comfortable with, say property A. Assume X has property A. Then you have to prove that if X has property A, it must also have property B. From there, show that if X has property B, then it must also have property D, and so on. At the end, you need to lead back to the property you started with — complete the circle. It doesn’t matter what order you go in as long as you can get from any one statement (A-D) to any other statement by the time you’ve finished.

If this is confusing, don’t worry. The whole point of this type of proof is this: Let’s say you come across a bear in the wild (yes, we’re back to the bear). You really want to know what blood type it has, but you are (understandably) hesitant about taking a blood sample. But because Blood Type = O Negative happened to be property D in the theorem you just proved, you know that since the bear has blue fur, it must have blood type O Negative. You saved yourself a lot of work (and maybe your life).

This style of theorem and proof has always fascinated me. Even though the real bulk of the proof might just be to find an easier way to show property D (like the example above), you still get to explore the other properties along the way, and maybe derive some neat consequences.

(If you are interested in these mathy things, please write to me - I can give examples that don’t involve bears, or more examples involving bears - or just talk about all these things. :) )

So, this is the type of proof we were tackling in Algebra at 8:00 this morning. Our professor rolled up his sleeves and wrote the whole theorem statement on the board, and then wrote those great letters: “Proof:” and we waited for what was to come next. “A implies B” he wrote. He turned to us. “This is a great proof,” he said. “Ready?” Collective, halfhearted nods from the crowd. Then he wrote “Obvious.”  on the board. And that was it.

After another one of the implications (say, C implies D) in the proof was also just “obvious” (or, another favorite word of mathematicians, “trivial”– oh, how this word sounds to math students. Can you see the self esteem shriveling into nothing?), we had to have a quick discussion about it. Here’s what he told us.

First of all, these statements “are only ‘obvious’ if you know which definitions or other theorems to apply and in which order”. But once you do, the answer is clear. Second of all, it’s not “supposed to be obvious to you today when you hear it in the lecture”. He continued: “The difficulties are hidden in the notation.” Quite clever and quite true, not only of mathematical things. 

 And finally, the real stuff:

“What I do here in the lecture is 10% of the work, of the material.” 

I think this is what a lot of people don’t realize about math courses and that is a reason that many don’t do well in them. Just the notes from the lecture, just passively writing them down isn’t enough. Even just mechanically doing the homework isn’t enough. There is much, much more work — hidden in the details. You have to work with it enough to understand it from the inside, not just from the outside.

And that is exactly why I need to stop writing this right now and go read over my notes from this morning. Oy vey!

"Similarly Complicated", and other mathematical niceties, Part 1

Today's a Monday, which for me means four lectures in a row, almost without break (there is generally a fifteen minute break between classes, but with going to the restroom, maybe going to another building or buying something to eat, this vanishes quickly). Each lecture is two hours long (roughly), so it's quite a long block of note-taking from 8 a.m. to 4 p.m. But today, each of my math professors have just been so...mathematician-y that I felt the need to use one of my fifteen minute breaks to write this.

One such hilarious (to me) moment today already made its appearance in the title of this post. In Analysis II (very abstract calculus, roughly explained) this morning, we worked our way at one point through a very tedious example. Lots of indices — let me say what I mean by indices. Indices (plural of index) are the subscripts we use to know which entry of a particular vector we are talking about. 

For example, the vector x:

The first entry in that vector is 5, the second 4, and the third -1. In math speak, we would write x₁=5, x₂=4, etc. This vector has three entries in it, so we say it has length 3 . Alright? Bear with me now, it’s going to get complicated. First, imagine you have a vector of length n. What’s n? Could be anything. Got that? Kinda? Good. Now imagine that you don’t have just one vector of length n but a whole series (or list) of them. So, now we might want to talk about the entry at index i  of the kth vector in that sequence…You get the idea. Subscripts of subscripts of indices. 

Most people in class were either pulling on bits of their hair as they tried to keep track of all the tiny k's and n's and x's, or they were just staring into space after having given up. (Let me note -these kinds of details are not, in my opinion, what makes math difficult. They are not abstract concepts that you need to wrap your head around. Rather, they are the tiny details (that are indeed very important) that we use to be incredibly precise about what it is we are calculating and though they are not in themselves difficult, these tedious indices and things like them certainly can, and do, make math problems discouraging). Finally, we finished the example in the lecture and the professor smiled at us and said we were about to do a different example now that was "ähnlich kompliziert", which means 'similarly complicated'. Also complicated, and in exactly the same way. Sometimes phrases just catch my eye (or ear) and I think they are worth noting in their perfect suitableness.