On Abstractions, Generalizations

I was battling through Walter Rudin’s Principles of Mathematical Analysis the other day when I came across the following theorem:

Theorem Suppose f is a continuous real function on a compact metric space X, and

Then there exist points such that and .

And then the proof is one line; it is a corollary of two other theorems:

1. If f is a continuous mapping of a compact metric space X into , then is closed and bounded. Thus, f is bounded.

2. Let E be a nonempty set of real numbers which is bounded above. Let y = sup E. Then . Hence if E is closed.

I was reminded of the same, albeit more specific theorem, in Spivak’s Calculus.

Theorem If f is continuous on [a, b], then there is a number y in [a, b] such that for all x in [a,b].

And then the proof

Proof f is bounded above on [a,b], which means that the set

is bounded. This set is not empty, so it has a least upper bound . Since for x in [a, b] it suffices to show that for some y in [a, b].

Suppose instead that for all y in [a,b]. Then the function g defined by

is continuous on [a, b] since the denominator is never 0. On the other hand, is the least upper bound of A; this means that

for every there is x in [a, b] with .

This, in turn, means that,

for every there is x in [a, b] with

But this means that g is not bounded on [a,b], contradicting continuous functions are bounded.

Phew. The more complicated proof from real analysis is significantly shorter than the one from calculus, even though the one from analysis is actually strictly stronger. I could do a similar writeup for the calculus proof of the fundamental theorem of algebra vs. the complex analytic one using Liouville’s theorem, but the calculus proof would be like 10 hours of typing.

Staedtler Riptide: A Review

Staedtler is the kind of company, like BMW, that you wish were an American company. Y’know, like something to be proud of? Like BMW — oh wait. Mercedez — oh wait. Budweiser? — right.

I’ve only owned a couple Staedtler pens in my life, so when I saw these Staedtler pencils in the teeny little university bookstore, I knew I had to try them. There are only a few criteria I look for in a writing utensil:

1. Comfort of grip

The grip must be comfortable! You have to be able to write out the Laplacian in spherical coordinates without your hand cramping up.

2. Ease of use of mechanical features

This includes things like clicking a pen, removing a cap, or in this case, advancing the lead.

3. Weight and shape

This is the most important. It is absolutely essential that my writing utensils be weighted and shaped so that they can be easily spun.

So how does it stack up? Well for (1) it’s pretty average. The grip is just a piece of rubber that wraps around near the head. It’s not particularly awful, but it’s nothing special.

(2) is the pencil’s weakest point. Clicking to advance the lead is an arduous process. It almost feels like there’s something rusty in there.

(3) is the pencil’s strongest point. It’s a natural width, and is shaped a lot like a standard Bic pen. But the weight—oh the weight. The weight is heavenly. These mofos are heavier than a normal mechanical pencil, and the weight is almost perfectly distributed. I daresay they compare to the Bic Softfeel Med.

Did I mention it was less than $2 for three of them?

On Collisions: Math and Colloquial Speech

So I was remembering this time in, say, fourth grade when the teacher posed the following question:

How many numbers are between ten and twenty?

Of course, this problem is well within the skills of a fourth-grader. Simply subtract ten from twenty and you get ten. In fact, the only appreciable difficulty is in the interpretation of the question. Students, particularly children, struggle with these so-called “word-problems”: problems where the math is relatively simple, but the problem is phrased as a question. You know, using words and stuff.

Being who I am, I completely failed at understanding the question; I answered, “Eleven.”

So I got laughed at… But to this day I maintain that 11 is just as accurate as—indeed more accurate than—10. Why? It all lies in the interpretation of the word “between.” If “between 10 and 20″ means

Then yes, there are 10 such numbers: . But who the hell uses “between” to mean “including the upper limit but excluding the lower limit”? In my opinion, there are only two reasonable ways to interpret “between ten and twenty”:

1. 
2. 

But then the correct answer is either or . I chose option (1) in fourth grade, but option (2) is perfectly reasonable.

The problem is that this becomes no longer strictly a subtraction problem. What the teacher wants is for the students to compute . But this is wrong, so how do we reinterpret the question?

I propose the following: first consider the question, “what are the numbers between ten and twenty?” This question still has the ambiguity of the word “between,” but no person in his right mind would answer:

or

Then we can ask, after the set of numbers has been identified, “how many of them are there?” I think if the problem were posed this way, no one would ever arrive at an answer of eleven (unless they miscount).

Further Rambling

I’ve decided that word problems like the ones we’re given in grade school don’t really count as “math.” I think they’re physics. Consider the topics covered in college- and higher-level mathematics. They’re abstract and almost completely disconnected from the real world. A mathematician doesn’t care if we live in Euclidean space or a Minkowski space; if they can prove something about both, then that is interesting.

A physicist (or statistician maybe) concerns himself with questions like these word problems—problems dealing with possible real-world scenarios. When we answer such questions, we don’t really learn a lot about math; rather, we learn about the connection between math and the world we live in.

That’s not to say word problems aren’t good or helpful, but I think school teachers should be very careful both in posing questions and in receiving answers. What if my teacher had let me explain my logic? It would have been nice to shut up the kids who laughed at me for a “stupid” answer, and they might learn something at the same time.