The letter ‘X’ is a problem. For all that the letter ‘x’ is important to mathematics there aren’t many mathematical terms starting with it. Mr Wu, mathematics tutor and author of the MathTuition88 blog, had a suggestion. Why not 90s it up a little and write about an Extreme theorem? I’m game.
The Extreme Value Theorem, which I chose to write about, is a fundamental bit of analysis. There is also a similarly-named but completely unrelated Extreme Value Theory. This exists in the world of statistics. That’s about outliers, and about how likely it is you’ll find an even more extreme outlier if you continue sampling. This is valuable in risk assessment: put another way, it’s the question of what neighborhoods you expect to flood based on how the river’s overflowed the last hundred years. Or be in a wildfire, or be hit by a major earthquake, or whatever. The more I think about it the more I realize that’s worth discussing too. Maybe in the new year, if I decide to do some A To Z extras.
Extreme Value Theorem.
There are some mathematical theorems which defy intuition. You can encounter one and conclude that can’t be so. This can inspire one to study mathematics, to understand how it could be. Famously, the philosopher Thomas Hobbes encountered the Pythagorean Theorem and disbelieved it. He then fell into a controversial love with the subject. Some you can encounter, and study, and understand, and never come to believe. This would be the Banach-Tarski Paradox. It’s the realization that one can split a ball into as few as five pieces, and reassemble the pieces, and have two complete balls. They can even be wildly larger or smaller than the one you started with. It’s dazzling.
And then there are theorems that seem the opposite. Ones that seem so obvious, and so obviously true, that they hardly seem like mathematics. If they’re not axioms, they might as well be. The extreme value theorem is one of these.
It’s a theorem about functions. Here, functions that have a domain and a range that are both real numbers. Even more specifically, about continuous functions. “Continuous” is a tricky idea to make precise, but we don’t have to do it. A century of mathematicians worked out meanings that correspond pretty well to what you’d imagine it should mean. It means you can draw a graph representing the function without lifting the pen. (Do not attempt to use this definition at your thesis defense. I’m skipping what a century’s worth of hard thinking about the subject.)
And it’s a theorem about “extreme” values. “Extreme” is a convenient word. It means “maximum or minimum”. We’re often interested in the greatest or least value of a function. Having a scheme to find the maximum is as good as having one to find a minimum. So there’s little point talking about them as separate things. But that forces us to use a bunch of syllables. Or to adopt a convention that “by maximum we always mean maximum or minimum”. We could say we mean that, but I’ll bet a good number of mathematicians, and 95% of mathematics students, would forget the “or minimum” within ten minutes. “Extreme”, then. It’s short and punchy and doesn’t commit us to a maximum or a minimum. It’s simply the most outstanding value we can find.
The Extreme Value Theorem doesn’t help us find them. It only proves to us there is an extreme to find. Particularly, it says that if a continuous function has a domain that’s a closed interval, then it has to have a maximum and a minimum. And it has to attain the maximum and the minimum at least once each. That is, something in the domain matches to the maximum. And something in the domain matches to the minimum. Could be multiple times, yes.
This might not seem like much of a theorem. Existence proofs rarely do. It’s a bias, I suppose. We like to think we’re out looking for solutions. So we suppose there’s a solution to find. Checking that there is an answer before we start looking? That seems excessive. Before heading to the airport we might check the flight wasn’t delayed. But we almost never check that there is still a Newark to fly to. I’m not sure, in working out problems, that we check it explicitly. We decide early on that we’re working with continuous functions and so we can try out the usual approaches. That we use the theorem becomes invisible.
And that’s sort of the history of this theorem. The Extreme Value Theorem, for example, is part of how we now prove Rolle’s Theorem. Rolle’s theorem is about functions continuous and differentiable on the interval from a to b. And functions that have the same value for a and for b. The conclusion is the function hass got a local maximum or minimum in-between these. It’s the theorem depicted in that xkcd comic you maybe didn’t check out a few paragraphs ago. Rolle’s Theorem is named for Michael Rolle, who proved the theorem (for polynomials) in 1691. The Indian mathematician Bhaskara II, in the 12th century, stated the theorem too. (I’m so ignorant of the Indian mathematical tradition that I don’t know whether Bhaskara II stated it for polynomials, or for functions in general, or how it was proved.)
The Extreme Value Theorem was proven around 1860. (There was an earlier proof, by Bernard Bolzano, whose name you’ll find all over talk about limits and functions and continuity and all. But that was unpublished until 1930. The proofs known about at the time were done by Karl Weierstrass. His is the other name you’ll find all over talk about limits and functions and continuity and all. Go on, now, guess who it was proved the Extreme Value Theorem. And guess what theorem, bearing the name of two important 19th-century mathematicians, is at the core of proving that. You need at most two chances!) That is, mathematicians were comfortable using the theorem before it had a clear identity.
Once you know that it’s there, though, the Extreme Value Theorem’s a great one. It’s useful. Rolle’s Theorem I just went through. There’s also the quite similar Mean Value Theorem. This one is about functions continuous and differentiable on an interval. It tells us there’s at least one point where the derivative is equal to the mean slope of the function on that interval. This is another theorem that’s a quick proof once you have the Extreme Value Theorem. Or we can get more esoteric. There’s a technique known as Lagrange Multipliers. It’s a way to find where on a constrained surface a function is at its maximum or minimum. It’s a clever technique, one that I needed time to accept as a thing that could possibly work. And why should it work? Go ahead, guess what the centerpiece of at least one method of proving it is.
Step back from calculus and into real analysis. That’s the study of why calculus works, and how real numbers work. The Extreme Value Theorem turns up again and again. Like, one technique for defining the integral itself is to approximate a function with a “stepwise” function. This is one that looks like a pixellated, rectangular approximation of the function. The definition depends on having a stepwise rectangular approximation that’s as close as you can get to a function while always staying less than it. And another stepwise rectangular approximation that’s as close as you can get while always staying greater than it.
And then other results. Often in real analysis we want to know about whether sets are closed and bounded. The Extreme Value Theorem has a neat corollary. Start with a continuous function with domain that’s a closed and bounded interval. Then, this theorem demonstrates, the range is also a closed and bounded interval. I know this sounds like a technical point. But it is the sort of technical point that makes life easier.
The Extreme Value Theorem even takes on meaning when we don’t look at real numbers. We can rewrite it in topological spaces. These are sets of points for which we have an idea of a “neighborhood” of points. We don’t demand that we know what distance is exactly, though. What had been a closed and bounded interval becomes a mathematical construct called a “compact set”. The idea of a continuous function changes into one about the image of an open set being another open set. And there is still something recognizably the Extreme Value Theorem. It tells us about things called the supremum and infimum, which are slightly different from the maximum and minimum. Just enough to confuse the student taking real analysis the first time through.
Topological spaces are an abstracted concept. Real numbers are topological spaces, yes. But many other things also are. Neighborhoods and compact sets and open sets are also abstracted concepts. And so this theorem has its same quiet utility in these many spaces. It’s just there quietly supporting more challenging work.
And now I get to really relax: I already have a Reading the Comics post ready for tomorrow, and Sunday’s is partly written. Now I just have to find a mathematical term starting with ‘Y’ that’s interesting enough to write about.