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January 22 - February 22, 2018
If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.
The edges are perfect because I want them to be—that is the sort of object I prefer to think about. This is a major theme in mathematics: things are what you want them to be. You have endless choices; there is no reality to get in your way.
On the other hand, once you have made your choices (for example I might choose to make my triangle symmetrical, or not) then your new creations do what they do, whether you like it or not. This is the amazing thing about making imaginary patterns: they talk back!
By concentrating on what, and leaving out why, mathematics is reduced to an empty shell.
Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity—to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs—you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.
Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions.
It would be bad enough if the culture were merely ignorant of mathematics, but what is far worse is that people actually think they do know what math is about—and are apparently under the gross misconception that mathematics is somehow useful to society! This is already a huge difference between mathematics and the other arts. Mathematics is viewed by the culture as some sort of tool for science and technology. Everyone knows that poetry and music are for pure enjoyment and for uplifting and ennobling the human spirit (hence their virtual elimination from the public school curriculum), but no,
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Mathematics should be taught as art for art’s sake. These mundane “useful” aspects would follow naturally as a trivial by-product.
The saddest part of all this “reform” are the attempts to “make math interesting” and “relevant to kids’ lives.” You don’t need to make math interesting—it’s already more interesting than we can handle! And the glory of it is its complete irrelevance to our lives. That’s why it’s so fun!
What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources—beautiful works of art by some of the most creative minds in history—in favor of third-rate textbook bastardizations?
The main problem with school mathematics is that there are no problems.
The trouble is that math, like painting or poetry, is hard creative work. That makes it very difficult to teach.
We learn things because they interest us now, not because they might be useful later. But this is exactly what we are asking children to do with math.
If I object to a pendulum being too far to one side, it doesn’t mean I want it to be all the way on the other side. But the fact is, people learn better when the product comes out of the process. A real appreciation for poetry does not come from memorizing a bunch of poems, it comes from writing your own.
If you don’t have enough of a feeling for your subject to be able to talk about it in your own voice, in a natural and spontaneous way, how well could you understand it?
A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light—it should refresh the spirit and illuminate the mind. And it should be charming.
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The essence of computer programming?
To make a definition is to highlight and call attention to a feature or structural property. Historically this comes out of working on a problem, not as a prelude to it.
That’s maybe the biggest reason why I do mathematics. Nothing I have ever seen or done comes close to having the transformative power of math. My mind gets blown pretty much every day.
We have discovered a pattern, and we think it continues. We could even verify that it works for the first trillion cases if we wanted. We could then say that it’s true for all practical purposes, and be done with it. But that’s not what mathematics is about. Math is not about a collection of “truths” (however useful or interesting they may be). Math is about reason and understanding. We want to know why. And not for any practical purpose.
Sometimes I like to imagine a Two-Headed Monster of mathematical criticism. The first head demands a logically airtight explanation, one with absolutely no gaps in the reasoning or any fuzzy “hand-waving.” This head is a stickler, and is utterly merciless. We all hate its constant nagging, but in our hearts we know it is right. The second head wants to see simple beauty and elegance, to be charmed and delighted, to attain not just verification but a deeper level of understanding. Usually this is the more difficult head to satisfy. Anyone can be logical (and in fact, the validity of a deduction
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To say that math is important because it is useful is like saying that children are important because we can train them to do spiritually meaningless labor in order to increase corporate profits.
Again, the right thing for me to do as your math teacher would be nothing. That’s a thing most teachers (and adults generally) seem to have a hard time doing. Were you my student (and assuming this problem interested you) I would simply say, “Have fun. Keep me posted.” And your relationship to the problem would develop in whatever way it would.
Is that great, or what? I only wish I could see your face—to see if your eyes light up, and to make sure that you get the joke, so to speak.
What I’m saying is that proofs are bigger than the problems they come from. A proof tells you what really matters and what is mere fluff, or irrelevant detail; it separates the wheat from the chaff. Of course, some proofs are better than others in this regard. Often a new argument is discovered that shows that what was previously thought to be an important assumption is in fact unnecessary. I suppose what I’m really trying to say here is that mathematical structures are designed and built not so much by us, as by our proofs.
The historical development of mathematics (especially in the past couple of centuries) exhibits a consistent, undeniable pattern: first come the problems, whose sources are many and varied, often inspired by the real world. Eventually, connections are made between diverse problems, usually due to common elements that appear in various proofs. Abstract structures are then devised that can “carry” the kind of information that forms the connection
You don’t need a license to do math. You don’t need to take a class or read a book. Mathematical Reality is yours to enjoy for the rest of your life. It exists in your imagination and you can do whatever you want with it.
And if you are a math teacher, then you especially need to be playing around in Mathematical Reality. Your teaching should flow naturally from your own experience in the jungle,
If you love working with children and you really want to be a teacher, that’s wonderful—but teach something that actually means something to you, about which you have something to say. It’s important that we be honest about that. Otherwise I think we teachers can do a lot of unintentional harm.
