Guessing the Teacher's Password
overcomingbias.com
overcomingbias.com
If you glance at my answers to problem sets 4, 5, and 8 then you'll get the general idea of what the course involves. The problem sets are really fun because you're forced to play with the numbers and really understand the material. It might not be as hard as a traditional physics course, but it's every bit as rigorous in the real sense of the word. It's brilliance is that it really stresses the fundamentals, probably more so than any other physics material in existence. This is important because, as one professional Go player said after observing a Japanese baseball game,
"In every confrontation with a real American professional team it seems that what we need to learn from them, besides their technique of course, is how uniformly faithful their players are to the fundamentals. Faithfulness to the fundamentals seems to be a common thread linking professionalism in all areas." (T. Kageyama)
It starts assuming nothing more than basic arithmetic, and builds up to differntial equations, tensors, string theory, quantum relativity, etc.
I'm warning you up front that it's very math heavy. It takes some work to get through, but its worth it.
"Oh," said I, "that's interesting. What does it mean for a form to be transcendent or immanent?"
"I don't know, I just know that's what they thought."
We then had a discussion about whether or not he really knew that, and whether or not he was OK with paying over $25k a year to learn how to be a parrot.
The sad thing: he was OK with it, and probably rightly so. He realized that mostly what people want are parrots and you can get paid quite a decent amount of money for being a good parrot. Not a bad gig if you're fine being a cog. There are certain comforts it provides.
High school science competitions (school science fairs, all the way upto Siemens-Westinghouse and Intel STS) particularly guilty of this.
I remember I just -couldn't- accept that we can't take square roots of negative numbers. I mean, sure, two negatives multiplied together is a positive, but...there HAS to be a way! Numbers, to my 6th grade mind, weren't arbitrarily going to deny your ability to manipulate them.
Fortunately for me, I happened to be reading a book on Mandlebrot and in the course of discussing his work the author introduces the concept of imaginary numbers. So the next day in class I asked: "Are you suuuure?"
"Yes, I'm sure."
"Well, I was reading in this book and so-and-so says you can take the square root of negative numbers, you get 'i' for the root of -1 and..."
"Why don't you go wait outside."
Where I got told that although I move to the beat of a different drummer, when I'm in her class, I really just need to be quiet. That's when public education and I had a falling out from which our relationship has never recovered.
All of that to say: I think part of the 'problem' is that kids who do try to think on their own are often in a situation where they're challenging accepted rules without the social grace to do so constructively. But rather than being taught the social skills, they're just shut down. Very sad, actually. They could be taught how to 'wander well'.
Am I wrong there?
i is defined as sqrt(-1), if that answers your question.
My point is that sqrt(-1) = i is an extra assumption that you don't need to make in order to derive the math you do in school. If you don't make that assumption, you can still derive a lot of math. If you do make that assumption, then you can also derive the theorems of complex analysis.
I think that's true, anyway. I could be wrong.
It just seems strange to me that someone would be bothered by the lack of negative square roots, since their existence is never derived, only assumed. But then again, people's minds work very differently, especially in Mathematics.
I was the opposite in Math class. I fought accepting "i" when they tried to teach it to us since it seemed so arbitrary and contrived to me. This was before I discovered that all Math is arbitrary and that there is no "real math", anyway.
sqrt(-1) = sqrt(e^Pi) = e^(Pi/2) = i
So the new concept was really a 2D plane for numbers, not a new definition for sqrt of negative integers.
e^(i)(pi/2) = i.
The simple assumption that sqrt(-1) = i is problematic because you can get something like
i^2 = (sqrt(-1))^2 = sqrt((-1)^2) = 1.
The wonder and surprise of complex numbers is that assuming seemingly arbitrary properties of a constant like i lead to a number of deep and beautiful results.
I hadn't reached my correct answers the "right" way.
I would also lose points for "not showing my work", but if you can't factor, say, x^2-9 in your head, maybe you have deeper issues.
I also once got an answer marked wrong because the teacher couldn't understand my intermediate steps, even though the answer (and steps) were actually correct. When I showed him, he refused to change my grade. That was at University.
I realized that math in school was more like one of those simon games where you just mindlessly press the arbitrary sequence of lights you just saw, than actual mathematics. This analogy helped me suffer through "school math" while I perused real math on my own.