I’ve recently seen many links to a site called “Better Explained”, purported to be a great resource on basic mathematics. I went there, looked at the topics offered, and chose what seemed like a tough one to explain well – the curl from vector calculus.
A conservative field is “fair” in the sense that work needed to move from point A to point B, along any path, is the same. For example, consider a river: it’s field is conservative. Sure, you can get a free ride downstream, but then you have to do work to get back to your starting point. Or, you can do work to move upstream, and get a free ride back. Either way, the amount of work you “put in” is the same as what you get back.
This is atrocious. For one thing, it misrepresents rivers, which flow more slowly by their banks than in their centers, and hence have non-zero curl. More seriously, it’s flapdoodle. As you float down a river, your speed stays the same, but the work you do to row back to your starting point increases. You don’t, in fact, put in the same as you get back. The passage takes an idea that’s correct – that fields of force with zero curl are conservative – and applies it inappropriately to a velocity field.
The article on the gradient was similarly bad, claiming incorrectly that the gradient is “just a direction”. (This is not taken out of context; it’s the article’s motif.) There is an article called “Understanding Exponents (Why Does 0^0 = 1?)” (0^0 is not 1. It is indeterminate.) An article on primes says as a bullet point “Primes appear randomly distributed” and later emends this with
We’ve tried for centuries to find a pattern, but we can’t. We have no idea where the gaps are or when the next prime is coming. (That’s not quite true — there’s interesting hypotheses and conjectures, but the riddle is not solved).
This ignores the prime number theorem, which certainly counts as a pattern, and is not a hypothesis or conjecture. Other articles, if not outright incorrect, are dubious. The articles on Bayes’ theorem and calculus, for example, confuse the idea being explained with an example of it.
I decided that site was a “do not like”. Once I made this decision, it seemed to compound. I disliked the presumption of a site calling itself “better”. I thought the proportion of metadiscussion was too high and self-indulgent. I summarized the site to myself as the work of an amateur, implying simultaneously the word’s most elevating sense and its most demeaning.
I continued, reading an article on the Pythagorean theorem, looking for something to rip apart. On finishing, I felt a deflated. It was a decent article. It wasn’t novel to me, but it did have an interesting insight. The pictures were nice. The language was clear. “Well, that’s okay,” I thought. “The errors in those other articles were bad enough to make up for it.”
That’s when I realized I had gone insane. I was upset because something was good when I wanted it to be bad.
I sat a while in perplexity, backtracking. I wanted to know how I arrived at such a ludicrous mental state. Of course I came to no firm conclusions. I never have, when thinking about such slippery psychological matters. (Actually, sometimes I come to firm and brilliant conclusions, which I then view as naive and stupid a few months later, when the insight losses its luster and proves not to have led me to the enlightenment I expected.)
Everyone seemed to like “Better Explained”. Whenever I saw it linked, it was with high encomiums. “Better Explained” is popular and lauded, and because the site is about something I care about – making mathematics intuitive – I wanted a bit of that attention for myself. I implicitly felt that praise is conserved, so more of it going to the other guy meant less for me.
It disturbed me that my opinion of the site was so strongly influenced by everyone else’s. I’m perfectly competent to judge the site for myself, at least at first blush. Reading others’ reviews makes sense if it leads you to interpret a work from a new viewpoint, but I had switched to a completely different realm. I was reacting directly to other people’s opinions, to the point where the content of the site itself was almost incidental.
It also felt that my estimation of the site as “bad” had too much inertia. Once I decided the articles were inexpert and misleading, I expected the same from upcoming articles, which is reasonable. But I also wanted that from new articles, which is not. It’s as if I had such a strong desire for the world to cleave cleanly into two camps that I swept aside any data that would have left me ambivalent.
I went back, continuing to browse. With effort, I was able to feel roughly neutral to most articles. Bored, even. That’s not highly-sought state, but it’s a significant improvement over supercilious.
I already knew the content of the math articles, and had my own intuition for them. (I had no interest in the computer articles). Mostly, I found Kalid’s intuition for each new subject superfluous. But I thought that if I described my own intuition about, say, the fundamental theorem of calculus, maybe Kalid would feel the same way.
I learned more from “Better Explained” than I expected, and more than was on the site. I learned that intuition is individual – perhaps one reason I have no consistent success in trying to teach it. I gained a wariness of my own opinions, and a general distrust of those of others, which I assume to be similarly tainted.
There is very little new here. I suspect any adult can cite many scenarios in which they like or dislike a thing because of what other people think of it, and equally many where they want a thing to be bad. But even though it’s no secret that this is happening, I don’t always realize it in real time, or if I do, don’t realize how weird it is.