Thanks, that was a lot of fun to go through. Great tool for building intuition.
Q: What does the B. in Benoit B. Mandelbrot stand for?
A: Benoit B. Mandelbrot.
That was a great article. I like the kickoff--that math is all about a series of consistent choices, and we just name some of them Algebra, Geometry, etc.
That explanation of complex numbers and square roots of complex numbers is just lovely.
They didn't mention that we get the word absurd from negative numbers or surds. Negative numbers were considered ab surd.
The big problem isn't that math isn't interesting. It's that math is like music or speaking a foreign language. You can appreciate it well enough, but to actually do it involves memorization, familiarization, conceptualization and practice, practice, practice. Still, an amusing enough page.
Maybe they didn't mention that because it isn't true:
Etymology: < Middle French, French absurde (adjective) unreasonable, contrary to common sense (late 14th cent.; beginning of the 13th cent. in Old French as absorde ), (noun) (with indefinite article) something absurd, an absurdity (16th cent.), (with definite article) the absurd, absurdity (17th cent.) and its etymon classical Latin absurdus out-of-tune, discordant, awkward, uncouth, uncivilized, preposterous, ridiculous, inappropriate < ab- ab- prefix + surdus surd adj. With use as noun compare classical Latin absurda , neuter plural (Quintilian). Compare earlier absurdity n.
Ok, what's up with "absurdity"?
Etymology: Originally (in sense 1) < post-classical Latin absurditat- , absurditas dissonance (4th cent.), perversity (5th cent.) < classical Latin absurdus absurd adj. + -tās (see -ty suffix1; compare -ity suffix). Subsequently (in sense 2) < Middle French absurdité (French absurdité ) absurd act or statement (1371-5) < post-classical Latin absurditat- , absurditas . Compare Spanish absurdidad (c1440 or earlier), Portuguese absurdidade (c1727), Italian assurdità (a1573).
The first example usage for "absurdity" is indeed musical, from 1429: " There ware difformitee the beutee of Absolon..There Dauid harpe and the musik of Jubal ware absurditee."
There is a usage of "absurd" applied to negative numbers from 1557 ("8−12 is an Absurde nomber. For it betokeneth lesse then nought by 4."), but the first usage given comes from a piece of writing called "The determinations of the moste famous and mooste excellent vniuersities of Italy and Fraunce, that it is so vnlefull [sic] for a man to marie his brothers wyfe" (1531), and there it just means "contrary to reason".
Still, both "surd" and "absurd" derive from Latin "surd," which means "deaf" via Arabic. Language Log was on the case (at great and hilarious length), as expected: Ab Surd
Includes bonus Lewis Carroll poem:
And what are all such gaieties to me,
Whose thoughts are full of indices and surds?
x2 + 7x + 53
= 11/3
AND bonus bonus Cordwainer Smith quatrain:
She wasn't the woman I went to seek;
I met her by the merest chance.
She did not speak the French of France,
But the surded French of Martinique.
As they say, "read the whole thing!"
At each step, we feel hoodwinked: we were only shown a part of the puzzle! As it turned out, there was a 'better' set of numbers waiting to be discovered, more comprehensive than the last.
This pretty much exactly describes how I reacted to my chemistry education. Every year they'd explain that actually, last year's model wasn't right, and this is how things really work. Only to do the same thing again the following year. Though, strangely enough, I didn't rebel against maths.
You think it's an accident they give the Nobels in Stockholm?
11: If they skipped right to the final model, would it have been comprehensible?
11: If they skipped right to the final model, would it have been comprehensible?
Probably not. I'm not saying 14-year old me was being reasonable. Just that that sense drove me away from science subjects at that age.
Statistics does that a lot. You learn the simple model that isn't actually properly specified before you learn the, usually more complex, model that works. It annoys me sometimes, but I do find that I usually run the simple model anyway as an error check because the coding is more simple.
The worst offender is philosophy. They start out with how they are going to answer questions about epistemology or what have you, drag you on for a whole semester, and then reveal the answer is a nicely worded version of "I don't know."
The problem with 11, AIMHMHB, is that it trains people to believe that the simple and intuitive explanation for anything cannot possibly be the correct one, and thus trains them to be Slate writers.
re: 17
To be fair, really advanced practitioners move on from, "I don't know" to "It depends".
I did a teaching evaluation of a chemistry teacher who I have a very high opinion of, and he used point 11 very explicitly as his teaching framework. (It was pitched as Inquiry Based Learning but it was very, very different from what that means in math.)
He took a historical approach and a knowledge-acquisition approach - with each model, they discussed what seemed to work, and then what new technology of the time caused failures in that model, and how the next model integrated the new information more accurately. I thought it worked nicely.
To be fair, really advanced practitioners move on from, "I don't know" to "It depends".
Some people also favor "it isn't really a problem at all".
This is a really nice link, neb! I shared it with other people.
20 is kind of how we were taught maths in the English equivalent of high school. The curriculum as a whole wasn't historical in a chronological sense, but we'd have regular "circus hours" where teachers would riff on, say, Euler and Gauss, or Euclidean geometry from first principles, or Russell and set theory.
Three cheers for robust, methodological, falsification.
we'd have regular "circus hours" where teachers would riff on, say, Euler and Gauss, or Euclidean geometry from first principles, or Russell and set theory.
what kind of terrible circus is this?
22. I thought "it needs more study" was the go-to phrase.
26: Proofs and Refutations Lakatos.
27: You haven't been to a circus until you've seen Tarski pull a rabbit into a hat.
30: What's funny about that book is that the process he described has ended. There are rigorous proofs of Euler's original theorem that no one questions. The way the topic has progressed has rather been in forms of greater and greater generality -- the modern version is in terms of the Euler characteristic in algebraic topology.
Education recapitulates discovery.
Or at least it should if you want to train people in how science actually works. If you want people to memorize facts and how to use equations and not actually believe that they really describe reality (a la Eric Mazur's student's illuminating response: "How should I answer these questions? According to what you taught me? Or according to the way I usually think about these things?") then by all means present the current theory as received wisdom.
I worked through the examples on that page, and, honestly, all I could think was "pretty pictures". I'm sure the key to it is thinking of all numbers as imaginary, but I can't. I'm an instinctive platonist. I'd sooner believe that real numbers are real than that mutable humans are. So, with imaginary numbers in those animations, a huge part of me thinks, "but why do the pushmepullyou? while another part can't do the vectors and work out why pushing this and pulling that should leave the green dot where it does.
I'm sure the key to it is thinking of all numbers as imaginary
No, no, just think of all numbers as real - it's just that some of them don't correspond to things you can see and perceive in the physical world. And this isn't just true for things like i; it's also true for things like "a trillion". I can't imagine a trillion anything. Not even otters. But I have no problem with the concept.
I bet I could picture at least 100,000 otters.
I had an argument with a Platonist on whether the quaternions "really existed".
Quantum mechanics seems to require complex numbers, which makes them seem pretty real to me.
The fact that the ratio of a circle's diameter to its circumference isn't a rational number despite circles being real suggests to me that '1' isn't real.
It's easy to get a perfect circle is you start with a perfect sphere.
One trillion is 10^12 (nowadays, anyway). 10^12 = (10^4)^3. 10^4 is ten thousand. Can you imagine a thousand otters in a row? Do ten of those. Now make an ottercube with that as a side.
Can we give them typewriters to test another theory?
Perfect circles aren't real but we can approximate them as closely as we'd like to. If you're willing to make it quite large I'm not even sure to what degree Planck units get in the way of creating a locus of points whose perimeter/diameter radius differs from pi to an arbitrarily small degree.
Can you imagine a thousand otters in a row? Do ten of those.
That's a lot of otter poop.
Perfect circles are all in the shoulder swing.
Also, always draw your circle first and then put your axes through the center, on the chalkboard. If you draw the axes first, it's much harder to make your circle on center.
With great shame I never learned the secret of how to draw circles well. The rare times I have to whiteboard one it's just embarrassing.
Now make an ottercube with that as a side.
You can describe it and you can imagine a cube with, effectively, a little label saying "it's ten thousand otters long, deep, and high", but that's different from being able to imagine a trillion otters.
I mean, the famous example of imagining a regular chiliagon can't be addressed (well) just by saying "well, it's a regular, thousand-sided polygon, done".
You're just bitter because you don't have a good imagination.
The halacha follows R. Nosflow. https://cabin-pressure.livejournal.com/60389.html
I can imagine it, it's just that the resolution of my imagination is fuzzy.
Picture a regular n-gon. Then take the case that n=1000.
Picture Germany being supreme over all for a year.