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January 3 - February 21, 2022
The ultimate reason for teaching kids to write a proof is not that the world is full of proofs. It’s that the world is full of non-proofs, and grown-ups need to know the difference. It’s hard to settle for a non-proof once you’ve really familiarized yourself with the genuine article.
What Lincoln had taken from Euclid (or what, already existing in Lincoln, harmonized with what he found in Euclid) was integrity, the principle that one does not say a thing unless one has justified, fair and square, that one has the right to say it.
another slogan of the always-quotable Poincaré: “Mathematics is the art of giving the same name to different things.”
A proof is crystallized thought. It takes that brilliant buoyant moment of “getting it” and fixes it to the page so we can contemplate it at leisure. More importantly, we can share it with other people, in whose mind it springs to life again.
Proof makes insight portable.
So chess, for all its lore and mystique, is the same kind of thing as Nim and tic-tac-toe. If two absolutely perfect players faced off, either white would always win, white would always lose, or the game would always end in a draw.
Fermat showed that, if n is prime, 2n is 2 more than a multiple of n.
When we say the lesson at hand is “easy” or “simple,” and it manifestly isn’t, we are telling the student that the difficulty isn’t with the mathematics, it’s with them. And they will believe us. Students, for better or worse, trust their teachers. “If I didn’t even get this and it was easy,” they’ll say, “why bother trying to understand something hard?”
If math class is easy, you’re doing it wrong.
there is no reason to know very many digits of π. There are real-world contexts where you’d want to know seven or eight digits, sure. But the hundredth digit? It’s hard to imagine what you’d need that for. Forty digits is already enough to compute the circumference of a circle the size of the Milky Way to within the size of a proton.
That’s the danger of extrapolation. It tends to get less reliable the further out you get from the known data your differences are anchored in.
Real modeling is always a dance between predictable dynamics and our unpredictable responses.
the golden ratio is the proportion between the diagonal of such a pentagon and its side.
There’s a lot of wiggle room, and the more wiggle room your theory has, the easier it is to describe what’s already happened with a confident Just as I thought!
All shortest paths on the sphere, it turns out, are pieces of “great circles,” so-called because they’re the biggest circles you can draw on a sphere, the ones that pass through two directly opposite points. And a great circle is what we mean by a line on the sphere.
Linear algebra allows one to extend one’s intuition about three-dimensional space to spaces of whatever dimension you like;
Maybe it’s not that entropy can’t increase, but that it is incredibly unlikely for it to increase. Shuffling a deck of cards is a time-reversible process, too. You have probably never shuffled a mixed-up deck of cards only to find it restored to perfect factory order. But that’s not because it’s impossible—it’s not! It’s merely improbable.
the twenty-six smallest states, whose fifty-two representatives make up a majority of the Senate, speak for just 18% of the population.
The convex hull of a shape is the union of every line segment joining every pair of points in the shape:
what makes a gerrymander work is that your party wins a lot of districts by a little and loses a few districts by a lot.
People often complain that no one likes facts and numbers and reason and science anymore, but as someone who talks about those things in public, I can tell you that’s not true. People love numbers, and are impressed by them, sometimes more than they should be. An argument dressed up in math carries with it a certain authority. If you’re the one who outfitted it that way, you have a special responsibility to get it right.
If you never guess wrong, you’re not guessing about hard enough things.
