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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.
if there’s one thing we can learn from the history of geometry, it’s that you shouldn’t admit a new axiom into your book until it really proves its worth. Always be skeptical when someone tells you they’re “just being logical.” If they are talking about an economic policy or a culture figure whose behavior they deplore or a relationship concession they want you to make, and not a congruence of triangles, they are not “just being logical,” because they’re operating in a context where logical deduction—if it applies at all—can’t be untangled from everything else. They want you to mistake an
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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. Geometry is a form of honesty.
Democratic governments are founded on the principle that every citizen’s views are to be represented in the decision-making of the state. Like all good principles, this is easy to state, difficult to make precise, and almost impossible to implement in a fully satisfying way.
Each state, however small, gets at least three votes in the Electoral College, which ultimately decides the presidency. Wyoming’s 579,000 people—about as many as live in greater Chattanooga—share three electoral votes among them, which means each electoral vote represents about 193,000 Wyomingites. California has almost 40 million people, so each of its fifty-five electoral votes stands for more than 700,000 Californians. This is, as your constitutional-originalist friends probably often remind you, by design.
The system we eventually arrived at shouldn’t be thought of as a brilliant encapsulation of the founders’ wisdom; it was a compromise reluctantly and wearily arrived at, nobody having been able to come up with anything better. If you have ever sat in a long meeting as day care pickup got nearer and nearer, knowing you couldn’t go home until the meeting produced a policy document everyone there could make themselves grumblingly sign, you have a pretty good idea of how the Electoral College came to be.
Even if you’re on board with the representational inequities baked into the Electoral College, you’d better be aware that they’ve gotten a lot more intense since the framers’ time. In the 1790 census, the largest state, Virginia, had eleven times the population of the smallest, Rhode Island. Right now, the ratio between Wyoming’s population and California’s is about 68. Would the constitutional convention have been game to assign Rhode Island so much power to appoint senators and electors if it had been six times smaller than it was?
A voter in Pennsylvania may have less influence in the presidential election than one in New Hampshire, but they have infinity times as much as an American who lives in Puerto Rico or the Northern Marianas or Guam.
You might think of the Senate and Electoral College as a kind of standardized test, a quantifiable proxy for whatever we think of as the popular will. Like any standardized test, it roughly measures the thing it’s supposed to; but it can be gamed, and the longer it persists in fixed form, the better people get at gaming it, getting more and more accustomed all the while to thinking of the test itself as the thing that really matters.
Sometimes I imagine a distant future where whole regions of the United States, ravaged by climate change and unchecked pollution, are inhabited solely by a handful of cyborg-people aged a hundred and up, kept in stasis in purified-air boxes and roused into consciousness by their machine parts once every even-numbered year, just long enough to mark a ballot for the congressional representatives the Constitution guarantees them. And there will still be opinion pieces in the newspapers praising the founders’ keen insight in designing a system of self-government that has served us so well, and for
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The superficially fair criterion that a 50–50 split of votes should result in a roughly 50–50 split of seats, Schimel argued, would actually be biased against Republicans, not just in Wisconsin but in every state whose dense cities were dominated by Democratic voters—which is to say, just about every state.
A false figure can be corrected. A true one chosen to make the wrong impression is a much harder poodle to muzzle.
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.
Brett Kavanaugh, with help from Paul Clement, collaborate on building a fictional version of the case in which the plaintiffs are asking the court to impose some form of proportional representation on the states.
The majority’s ruling, in the end, turns on a technical point—that partisan gerrymandering is a “political question,” which means that, even if the Constitution has been violated, the Supreme Court is forbidden to intervene. That the results of gerrymandering “reasonably seem unjust”—that they are, in fact, “incompatible with democratic principles”—isn’t in dispute; those are in fact quotes from the majority decision!
But just because something is unjust and incompatible with democratic principles and fiendishly effective, Justice Roberts writes, doesn’t mean it’s within the purview of the court to find a constitutional violation. Gerrymandering stinks, but not so badly the Constitution can smell it.
You can feel the discomfort in the decision: not just in the concessions that gerrymandering impedes democracy, but in the fervently expressed wish that somebody other than the justices of the Supreme Court will do something about it.
In Arkansas, redistricting reform group Arkansas Voters First gathered more than a hundred thousand signatures in the middle of a pandemic to get a constitutional amendment on the November ballot; the secretary of state declared the petitions void, because the certification that paid canvassers had undergone criminal background checks was misworded on the relevant form. State politics is full of veto points, so political factions with turf to protect have many ways to shield themselves from the public.
Talking to people about the dark art of gerrymandering is a form of teaching math, and math has intrinsic purchase on the human mind, especially when it’s twined up with other things we care deeply about: power, politics, and representation. Gerrymandering was a huge success when it was done behind locked doors. I’d like to believe it can’t persist in an open, well-lighted classroom.
Where arithmetic is a slog, geometry is a kind of liberation. Insight so powerful it blows the walls out sideways
The Pythagorean Theorem isn’t true because Pythagoras said it was; it’s true because we can, ourselves, prove that it’s true. Behold!
it is not order only, but unexpected order, that has a value.
We use formal proof as a scaffold, to extend our intuition’s reach, but it would be useless, a ladder to nowhere, if we weren’t using it to get to a point we could somehow, inexplicably, see.
Proof is an essential tool for us—the measure of our certainty, just as it was for Lincoln. But it is not the point. The point is to understand things. We want not just the facts but the souls of the facts. It’s at the moment of understanding that the walls go transparent, the ceiling flies off, and we’re doing geometry.
If you never guess wrong, you’re not guessing about hard enough things.
like Lincoln, rigorously criticizing our own beliefs and assumptions—we
Everyone does geometry differently, but everyone does it. It is, just as its name says, the way we measure the world, and therefore (only in geometry do we get to say “therefore”) the way we measure ourselves.
