Portraits of Wildflowers

Perspectives on Nature Photography

Archive for September 19th, 2021

From snow to fire

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Yesterday’s post showed you a happy colony of snow-on-the-prairie, Euphorbia bicolor. Now here’s one of its genus-mates, Euphorbia cyathophora, known as fire-on-the-mountain for the bright red of its bracts. Another common name is wild poinsettia, a reference to a more-familiar genus-mate, Euphorbia pulcherrima, that people decorate their places with during the Christmas season.

I photographed this fire-on-the-mountain on the morning of September 11th at the Lady Bird Johnson Wildflower Center. It hadn’t rained, but the staff waters the plants in the central courtyard, and that accounted for the droplets in the photograph.


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I’m reading mathematician Jordan Ellenberg’s new book Shape. Here’s a passage from the first chapter.

We encounter non-proofs in proofy clothing all the time, and unless we’ve made ourselves especially attentive, they often get by our defenses. There are tells you can look for. In math, when an author starts a sentence with “Clearly,” what they are really saying is “This seems clear to me and I probably should have checked it, but I got a little confused, so I settled for just asserting that it was clear.” The newspaper pundit’s analogue is the sentence starting “Surely, we can all agree.” Whenever you see this, you should at all costs not be sure that all agree on what follows. You are being asked to treat something as an axiom*, and 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 assertively expressed chain of opinions as the proof of a theorem. But once you’ve experienced the sharp click of an honest-to-goodness proof, you’ll never fall for this again. Tell your “logical “opponent to go square a circle.**

A big reason we surely can’t all agree is that people often use a given term to mean different things. What one person considers a “fair share,” another person takes to be a disproportionate burden. One person uses “justice” to mean a desired outcome, while for another person “justice” means due process and equal treatment. If we don’t start from the same definitions, why would we expect to reach the same conclusions?

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* An axiom is a principle that everyone will take to be true and will use as a starting point to figure out other facts or relationships. For example, one axiom of mathematics is that a whole thing is equal to the sum of its parts. Another axiom is that two things that are each equal to a third thing are equal to each other.

** “Squaring the circle” was a geometric challenge that meant: Using only a compass, a straightedge, and a finite number of steps, construct a square that has the same area as a given circle. For centuries mathematicians tried and failed to figure out how to do that. In 1882, Ferdinand von Lindemann finally proved that squaring the circle is impossible. (So much for the common notion that “You can’t prove a negative. Sometimes you can.)

© 2021 Steven Schwartzman

Written by Steve Schwartzman

September 19, 2021 at 4:33 AM

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