Math Variation Quotes

He walked straight out of college into the waiting arms of the Navy. They gave him an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back? Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn't prove his intelligence, what would? Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal. Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.

Fairy tales, fantasy, legend and myth...these stories, and their topics, and the symbolism and interpretation of those topics...these things have always held an inexplicable fascination for me," she writes. "That fascination is at least in part an integral part of my character — I was always the kind of child who was convinced that elves lived in the parks, that trees were animate, and that holes in floorboards housed fairies rather than rodents. You need to know that my parents, unlike those typically found in fairy tales — the wicked stepmothers, the fathers who sold off their own flesh and blood if the need arose — had only the best intentions for their only child. They wanted me to be well educated, well cared for, safe — so rather than entrusting me to the public school system, which has engendered so many ugly urban legends, they sent me to a private school, where, automatically, I was outcast for being a latecomer, for being poor, for being unusual. However, as every cloud does have a silver lining — and every miserable private institution an excellent library — there was some solace to be found, between the carved oak cases, surrounded by the well–lined shelves, among the pages of the heavy antique tomes, within the realms of fantasy. Libraries and bookshops, and indulgent parents, and myriad books housed in a plethora of nooks to hide in when I should have been attending math classes...or cleaning my room...or doing homework...provided me with an alternative to a reality I didn't much like. Ten years ago, you could have seen a number of things in the literary field that just don't seem to exist anymore: valuable antique volumes routinely available on library shelves; privately run bookshops, rather than faceless chains; and one particular little girl who haunted both the latter two institutions. In either, you could have seen some variation upon a scene played out so often that it almost became an archetype: A little girl, contorted, with her legs twisted beneath her, shoulders hunched to bring her long nose closer to the pages that she peruses. Her eyes are glued to the pages, rapt with interest. Within them, she finds the kingdoms of Myth. Their borders stand unguarded, and any who would venture past them are free to stay and occupy themselves as they would.

For our afternoon, I’ve decided to attempt to give Rhiannon the satisfaction of being fully alone. Not the lethargy of lying on the couch or the dull monotony of drifting off in math class. Not the midnight wandering in a sleeping house or the pain of being left in a room after the door has been slammed shut. This alone is not a variation of any of those. This alone is its own being. Feeling the body, but not using it to sidetrack the mind. Moving with purpose, but not in a rush.

the satisfaction of being fully alone. Not the lethargy of lying on the couch or the dull monotony of drifting off in math class. Not the midnight wandering in a sleeping house or the pain of being left in a room after the door has been slammed shut. This alone is not a variation of any of those. This alone is its own being. Feeling the body, but not using it to sidetrack the mind. Moving with purpose, but not in a rush. Conversing not with the person next to you, but with all of the elements. Sweating and aching and climbing and making sure not to slip, not to fall, not to get too lost, but lost enough.

How was one to communicate an idea to another person if when one had something to say, one could choose from ten different words for every word in a sentence? The number of potential variations boggled the mind. So much so that shortly after arriving at St. Paul’s, Woolly had gone to his math teacher, Mr. Kehlenbeck, and asked him if one had a sentence with ten words and each word could be substituted with ten other words, then how many sentences could there be? And without a moment’s hesitation, Mr. Kehlenbeck had gone to the chalkboard, scratched out a formula, and done a few quick calculations to prove incontrovertibly that the answer to Woolly’s question was ten billion. Well, when confronted with a revelation like that, how was one to even begin writing an

The radial patterning of Protestantism allows us to use a county’s proximity to Wittenberg to isolate—in a statistical sense—that part of the variation in Protestantism that we know is due to a county’s proximity to Wittenberg and not to greater literacy or other factors. In a sense, we can think of this as an experiment in which different counties were experimentally assigned different dosages of Protestantism to test for its effects. Distance from Wittenberg allows us to figure out how big that experimental dosage was. Then, we can see if this “assigned” dosage of Protestantism is still associated with greater literacy and more schools. If it is, we can infer from this natural experiment that Protestantism did indeed cause greater literacy.16 The results of this statistical razzle-dazzle are striking. Not only do Prussian counties closer to Wittenberg have higher shares of Protestants, but those additional Protestants are associated with greater literacy and more schools. This indicates that the wave of Protestantism created by the Reformation raised literacy and schooling rates in its wake. Despite Prussia’s having a high average literacy rate in 1871, counties made up entirely of Protestants had literacy rates nearly 20 percentile points higher than those that were all Catholic.18 FIGURE P.2. The percentage of Protestants in Prussian counties in 1871.17 The map highlights some German cities, including the epicenter of the Reformation, Wittenberg, and Mainz, the charter town where Johannes Gutenberg produced his eponymous printing press. These same patterns can be spotted elsewhere in 19th-century Europe—and today—in missionized regions around the globe. In 19th-century Switzerland, other aftershocks of the Reformation have been detected in a battery of cognitive tests given to Swiss army recruits. Young men from all-Protestant districts were not only 11 percentile points more likely to be “high performers” on reading tests compared to those from all-Catholic districts, but this advantage bled over into their scores in math, history, and writing. These relationships hold even when a district’s population density, fertility, and economic complexity are kept constant. As in Prussia, the closer a community was to one of the two epicenters of the Swiss Reformation—Zurich or Geneva—the more Protestants it had in the 19th century. Notably, proximity to other Swiss cities, such as Bern and Basel, doesn’t reveal this relationship. As is the case in Prussia, this setup allows us to finger Protestantism as driving the spread of greater literacy as well as the smaller improvements in writing and math abilities.

If adding two numbers produced a random result each time, we could never rely on math. Fortunately there are definite answers with no variation. Similarly, there is nothing random about the study of science. If each iteration of an experiment yielded a different result from the same variables, we would not be able to conclude anything with certainty. The scientific method is not compatible with randomness. If the universe were truly random, the study of science itself would not be possible. The laws of nature stand in direct opposition to the notion that all is born of chance.

Patience is a word we use a lot to describe great teachers at work. But what I saw was not patience, exactly. It was more like probing, strategic impatience. The master coaches I met were constantly changing their input. If A didn't work, they tried B and C; if they failed, the rest of the alphabet was holstered and ready. What seemed like patient repetition from the outside was actually, on closer examination, a series of subtle variations, each one a distinct firing, each one creating a worthwhile combination of errors and fixes that grew myelin.

Taking a superior variation always makes sense, but we would only take inferior ones when the die shows, say, a 2 or more. After a while, we’d cool it further by only taking a higher-price change if the die shows a 3 or greater—then 4, then 5. Eventually we’d be mostly hill climbing, making the inferior move just occasionally when the die shows a 6. Finally we’d start going only uphill, and stop when we reached the next local max. This approach, called Simulated Annealing, seemed like an intriguing way to map physics onto problem solving. But would it work? The initial reaction among more traditional optimization researchers was that this whole approach just seemed a little too … metaphorical. “I couldn’t convince math people that this messy stuff with temperatures, all this analogy-based stuff, was real,” says Kirkpatrick, “because mathematicians are trained to really distrust intuition.

Put it this way. If you or I were to build a machine, we'd go about it logically, with the fewest necessary parts moving in clean, efficient ways. But living nature doesn't work that way at all. It builds via the most fantastic redundancies and curlicues, millions of little variations around a theme, so that if three-quarters go haywire, life survives. The results are Rube Goldberg devices, but sturdy Rube Goldberg devices, unimaginably weird and densely layered Rube Goldberg devices, literally unimaginable in that our brains aren't adequate to comprehend the sort of microscopic megacities hidden within the tiniest cell. I thought that was neat.

Many people walk through life in a cognitive slumber, blindly following the norms attached to their identity. ■ “I’m terrible with directions.” ■ “I’m not a morning person.” ■ “I’m bad at remembering people’s names.” ■ “I’m always late.” ■ “I’m not good with technology.” ■ “I’m horrible at math.” … and a thousand other variations. When you have repeated a story to yourself for years, it is easy to slide into these mental grooves and accept them as a fact. In time, you begin to resist certain actions because “that’s not who I am.