According to Eugenia Cheng, what is the definition of math?
Basics
Eugenia Cheng Makes Math a Slice of Cake
Information technology can also be a slice of pie, or custard — so says a professor and author who spreads the magic of numbers through dessert recipes.
Eugenia Cheng in her kitchen at work on 1 of her mathematically inspired desserts. Credit... Whitten Sabbatini for The New York Times
CHICAGO — We had just finished the mathematician Eugenia Cheng's splendid demonstration of nonassociativity where the society of operations counts — as it does in, say, subtraction.
At present she wanted to forge ahead with the adjacent lesson, in knot theory.
I suggested we wait until later. "Why?" she asked.
"Well, we shouldn't eat ii desserts earlier dinner, should we?" I said, and giggled nervously.
"Why not?" she replied, not giggling. She tightened her apron strings and walked over to her stove.
Of course. What was I thinking? Hadn't Dr. Cheng already made clear her conviction that in mathematics, rules are like eggs: meant to be broken, stirred, flipped over and sense of taste-tested? And that day, we had broken a lot of eggs.
"You're absolutely right," I said, rushing to her side for the m unveiling of another mathematically themed confection.
Dr. Cheng pulled from the oven a perfectly baked specimen of what she calls Bach pie, named for the swell composer beloved by mathematicians everywhere: an oblong rectangle of creamy dark chocolate studded with banana slices and topped past an Escher-like braid of four glazed pastry plaits that followed divergent trajectories, never quite crisscrossing where you expected them to.
The filling was a clever concatenation — "BAnana added to CHocolate gives you Bach," Dr. Cheng said. The braiding illustrated the structure of a Bach prelude and the sorts of patterns that knot theorists study "to encounter how looped upwards the braids are," Dr. Cheng said, "and whether you can transform one braid into another by wiggling the different strings."
The pie was a true wedlock of art and math, too beautiful to asperse, and besides, y'all're not supposed to untie knots with your teeth, are you?
Some other rule, hands cleaved.
Dr. Cheng, 39, has a knack for brushing bated conventions and edicts, like so many pie crumbs from a cutting lath. She is a theoretical mathematician who works in a rarefied field chosen category theory, which is and then abstract that "even some pure mathematicians call up information technology goes likewise far," Dr. Cheng said.
At the same time, Dr. Cheng is winning fame as a math popularizer, convinced that the pleasures of math tin can be conveyed to the legions of numbers-averse humanities majors still recovering from high school algebra. She has been featured on shows like "Late Nighttime With Stephen Colbert," and her online math tutorials have been viewed more than than a million times.
The hardcover edition of her commencement book, "How to Bake π: An Edible Exploration of the Mathematics of Mathematics," has sold nearly 25,000 copies in this country and been translated into six languages, a surprising hit for a text visibly if judiciously seasoned with numbers, graphs and equations. The book is beingness released in paperback this month.
Epitome 
"I spend a lot of fourth dimension explaining mathematics on blogs, and I try to cut through the technicalities and make things easier to empathize," said John Baez, a professor of math at the University of California, Riverside (and yes, a cousin of Joan). Still, his posts are aimed at scientists and others with some quantitative groundwork.
"Eugenia has gone all the way in," he said. "She's trying to explicate math to everybody, with or without pre-existing expertise, and I retrieve she'due south doing wonderfully."
A New Mission to Serve
Then committed is Dr. Cheng to mass math demystification that she recently left a tenured professorship at the University of Sheffield in Uk to take a position at the Schoolhouse of the Art Institute of Chicago, where she teaches math to fine art students, lectures widely and continues her inquiry in category theory on the side.
Dr. Cheng adopts a literal approach to making math more flavory. "Math is about taking ingredients, putting them together, seeing what you lot can make out of them, and and so deciding whether information technology'south tasty or not," she said.
Every chapter in "How to Bake π" offers recipes for desserts and other dishes that encapsulate mathematical themes. To demonstrate how math seeks to identify underlying similarities across a broad prepare of issues, for example, Dr. Cheng starts with a recipe that can be readily tweaked to brand mayonnaise instead of hollandaise sauce.
"Books might tell you that hollandaise sauce needs to be done differently," she writes, "only I ignore them to make my life simpler. Math is as well in that location to make things simpler, by finding things that wait the same if y'all ignore some modest detail."
Her recipe for lasagna illuminates the importance of context to math. Dr. Cheng lists among the bones ingredients "fresh lasagna noodles," and and then points out that another cookbook might deem the noodles not truly basic and instead describe their preparation from scratch.
So, as well, do numbers change their character and degree of basicness depending on context. The number 5, for example, when viewed amongst the natural, or counting, numbers is one of those elemental creatures: a prime, divisible only past 1 and itself.
But in the context of the and so-called rational numbers, which include fractions, five loses its prime identity and gains versatility, able to be divided into ever tinier slivers, similar a cake at a dieters' convention.
The number 1 in its multiplicative identity is practically crippled, leaving other numbers unchanged: 6 times 1 equals 6. In its additive capacity, however, 1 is unstoppable: if yous go on adding 1 to itself, Dr. Cheng noted, you lot tin generate all the natural numbers, out to infinity.
Context tin can prod numbers to defy form-school verities: ii plus 2 equals 4, and that's that. Only not if you're talking about a clock face with only three numbers: 1, 2 and 3. In that instance, two plus 2 equals 1 – if you lot get-go at the 2 and move clockwise by 2, you attain ane.
"I admit I was skeptical at first about her analogies to cooking, simply I ended up being completely sold," said Steven Strogatz, a professor of applied mathematics at Cornell University who likewise writes pop books.
"She conveys the spirit of creativity and creativity in math that all mathematicians feel merely practise a very poor job communicating when didactics math. Refreshing is the word that keeps coming to heed."
Dr. Cheng wears glasses, has long black hair and walks with an elegant, upright posture, similar a dancer. She is alternately outgoing and reserved, passionate and matter-of-fact. She is a classically trained musician and owns a 1910 Steinway m piano.
She is also an extremely expert cook. On a recent afternoon, nosotros are in her well-stocked kitchen, and she is stirring a mix of egg yolks, carbohydrate and foam — my lesson in nonassociativity, otherwise known equally custard. This is something where the fashion you combine ingredients counts, she said.
If you're making a cake, you tin can throw together the flour, sugar, butter and eggs however you delight, and the cake will come out fine — that's an associative process. Not so for preparing custard. You must offset combine the carbohydrate and egg yolks and whisk them into a barm earlier you cascade in the cream.
Blend the ingredients in a different order, she said, "and you lot end up with a runny mess."
Math's Ingredients
In math, too, you have associative operations like improver and multiplication, where yous can modify how your numbers are whisked together as you calculate and still emerge with the same answer: (4+5) + 6 is the same as four+(5+6).
Subtraction, division and exponentiation are nonassociative operations: Who's clumped with whom matters. Then, while 10-(6-4) will requite y'all 8, (10-half-dozen)-4 leaves yous with cypher.
Dr. Cheng continues stirring the custard, peering into the pot to ensure that lumps aren't forming. "People ask me, do you eat your custard warm or cold?" she said. "I tell them, I eat information technology all at one time."
She adjusts the temperature of the burner, up and down, upward and downwards. Getting the custard to come out just correct can be tricky, she said — harder than doing math, and more like living life.
Dr. Cheng insists that the public has it all wrong about math being difficult, something that just the gifted mathletes amidst usa tin can do. To the contrary, she says, math exists to make life smoother, to solve those issues that tin be solved past applying math'southward most powerful tool: logic.
Science may depend on forming hypotheses, doing experiments and gathering evidence that support or refute your hypothesis, merely math is simply a thing of stating the terms of your statement and and so defending those statements using logic.
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"The great matter well-nigh math is you don't need much to start exploring it," Dr. Baez said. "No expensive equipment, simply pencil and newspaper, and you can starting time footling effectually with patterns and numbers."
The key to thinking logically is to get comfortable with abstraction. Numbers are one type of abstraction that nosotros take for granted: The number 3 tin stand in for 3 bananas, 3 pies, 3 Bach partitas.
The next step up in abstraction is to replace specific numbers by symbols like ten and y, a powerful maneuver that allows you to apply your argument more broadly, just as a full general-purpose recipe for pie can be the vehicle for banana, blueberries, pumpkin, nuts, knots, you proper name it.
As an case of abstraction at work, "How to Bake π" presents this little encephalon teaser: My male parent is three times as old as I am now, but in ten years' time, he volition be twice as old as me. How old am I? It then takes the reader through the solution by way of x-y abstraction.*
Dr. Cheng recognizes that people tin feel uncomfortable with some of the abstractions required past mathematical thinking, by the need to ignore the particulars of, say, this green round pillow and that square purple pillow in favor of an abstruse ideal of a pillow that you're going to telephone call x.
Merely it's just a matter of do, she said, before the idea starts to experience similar a real object that you can manipulate with ease. "You lot become very good at separating what's relevant from what isn't, and that tin can be very useful in daily life," she said.
Sometimes, she finds it "oddly satisfying" to mentally shave a disguised man or imagine how a furry domestic dog would look similar after a swim in a lake. "That'southward what abstraction is," she said. "You reveal the structure underneath."
A Number Chef's Roots
Dr. Cheng's upbringing in rural Sussex as one of 2 daughters led her readily to math love. Their mother, a statistician who had been considered the most bright math student at her Hong Kong high school, would proceed her girls entertained at the beach for hours with books of logic puzzles.
And because their mother commuted each twenty-four hour period to a London bookkeeping firm while their male parent had a child psychiatry practice near home, the girls grew accustomed to the idea of a briefcase mother and a male parent who made dinner.
"I think Eugenia and I both benefited from the atypical setup," said Alethea Cheng Fitzpatrick, Dr. Cheng'due south older sis and now an architect in New York. "We didn't go into machismo with whatever preconceptions about gender roles or gender abilities."
Later, during graduate schoolhouse at Cambridge University and 3 postdoctoral fellowships in Europe and the The states, Dr. Cheng would ofttimes find herself in a tiny minority. "I've been to conferences where at that place were more cubicles in the women's bathroom than there were women to use them," she said.
Today, in the United states of america, less than 30 percent of graduate students and about 12 percent of tenured faculty members in math are women. Dr. Cheng is striving to change attitudes across the board. "I want to be a role model for men equally well as women," she said.
The custard is ready. The ideal becomes real. I finish the get-go bowl, and ask for more.
* Allow 10 be me and y be my father. Today, y=3x. In x years, my male parent will exist y+10 and twice my age, then y+10=2(ten+10). Since 3x=y, we can get rid of the y in that equation, as 3x+10=2(x+10). And since 2(ten+ten) is the aforementioned as iix+xx, we can now say 3x+10=2x+20. Finally, decrease twox from both sides of the equation and y'all get x+x=20. Therefore, x=10, and I am x years old.
Source: https://www.nytimes.com/2016/05/03/science/eugenia-cheng-math-how-to-bake-pi.html
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