Cucina Matematica

Jim Henle, Editor

What Kind of … Are You? JIM HENLE

This is a column on mathematics and food. It’s about similarities between the two: the surprising cultural, structural, philosophical, and mystical features common to mathematics and gastronomy.

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Jim Henle Department of Mathematics and Statistics Burton Hall, Smith College Northampton MA 01063, USA e-mail: [email protected]

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e’re not all the same. There are different kinds of cooks and different kinds of mathematicians. I’m not talking about fields, specialties, or ethnicities. I mean different in the way we approach a problem. I mean different in style, philosophy, or prejudices. Cooks: A few years ago I read a recipe from a fairly sophisticated chef that called for ketchup. That seemed strange to me. If you’re fussy about ingredients—if the eggs have to be fresh, if the butter has to be unsalted, if the cheese has to be parmesan, from Parma, and freshly grated—how can you just say ‘‘ketchup’’? There are different ketchups. Some are on the sweet side, some are more acidic. Some have ingredients that are difficult to spell. Of course a recipe can be more precise and call for, say, Heinz ketchup. But I found that I still had problems. Any ketchup has ingredients. It was made by a method. Can I trust the ingredients and the method?1 Am I cooking this dish or am I outsourcing it? In thinking about these issues, I discovered that I have a prejudice against ingredients with ingredients. I’m that kind of cook. Mathematicians: When I took this prejudice apart last year, it suddenly made sense. I began my mathematical life as a set-theorist and logician. I am drawn to logic, because I appreciate the certainty, or at least the stability of first principles, of a system of axioms. The idea of working for days, or maybe months, from a set of primitive postulates to prove that two plus two equals four is sort of attractive to me. In general, I want to see all the steps of a proof. I don’t want to use theorems that I haven’t seen proved. I’m that kind of mathematician. Or that’s the kind I was. After an extended stay in the axiomatic trenches, I’ve become comfortable with arithmetic. After working with infinite processes, I’m comfortable with analysis, and so on. I’m willing to use results that I didn’t prove or see proved. But I still enjoy getting down to first principles from time to time. And I regularly put my students through the process. Similarly, having made mayonnaise, mustard, and ketchup, I can accept the commercial versions of these most of the time. But I’ll never forget how excited I was to find a recipe for ‘‘catsup’’ in Joy of Cooking (1967). I made it once. It called for a peck of tomatoes. Are ‘‘pecks’’ in the Common Core? I still make my own brown sugar, partly because it’s so easy. Far from being a less-refined sugar, commercial brown sugar is made (I read somewhere) by combining white sugar and molasses. You and I can do that too. Try it; experiment with a half-cup of sugar and a teaspoon or two of

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Heinz currently makes seven different versions of its ketchup. With high-fructose corn syrup, with sugar, with less sugar, with less salt, etc. All seem to have ‘‘natural flavoring.’’ I spend a lot of time in the grocery store reading labels.

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THE MATHEMATICAL INTELLIGENCER  2014 Springer Science+Business Media New York

DOI 10.1007/s00283-013-9437-x

molasses; add more molasses for darker sugar. And for most recipes calling for brown sugar you don’t have to make the mix, you can just dump in white sugar and molasses. I make my own sour milk/yogurt/sour cream. It’s really filmo€lk, a Scandinavian cultured substance. Like brown sugar, it’s easy, and perfect for my style of cooking (lazy but pretentious). After you get it going, you produce more by putting a teaspoon or two of the soured milk into fresh milk and letting it stand at room temperature until it sets—no heating, no electric gadget. ‘‘Room temperature’’ at my house is unstable. Filmo€lk in my kitchen can take anywhere from 6 to 24 hours to set. You can make filmo€lk with fat-free milk. You can make it with lactose-free milk. You can make it with heavy cream. I was lucky to be given a sample by a friend; several companies sell starter packages. I also make my own vinegar. A friend gave me some red wine vinegar with an active ‘‘mother.’’ That’s a slimy blob with bacteria that turns alcohol and sugar to vinegar. The mother sits at the bottom. I use my vinegar for all sorts of purposes. From time to time I add to it the last inch or two of a bottle of red wine. The acidity of my vinegar must vary enormously. I don’t think about it. Mothers can grow spontaneously, but it took me a long time to get a white wine vinegar started. Now I’m trying to start some champagne vinegar. You can buy cider vinegar with a mother. Bragg’s apple cider vinegar is proud of its mother. Perhaps that can be used to get wine vinegar going. But most cooks are comfortable with composed ingredients. That’s a style too. In fact, cooking with commercial products was big when I was a kid: cookies made with corn flakes, casseroles made with canned mushroom soup, meatloaf made with bottled spaghetti sauce, etc. My favorite convenience dish was the clam dip that was probably an invention of Kraft Foods. My mother made it for parties. On the morning after one of those parties, my brothers and I would scoop up the remains with leftover Fritos. I fell in love. I went on the web recently looking for a recipe. I found many, but none matched my memory—a can of minced clams, a couple of packages of cream cheese with chives, and some Worcestershire sauce. Kraft no longer seems to make the 3-oz cream cheese with chives. Their cream cheese with chives and onion is gooey; perhaps it has less cheese and more water. Given all that, the recipe below is close to what I remember. (KRAFT) CLAM DIP a 6 1 1

6.5-oz can of minced clams, oz cream cheese 1/2 tsp. Worcestershire sauce 1/2 tsp. chopped chives

Reserve the liquid in the can. Mix the rest together. Add enough of the reserved liquid to make a good dipping consistency. Chill. Serve with Original Fritos, the real thing.

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Like cooks, most mathematicians are happy to build on the achievements of others. Indeed, it’s hard to imagine doing mathematics without the structures, insights, and monumental theorems that are our birthright. The real number system alone is an incredibly sophisticated gadget. I’ll bet many mathematicians haven’t gone through a construction of the reals, step-by-step. And some haven’t constructed Lebesgue measure. Few, surely, have seen a complete proof of the Jordan Curve Theorem. And how many do you suppose have read a proof of the Four Color Theorem?2 Here’s another pair of contrasting styles. Some mathematicians are problem solvers. They hear about a problem, they become intrigued, and they try to solve it. Other mathematicians might be called dreamers. They dream up ideas, concepts, fields to explore, and problems to solve. Of course, many mathematicians do both. The same is true in cooking. Here are the stories of two dreamers. A mathematician: Benoit Mandelbrot discovered a class of self-similar geometric objects. His discovery opened up a new field for mathematicians to explore. True, the mathematics had roots in the work of earlier researchers, Gaston Julia, for example. Mandelbrot called the objects he found ‘‘fractals.’’ He wrote papers. Mathematicians became excited and worked on his ideas. And: A cook: Paul Prudhomme had an amazing idea: a stuffed, boned chicken inside a stuffed, boned duck and then all of that inside a stuffed, boned turkey. The whole thing is roasted. True, the idea had roots in Europe, especially Britain, where Victorian cooks would nest carcasses for overthe-top feasts. Prudhomme called his composition ‘‘turduckhen.’’ Cooks became excited and tried out the recipe. Where dreamers lead, problem solvers follow. Plenty of mathematicians now work on fractals, singly and in groups. And an industry (small industry) has grown up around turduckhen. You can order them over the internet. I have to say that fractals may be more important than turduckhen. But the two stories follow the same arc. A few years ago, some friends of mine and I cooked Prudhomme’s turduckhen. It took us a weekend of work. Boning can be tricky, especially if you don’t know what you’re doing. And Prudhomme has separate stuffings and/ or coatings for each bird plus a complex sauce for the finished product. The result was impressive, but disappointing. The many tastes and textures produced a gastronomic cacophony. And the turkey meat was pretty dry, because the huge mass had to roast for hours and hours and hours to make sure that the duck and the chicken were completely cooked. Dreamers need problem solvers. J. Kenji Lo´pez-Alt is a gastronomic problem solver. He tackled the problem of the turduckhen with energy and intelligence. I haven’t tried his solution yet, but I believe in it; it is clever and convincing. In brief, he roasts the stuffed chicken before it goes into the duck and he roasts the duck before it goes into the turkey. You can check it out at www.seriouseats.com.

None. At present, every proof of the 4CT requires the cooperation of a computer.

 2014 Springer Science+Business Media New York, Volume 36, Number 1, 2014

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As in mathematics, every solution begets a problem. Here’s a gastronomic problem: Is a vegetarian turduckhen possible? I mean here a roasted dish with no meat whose component parts (at least three) are well ordered by inclusion. I want a good one, of course. This seems pretty difficult. There are other styles to think about, but I’ll leave them to future columns. I’ll close with one more analogy and a recipe. Cooks: We want our food to be healthy and edible. We also want it to be delicious. After all that, it’s nice if it looks good on the table. Mathematicians: We want our math to be correct. If possible, we also want the reasoning and structures to be elegant. After all that, it’s nice if it looks good in In each case, the third criterion is the least important. But sometimes it’s a reflection of the first two. Leonhard Euler took the formula for a geometric series with -1 \ a \ 1, 1 þ a þ a2 þ a3 þ . . . ¼

Slice the papaya in half, remove the seeds and cut away the rind. Slice the avocado in half, remove the pit and scoop out the flesh. Chop the fruits into pieces roughly the same size and shape. Drizzle with oil and mix gently. Sprinkle with lime juice. Don’t be stingy with the lime; it’s not as sour as lemon juice and it has more flavor. It counters the sweetness of the papaya and the unctuousness of the avocado. I like walnut oil, but any neutral oil is fine, grape-seed oil, for example. You have to be careful with walnut oil. If not refrigerated it becomes rancid. A good papaya is soft, mildly sweet, and juicy. They can be difficult to find. The large, red-fleshed maradol papayas can be wonderful, but they are mostly grown as a vegetable and often don’t achieve juiciness. I’ve had better luck with the smaller, yellow-fleshed papaya. Both varieties are usually unripe at the store.

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and felt the desire to look at it with a outside the range of validity, in particular, for a = 2. The resulting equation was attractive, despite its obvious absurdity. 1 þ 2 þ 22 þ 23 þ . . . ¼ 1 Euler had a feeling that the equation contained some sort of truth. You could argue that his instinct was based on form—it would be cool if it were true. This is not what most people think of as ‘‘mathematical reasoning.’’ But Euler was right. He anticipated, among other developments, the 2-adic numbers. With Euler as my inspiration (maybe) I considered a similarity between avocados and papayas. Cut in pieces, they have roughly the same texture. I wondered if they could be combined in a salad. I wasn’t thinking of taste. Only form. It would be cool, I thought, if it worked. It did work, with the addition of a dressing. It’s a refreshing salad that brings together two fruits that by themselves are less interesting.

I let them sit around until they are yellow and becoming soft. They may even appear to be rotting in spots.

PAPAYA-AVOCADO SALAD roughly equal amounts of ripe papaya and avocado walnut oil lime juice to taste salt and freshly ground pepper to taste

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Use just a little salt. The pepper is for its fragrance. For that fragrance, the pepper should be freshly ground.