The Proof and the Pudding: What Mathematicians, Cooks, and You Have in Common

by Jim Henle

Overview

Tantalizing math puzzles and cooking recipes that show how mathematical thinking is like the culinary arts

Tie on your apron and step into Jim Henle's kitchen as he demonstrates how two equally savory pursuits—cooking and mathematics—have more in common than you realize. A tasty dish for gourmets of popular math, The Proof and the Pudding offers a witty and flavorful blend of mathematical treats and gastronomic delights that reveal how life in the mathematical world is tantalizingly similar to life in the kitchen.

Take a tricky Sudoku puzzle and a cake that fell. Henle shows you that the best way to deal with cooking disasters is also the best way to solve math problems. Or take an L-shaped billiard table and a sudden desire for Italian potstickers. He explains how preferring geometry over algebra (or algebra over geometry) is just like preferring a California roll to chicken tikka masala. Do you want to know why playfulness is rampant in math and cooking? Or how to turn stinky cheese into an awesome ice cream treat? It’s all here: original math and original recipes plus the mathematical equivalents of vegetarianism, Asian fusion, and celebrity chefs.

Pleasurable and lighthearted, The Proof and the Pudding is a feast for the intellect as well as the palate.

Product Details

ISBN-13:	9781400865680
Publisher:	Princeton University Press
Publication date:	04/27/2015
Sold by:	Barnes & Noble
Format:	eBook
Pages:	176
File size:	20 MB
Note:	This product may take a few minutes to download.

About the Author

Jim Henle is the Myra M. Sampson Professor of Mathematics and Statistics at Smith College. His books include Sweet Reason: A Field Guide to Modern Logic and Calculus: The Language of Change. He lives in Northampton, Massachusetts.

Read an Excerpt

The Proof and the Pudding

What Mathematicians, Cooks, and You Have in Common

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By Jim Henle

PRINCETON UNIVERSITY PRESS

Copyright © 2015 Princeton University Press
All rights reserved.
ISBN: 978-1-4008-6568-0

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CHAPTER 1

THE MAP SCIENTIST

WE'RE GOING to start with two investigations, one in mathematics and one in cooking. On the surface, they have little in common. There are similarities, though, similarities of "spirit," for want of a better word. I'll say more at the end.

DOODLES

A few years ago, I was doodling, that is, I was sitting with pen and paper, drawing lines with no purpose in mind. I drew a square with grid lines.

[ILLUSTRATION OMITTED]

I put diagonals in some of the boxes.

[ILLUSTRATION OMITTED]

I imagined the lines as mirrors. I wondered what would happen if a ray of light entered the square and started bouncing around.

[ILLUSTRATION OMITTED]

I noticed that it was possible for a ray of light to visit the same box twice, that is, it could bounce twice off the same mirror.

[ILLUSTRATION OMITTED]

That made me wonder how long I could keep things going. Given a square, how long a path could I make, assuming I could place the mirrors wherever I wanted?

I started small. The 2 × 2 square allowed a path of length 5.

[ILLUSTRATION OMITTED]

(I count each square I enter. I count a square twice if I enter it twice.)

[ILLUSTRATION OMITTED]

For the 3 × 3 square, I first got a path of length 9,

[ILLUSTRATION OMITTED]

then one of length 10,

[ILLUSTRATION OMITTED]

then one of length 11.

[ILLUSTRATION OMITTED]

That seemed the best I could do.

Do? Do? What was I doing?

I was just having fun. First there was a grid. Then there were mirrors. Without a plan, I found myself drawing lines, scratching in diagonals, and tracing beams of light.

I was curious. I wanted to know how long a path I could make. I wanted to know the longest path in a 4 × 4 square, the longest path in a 5 × 5 square, and so on. After a while I got curious about rectangles, too.

You should understand that you're dealing with someone who gets very excited about primitive pen-and-paper activities. When I first thought of mirrors in a square, I drew grid after grid after grid after grid after grid.

The best I could do for the 4 × 4 was a path of length 22.

[ILLUSTRATION OMITTED]

The best I could do for the 5 × 5 was a path of length 35.

[ILLUSTRATION OMITTED]

But that looks a little unsatisfying, doesn't it? There s a square (upper right-hand corner) I never visit. If I visited every square, could I get a longer path, couldn't I? But I've tried. I don't think I can!

After some thought, I was able to prove that my answer for the 4 × 4 square was the best that was possible.

You may be wondering: "Is this really mathematics?"

It is indeed mathematics. I take an expansive view of the subject. For me, any structure that can be described completely and unambiguously is a mathematical structure. And any statement about that structure that can be proved beyond doubt is a mathematical statement. The proof is a mathematical achievement. Inventing such a structure, making discoveries about it, proving statements about it—that's mathematics.

The structure of the square, the mirrors, and the rays of light can be described completely and unambiguously. And the fact that the longest possible path on a 4 × 4 is 22 is a genuine mathematical statement.

Here's my proof that the longest path on a 4 × 4 is 22:

Since we have an example of a path of length 22, all we have to do is show that no longer path is possible.

Now it's clear that we can visit a square at most twice, like this—

[ILLUSTRATION OMITTED]

But we can visit an edge square only once,

[ILLUSTRATION OMITTED]

unless we are entering or exiting,

[ILLUSTRATION OMITTED]

so the best we can do is

1. visit the four interior squares twice,

2. visit the edge squares once, except

3. visit two edge squares twice.

That makes a total of 22, as in the case of the example earlier.

[ILLUSTRATION OMITTED]

And that's a proof that 22 is the best we (or anyone) can do.

I call this a "doodle." There are more doodles. Anyone can invent a doodle—you just invent your own rules. I've had fun, for example, with a one-way mirror doodle. It looks like this.

[ILLUSTRATION OMITTED]

In one direction the mirror reflects and in the other it doesn't. Here's a little example.

[ILLUSTRATION OMITTED]

This can be a lot of fun. I think I can get a path 33 squares long in a 4 × 4 square.

There are doodles and doodles. I've set up a website, with notes for most chapters in this book:

press.princeton.edu/titles/10436.html

I invite the reader to visit the site. In particular, there are more doodles there.

I also invite you to share your doodle ideas with me.

[e-mail]

NOOPLES

This may seem a little abrupt. Noodles and doodles appear to have almost nothing in common. I'll say something at the end of the chapter.

It started a few years ago. I was preparing a dinner for friends. I had planned to cook some sort of spaghetti dish.

But my plans were confounded when I learned that one of the guests had celiac disease, an allergy to wheat gluten. She couldn't eat wheat pasta.

That was bad news. But I was determined to cook that dish. At the grocery store I found corn spaghetti. I bought a box of it. I cooked it—and served my puzzled friends a gooey mess.

I may have overcooked it. But I suspect that the only way to undercook corn spaghetti is to leave it at the store.

My guest was embarrassingly grateful. I could have left it there but I saw a challenge that intrigued me. Is there something to take the place of pasta? Could I find a substance that

• is gluten-free, and

• is functionally equivalent to pasta?

No dumplings. No gnocchi. No couscous. All of these contain wheat.

What I wanted was a sort of multipurpose faux pasta, something that could comfortably take the place of macaroni or penne. I tried practically everything. French fries. Brussels sprouts. Corn flakes.

The challenge may seem difficult and maybe even pointless. But it attracted me.

And it entertained me. The reader would be alarmed to know what, over several years, I put on the table in lieu of pasta. I had successes. I had failures. No one outside my immediate family was seriously harmed.

My happiest experiments were with corn-off-the-cob. Here is a reasonable example.

One could complain, "Corn isn't soft like pasta. And corn doesn't absorb flavors like pasta." Well, that's true. But this is a great dish.

If you want a soft faux pasta, rice works. I don't mean risotto, though. Risotto isn't faux pasta. The process of cooking real pasta is the same, mostly, no matter what sauce you use. A proper faux pasta should be something you just cook and then mix with sauce. With risotto, different recipes differ at the start.

The type of rice is important. Good jasmine rice can give you a soft but chewy grain that works well with many pasta sauces.

I hope I haven't made cooking rice sound difficult. It's not difficult. You have a few minutes leeway in turning down the heat. And if the rice boils over, that's okay. It just makes a mess.

You also have a few minutes leeway in turning off the rice. And if the rice burns, most of it is still good. And I have a great recipe for the brown stuff at the bottom.

Other ideas? Chickpeas? Zucchini? Scallopini? There's no end to this.

There are more recipes on the website:

press.princeton.edu/titles/10436.html.

And if you have ideas, I'm interested.

NOODLES AND DOODLES

Apart from rhyming, noodles and doodles have nothing in common. I chose them to illustrate some shared features of mathematics and gastronomy, features that appear in this book again and again.

First of all, they are pleasures. Of course, sometimes we cook because we're hungry. And sometimes we calculate because we have to pay our taxes. But real cooks and real mathematicians play. They play with structures, they play with ingredients, they play with the ideas and the flavors that attract them strongly.

Second, while the attraction is aesthetic, it's also intellectual. We're curious. We want to taste; we want to tinker; we want to explore; we want to find out. We savor the unknown.

Third, and this may be the most important point, we often don't know what we're doing. We stumble around. Mathematics and gastronomy are mysteries. We have to stumble to make progress. We experiment. We try one thing. We try another. We may appear to have no method. But that's not true. Stumbling around is a method. It's the go-to method, surprisingly, of the best cooks and the best mathematicians.

Stumbling (and making progress) is the focus of the next chapter.

Hundreds of books are devoted to solving math problems. Thousands of books are devoted to cooking techniques. In the next chapter I will convince you (maybe) that the key to one is the key to the other.

CHAPTER 2

THE ARROGANT CHEF

I SOMETIMES hear people say

"I can't bake bread."

The way they say it is familiar to me. The words are different, but the tone is the same.

"I can't do math."

The confessions are similar and similarly sad. They're not about weakness; they're about anxiety. In truth, everyone can do math and everyone can bake bread. Both acts are exercises in problem-solving. The remarkable fact is that the best method for solving math problems is also the best method for solving problems in the kitchen.

I have a simple theory about problem-solving. What you need to solve problems is a split personality. You need, first of all, confidence. Good problem-solvers are sure they can solve anything. Given a problem, successful problem-solvers dive in fearlessly, certain that they'll crack it right away.

But you also need doubt. Once you have a solution, the confidence has to step back. You need to question your answer, worry about it. Test it, tweak it. At the final stage, good problem-solvers act as though they're sure there's something wrong with their answers.

I think you can see the usefulness of both these personalities. You need the confident one at the outset. Without confidence, it's hard to begin. And if you do get started, lack of confidence can sap your strength and will.

And then, when you have an answer, you want the humble, doubting personality. You want to analyze your answer, understand it, check it for flaws.

The first personality is the most difficult to assume. You can't simply decide to be confident. I offer my students a substitute personality which seems to work pretty well. I tell my students to be arrogant.

BAKING BREAD

As an example, let's see how arrogance works in the kitchen. Let's talk about bread. So much has been written about bread that trying it for the first time can be scary. The difficulties are endless; there's so much to know, and so many ways to fail, or so it seems. How do you deal with yeast? How do you knead? How do you shape the loaves? How do you know when it's done? It takes years to learn!

Let's look at a simple recipe, and see how an arrogant (and ignorant) cook can bake bread successfully on the first try.

That's all. It's a minimalist recipe.

The meek cook says: "It doesn't tell me how much flour to add or how to knead. It expects me to know these things already. I don't know these things. It's hopeless."

But the arrogant cook thinks: "If the recipe doesn't tell me things, it must mean they aren't very important. I'm going ahead to see what happens."

Let's be arrogant and try it.

We have no problems until we add the flour. We don't have bread flour. We're using all-purpose flour (this is a purpose, isn't it?). We add flour and after a while it gets hard to stir. Maybe it's time to knead.

We have a hazy idea of kneading. Maybe we've seen an "iron chef" do it, or maybe we saw something on YouTube. In any case, we put our hands into the bowl and start to mess around. It's pretty sticky. It doesn't seem right. Maybe it could use more flour. We put in more flour. We wrestle with it some more. Maybe we dump it out on the counter and maybe we don't. After a while, we decide that's all the kneading it's going to get. The dough looks untidy. We set it aside to rise.

Rise where? In the bowl? On the counter? Under the bed? The bowl looks bad. There are bits of flour sticking to it, so we leave the dough on the counter.

Somehow, we stumble through the recipe. It doesn't exactly rise, it sort of spreads out. It takes hours. We keep going. We make crude loaves. We bake them. How did they turn out?

Actually ... the bread came out pretty good.

Why? In all honesty, we did a lousy job. We didn't knead the dough enough. Then the dough dried out when we left it on the counter. The recipe we used was basic, almost primitive. We put nothing exciting or flavorful into the dough. Why did it work?

Well first of all, supermarket bread is terrible. We have to fail very badly indeed to produce something worse. No matter what we do, our bread has the incomparable taste and aroma of yeast. It's real.

Second, bread is extremely forgiving. You can use more yeast or less yeast, more sugar or less sugar, more oil or less oil. You can knead it longer or knead it less, let it rise longer or less, bake it longer or bake it less. You'll still make pretty good stuff.

Strange things can happen, of course. I've made bread that flowed out and over the rim of the bread pan. Prying it loose from the pan was a battle. It looked ridiculous. But it tasted great.

And what if we had failed?

In the event of failure, we would have wasted ingredients worth at most a dollar. We didn't spend much time. We didn't disappoint anyone (we weren't so arrogant that we invited guests). And we learned something.

Every time we cook we learn. Wearing our humble personality, we reflect on what we liked and what we didn't like. We look for more information, recipes, and descriptions. We plan what we might do to improve the bread.

I bake bread several times a week. I've been doing it for 25 years. My basic recipe took ages to develop. It's fabulous and I say so myself.

I'm going to give you the recipe here and will humbly accept your praise. But part of me hopes that you will ignore it. That part of me wants you to spend years in your own quest for bready perfection. If you do, you'll have fun (and eat well). And you'll end up with a loaf that is new and wonderful.

A little explanation before I give you the recipe. I wanted my bread wheaty but light. I wanted it healthy but entertaining. I wanted it flavorful but undemanding. It's mostly white flour, so it has a light texture. The wheat germ gives it the protein of whole wheat bread. The other items provide additional wheatiness.

The recipe is designed for taste and convenience. It takes me 15 minutes of work altogether, time that can be scattered throughout the day at my convenience. It's no trouble to supply a household with fresh bread every day.

The Nobel Prize–winning physicist Richard Feynman was a paragon of confidence. He was intellectually curious but curiosity is not uncommon. He was brilliant but many people are brilliant. What set Feynman apart was his fearlessness. He felt he could tackle any problem. To him, ignorance was no obstacle. His wonderful memoir Surely You Must Be Joking, Mr. Feynman, contains many examples of this—cognition, myrmecology (ants), sexology, locksmithing, urology.

That's the difference between a great scientist and an ordinary genius:

Confidence.

(Continues...)

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Excerpted from The Proof and the Pudding by Jim Henle. Copyright © 2015 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Preface vii
Acknowledgments ix
1 The Mad Scientist 1
2 The Arrogant Chef 11
3 Simple Tastes 21
4 Complex Flavors 30
5 The Discriminating Eater 40
6 The Persistent Cook 44
7 Gluttony 54
8 Vanity, Sloth, Parsimony, and Lust 62
9 On the Edge, and Over 70
10 Thinking Globally 79
11 Eating Locally 87
12 The Humble Cook 92
13 The Clueless Geek 99
14 Elegant Dishes 103
15 Food for the Masses 110
16 Good Food for the Masses 111
17 Just for Fun 112
18 Just to Be Weird 117
19 Celebrity Chefs 124
20 Economy 128
21 Ethics 135
22 Fusion 138
23 It All Comes Together 149
24 It All Falls Apart 151
25 A Proof and a Pudding 161
Index 163

What People are Saying About This

From the Publisher

"If you're a fan of Julia Child or Martin Gardner—who respectively proved that anyone can have fun preparing fancy food and doing real mathematics—you'll enjoy this playful yet passionate romp from Jim Henle. It's stuffed with tasty treats and ingenious ideas for further explorations, both in the kitchen and with pencil and paper, and draws many thought-providing parallels between two fields not often considered in the same mouthful."—Colm Mulcahy, author of Mathematical Card Magic: Fifty-Two New Effects

"The Proof and the Pudding challenges mathematicians to be chefs and chefs to be mathematicians. Using a pencil in his mathematical kitchen, Henle explores the natural connections between mathematics and cooking and reveals how both can be creative, fun, and memorable. So pull up a plate and enjoy helping after helping of insight into gastronomy, math, and problem solving."—Tim Chartier, author of Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing

"Henle has written an enjoyable book that does an excellent job of relating mathematics to something we all do every day. Part cookbook and part popular-math book, The Proof and the Pudding is well written, insightful, and enlightening."—Oscar E. Fernandez, author of Everyday Calculus: Discovering the Hidden Math All around Us

Editorial Reviews

Shortlisted for the 2016 Gourmand World Cookbook Awards, in Digital E-Books

From the Publisher