Burn Math Class: And Reinvent Mathematics for Yourself

443 Pages·2016·15.79 MB·English

by  Jason Wilkes

Copyright © 2016 by Jason Wilkes Published by Basic Books, A Member of the Perseus Books Group All
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Library of Congress Control Number: 2015956108
ISBN: 978-0-465-07381-8 (e-book) 10 9 8 7 6 5 4 3 2 1
Contents
Preface
Prefacer
Act I
1 Ex Nihilo
1.1 Forgetting Mathematics
1.2 How to Invent a Mathematical Concept
1.3 Reunion
Interlude 1: Dilating Time
Things You Never See
Shortcut Distances
The Fiction of Absolute Time
2 The Infinite Power of the Infinite Magnifying Glass
2.1 Turning Hard Stuff into Easy Stuff
2.2 Inventing Calculus
2.3 Understanding the Magnifying Glass
2.4 Hunting Extremes in the Dark
2.5 A Brief Sermon on Rigor
2.6 Reunion
Interlude 2: How to Get Something from Nothing
Ex Nihilo Redux: From an Abbreviation to an Idea?
Why This Forces Zero Powers to Be 1
Why This Forces Negative Powers to Be Handstands
Why This Forces Fractional Powers to Be n-Cube Side Lengths
3 As If Summoned from the Void
3.1 Who Ordered That?
3.2 !scitamehtaM eht gnikcirT
3.3 An Attractive New Character Enters
3.4 Hammers, Patterns, and Hammer Patterns
3.5 A Phase Transition in the Creation Story
3.6 Hammers and Chains
3.7 Reunion
Interlude 3: A Backωαrd Look to the Future
Part I: The End
Part II: The Beginning
Act II
4 On Circles and Giving Up
4.1 A Conceptual Centrifuge
4.2 Impostor Syndrome
4.3 Equivalences of Our Ignorances
4.4 The Mixture Separates
4.5 What Is Meaningful
4.6 The Dilemma
4.7 Molière Is Dead! Long Live Molière!
4.8 A Tiresome Cacophony of Superfluous Names
4.9 Calculus of the Incalculable
4.10 Reunion
Interlude 4: The Nostalgia Device
The Void Is No Place for an Entity
Nostalgia for Plus-Times Machines
5 Aesthetics and the Immovable Object
5.1 Reaching into the Void
5.2 The Four Species
5.3 The Formal and the Informal
5.4 The Zoology of the MA Species
5.5 The MM Species
5.6 Reunion
Interlude 5: Two Clouds
No More Mysteries?
Intralude: Introduction to Meta and Steve
SubIntralude: The Beweis Bar
Extralude: The Existence Dilemma
Act III
6 Two in One
6.1 Two is One
6.2 Field-Testing the Fundamental Hammer
6.3 Forging the Anti-Hammers
6.4 The Second Cloud
6.5 Reunion
Interlude 6: Slaying Sharp
Best Served Cold
A Plan of Attack
An Excessive Pile of Weaponry
Hammering with Reabbreviation
Sharp Resists
Back to the Drawing Board
Unearthing Something We Buried
Unleashing the Nostalgia Device
Infinite Shattering with Anti-Hammers
Sharp Yields
N New Is Old
N.1 A Bridge
N.2 The Notational Minefield of Multivariable Calculus
N.3 Enough Symbol Games! How Can We Picture This?
N.4 Is That All?
N.5 Reunion
Interlude N: (Mis)interpreted (Readings) (Re)interpreted
Misinterpretation
Reinterpretation
The Infinite Beauty of the Infinite Wilderness
.1 The Surprising Unity of the Void
.2 Crossing the Rubicon
.3 Inventing Cannibal Calculus
.4 The Pedagogical Mutilation of Infinite-Dimensional Calculus
.5 The Infinite Jackpot: Putting Our Ideas to Work for Us
.6 Reunion
.ω Actions Speak Louder
Chapterlude Ω: In Nihilo
Eponymous
MetaIntervention
The Book
Dictionary
From Our Terminology to Standard Terminology
From Standard Terminology to Our Terminology
Index
Dedication
This is harder than I expected. Let’s see. . .
Let J ≡ “always being himself.”
Let W ≡ “the same reason.”
Let R ≡ “all the help. Hopefully that will make sense eventually.”
Okay. . .
(To E. T. Jaynes, for J) (To David Foster Wallace, for W) (1 − α)(To Reader,
1
for R) where α ∈ [0, 1] is determined by the final dedicatee.
. . . (
1. I promise the book isn’t this confusing.
Preface
Good fiction’s job [is] to comfort the disturbed and disturb the
comfortable.
—David Foster Wallace, interview with Larry McCaffery
Fiction and nonfiction are not so easily divided.
—Yann Martel, Beatrice and Virgil
Burn Math Class
Okay, don’t really burn math class. Or anything. Arson is mean, and extremely
illegal. I. . . No, nevermind. I don’t want to start the book like that.
(Author thinks for a moment.)
Alright, I think I’ve got it. Sorry about that.
(Ahem.)
We should all be very angry. Something beautiful has been stolen from us, but
we’ve never felt its absence because the theft happened long before we were
born. Imagine that by some massive historical accident, we had all been
convinced that music was a dull, tedious, rigid enterprise to be avoided except
when absolutely necessary. Suppose that we had all attended music classes for
more than a decade during our youth, and that through some feat of brilliant
sadism among the instructors, we all left these classes with a firm belief that
music is at most a means to an end. We might agree that everyone should have a
basic level of familiarity with the subject, but only for reasons of practicality:
you need music because it might — rarely — help you with other things. But the
consensus view would hold that music more closely resembles plumbing than an
art form.
The world would still be full of artists, of course. Just as it’s full of them
now. By artists, I don’t necessarily mean art-school students, or professional
artists, or the guy who wrote some stuff on a toilet and put it in a museum. I
mean the people who pull things into existence out of nothingness; who refuse
not to be themselves; who have their own way of fracturing reality that’s so
authentic you can feel it in your nerve endings; who enter the world on fire and
too often die young. “Music is not for them,” we’d agree. “Music is for the
accountants among us, and it’s best if we leave it to them.” As farfetched as such
a situation might seem, this is exactly what has happened to mathematics.
Mathematics has been stolen from us, and it is time we take it back.
With this book, I am advocating a process of conceptual arson. The state of
mathematics education all over the world has degenerated to a point where it no
longer makes sense to do anything but burn it all down and start over. We begin
by doing just that. In this book, mathematics is not approached as a preexisting