Table Of ContentMATHEMATICS
IS ABOUT THE WORLD
MATHEMATICS
IS ABOUT THE WORLD
______________________
HOW AYN RAND’S THEORY OF CONCEPTS
UNLOCKS THE FALSE ALTERNATIVES
BETWEEN PLATO’S MATHEMATICAL
UNIVERSE
AND HILBERT’S GAME OF SYMBOLS
Robert E. Knapp
Copyright © 2014 Robert E. Knapp
No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form, or
by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
consent of the author.
First Edition
Visit the website for the book: www.MathematicsIsAboutTheWorld.com to find out more about the book
and to contact the author.
CONTENTS
Acknowledgements 7
Preface 9
PART 1: ELEMENTARY
1. Euclid’s Method
23
2. Measurement and the Geometry of Magnitudes 101
3. Geometric Area, Proportion, and the Parallel Postulate 177
4. Numbers as a System of Measurements 229
5. Geometry and Human Cognition
277
PART 2: ADVANCED
6. Set Theory and Hierarchy in Mathematics 293
7. Vector Spaces: A Study in Mathematical Abstraction 357
8. Abstract Groups and the Measurement of Symmetry 425
Index 489
Acknowledgements
My greatest debt is to Ayn Rand’s Introduction to Objectivist Epiestemology
without which this book could not have been conceived. Published during my
college years, it has illuminated my understanding and appreciation for
mathematics across five decades.
I have been stimulated by a number of extended exchanges with Harry
Binswanger on a variety of mathematical topics. In regards to my own work,
Harry has provided welcome encouragement and valuable ciriticism that have
helped me sharpen my formulations and, indeed, saved me from several serious
mistakes.
Pat Corvini’s fascinating and stimulating lectures on Zeno’s paradox, together
with the reactions of its audience first convinced me that there was a potential
audience for my own ideas and triggered my first essays on how to look at
mathematics.
Objectivist Conferences (OCON) provided my first opportunity for a public
presentation of my ideas and I am grateful for the questions and comments from
my audience, and especially to Ray and Rebecca Girn, Evan Picoult, and Irene
Knapp who previewed my courses and offered many valuable comments and
suggestions.
I have received enouragement and valuable comments on chapters and/or earlier
formulations of these ideas from Harry Binswanger, Raymond Knapp, Evan
Picoult, Ray Girn, Rebecca Girn, Robert Rubinstein, Kenneth and Sandy Landin,
Judy Knapp, and Shrikant Rangnekar.
Shrikant was particularly helpful in the earlier stages of this book, reading and
critiquing every chapter as it was written, and making a second careful reading
of the initial draft. His helpful observations and suggestions, interest, and
encouragement was at once helpful and very motivating. His guidance through
my early attempts to find a publisher or an agent, though unsuccessful, netted
much of the material that has made it into my preface. Throughout, his
knowledge and judgement of the publishing industry has been invaluable and his
enthusiasm for my book has been inspiring.
Finally, it should go without saying that all responsibility for both the content
and presentation of that content is entirely my own.
Preface
What is mathematics about? Is there a world of mathematics in which straight
lines are infinitely straight, are infinitely thin, and contain an uncountable
infinity of points? Or, on the contrary, is mathematics a purely formal system of
deductions from axioms, a system that is beautiful to some of us, but boring and
just too difficult for the rest of us? Is it devoid, in either case, of any worldly or
other worldly content, yet
mysteriously applicable to the world we inhabit? Or, finally, contrary to the
nearly universal range of acceptable academic views, is mathematics about the
world?
This book is written for people who care about such questions, people who are
still looking for answers and are not satisfied with the standard suggestions,
people who are open to a common sense-yet radical-alternative.
This is the book I would have loved to find when I encountered abstract
mathematics in my high school years and later, in college, began the process of
becoming a mathematician.
It’s the alternative perspective I needed when I read, also in high school,
Russell’s Introduction to Mathematical Philosophy. Though from a far different
perspective, like Russell’s, mine is a book on how to think about mathematics.
My own fascination with mathematics began with a book by Irving Adler
entitled Magic House of Numbers, which I checked out of the library in sixth
grade because of the word “magic” in the title. Thus began a lifetime of interest,
study, and for a brief time research, in mathematics. By the end of ninth grade,
with the aid of old textbooks lying around the house, I was learning
trigonometry, analytic geometry, and the beginnings of calculus.
In my sophomore year of high school, I found a copy of Birkoff and MacLane’s
highly regarded A Survey of Modern Algebra in my school library. And I was
struck by the opening of Chapter III, “Polynomials”, which reads “Let D be any
integral domain, and let “x” be any symbol. Suppose one forms sums,
products, and differences of x with the elements of D and with itself, subject to
products, and differences of x with the elements of D and with itself, subject to
the rules of ordinary algebra…”
Now it is true that a polynomial defines a rule that can be applied to any
mathematical domain that employs a sensible notion of addition and
multiplication. One can, for example, apply
polynomials to matrices and even to other polynomials. So a
polynomial has a meaning and significance as a set of rules that transcends any
particular mathematical domain to which it might be applied.
But this is not what I took the authors to mean, nor what they were actually
saying. Instead, I heard: Don’t think of
polynomials as meaning anything in particular. Look at them as manipulations of
meaningless symbols. Well, I continued for many more pages before I returned
the book to the library, but I got nothing out of it. Such was my first encounter
with abstract mathematics!
By the end of my junior high school year I had read about matrices and vector
spaces from an interesting perspective that made sense to me. Then, at some
point near the end of that year or, perhaps, that summer I started learning about
“rings” and “ideals”
as an abstract study. I had no idea why “ideals” were important or what they
were about. That “rings” were a general framework for studying divisibility and
factorization never occurred to me. But I remember thinking, “Rings are just a
way of thinking about
numbers, matrices, and polynomials.” From that moment, I
embraced mathematical abstraction. Yet, simultaneously, I
embraced the view that mathematics is about the world. I had glimpsed that
mathematical abstractions are a way of looking at things; neither a path to a
separate mathematical world nor a play with meaningless symbols.
During my senior year in high school I started reading, with great appreciation,
Halmos’ great text on finite dimensional vector spaces, presented on an abstract
level, but very readable and inspiring. Later, I began, again with appreciation,
my freshman year in college reading Dieudonne’s highly abstract Foundations of
Modern Analysis. Both books reinforced my belief in the power of mathematical
abstraction; neither deterred me from my view that mathematics is about the
world.
At some point during my college freshman year, I realized that neither
mathematicians nor philosophers of mathematics
shared my perspective, offering only the alternatives of formalism (a game of
symbol manipulation), Platonism (a separate world of mathematics), or, as a
third, the Fregean view that mathematics is essentially a branch of logic. I could
accept none of these choices.
However, I had discovered Ayn Rand and during the following summer she
began her epochal series of articles on Objectivist epistemology. It struck me,
from her first essay linking mathematics and concept formation, that I had found
my key to understanding mathematics. In regards to its specific application to
mathematics, however, I also realized that I was on my own.
The purpose of this book is to share my present understanding and perspective,
to offer a distinctive view of mathematics that was made possible for me by Ayn
Rand’s essays and extemporaneous comments relating to her theory of
universals.
And my central message is this: You do not have to choose between
mathematical abstractions and reality. Mathematical abstractions are a way of
understanding the world, of deepening and enriching one’s perspective on the
world. One understands the essence of a mathematical discipline when one
grasps what it is trying to measure. Mathematics is about the world.
Though I accept neither alternative, I find an important difference between the
ways that mathematicians and philosophers each typically look at mathematics.
Mathematicians in general, operating in a world of “complex manifolds”,
“homological
algebra”, infinite-dimensional “function spaces”, and “fiber bundles” tend to
believe in some kind of Platonic mathematical world suggested by, but distinct
from, the world we live in and taking on a life of its own. They know that
mathematics is hard work and must involve something more than arbitrary
symbols and rules. Philosophers, on the other hand, generally dismiss a world of
symbols and rules. Philosophers, on the other hand, generally dismiss a world of
mathematics. They differ on what should take its place. Some of the best,
certainly, have attempted a naturalistic, reality-based, approach to understanding
mathematics, starting with ordinary numerical concepts of the sort we form in
childhood. But they typically try to incorporate formal set theory as it was
developed during the 20th century and consequently, at some point along the
way, they leave the world behind.
But is there a third alternative? When I say to someone who is neither a
mathematician nor a philosopher, “Mathematics is about the world,” the usual
response is, “Of course!” But such responses quickly change when I bring up
infinitely straight infinitely thin straight lines and, in general, mathematical
infinity.
How can there be an infinite number of counting numbers? Or, on the contrary,
is there a finite number? Is there a point, a large number “N” such that N is
meaningful, but N + 1 is meaningless?
And, if so, what about the immediate implication that the equation x – 1 = B has
a solution for B = 5 or for B = 500 trillion, but does not have a solution for this
“largest” number B = N? Or what about the fact that, for this largest number I
am calling N, we would not be able to express a length of N feet in inches,
because we would have to multiply N times 12 to find the number of inches
resulting in a number larger than N?
It appears that a number system, to be manageable, must be open-ended.
Mathematical infinities that involve numbers, infinities that do not exist in the
world,
are,
indeed,
an
indispensable part of mathematics. Similarly, in geometry, the infinite precision
of Euclid’s geometric reasoning is necessary to the entire structure and the
relationships that Euclid uncovers must be utilized in any attempt to measure
degrees of imprecision in one’s measurements of objects in the world.
In sum, it seems that the more one thinks about mathematics and its methods
In sum, it seems that the more one thinks about mathematics and its methods
(and the more advanced one’s
mathematical studies) the harder it becomes to maintain, as I nonetheless do, that
mathematics is about the world. These
paradoxes are obvious and must be answered: If mathematical
infinities do not exist in the world, how can a mathematics that includes
mathematical infinities be about the world? Or, if all geometric measurements
have finite precision, yet geometric
concepts and arguments are infinitely precise, how can geometry be about the
world?
So this book is, necessarily, a defense of an unpopular viewpoint; but more
fundamentally it is a book about how to think about mathematics.
As a pursuit, mathematics is about discovering relationships between quantities,
about discovering and solving algebraic and differential equations or finding
geometric
relationships. It is, as Ayn Rand put it, the science of measurement.
But why is measurement important? And why do the demands of
measurement require the amazingly abstract complex science that mathematics
became during the 19th and 20th centuries? Do all these abstractions really have
something to do with measurement?
I offer my affirmative answer, in part, as the key to relating mathematics to the
world. But I also believe that the measurement viewpoint is the key to
understanding the underlying purpose and the logical structure of mathematics.
In this book, I apply my viewpoint, first to elementary and then to more
advanced mathematical topics. I start with elementary topics to show how
mathematics begins with measurement and to show how mathematical progress
is driven by the needs of
measurement. I proceed to more advanced topics to indicate, by selected
examples, how measurement remains the underlying
