###### Basics

# The Life of Pi, and Other Infinities

Matt Dorfman

###### By NATALIE ANGIER

###### Published: December 31, 2012

On this day that fetishizes finitude, that reminds us how rapidly our
own earthly time share is shrinking, allow me to offer the modest
comfort of infinities.

This week: we take up infinity (though this podcast will be finite) and a
secret weapon against the bloodsuckers of the night — you, on deworming
pills.

An Endless Subject, and a Way to End Bedbugs?

Yes, infinities, plural. The popular notion of infinity may be of a
monolithic totality, the ultimate, unbounded big tent that goes on
forever and subsumes everything in its path — time, the cosmos, your
complete collection of old Playbills. Yet in the ever-evolving view of
scientists, philosophers and other scholars, there really is no single,
implacable entity called infinity.

Instead, there are infinities, multiplicities of the limit-free that
come in a vast variety of shapes, sizes, purposes and charms. Some are
tailored for mathematics, some for cosmology, others for theology; some
are of such recent vintage their fontanels still feel soft. There are
flat infinities, hunchback infinities, bubbling infinities, hyperboloid
infinities. There are infinitely large sets of one kind of number, and
even bigger, infinitely large sets of another kind of number.

There are the infinities of the everyday, as exemplified by the figure
of pi, with its endless post-decimal tail of nonrepeating digits, and
how about if we just round it off to 3.14159 and then serve pie on March
14 at 1:59 p.m.? Another stalwart of infinity shows up in the
mathematics that gave us modernity: calculus.

“All the key concepts of calculus build on infinite processes of one
form or another that take limits out to infinity,” said Steven Strogatz,
author of the recent book “The Joy of x: A Guided Tour of Math, From
One to Infinity” and a professor of applied mathematics at Cornell. In
calculus, he added, “infinity is your friend.”

Yet worthy friends can come in prickly packages, and mathematicians have learned to handle infinity with care.

“Mathematicians find the concept of infinity so useful, but it can be
quite subtle and quite dangerous,” said Ian Stewart, a mathematics
researcher at the University of Warwick in England and the author of
“Visions of Infinity,” the latest of many books. “If you treat infinity
like a normal number, you can come up with all sorts of nonsense, like
saying, infinity plus one is equal to infinity, and now we subtract
infinity from each side and suddenly naught equals one. You can’t be
freewheeling in your use of infinity.”

Then again, a very different sort of infinity may well be freewheeling
you. Based on recent studies of the cosmic microwave afterglow of the
Big Bang, with which our known universe began 13.7 billion years ago,
many cosmologists now believe that this observable universe is just a
tiny, if relentlessly expanding, patch of space-time embedded in a
greater universal fabric that is, in a profound sense, infinite. It may
be an infinitely large monoverse, or it may be an infinite bubble bath
of infinitely budding and inflating multiverses, but infinite it is, and
the implications of that infinity are appropriately huge.

“If you take a finite physical system and a finite set of states, and
you have an infinite universe in which to sample them, to randomly
explore all the possibilities, you will get duplicates,” said Anthony
Aguirre, an associate professor of physics who studies theoretical
cosmology at the University of California, Santa Cruz.

Not just rough copies, either. “If the universe is big enough, you can
go all the way,” Dr. Aguirre said. “If I ask, will there be a planet
like Earth with a person in Santa Cruz sitting at this colored desk,
with every atom, every wave function exactly the same, if the universe
is infinite the answer has to be yes.”

In short, your doppelgängers may be out there and many variants, too,
some with much better hair who can play Bach like Glenn Gould. A far
less savory thought: There could be a configuration, Dr. Aguirre said,
“where the Nazis won the war.”

Given infinity’s potential for troublemaking, it’s small wonder the ancient Greeks abhorred the very notion of it.

“They viewed it with suspicion and hostility,” said A. W. Moore,
professor of philosophy at Oxford University and the author of “The
Infinite” (1990). The Greeks wildly favored tidy rational numbers that,
by definition, can be defined as a ratio, or fraction — the way 0.75
equals ¾ and you’re done with it — over patternless infinitums like the
square root of 2.

On Pythagoras’ Table of Opposites, “the finite” was listed along with
masculinity and other good things in life, while “the infinite” topped
the column of bad traits like femininity. “They saw it as a cosmic
fight,” Dr. Moore said, “with the finite constantly having to subjugate
the infinite.”

Aristotle helped put an end to the rampant infiniphobia by drawing a
distinction between what he called “actual” infinity, something that
would exist all at once, at a given moment — which he declared an
impossibility — and “potential” infinity, which would unfold over time
and which he deemed perfectly intelligible. As a result, Dr. Moore said,
“Aristotle believed in finite space and infinite time,” and his ideas
held sway for the next 2,000 years.

Newton and Leibniz began monkeying with notions of infinity when they
invented calculus, which solves tricky problems of planetary motions and
accelerating bodies by essentially breaking down curved orbits and
changing velocities into infinite series of tiny straight lines and tiny
uniform motions. “It turns out to be an incredibly powerful tool if you
think of the world as being infinitely divisible,” Dr. Strogatz said.

In the late 19th century, the great German mathematician Georg Cantor
took on infinity not as a means to an end, but as a subject worthy of
rigorous study in itself. He demonstrated that there are many kinds of
infinite sets, and some infinities are bigger than others. Hard as it
may be to swallow, the set of all the possible decimal numbers between 1
and 2, being unlistable, turns out to be a bigger infinity than the set
of all whole numbers from 1 to forever, which in principle can be
listed.

In fact, many of Cantor’s contemporaries didn’t swallow, dismissing him
as “a scientific charlatan,” “laughable” and “wrong.” Cantor died
depressed and impoverished, but today his set theory is a flourishing
branch of mathematics relevant to the study of large, chaotic systems
like the weather, the economy and human stupidity.

With his majestic theory of relativity, Einstein knitted together time
and space, quashing old Aristotelian distinctions between actual and
potential infinity and ushering in the contemporary era of infinity
seeking. Another advance came in the 1980s, when Alan Guth introduced
the idea of cosmic inflation, a kind of vacuum energy that vastly
expanded the size of the universe soon after its fiery birth.

New theories suggest that such inflation may not have been a one-shot
event, but rather part of a runaway process called eternal inflation, an
infinite ballooning and bubbling outward of this and possibly other
universes.

Relativity and inflation theory, said Dr. Aguirre, “allow us to
conceptualize things that would have seemed impossible before.” Time can
be twisted, he said, “so from one point of view the universe is a
finite thing that is growing into something infinite if you wait
forever, but from another point of view it’s always infinite.”

Or maybe the universe is like Jorge Luis Borges’s fastidiously imagined
Library of Babel, composed of interminable numbers of hexagonal
galleries with polished surfaces that “feign and promise infinity.”

Or like the multiverse as envisioned in Tibetan Buddhism, “a vast system
of 1059 universes, that together are called a Buddha Field,” said
Jonathan C. Gold, who studies Buddhist philosophy at Princeton.

The finite is nested within the infinite, and somewhere across the
glittering, howling universal sample space of Buddha Field or Babel,
your doppelgänger is hard at the keyboard, playing a Bach toccata.

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