Infinity Is a Beautiful Concept – And It’s Ruining Physics
I was seduced by infinity at an early age. Georg Cantor’s diagonality proof that some infinities are bigger than others mesmerized me, and his infinite hierarchy of infinities blew my mind. The assumption that something truly infinite exists in nature underlies every physics course I’ve ever taught at MIT—and, indeed, all of modern physics. But it’s an untested assumption, which begs the question: Is it actually true?
A Crisis in Physics
There are in fact two separate assumptions: “infinitely big” and “infinitely small.” By infinitely big, I mean that space can have infinite volume, that time can continue forever, and that there can be infinitely many physical objects. By infinitely small, I mean the continuum—the idea that even a liter of space contains an infinite number of points, that space can be stretched out indefinitely without anything bad happening, and that there are quantities in nature that can vary continuously.
The two assumptions are closely related, because inflation, the most popular explanation of our Big Bang, can create an infinite volume by stretching continuous space indefinitely. The theory of inflation has been spectacularly successful and is a leading contender for a Nobel Prize. It explains how a subatomic speck of matter transformed into a massive Big Bang, creating a huge, flat, uniform universe, with tiny density fluctuations that eventually grew into today’s galaxies and cosmic largescale structure—all in beautiful agreement with precision measurements from experiments such as the Planck and the BICEP2 experiments. But by predicting that space isn’t just big but truly infinite, inflation has also brought about the socalled measure problem, which I view as the greatest crisis facing modern physics.
Physics is all about predicting the future from the past, but inflation seems to sabotage this. When we try to predict the probability that something particular will happen, inflation always gives the same useless answer: infinity divided by infinity. The problem is that whatever experiment you make, inflation predicts there will be infinitely many copies of you, far away in our infinite space, obtaining each physically possible outcome; and despite years of teethgrinding in the cosmology community, no consensus has emerged on how to extract sensible answers from these infinities. So, strictly speaking, we physicists can no longer predict anything at all!
This means that today’s best theories need a major shakeup by retiring an incorrect assumption. Which one? Here’s my prime suspect: ∞.
Infinity Doesn’t Exist
A rubber band can’t be stretched indefinitely, because although it seems smooth and continuous, that’s merely a convenient approximation. It’s really made of atoms, and if you stretch it too far, it snaps. If we similarly retire the idea that space itself is an infinitely stretchy continuum, then a big snap of sorts stops inflation from producing an infinitely big space and the measure problem goes away. Without the infinitely small, inflation can’t make the infinitely big, so you get rid of both infinities in one fell swoop—together with many other problems plaguing modern physics, such as infinitely dense blackhole singularities and infinities popping up when we try to quantize gravity.
In the past, many venerable mathematicians were skeptical of infinity and the continuum. The legendary Carl Friedrich Gauss denied that anything infinite really exists, saying “Infinity is merely a way of speaking” and “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics.” In the past century, however, infinity has become mathematically mainstream, and most physicists and mathematicians have become so enamored with infinity that they rarely question it. Why? Basically, because infinity is an extremely convenient approximation for which we haven’t discovered convenient alternatives.
Consider, for example, the air in front of you. Keeping track of the positions and speeds of octillions of atoms would be hopelessly complicated. But if you ignore the fact that air is made of atoms and instead approximate it as a continuum—a smooth substance that has a density, pressure, and velocity at each point—you’ll find that this idealized air obeys a beautifully simple equation explaining almost everything we care about: how to build airplanes, how we hear them with sound waves, how to make weather forecasts, and so forth. Yet despite all that convenience, air of course isn’t truly continuous. I think it’s the same way for space, time, and all the other building blocks of our physical world.
We Don’t Need the Infinite
Let’s face it: Despite their seductive allure, we have no direct observational evidence for either the infinitely big or the infinitely small. We speak of infinite volumes with infinitely many planets, but our observable universe contains only about 10^{89} objects (mostly photons). If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information, specified by a number with infinitely many decimal places. In practice, we physicists have never managed to measure anything to more than about seventeen decimal places. Yet real numbers, with their infinitely many decimals, have infested almost every nook and cranny of physics, from the strengths of electromagnetic fields to the wave functions of quantum mechanics. We describe even a single bit of quantum information (qubit) using two real numbers involving infinitely many decimals.
Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too—in a way that’s more deep and elegant than the hacks we use for our computer simulations.
Our challenge as physicists is to discover this elegant way and the infinityfree equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I’m betting that we also need to let go of it.
Excerpted from This Idea Must Die, edited by John Brockman. Used with permission.
Image by Mark Ramsay via Flickr

Rich

atanu nath

Jeff Kesner


LEK56

7eggert


http://www.mazepath.com/uncleal/qz4.htm Uncle Al

Jeff Kesner


http://philosophyinfigures.tumblr.com/ Ryan Reece

http://www.nukingpolitics.com Keln

Ramome



Phillip Rulon

Henry Massey

7eggert


7eggert

minstrelmike

GuestWhom

7eggert

daqu

diogenes


diogenes

bt




Franck Merlot

diogenes

Franck Merlot




Mark Byron

Jeff Kesner


P. J. Dillon

Franck Merlot


Andrew Howard

7eggert

Andrew Howard

7eggert




gtrmath

silver fox

David S. Warren

Juror #8

GridUser

gtrmath


DarkMatterExplainer

Juror #8



Mike 3/505

Juror #8

Mike 3/505

Gary Ronan


7eggert

Mike 3/505

RedStatePatriot


MaximumOvertroll

Dingo

MaximumOvertroll

Dingo


Andy

MaximumOvertroll

Anechidna


7eggert

MaximumOvertroll

7eggert

MaximumOvertroll

VendicarDecarian0

Jason



protn7

Dingo

protn7




Jeff Kesner

Dingo


Aeffesstoo

the truth


joshuagenes

Yoikes

joshuagenes



Michael Ryan

http://plus.google.com/CadeJohnson delbarco

nfojunky


modmans2ndcoming

Robert Anweiler

Nate

Mike 3/505

VendicarDecarian0



ericlipps

VendicarDecarian0


Dan Poisson

VendicarDecarian0


diogenes

VendicarDecarian0

Andrey Lalev


jhs

Andy

jhs

DarkMatterExplainer

VendicarDecarian0

jhs



jhs


VendicarDecarian0


http://www.nukingpolitics.com Keln

7eggert

http://www.nukingpolitics.com Keln

Jeff Kesner

7eggert

waffleironmarch

7eggert



Dan Poisson


albertobladerunner


Mrowley

Jeff Kesner

wikiwhat

Jeff Kesner

Steve




Dingo

Jason

VendicarDecarian0


bt

seescaper

Jeff Kesner


CrazyHungarian

johnwerneken

JoeThePimpernel

fedup

putupjob

http://charlesrrkiss.info/ Charles R. Kiss

Dan

http://charlesrrkiss.info/ Charles R. Kiss


Bee Farms

f_this_state

YeahRight

Franck Merlot


Jason

Franck Merlot


blair houghton

YeahRight


Douglas J. Bender

YeahRight


YeahRight

Matthew

Frank_Truth

GorgonPolus577

davidt

DJ

John Dodero

johnnysy

ronaldmsonntag

YeahRight

ronaldmsonntag

YeahRight




Anton Szautner

Jim

Randall Wakefield

Randall Wakefield

Jacobo

Joshua Kellerman

YeahRight

Joshua Kellerman

YeahRight




Ken Fritz

Artur Sixto

modmans2ndcoming

YeahRight

DarkMatterExplainer

YeahRight

albertobladerunner


albertobladerunner


albertobladerunner


Duke Eyman

YeahRight

albertobladerunner


ThatCat

YeahRight


http://posdarwinista.blogspot.com.br Enézio E. De Almeida Filho

Ken Weiss

God’swill Doctore Abang

YeahRight

God’swill Doctore Abang

YeahRight




GarretKadeDupre

Lawrence Cooper

Tony Smith

FUBARISME

ericlipps

YeahRight


http://www.thegardener.org/ David Kline

j2saret

YeahRight

waffleironmarch



daqu

boonteetan

YeahRight


Barry Rudd

Randall Lusk

YeahRight

Randall Lusk



Randall Lusk

YeahRight

waffleironmarch



https://plus.google.com/+DrT%C3%B3thL%C3%A1szl%C3%B3Zolt%C3%A1n/posts László Tóth

EquusMtn

Michael Duke

himanshu khatri

Thomas Foster

waffleironmarch


Muhammad Shatry

j3relowf

daqu

Douglas Broccone

Franck Merlot

Paul Bristol

disqus_zXLbNfw1Yi

Rocky

Franck Merlot

the truth

the truth

bt

Jim

Wizardofmann


Ravi Sadana

Recidivist

antonio carlos motta

antonio carlos motta

ilduderino