Connecting [Barney] Google – Part 2—Infinity, Economics and the Barney Google Universal Tour

Posted on February 8, 2010

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Continued from Connecting [Barney] Google, Part 1

We looked at Barney Google historically.  We met the Yellow Kid, Mutt and Jeff, some yellow journalists and others.

We examined Barney psychologically.  We found him to be a purely human artefact, part of the secret language of humans.

(Unlike music, for instance, which birds and other beasts appreciate.  Your cat might bop to Charlie Parker, but cartoons are abstract in a purely human way.  A cat is tone deaf to Barney’s level of abstraction.)

We examined humour.  You know how Barney likes to make us laugh.  We met Snuffy Smith and other hillbillies and some eugenicists in white coats, and found out that being laughed at isn’t always so jolly, although not everybody always minds.

Then we met the mathematician Edward Kasner and his nephew Milton, and they introduced us to Barney Google’s namesake, the googol.

So let us continue this exploration—the Barney Google Universal Tour—by examining what a googol is.  Why? you ask.  Because Edward Kasner invented two numbers, and you can’t understand the second one until you understand the first.

A hundred looks like this:  100.  A hundred is a one followed by two zeros.  In scientific notation it looks like this 102

A google looks like this:

10,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000

A google is a one followed by one hundred zeroes.  In scientific notation a googol looks like this:  10100.

Each additional zero added to the end of a number increases its value tenfold.  That makes the journey from one hundred, with two zeros, to a googol, with a hundred zeroes, a very long distance indeed.  But we need to make the journey in order to see what Dr. Kasner’s next number looks like.

That next number is also named after Barney Google.  It’s called the googolplex.  A googolplex is a one followed by a googol zeroes.

Written out in scientific notation, a googolplex looks like this:

1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

I can’t show you what a googolplex looks like in longhand, because I would then have to write out a googol’s worth of zeroes.  And that would present more problems than merely  a bad case of carpal tunnel syndrome.

You see, a googol is big, much bigger than 100 zeroes can adequately convey.  To measure a googol, we’ll need a big yardstick, which means we’ll have to go cosmic.  Cosmic distances are measured in light years, so let’s start there.

Light travels 300,000 kilometres per second, and in a year travels approximately 10 trillion kilometres (10,000,000,000,000 k.)  The Sun is the nearest star, but at distance of a mere 150 million kilometres, it’s too close to be a useful measuring stick.  The next closest star, Alpha Proxima, is 4.3 light years away, but that’s still too close.

Our sun and Alpha Proxima, and something like 200 to 400 billion other stars, belong to the Milky Way, an elegant spiral galaxy a hundred thousand light years across.  A hundred thousand light years is equal to a million trillion kilometres, a nineteen digit number.

Which is too small to bring us close to googol.

The Milky Way, our quaint little galactic home, is a member of the Local Group, a cluster of galaxies which includes the Clouds of Magellan, Andromeda, and twenty-five or twenty-six other galaxies.  The Local Group measures about 100 million light years across, which is a 100 million trillion kilometres, a twenty-one digit number.

The Local Group itself belongs to the Virgo Supercluster.  The supercluster we belong to is 110 million light years across, which is 1.1 billion trillion kilometres, a twenty-two digit number.  Still seventy-nine digits short.  Superclusters are the largest structures in the universe, but they are much too small for our purposes.

As the Friendly Giant used to say, “Look up, waaaay up.”

The universe itself is about 13.5 billion years old, give or take.  It came into the existence with the Big Bang.  Presuming that it has been propagating outwards at the speed of light ever since (and not accounting for the subsequent expansion of space, which we don’t even want to go into), that would make the universe about 27 billion light years wide.  Using kilometres, the width of the universe can thus be expressed with a twenty-four digit number.  Measured in centimetres instead, we can get a 29 digit number.  Measured in microns—a millionth of a metre—the width of the universe can still be expressed in a 33 digit number.  Unfortunately microns are too big and the universe is too small, if we want to find a googol.

Cosmic distances won’t serve, so what about cosmic stuff?  It is estimated that, adding all the stars, planets, galaxies, superclusters, interstellar dust and whatnot together, there are somewhere between 1079 and 1081 atoms in the universe.  Assuming the greater number is correct, let’s say we transformed every one of these atoms into graphite for your pencil.  Every atom (since we want to write very small) represents a zero in the number googolplex.

Almost right away, we run out of lead.  Even using only a single atom to represent every zero in the number googolplex, we would still need 10 billion billion universes to write out the number in full.

A hundred with only two zeros is easy to write and the quantity it represents is easy to understand.  A googol with a hundred zeroes is easy to write, but the quantity it represents is pretty much beyond human comprehension.  But at least it bears a passing resemblance to the universe we live in.

The googolplex, on the other hand, represents a quantity so large that it makes everything we’ve been using for measurement so far appear vanishingly small.  Comparing the googolplex—even written out the long way—with the quantity it represents, is like comparing a freckle on your nose to the Virgo Supercluster.

Except more extreme than that.

To find a googol, we had to step a long way outside of the universe itself.  We haven’t even attempted a googolplex, because the universe and everything in it is too small a conceptual yardstick for us even to begin.  Suffice it to say, these are very large numbers indeed.  And the question naturally arises, what use could such numbers be?

The mathematician Edward Kasner invented the googol and googolplex to make a simple point about infinity.  You’ve heard about infinity.  That’s the quantity economists use to express the total amount of growth possible in a capitalist economy.  There are many infinities, but that infinity will do.

So how big is infinity?

The Pacific Ocean is big.  And if you had a hundred Pacific Oceans you could get pretty wet.  And if you looted a few trillion trillion neighbouring universes, you might be able to find a googol of Pacific Oceans, and that would make you very wet indeed.  And if you went up several orders of magnitude to a googolplex of Pacific Oceans—and even counting the number of universes it would take to do such a thing would not be possible—and you put it all these Pacific Oceans in a bucket and dumped it all on the floor of infinity’s kitchen, how annoyed would infinity be if he or she stepped in the puddle?  Would infinity sink down to her knees?  No.  Would it cover infinity’s toes?  No.  The fact is, infinity wouldn’t even notice a damp spot on the floor.  A googolplex of Pacific Oceans wouldn’t make infinity’s socks wet.

The issue is that infinity isn’t really a number at all, or a quantity, because numbers and quantities are finite by definition.  Infinity is a concept with no relation to the real world.

You could say that about a googolplex too, I suppose, but being finite, it at least has the possibility of being real—for instance, as a number representing all the possible interactions between all the atoms in the universe—but that distinction cannot be extended to the concept of infinity.

Infinity—except as a concept—couldn’t fit into any universe unless that universe was itself infinite.  And we don’t know of any infinite universes.

So what is infinity doing in the equations of economists? Barney asks.  If infinity doesn’t exist, how can we have infinite growth under capitalism?  Does this mean that the equations underlying capitalist economics are sheer fantasy?

Such questions, Barney.  You are only a cartoon character.  How dare you question the entire basis of capitalist economics!  Shame on you for undermining people’s faith.

Everybody knows that capitalist economics is a religion.

Now go to your room, and we’ll finish our talk next time.

——-

These essays conclude here

Connecting [Barney] Google, Part 3 – Barney Becomes a Verb