The paradoxical twists and turns of infinity
have baffled many great thinkers. The first person to truly come
to grips with the concept was the remarkable Galileo Galilei.
by Brian Clegg
Infinity has long been treated with
a mixture offascination and awe. Some have equated it with godhead
 others see it as a concept with no practical value in the real
world, arguing that even mathematics apparently dependent on infinity
such as calculus could be made to work by resorting to inexhaustible
but finite quantities. The ancient Greeks were uncomfortable with
the concept, giving their word for it, apeiron, the same sort of
negative connotations we now apply to the word chaos. Apeiron was
out of control, wild and dangerous.
It took Aristotle to put infinity in its place so firmly that
hardly anyone would give it direct consideration again until the
nineteenth century. The approach he took was surprisingly pragmatic.
Infinity, Aristotle decided had to exist, because time appeared
to have no beginning and no end. Nor was it possible to say that
the counting numbers ever finished. If there were a biggest number
 call it 'max', then what was wrong with max+1 or max+2? But on
the other hand infinity could not exist in the real world. If there
were, for example, a physical body that was infinite, he argued,
it would be boundless  yet to be a body, by definition an object
has to have bounds.
The compromise Aristotle developed  and it's a clever one  was
to say that infinity both existed and didn't exist. Instead of
being a true property of anything real, he argued, there was just
potential infinity. Infinity that could in principle be, but in
practice never was. Aristotle gives us an excellent example to
illustrate this. The Olympic Games exist  it is impossible to
deny this. Yet were an alien to beam in (Aristotle didn't actually
include an alien in his example) and ask us 'show me this Olympic
Games of which you speak', we couldn't do it. At the moment they
don't exist in reality but they do exist as a potential. And infinity,
Aristotle argued, was in exactly the same potential state.
Galileo
Galilei

It was this form of
infinity that would spawn calculus and would be included in practically
all mathematical considerations of the infinite until Georg Cantor
blew the whole topic apart with the revelations that would eventually
contribute to his decline into madness. But one man bucked the trend
early, considering the implications of the real thing  and that
was a man who was no stranger to standing out from the crowd, Galileo
Galilei.
After Galileo's house arrest began in 1634, following the trial
over his heretical work on the motion of the Earth around the Sun,
Galileo was anything but inactive. It was then that he produced
the book that was arguably his greatest work of science, his equivalent
of Newton's Principia, named Discorsi e dimostrationi matematiche,
intorno a due nuove scienze, or Discourses and Mathematical Demonstrations
Concerning Two New Sciences. Galileo had considerable trouble getting
this published  the Inquisition made it clear that no work by
this heretic would be published in any country where it held sway.
When the book was eventually taken up by the great Dutch publisher
Elsevier, Galileo expressed his great surprise that it had been
published at all, which he claimed had never been his intention.
He commented in his introduction:
I was notified by the Elzevirs that they had these works of mine
in press and that I ought to decide upon a dedication and send
them a reply at once. This sudden and unexpected news led me to
think that the eagerness of your Lordship to revive and spread
my name by passing these works on to various friends was the real
cause their falling into the hands of printers who, because they
had already published other works of mine, now wished to honour
me with a beautiful and ornate edition of this work.
Hardly "sudden and unexpected news" 
Galileo was closely involved in the publication, but at least
he was protecting his back.
The book (as did his nearfatal work Dialogue Concerning the Two
Chief World Systems on the motion of the Earth) took the form of
a conversation between a number of characters, largely over serious
matters. But after wondering about what holds matter together (Galileo
thought it was tiny pockets of vacuum between the particles of
matter), they have a diversion, just for the fun of it, into the
nature of infinity.
Galileo brings out a number of points, but two are particularly
worthy of note. The first involves the rotation of a pair of wheels.
He starts with wheels with a few sides  for example, they could
be hexagons. These are three dimensional shapes  imagine the hexagons
are cut out of sheets of marble. The smaller hexagon is stuck to
the larger one, and each of them rests on its own horizontal rail.
Now we roll the combined wheel along so that it moves onto its
next side. As the big wheel turns it pivots on the corner and moves
along the track by the length of one side. But what has happened
to the smaller wheel? Not only has the big wheel moved on by that
distance, so has the small one. It has to: they're fixed together.
Yet in turning 1/6 of a rotation, the small one should only have
rolled along the track by the length of its own side  a much smaller
distance, marked in red on the diagram. To achieve the extra movement,
the smaller wheel was lifted entirely off the track.
Now here's the clever bit. Galileo imagined increasing the number
of sides. The more sides, the more sets of small movements along
the rail and small jumps you get as the wheels rotate. Finally,
let's imagine, were it possible, that we take that number of sides
to infinity. We end up with circular wheels.
Again we roll the two wheels, joined together, along their respective
rails. Again they both travel the same distance  in this case
a quarter of the circumference of the big wheel. But now something
strange has happened. The rim of the big wheel has rolled out a
quarter of its circumference on its track. The rim of the smaller
wheel has only rolled out its smaller quarter circumference, but
the small wheel still has to travel the same distance as the bigger
one, without ever leaving the track. There were no jumps, or at
least so it seems.
What Galileo imagined had happened was that as the smaller wheel
turns there are an infinite number of infinitesimally small gaps,
which add up to make the difference between the circumference of
the wheel and the distance it moves. Infinity has come into play
in a physical device to make the seemingly impossible happen.
After letting this
percolate through his brain in the background, Galileo's traditionalist
and frankly rather dim character, Simplicio, has a complaint.
What Galileo seems to be saying (or technically Salviati, the
character that represents Galileo's voice in the book) is that
there are an infinite number of points in one circular wheel
and an infinite number of points in the other. But somehow, though
each had the same infinity of points, one added up to a greater
distance than the other. One infinity was both the same as the
other and larger.
Salviati is rueful. That's the way it is with infinity. It is
a problem, he reckons, of dealing with infinite quantities using
our finite minds. And he goes on to show how this is perfectly
normal behaviour once you are dealing with infinity.
The simple mathematical tool he uses to demonstrate this is the
square  that's the square of a number, not the shape. Salviati
makes sure Simplicio knows what a square is  any number multiplied
by itself. So, he imagines going through the integers, multiplying
each one by itself. It's not rocket science. For every single integer
there is a square. We have an infinite number of integers, and
there's an infinite number squares in a onetoone correspondence.
But here's the rub. There are lots of numbers that aren't squares
of anything. So though there's a square for every single integer
 an infinite set of them  there are even more individual numbers
than there are squares. Arggh. Simplicio's brain hurts, and it
doesn't surprise us. Galileo has spotted something very special
about infinity. The normal rules of arithmetic don't really apply
to it. You can effectively have 'smaller' and 'bigger' infinities,
one a subset of the other, that are nonetheless the same size.
The true implications
of Galileo's ponderings would take more than 300 years to come
out, but he had sowed the seeds of all that was to come  and
clearly enjoyed a delight and fascination with the paradoxical
nature of infinity. A fascination we can all share.
