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180 Are the Androids Dreaming Yet?

the number zero behaves just like any other counting number. It also
works with multiplication. If I have zero lots of 4 cakes, I have no cakes.
Zero times four is zero, so multiplication with zero works. There is one
embarrassing exception, if I divide by zero I seem to get infinity. When I
was a child this was a definition for infinity, but nowadays mathematicians
simply forbid the operation. Division by zero is not allowed and if you
try it on your computer, you will get the not terribly useful, #DIV/0!
Error. That’s progress I guess!

Zero had been tamed. What about infinity?
Cantor showed that while you could think of infinity as a number,
it might not be just one number. He proposed there are many infinities.
In fact, there are a greater than infinite number of them! He did this
through a rigorous analysis of a new branch of mathematics called set
theory.

Set theory is now the cornerstone of modern mathematics, but it
was treated with suspicion in Cantor’s time. Rather than embrace the new
thinking, many mathematicians ridiculed it; Poincaré wrote that Cantor’s
ideas were a grave disease infecting the discipline of mathematics! This
seems odd given our modern propensity to embrace innovation, but the
tone of science back then was different: innovation was not necessarily
considered a good thing.

At the turn of the 20th century, scientists were on a mission to tidy
things up. Lord Kelvin announced in 1890 that mankind had discovered
everything there was to know and the role of future scientists was simply
to catalogue and observe the consequences of these laws, and to improve
the accuracy of measurement. The last thing scientists wanted was a
completely new set of numbers that behaved in strange ways. Cantor was
upsetting the apple cart, but he was in good company. Just a few miles
away in Berlin, a young Albert Einstein was beginning to study physics
in his spare time. Those studies would culminate in his four papers of
1905, two on Quantum Mechanics and two on Relativity, ushering in the
modern age of physics.

How to Count
To understand infinity you need to count in a particular way. You’re
probably used to counting with numbers. You count apples: one, two,
three, and say, “I have three apples.” You can do the same with oranges.
If you have three apples and three oranges, the totals are the same and
you can declare you have the same number of fruits. This is the first way
to count.

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