From 0 to Infinity in 26 Centuries (16 page)

The Digital Age

Although they didn’t become commonplace in homes until the 1980s, one can’t now imagine a life without computers. The history of the computer is intimately linked
with the story of mathematics. We have seen already that pioneers like Leibniz and Pascal blazed (sorry...) a trail with mechanical calculators, but the need for faster and more flexible machines
sparked off the digital revolution.

C
HARLES
B
ABBAGE
(1791–1871)

The English mathematician Charles Babbage was the son of a banker, and is best known as the ‘Father of the Computer’ because of his work in mechanical computing.
Babbage read mathematics at Cambridge University but found his studies unfulfilling, which led him to found the Analytical Society in 1812 along with a group of fellow students. Apparently,
Babbage’s moment of inspiration came when he was one day poring over a group of logarithm tables that were known to contain many errors. Working out the entries for log
tables is a very tricky process and inevitably human ‘computers’ – as they called people who calculated sums in those days – made mistakes. Babbage maintained that the
tedious calculations, which require little thought, only accuracy, could be performed by an elaborate calculating machine.

Machine takes over

Babbage secured government funding and produced plans for his
Difference Engine
. However, such was the complexity and size of the machine that it was never built in his
lifetime. (A Difference Engine was built in the 1980s and today resides in London’s Science Museum. At just over 2 metres high and 3.5 metres long it’s a fairly sizeable machine.)

As interest in the Difference Engine project began to wane, Babbage started work on the Analytical Engine using the knowledge he had gleaned while designing his first computer. This machine was
intended to be much more like a computer as we know it. It could be programmed to perform particular combinations of mathematical functions. One set of punched cards would be used to programme the
engine, another set would introduce data to the engine and then the machine itself could punch blank cards with the results, effectively saving them in its memory for future use.

Babbage was still working on the Analytical Engine when he died. Because of the immense cost and complexity involved, a
working model has never been built, although, at
the time of writing, British programmer and mathematician John Graham-Cumming is leading a project to build one for the first time.

A
DA
L
OVELACE
(1815–52)

One of the few notable pre-twentieth-century female mathematicians, Ada Lovelace was the daughter of the famous poet and rake Lord Byron and Anne Isabella Milbanke, to whom
Byron was briefly married. Milbanke was convinced Byron’s excesses had been the result of a kind of insanity and she sought to shield her daughter from a similar fate by encouraging her to
pursue a persistent education, especially in the logical discipline of mathematics.

Get with the programme

Despite being unable to enter university, Lovelace was privately tutored and she continued with mathematics throughout her life. She encountered Charles Babbage, who asked her
to translate into English an article on his Analytical Engine that had been written by an Italian mathematician. Lovelace duly did so, adding extensive notes on the machine, including a set of
instructions that would have had the machine produce the
Bernoulli numbers
, a sequence of numbers named in honour of Jacob Bernoulli. For this, Lovelace is considered to be the first ever
computer programmer. Sadly she died from cancer at the age of thirty-six.

G
EORGE
B
OOLE
(1816–64)

An Englishman who became a mathematics professor in Ireland, George Boole published major works on
differential equations
(equations that involve derivatives of a
function), but he is most remembered for his work on logic.

Boole sought to set up a system in which logical statements could be defined mathematically and then used to perform mathematical operations on the statements; the results would be generated
without the need to think through the problem
intellectually. Boole’s system aimed to take a raft of logical propositions and see how they combined together with the
aid of maths rather than philosophy.

A logical step

In order to set up the system, Boole developed what became known as
Boolean algebra
: letters defined either as logical statements or as groups of things. The letters can
have a value of 1, meaning ‘true’, and 0, meaning ‘false’.

So imagine you are considering dogs as a group of things, and you let x represent shaggy dogs and y represent yappy dogs. You can then make a table for the dogs using values for x and y:

Boole then defined three simple mathematical operations we could conduct with the results – AND, OR and NOT. AND is defined as the multiple of the two values. As the table
below shows, if x and y are ‘true’, we expect the answer ‘true’ or 1, and everything else to be 0 or ‘false’:

This seems rather self-evident – a dog can only be shaggy AND yappy if the dog is shaggy and yappy, but the example is useful because it shows you can come to the same
conclusion using very simple arithmetic.

There are a host of other Boolean operations that can be reduced to arithmetic.

In the 1800s Boole’s work had implications for mathematical logic and set theory, but it was not until the twentieth century that a more practical use was found. An electronics researcher
called Claude Shannon discovered that he could use Boolean operations in electrical circuits – he could generate a simple electrical circuit to take logical steps and therefore make decisions
based on them. As Boole’s work uses only values of 0 or 1, ‘true’ or ‘false’, or ‘on’ or ‘off’ it is known as a
digital
method. In
essence, Boole paved the way for the first electronic digital computers.

Setting things out

Set theory
is the branch of mathematics concerned with placing objects into groups called sets. Sets can be defined as containing:

Boolean algebra works in set theory, like the example of the shaggy and yappy dogs on page 145.

John Venn (1834–1923) was a British priest and logician who, besides designing and building a cricket-ball-bowling machine, is best known for
Venn diagrams
, which
help to show set theory in a visual way. To use our earlier example, if you are considering all types of dog, then the rectangular box should be receptive to all types of dog. I introduced two
sets: x, the set of shaggy dogs and y, the set of yappy dogs. We saw that some shaggy dogs are also yappy, so those two sets need to overlap.

The overlap in the middle represents x AND y. x OR y would be any dog within the two circles. NOT x would be any unshaggy dog.

To bring things back into more mathematical territory, the German mathematician Georg Cantor (1845–1918) was very interested in infinite sets, which, as the name
suggests, are sets containing an infinite number of things.

Consider the set of positive whole numbers (1, 2, 3…) which mathematicians call the natural numbers. This is an infinite set because we can keep on counting for ever. Then consider the
set of numbers in which the numbers can be anything at all, including fractions, negative numbers and those irrational numbers like π, e and ϕ that we saw earlier (mathematicians call this
continuous list of numbers real numbers). This is also an infinite set, and it contains more numbers than the set containing the natural numbers. By comparing the set of natural numbers, which is
infinite, with the set of real numbers, which is
also infinite, Cantor was able to show that, although both were infinite, they were not the same size. Therefore we have the
idea that there are different infinities for different infinite sets. This sort of business is the reason many mathematicians consider infinity to be a concept, rather than a number.

Cantor said that the natural numbers were
countably infinite
because we make progress towards infinity as we count. The real numbers are
uncountably infinite
because, no matter
where you start counting from, you will not make much progress as you add on an infinitesimal amount each time.

Cantor developed the
continuum hypothesis
, which states:

So far, it has been proved mathematically that this hypothesis cannot be proved or disproved within the current limits of standard set theory, and it remains one of the greatest
unsolved problems in mathematics.

A
LAN
T
URING
(1912–54)

Turing was a brilliant British mathematician and scientist who is well known for the instrumental role he played in the Allied effort to break the German Enigma cipher, a coding
machine used by the Nazis during the Second World War.