# How much maths can you buy with £50?

The Bank of England has recently announced their decision for who to feature on the newly redesigned £50 note, and I’m pleased to say it’s a mathematician! They’d already decided it was going to be a scientist, and hundreds of options were discussed as part of a public consultation, resulting in a shortlist of 12 names including Srinivasa Ramanujan and Ada Lovelace. But the winner was chosen as mathematician, codebreaker, computer scientist and LGBT+ icon Alan Turing, who will feature on the new notes from 2021 onwards.

Turing is mostly known for his work on the Enigma code during World War II. As one of GCHQ’s codebreakers stationed at Bletchley Park, he was part of the team who developed a method of deciphering the German ciphers, and his insights were invaluable. As part of the codebreaking effort, Turing helped design and build what’s widely considered to be one of the world’s first real computers – a mechanical computing machine called Colossus, which took in input by punched tape and crunched through millions of possibilities to determine the machine settings each day, and allowed the codebreakers at Bletchley to read top-secret military messages.

Much of the work done at Bletchley Park during the war was classified, and the contributions made by Turing and hundreds of others weren’t known until many years later – in some cases, after people had died, meaning they could never talk to their friends and family about what they really did there. But Turing’s legacy extends beyond the walls of Bletchley Park, and as well as his amazing work there, he contributed to mathematics and computer science in myriad other ways during his lifetime.

#### Computing and Artificial Intelligence

In 1936, before the Second World War, Turing published a highly significant paper on a topic crossing over between maths and computer science. “On Computable Numbers, with an Application to the Entscheidungsproblem” (PDF) discussed computable numbers – ones which can be calculated using a given algorithm, to a degree of precision of your choosing, such that the algorithm will terminate in a finite amount of time. This includes all whole numbers and fractions, and in fact all the numbers you’ve ever really heard of, including irrational numbers like e and π. Numbers which are non-computable are known to exist, but (as you would expect) they’re not easy to describe or define.

Turing’s paper concerned the Entscheidungsproblem – also known as the “decision problem” – the question of whether or not a given statement can be proved using logic, from the axioms of the system. His proof involved finding algorithms that would construct a valid proof of a statement, and if no such algorithm was possible, the statement could not be proved.

In order to do this, he introduced the concept of a Turing Machine – a universal calculator that takes inputs and computes them in a specified way to produce outputs, and which can be applied to any logical problem or computation. Turing Machines don’t necessarily exist in reality, partly because they would require an infinitely long input tape and might take longer than the life of the universe to run a program. Although that hasn’t stopped some people from trying to make one – several beautiful examples have been built, including this one which illustrates the concept perfectly.

Turing’s work on the conceptual processes of computation preceded real computing machines by several years – and may have been a strong influence in his decision to build a computer to solve the Enigma – but it also laid the foundations for much of computing theory for the years to come. After the war, he worked in Manchester helping to develop some of the earliest commercial computers, including the Manchester Baby – the world’s first electronic stored-program computer.

Another computing-related idea we owe to Turing is that of the Turing Test, which is designed as a test for true artificial intelligence. While the question of whether an artificial being can be considered to be truly intelligent is somewhat fuzzy, Turing came up with a way to define a true computer intelligence.

The Turing Test says that if you could set up a situation where a human can have a conversation with a computer intelligence, but in such a way they don’t know whether they are talking to a computer or a person (through text messages relayed on a screen, or similar), the machine can be considered truly intelligent if it can fool the human into thinking it’s another human.

#### Morphogenesis

Later in his life, Turing chose a different topic to study – morphogenesis. Morphology is a term used to describe physical characteristics of living creatures in biology, and morphogenesis is the study of how those physical characteristics are generated. As well as spending some time studying the patterns of seed arrangements in sunflower heads, Turing looked at coloured patterns on animal fur, and proposed a model for how such patterns might occur.

Reaction-diffusion systems involve chemical reactions taking place in liquids, where the reagents involved are mixing and diffusing within the system. The model Turing developed considers two substances – an activator chemical, which causes a colour change, and an inhibitor chemical, which suppresses the activator chemical. Both chemicals will obey their natural tendency for diffusion, and try to spread out and equalise their concentrations.

Given a system in which the activator chemical produces more of itself (except when it’s in the presence of inhibitor), and assuming the rates at which the two different chemicals diffuse are known, it’s possible to model mathematically what will happen over time. The initial starting arrangement of chemicals can be fed into the model and the system allowed to evolve over time, to see what happens.

Interestingly, given different starting conditions, and choosing rates of reaction and diffusion for each of the chemicals, a range of different patterns can be created – resulting in a stable arrangement of areas of high and low concentration of the activator chemical. Depending on the settings, patterns of spots and stripes can be created, some of which look delightfully similar to the patterns visible on animals – if these chemical reactions were taking place while the embryonic leopard, or zebra, was developing, it could explain how their colouration formed.

Of course, any model has its limitations, and while Turing’s reaction-diffusion idea gives some tantalising hints as to what might be happening in these situations, there are plenty of examples it can’t explain. Turing himself said in his paper, “This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge.”

#### Turing’s Legacy

Alan Turing is remembered by mathematicians and computer scientists as someone whose genius was the catalyst for huge changes in the way we think about and perceive the world, and who made huge contributions to mathematics in a number of different fields – because as disparate as the different topics he worked on were, they all fall under the umbrella of mathematics. He’s recognised by computer scientists in the form of the ACM’s A.M. Turing Award, whose Laureates are invited to the HLF every September.

He’s also remembered by many as an icon in the LGBT+ community, due to the cruel treatment, discrimination and prosecution he faced as a gay man during a time when such a thing was seen as wrong. His inclusion on the £50 note is seen by some as an attempt to gloss over this injustice – along with the treatment of hundreds of other people who lived through the same thing, and who have not received a formal pardon. But it’s still wonderful to see him recognised in this way, and if future generations remember him as a result, his story and legacy will live on.

Question: How much maths can you buy for £50?

Answer: A lot – except that you can’t buy maths (any more than you can buy love?).