Maths, but not as you know it…
The Norwegian Academy of Science and Letters kindly provided me with a scholarship to attend the Abel Prize week in Oslo earlier this year where I interviewed the 2018 Abel Laureate Robert Langlands.
In the first of a series of videos documenting my experience, Robert describes how he came to do Mathematics at university…
Incredibly excited to announce that I’ve won an award courtesy of OxTALENT at the University of Oxford in the category of Outreach and Widening Participation. Thanks to you all for the support and here’s to many more exciting times for Tom Rocks Maths!
Outreach and widening participation activities deliver an important dimension of the University’s work in raising aspirations, promoting diversity and encouraging people from non-traditional backgrounds to enter higher education. This category awards staff and students who have made innovative use of technology to deliver exceptional widening participation activities and to support learners from disadvantaged backgrounds.
Tom Crawford (St Hugh’s College) for ‘Tom Rocks Maths’
Approximating Pi was a favourite pastime of many ancient mathematicians, none more so than Archimedes. Using his polygon approximation method we can get whole number bounds of 3 and 4 for the universal constant, with only high-school level geometry.
This is the latest question in the I Love Mathematics series where I answer the questions sent in and voted for by YOU. To vote for the next question that you want answered next remember to ‘like’ my Facebook page here.
Stripping back the most important equations in maths so that everyone can understand…
The Normal Distribution is one of the most important in the world of probability, modelling everything from height and weight to salaries and number of offspring. It is used by advertisers to better target their products and by pharmaceutical companies to test the success of new drugs. It seems to fit almost any set of data, which is what makes it SO incredibly important…
You can watch all of the Equations Stripped series here.
Time for the next puzzle in the new feature from Tom Rocks Maths – check out the question below and send your answers to @tomrocksmaths on Facebook, Twitter, Instagram or via the contact form on my website. The answer to the last puzzle can be found here.
You are walking through the jungle with two friends when all of a sudden you are attacked by a group of cannibals. Fortunately, they do not eat you straightaway, but instead devise a puzzle that you must solve to avoid being eaten. The setup is as follows:
The questions is: what colour hat is the person at the front wearing and how did they know the answer?
The answer will be posted in a few weeks along with the next puzzle – good luck!
If all the maths you’d ever seen was at school, then you’d be forgiven for thinking numbers were boring things that only a cold calculating robot could truly love. But, there is a mathematical proof that you’d be wrong: Gödel’s incompleteness theorem. It comes from a weird part of maths history which ended with a guy called Kurt Gödel proving that to do maths, you have to take thrill-seeking risks in a way a mindless robot never could, no matter how smart it was.
The weirdness begins with philosophers deciding to have a go at maths. Philosophers love (and envy) maths because they love certainty. No coincidence that Descartes, the guy you have to thank for x-y graphs, was also the genius who proved to himself that he actually existed and wasn’t just a dream (after all, who else would be the one worrying about being a dream?). Maths is great for worriers like him, because there’s no question of who is right and who is wrong – show a mathematician a watertight proof of your claim and they’ll stop arguing with you and go away (disclaimer: this may not to work with maths teachers…).
However, being philosophers, they eventually found a reason to worry again. After all, maths proofs can’t just start from nothing, they need some assumptions. If these are wrong, then the proof is no good. Most of the time, the assumptions will have proofs of their own, but as anyone who has argued with a child will know, eventually the buck has to stop somewhere. (“Why can’t I play Mario?” “Because it’s your bedtime.” “Why is it bedtime?!” “BECAUSE I SAY SO!”) Otherwise, you go on giving explanations forever.
The way this usually works for maths, is mathematicians agree on some excruciatingly obvious facts to leave unproved, called axioms. Think “1+1=2”, but then even more obvious than that (in fact, Bertrand Russell spent hundreds of pages proving that 1+1=2 from these stupidly basic facts!). This means that mathematicians can go about happily proving stuff from their axioms, and stop worrying. Peak boring maths.
But the philosophers still weren’t happy. Mostly, it was because the mathematicians massively screwed up their first go at thinking of obvious ‘facts’. How massively? The ‘facts’ they chose turned out to be nonsense. We know this because they told us things which flat-out contradicted each other. You could use them to ‘prove’ anything you like – and the opposite at the same time. You could ‘prove’ that God exists, and that He doesn’t – and no matter which one of those you think is true, we can all agree that they can’t both be right! In other words, the axioms the mathematicians chose were inconsistent.
Philosophers’ trust in maths was shattered (after all, it was their job to prove ridiculous stuff). Before they could trust another axiom ever again, they wanted some cast-iron proof that they weren’t going to be taken for another ride by the new axioms. But where could this proof start off? If we had to come up with a whole other list of axioms for it, then we’d need a proof for them too… This was all a bit of a headache.
The only way out the mathematicians and philosophers could see was to look for a proof that the new axioms were consistent, using only those new axioms themselves. This turned out to be very, very hard. In fact (and this is where Gödel steps in) it turned out to be impossible.
Cue Gödel’s incompleteness theorem. It says that any axioms that you can think of are either inconsistent – nonsense – or aren’t good enough to answer all of your maths questions. And, sadly, one of those questions has to be whether the axioms are inconsistent. In short, all good axioms are incomplete.
This may sound bad, but it’s really an exciting thing. It means that if you want to do maths, you really do have to take big risks, and be prepared to see your whole house of cards fall down in one big inconsistent pile of nonsense at any time. That takes serious nerve. It also means mathematicians have the best job security on the planet. If you could just write down axioms and get proof after proof out of them, like a production line, then you could easily make a mindless robot or a glorified calculator sit down and do it. But thanks to Gödel’s incompleteness theorem, we know for sure that will never happen. Maths needs a creative touch – a willingness to stick your neck out and try new axioms just to see what will happen – that no robot we can build will ever have.
Joe Double
The latest episode from Tom Rocks Maths on Oxide Radio – Oxford University’s student radio station. Featuring pirates that can’t count, the best way to carry a bundle of sticks, and special guest Toby, who talks about his favourite part of maths, his taste in music and tries out one of the infamous Tom Rocks Maths quizzes! Not forgetting the usual maths puzzle and great music from the Arctic Monkeys, Paramore and All Time Low…
Wherever we look in the world, we see competition between different groups or beings. Whether it’s two animals trying to earn the right to a watering hole, people trying to assert their social influence, or simply two sports teams playing against each other, this sort of interaction appears in many different situations. As humans, we have a natural desire to rank things that are in direct competition: which is better? Who would win if they faced each other? How does their rivalry compare to others?
We want to know the answers to these questions because it makes us enjoy the competition more, and we feel that we learn more about it. Imagine being able to correctly predict who would win every football match for the rest of the season, you’d probably feel pretty pleased with yourself… But, apart from the inevitable bragging rights, being able to rank competing entities and predict outcomes is an extremely useful skill in many different areas of research, including sociology, economics and ecology.
Of course, you need a bit of maths if you’re going to rank things reliably; you can’t just trust a hunch! There are many different methods that have been used before for rankings, but a group of scientists at the Santa Fe Institute in the USA have come up with a new way of doing it using springs!
So, the ranking system is… a trampoline?! Not exactly. This ingenious method, called SpringRank, treats each interaction as a physical spring, so the model is a whole system of connected springs. Think of a football league: between each pair of teams there is a spring in each direction, and the force of each spring is determined by how many times they have beaten each other in the past. For example, Manchester United have played Liverpool 200 times, winning 80 matches and losing 65. In our spring system, this means that the spring connecting the two teams is biased towards Manchester United – it requires more force to move closer to Liverpool than it does to move towards Manchester United. With this setup, it turns out that the best ranking of the teams is found when you make the total energy in all of the springs as low as possible.
But why use springs? The bonus is that we’ve been studying springs for hundreds of years and so we know the physics behind how they work, which makes it easy to do the calculations. We can use the positions of the springs to work out the rankings of millions of different teams in just seconds! Not only is the maths simple, but it’s also very effective, especially compared to other methods currently used for ranking. In tests run by the researchers, SpringRank performed much better at ranking competitors, as well as predicting the outcomes of future clashes, than existing methods. The data set covered topics as varied as animal behaviour, faculty hiring and social support networks, demonstrating just how versatile the method can be.
This research is a wonderful example of how different areas of science can be combined to create a tool that can actually be put to use in the real world. When learning the subjects separately at school, it’s hard to imagine that you could take centuries-old ideas from physics, turn them into mathematical models, and stick them into a computer program! But here we are, able to work out who is likely to become friends (and enemies), which animals will make it through the heatwave, and whether it’s worth bragging about your favourite team before the game has even happened. So next time you’re challenged to guess the league winner, reach for SpringRank and jump ahead of the competition!
Kai Laddiman
Cocaine used to be the drug of the rich and famous, but over recent years it has become cheaper and more readily available, and as a result more and more people are becoming addicted to this highly dangerous substance. A report last year from the UK Government Advisory Council found that 1 in 10 people between the ages of 16 and 59 had used the drug at some point. The current treatment for cocaine addicts is through therapy, but relapse rates remain high. Now a new study has linked cocaine addiction with a build up of iron in certain parts of the brain, and particularly areas known to control our inhibitions, although the team don’t yet know what the iron is doing there. I spoke with lead author Dr Karen Ersche…
You can listen to the full interview for the Naked Scientists here.
The 1900’s saw inventions that made a BIG change to our lives. Aeroplanes in 1903 changed the way we travel, TVs in 1925 changed home entertainment, and Microwaves in 1946 changed the way we eat. Nineteen also played an important role in the British Civil War and was the title of Adele’s first album…
You can listen to all of the Funbers episodes from BBC Radio Cambridgeshire and BBC Radio Oxford here.
Listen to me being tattooed whilst attempting to describe the process, and hear from my artist Nat on his experience as a tattooist…. all in the name of science.
You can also watch a short video below of the tattoo being done from the perspective of the artist.
Audio edited by Joe Double.
Time to celebrate with a glass of bubbly as we’ve reached the number 18! The legal drinking age in most countries around the world, unless you’re the US, Saudi Arabia or Haiti. In fact, in Haiti you only need to be ‘of school age’ to get your hands on the devil’s nectar…
You can listen to all of the Funbers episodes from BBC Radio Cambridgeshire and BBC Radio Oxford here.
