The Nobel Prize winning economist Kenneth J. Arrow died on Tuesday at his home in California. He was 95.
Arrow's contributions to economics were wide-ranging, but our favourite concerns something that's of immediate importance to all of us: democracy. As the recent US elections have shown yet again, the outcome of an election is not always entirely democratic. It was Hilary Clinton who won the popular vote, but Trump is president. Once you start thinking about voting systems, you soon realise that designing a good one is tricky.
So is there a perfect voting system? Arrow asked himself this question in the 1950s and found that the answer is no — even if you only make the most basic demands of the system.
Kenneth defined a voting system in a very mathematical way, as follows. There is a population of voters each of whom has a preference ranking of the candidates. A voting system takes these millions of preference rankings as input and by some method returns a single ranking of candidates as output. (If people only have one vote, then an input ranking would involve ties, as in "Clinton first, all the rest second".) The government can then be formed on the basis of this single ranking.
For a voting system to make any democratic sense, Kenneth required it to satisfy each of the following, fairly basic constraints:
- The system should reflect the wishes of more than just one individual (so there's no dictator).
- If all voters prefer candidate x to candidate y, then x should come above y in the final result (this condition is sometimes called unanimity).
- The voting system should always return exactly one clear final ranking (this condition is known as universality).
He also added a fourth, slightly more subtle condition:
- In the final result, whether one candidate is ranked above another, say x above y, should only depend on how individual voters ranked x compared to y. It shouldn't depend on how they ranked either of the two compared to a third candidate, z. Arrow called this condition independence of irrelevant alternatives.
Arrow proved mathematically that if there are three or more candidates and two or more voters, no voting system that works by taking voters' preference rankings as input and returns a single ranking as output can satisfy all the four conditions. His theorem, called Arrow's Impossibility Theorem helped to earn him the 1972 Nobel Prize in Economics.
You can find out more about the maths of voting in these Plus articles.
Where would the world be without mathematics?
From computer games to smart phones, and from the weather forecast to our solar system — mathematics is essential in describing and understanding the world around us. Have you ever wondered what the world would be like without mathematics?
Our solar system is described by maths. Image: NASA.
Mathematics has been part of human culture for millennia. This competition is your chance to explore how mathematics has developed and achieved its status. Where does mathematics come from? How do we know it's true? What is the contribution of a particular person or culture?
The British Society for the History of Mathematics (BSHM) believes that understanding where mathematics comes from and who has contributed to the development of mathematical ideas is an important part of understanding mathematics today. BSHM (with a little help from us) invites secondary school students to explore this question and communicate their findings for a wide audience.
You could write an article (maximum 1500 words), make a short video (maximum ten minutes) or a multi-media project (maximum ten minutes).
The competition is open to all young people aged 11 to 19 who are in secondary education. A prize of £100 will be awarded for the best entrant in each of the age categories 11-15 and 16-19. Winners will be announced on this web page by 27th May 2017.
The deadline for entries is Friday 24 March 2017. For details on hoe to enter, rules and guidelines, visit the BSHM website.
Srinivasa Ramanujan (1887 - 1920).
December 22nd would have been the 129th birthday of the legendary Indian mathematician Srinivasa Ramanujan, who recently achieved wider fame through the film The man who knew infinity. His story really is remarkable. Born in 1887 in a small village around 400km from Madras (now Chennai), Ramanujan developed a passion for maths very early on. By age 15 he routinely solved maths problems that went way beyond what his classmates were dealing with. He worked out his own method for solving quartic equations, for example, and even had a go at quintic ones (and failed of course, since the general quintic is unsolvable). But since he neglected all other subjects apart from maths, Ramanujan never got into university, and was forced to continue studying maths alone and in poverty. Only after a plea to an eminent mathematician, who described Ramanujan as "A short uncouth figure, stout, unshaven, not over clean," did Ramanujan eventually get a job as a clerk at the Madras Port Trust.
It was during his time at the Port Trust that Ramanujan decided to write a letter that was to change his life. It was addressed to the famous Cambridge number theorist G. H. Hardy who, accustomed to this early-twentieth-century form of spam, was irritated at first: a letter from an unknown Indian containing crazy-looking theorems and no proofs at all. But as he went about his day, Hardy couldn't quite forget about the script:
At the back of his mind [...] the Indian manuscript nagged away. Wild theorems. Theorems such as he had never seen before, nor imagined. A fraud of genius? A question was forming itself in his mind. As it was Hardy's mind, the question was forming itself with epigrammatic clarity: is a fraud of genius more probable than an unknown mathematician of genius? Clearly the answer was no. Back in his rooms in Trinity, he had another look at the script. He sent word to Littlewood that they must have a discussion after hall...
Apparently it did not take them long. Before midnight they knew, and knew for certain. The writer of these manuscripts was a man of genius.
From the foreword by C. P. Snow to Hardy's A Mathematician's Apology
Hardy invited Ramanujan to Cambridge, and on March 17, 1914 Ramanujan set sail for England to start one of the most fascinating collaborations in the history of maths. Right from the start the pair produced important results and Ramanujan made up for the gaps in his formal maths education by taking a degree in Cambridge. Perhaps the most famous story to emerge from this period has Hardy visiting Ramanujan as he lay ill in bed. Hardy complained that the number of the taxi he had arrived in, 1729, was a boring number, and that he worried this was a bad omen. "No," Ramanujan replied, apparently without hesitation. "It is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways":
Unfortunately, Ramanujan's sickness wasn't a one-off. His health had always been feeble, and the cold weather and unaccustomed English food didn't help. Ramanujan decided to return to India in 1919 and died the following year, aged only 33. He is still celebrated as one of India's greatest mathematicians.
You can find out more about Ramanujan's mathematics in these Plus articles:
- Ramanujan surprises again
- A disappearing number (accompanied by a podcast)
- Numbers, toys and music: A conversation with Manjul Bhargava
- Chaos in numberland
We have also recently reviewed the famous essay A mathematician's apology by Ramanujan's collaborator G.H. Hardy.
If you're in or near London, then you might want to get yourself to the Museum of London on Tuesday to see the first in a series of public lectures about maths, given by one of our favourite mathematicians, Chris Budd. The lectures will show how relevant mathematics is to all of our lives, and the process by which mathematical ideas move from the abstract to the practical, and also transfer technology between very different disciplines. Whilst introducing you to some advanced modern mathematical ideas, these lectures will start from an elementary level, be accessible to all, and will be packed with examples, many of which will be drawn from directly from Budd's own experience as an applied mathematician.
The lectures will take place on Tuesdays at 1pm, they are free and no reservation is required. See the Gresham College website for more details.
Click here to see Budd's Plus articles and podcasts.
Good news for all those who love maths and trying out puzzles, games, problems and generally cool and interesting maths type things! The Annual MathsJam Conference is on this November! A number of talks have already been proposed including teaching tiny horses to count, cheese, pizza and other food based problems, unreal real numbers, and a newly discovered thing. There's also a baking competition, a competition competition (yes, and it makes total sense), a t-shirt competition, and lots of good stuff. You can register here, as well as find out how to offer a talk, an activity, competition or cake if so inspired – have fun Jammers!