It may come as a surprise that your average proof in an academic journal is riddled with holes. Authors gloss over details, appeal to pictures, even intuition, and take hidden leaps of logical faith that, philosophically speaking, aren't entirely justified. These days mathematics contains proofs so long and complex that few people are able to check and understand them in full, yet once a
result has made it through the peer review process and into a journal, its truth is taken as read.
All this is a far cry from the mathematical dream which started with Euclid over 2000 years ago: that every mathematical statement should be derived from the very axioms of mathematics in a sequence of verifiable logical steps. Proofs which do this are known as formal proofs, and they are the focus of a special issue of the Notices of the American Mathematics Society, which is
now freely available online.
According to media reports there are two suspects in the dock: the rocket scientists' (a.k.a. the financial mathematicians) who provided the information behind the market's decisions, or the greedy bankers who only thought about quick profits and their end-of-year bonuses. In our latest podcast, we talk to David Hand, Chris Rogers and John Coates to
find out who is guilty.
Zero, and infinity — more than just numbers, these two mathematical concepts have worried and inspired mathematicians for centuries. But they have also inspired philosophers and artists. What is zero? What is infinity? What do they look like: A black hole or a blank page? A spiral or the horizon? Mathematician Marcus du Sautoy, historian of mathematics Eleanor Robson, Science Museum curator
Jane Wess and artist Paul Prudence will be discussing Zero to Infinity at the Dana Centre in London. You can hear their perspectives, take part in the discussion, and have a drink at the bar on Thursday 20 November at 7pm.
For further information visit the Dana Centre website. And while you're waiting, you can read more about infinity on Plus.
Born on the 12th of November 1927 in Kisai, Japan
Died on the 17th of November 1958 in Tokyo, Japan
Taniyama's name is associated with one of the most famous problems in mathematics: Fermat's last theorem. The theorem says that for any whole number n strictly greater than 2, there are no three non-zero whole numbers x, y and z such that xn + yn = zn. Almost 400 years ago Fermat scribbled in the margin of a book that he had a
proof for this assertion, which didn't fit in the margin. It wasn't until the 1990s that the theorem was finally proved by Andrew Wiles — and the proof definitely couldn't have been written in any margin!
Andrew Wiles didn't actually prove Fermat's last theorem, but a conjecture which now carries Taniyama's name. In 1955 Taniyama, who was working in algebraic number theory, posed a problem concerning so-called elliptic curves — these are curves defined by points in the plane whose co-ordinates satisfy a particular type of equation. Goro Shimura and André Weil mused over the question and
formulated a conjecture: that every elliptic curve should come with a modular form, a mathematical object that is symmetrical in an infinite number of ways. This is now known as the Taniyama-Shimura-Weil conjecture.
The link to Fermat's last theorem was made around 30 years later, when the mathematician Ken Ribet realised that if Fermat's last theorem were false, then this would mean that a particular elliptic curve comes without a modular form. In other words, if Fermat's last theorem were false, then the Taniyama-Shimura-Weil conjecture would be false also. Reformulating this yet again, if the
Taniyama-Shimura-Weil conjecture is true, then so is Fermat's last theorem. What Andrew Wiles proved is that the Taniyama-Shimura-Weil conjecture is indeed true for a class of examples that is sufficient to prove Fermat's last theorem. QED.
In 1958, just days after his 31st birthday and not long before he was meant to get married, Taniyama committed suicide. His explanation was this: "Until yesterday I had no definite intention of killing myself. ... I don't quite understand it myself, but it is not the result of a particular incident, nor of a specific matter."
Why are some people generous and others selfish? There's no doubt that both strategies pay off under certain circumstances, but research (as well as everyday experience) shows that we are not mere opportunists — some people simply are nicer than others. This raises a question which intrigues evolutionary psychologists: is there a selective force that works in favour of a wide range of
personalities, preventing us from all evolving the same optimal character trait? A possible answer has recently been published by mathematicians from the universities of Bristol and Exeter.
There are three new singing stars on the block, but you're unlikely to hear them on Radio 1. The musical talents of HD49933, HD181420 and HD181906, three nearby stars which are hotter and larger than our Sun, were discovered by a group of scientists, led by Eric Michel, using data from the CoRoT space-based telescope. Michel and his colleagues accurately measured accoustic vibrations in these
stars that not only make for eerie listening, but also could reveal important information about how all stars evolve.