I’ve just found an amazing 2002 article [pdf] from the American Mathematical Societyabout blind mathematicians.
I was surprised to learn that the majority work in geometry, supposedly the most ‘visual’ discipline, and fascinated to learn that they generally believe the experience of sight puts people at a disadvantage because it locks us into a perception-led view of space.
This can be a problem in geometry because it regularly works in problems that involve more than three dimensions or requires an understanding of objects from all ‘angles’ simultaneously.
Alexei Sossinski points out that it is not so suprising that many blind mathematicians work in geometry. The spatial ability of a sighted person is based on the brain analyzing a two-dimensional image, projected onto the retina, of the three-dimensional world, while the spatial ability of a blind person is based on the brain analyzing information obtained through the senses of touch and hearing. In both cases, the brain creates flexible methods of spatial representation based on information from the senses. Sossinski points out that studies of blind people who have regained their sight show that the ability to perceive certain fundamental topological structures, like how many holes something has, are probably inborn…
Sossinski also noted that sighted people sometimes have misconceptions about three-dimensional space because of the inadequate and misleading twodimensional projection of space onto the retina. “The blind person (via his other senses) has an undeformed, directly 3-dimensional intuition of space,” he said.
There is not any maths in the article but it is written for mathematicians so it contains lots of mysterious sentences like “Morin first exhibited a homotopy that carries out an eversion of the sphere in 1967″.
However, the article is also a fantastic history of blind mathematicians and has lots of quotes from current leaders in the field who explain who their supposed disability lets them better understand the maths of three and more dimensions.
Even for those without a maths background it’s an amazing insight into some remarkable people.
I used to do a lot of counting as a trumpeter in my local youth orchestra. Sitting in the brass section, counting out rests so I didn’t crash in early with a fanfare, I began to realise that mathematics and music had even deeper links. It is certainly a connection people have commented on throughout the ages.
"Music," wrote the great 17th-century German mathematician Gottfried Leibniz, “is the sensation of counting without being aware you were counting.” But there is more to this connection than counting. As the French baroque composer Rameau declared in 1722: “I must confess that only with the aid of mathematics did my ideas become clear.”
So is there really a link? Or is it crazy to try to connect the creative art of music with the steely logic of mathematics? Certainly the grammar of music – rhythm and pitch – has mathematical foundations. When we hear two notes an octave apart, we feel we’re hearing the same note, so much so that we give them the same name. (This is because the frequencies of the two notes are in an exact 1:2 ratio.)
Yet, while the combinations of notes we have been drawn to over the centuries can all be explained through numbers, music is more than just notes and beats – just as Shakespeare is more than just words from a dictionary. And it is in putting the notes together to create, say, the Goldberg Variations or Don Giovanni that I believe the true connection between mathematics and music reveals itself.
Many people react angrily to such a claim, believing music to be so much richer and more emotional than mathematics, and that to make such a comparison is to misunderstand what music is truly about. But, as a professor of mathematics, I think this argument misunderstands what mathematics is truly about.
Just as notes and rhythms are not all there is to music, so arithmetic and counting are not all there is to mathematics. Mathematics is about structure and pattern. As we’ve explored the universe of numbers, we’ve discovered strange connections and stories about numbers that excite and surprise us. Take the discovery by Fermat, the 17th-century French mathematician, that a prime number that has a remainder of 1 after division by 4 (like 41) can always be written as the sum of two square numbers (41=16+25). It was a realisation that linked the seemingly separate worlds of primes and squares.
Just as music is not about reaching the final chord, mathematics is about more than just the result. It is the journey that excites the mathematician. I read and reread proofs in much the same way as I listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. What people don’t realise about mathematics is that it involves a lot of choice: not about what is true or false (I can’t make the Riemann hypothesis false if it’s true), but from deciding what piece of mathematics is worth “listening to”.
I can get a computer to churn out endless true statements about numbers, just as a computer can be programmed to create music. The art of the mathematician lies in picking out what mathematics will excite the soul. Most mathematicians are driven to create not for utilitarian goals, but by a sense of aesthetics. The 19th-century French mathematician Henri Poincaré summed up this creative role thus: “To create consists precisely in not making useless combinations. Invention is discernment, choice … The sterile combinations do not even present themselves to the mind of the inventor.”
But for me, what really binds our two worlds is that composers and mathematicians are often drawn to the same structures for their compositions. Bach’s Goldberg Variations depend on games of symmetry to create the progression from theme to variation. Messiaen is drawn to prime numbers to create a sense of unease and timelessness in his famous Quartet for the End of Time. Schoenberg’s 12-tone system, which influenced so many of the major composers of the 20th century, including Webern, Berg and Stravinsky, is underpinned by mathematical structure. The organic sense of growth found in the Fibonacci sequence of numbers 1,2,3,5,8,13 … has been an appealing framework for many composers, from Bartók to Debussy.
Rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflects the world of symmetry and geometry. As Stravinsky once said: “The musician should find in mathematics a study as useful to him as the learning of another language is to a poet. Mathematics swims seductively just below the surface.”
• Marcus du Sautoy is speaking at the Cheltenham music festival on Saturday. Details: cheltenhamfestivals.com
A mathematician’s perspective on finding the love of your life. I burst out laughing several times. Even if you don’t read the whole thing, please at least skim this gem.
Let the “love of your life” be the currently living person who (if you got to know them) would:
- Fall romantically in love with you for as long as he or she lives.
- Cause you to fall romantically in love with her or him for as long as you live.
- Increase your average happiness at least as much as any other person satisfying both (1.) and (2.), if you were to become life partners.
In any event, I am sad to report that when applying the above definition for “the love of your life”, finding “the one” is essentially impossible. I strongly urge you not to try it. The probability that you meet the single person that would make you happiest of all is extremelysmall. There are about 7 billion people on earth, and (let’s say) more than a billion within a reasonable dating age range of you. That implies that there are at least 500 million people of the appropriate gender (a bit more if you are bisexual).
It is much smarter to view the search for love as an attempt to maximize your total lifetime romantic happiness over the rest of your life. This viewpoint leads to a very different optimal strategy than one would use to try to find the single best person. The total romantic happiness maximizing approach implies working to increase the moment to moment satisfaction you feel due to your romantic life, added up (or if time is continuous, integrated) over all of your remaining moments.
From now on, we will refer to a person employing such a romantic happiness maximizing strategy as a “Romaximizer”, and will kick the mathematics into high gear, introducing the Romaximizer equation:
- L is the number of years you have left to live
- H is the average amount of happiness per relationship year (that is, per year of time spent in relationships) that you will derive from your future relationships
- T is the average number of years that you will spend in each future relationship
- P is the average number of new people that you will meet per year
- F is the fraction of the people that you will meet that you will find sufficiently physically and personally attractive to consider dating
- D is the fraction of those people you will want a relationship with who will actually be willing to have a relationship with you
- M is the fraction of those people that you will find sufficiently attractive to consider dating that you will decide to try to actually begin a romantic relationship with