There are degrees of infinity

I suspect that infinity is something that the human mind cannot grasp.

I am certain that infinity is something that my particular mind cannot grasp.

How many times can you fold a piece of paper?

Hat tip to kottke.org.

It’s said that you can’t fold a piece of paper in half more than eight times.

High school student Britney Gallivan proved this wrong back in 2002.

To see how remarkable her achievement was, take a look at a brute-force approach to the problem below.

Ramanujan and the nature of genius

Srinivasa Ramanujan Iyengar was a poor Hindu with only a basic mathematical education who, as a young man, made important mathematical discoveries.  He impressed the great British mathematicial, G.H. Hardy, who invited him to join him at Cambridge University in England, where the two had a brilliant and fruitful collaboration, cut short when Ramanujan died young.

I read Robert Kanigel’s The Man Who Knew Infinity: A Life of the Genius Ramanujan after seeing the movie based on the book.  The movie does justice to the spirit of the book and mostly conforms to fact, but cannot duplicate Kanigel’s richness of detail.

Both the movie and the book gave me food for thought on the nature and sources of genius.  I once thought of mathematical discovery as a logical, step-by-step process, but I now realize it depends as much on inspiration as anything else.

Some of Ramanujan’s theorems came to him in dreams, sometimes on scrolls held by Hindu gods.

Since I do not believe in the Hindu gods myself, how do I explain the fact that Ramanujan’s visions of the gods have him true mathematical theorems and also good advice on major life decisions.

I have to believe that his visions were manifestations of his subconscious mind.  Brain scientists tell us that most cognitive activity takes place below the level of consciousness.  I believe that most inspiration and creative thought arises from subconscious  sources, and that the conscious mind performs an executive function—deciding which intuitions have a basis in reality.

Ramanujan: ‘the man who knew infinity’

I saw this movie a week or so ago.  I liked it a lot.  It is about the untutored Indian genius, Srinivasa Ramanujan, and how the famous British mathematician, G.H. Hardy, invited him to study with him at Cambridge University in England.

It begins with an epigraph quoting Bertrand Russell:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

The movie shows the interesting and quirky characters of Ramanujan and Hardy as interesting and quirky characters, products of two very different cultures, and the backgrounds of life in Madras, India, in the early 1910s and in Cambridge during World War One.

The two men represented very different ways of knowing.  Ramanujan, the deeply religious Hindu, saw things holistically, as a kind of mystic vision.  The movie shows him in his job as clerk, writing in the sum of a column of numbers without adding them up, yet getting the correct figure.

G.H. Hardy was an atheist.  He didn’t believe in anything that couldn’t be proved.  Ramanujan didn’t want to bother with proofs.  He thought Hardy should just be able to see that his mathematical discoveries were right.

After all, his theorems appeared to work.  You can use the Pythagorean Theorum for estimating measurements without knowing Euclid’s proof.  Except, according to the movie, there was at least one occasion in which Ramanujan was wrong.

Mathematics is an example of a reality that is intangible, yet real.   For Ramanujan, the study of mathematics was a kind of spiritual discipline.

He made a great sacrifice for his love of mathematics.  As a high-caste Hindu, he was considered defiled for crossing the ocean.  He separated from his wife, whom he deeply loved.   He had a hard time sticking to his vegetarian diet, and he suffered from the damp, cold English winters.  Eventually he caught tuberculosis and nearly died.   In fact, he did die, at the age of 32, shortly after he returned to India.

One good thing about life today is that institutions such as Cambridge are sensitive to cultural differences.  A contemporary Ramanujan would be provided with food that he could eat.

Bertrand Russell is a minor character in the movie, and it is interesting to see him in the prime of life, with dark hair and a dark mustache, and not the elderly, white-haired image I hold in my mind.

How the Enigma code was broken

This pair of videos explains exactly how the World War Two German Enigma code worked, why it was so hard to break, and what the British Bletchley Park team did to break it.