Monday, October 08, 2007

Mathematics and faith, God and infinity

A couple of early passages from David Leavitt’s The Indian Clerk, about the relationship between the great British mathematician G H Hardy and the untrained genius Srinivasa Ramanujan, whom Hardy invited to study under him at Cambridge in 1913. The first passage is from Hardy’s childhood, with a young vicar attempting to teach him the importance of religious faith.
“As I have been trying to explain to your son,” the vicar said, “belief must be cultivated as tenaciously as any science. We must not allow ourselves to be reasoned out of it.”

“Harold is very good at mathematics,” his mother said. “At three he could already write figures into the millions.”

“To calculate the magnitude of God’s glory, or the intensities of hell’s agonies, one must write out figures far larger than that.”

“How large?” Harold asked.

“Larger than you could work out in a million lifetimes.”

“That’s not very large, mathematically speaking,” Harold said. “Nothing’s very large, when you consider infinity.”

The vicar helped himself to some cake. “Your child is gifted,” he said, once he had swallowed. “He is also impudent.” Then he turned to Harold and said, “God is infinity.”
“God is infinity”: the linking of a mathematical concept with something that is built on faith rather than reason. This is interesting, given that most people see math as a cold, staunchly rational science based on numbers that confer a sense of order. Not much scope here for the “what if”, it would seem. But as this next passage suggests, the very conflict between rationalism and faith can be expressed in mathematical terms.
Often the simplest theorems to state were the most difficult to prove. Take Fermat’s last theorem, which held that for the equation x^n + y^n = z^n, there could be no whole number solutions greater than 2. You could feed numbers into the equation for the rest of your life, and show that for the first million n’s not one n contradicted the rule. Perhaps, if you had a million lifetimes, you could show that for the first billion n’s, not one contradicted the rule – and still, you would have shown nothing. For who was to say that far, far down the number line, far past the magnitude of God’s glory and the intensity of hell’s agonies, there wasn’t that one n that did contradict the rule? Who was to say there weren’t an infinite number of n’s that contradicted the rule?
In a way I think this is a good encapsulation of the agnostic dilemma – the minuscule doubt that continues to exist in the face of all reason; the impossibility of reaching a definite conclusion (or the unwillingness to reach a definite conclusion).

(Will write about the book at length once I’ve finished it - it’s been very absorbing so far, though Ramanujan hasn’t made an appearance yet and most of the early chapters deal with the Cambridge Apostles, a society that Hardy was a member of and that included, at the time, such intellectuals as Bertrand Russell, Lytton Strachey, John Maynard Keynes and Ludwig Wittgenstein. Apparently they spent much of their off-time in lengthy discussions of subjects such as “Should a Picture be Intelligible?” and “Does Absence Make the Heart Grow Fonder?”, and in the contemplation of sodomy.)


  1. I didn't understand the connection between that proof and agnosticism. The mathematical proof shows that the statement is valid for ANY n, as long it satisfies that basic criteria, whatever you choose, no matter how far you can go. In fact that's the idea of a mathematical proof. It is FINAL.

    A better example would be scientific statements like "All Crows are Black." No matter how many crows you see, you can never prove it to be final. There will always be some doubt remaining that there is some crow somewhere which is not black that you have not seen yet. Basically it is the difference between deductive and inductive logic.

    Read a few reviews of the book and wanted to read it too but when I saw it looked too heavy and too long. Will probably read it when it comes out in paperback.

    This connection between mathematics and human psychology is also brilliantly explored in Musil's Confusions of Young Torless. Here it is the concept of Irrational numbers... it is basically about human being's ability to think about and invent abstractions and how it alienates them from the world of concretes and real experience often leading to perversion of their moral judgments. It has a lot of many more other very complex things too. It is quite small unlike his masterpiece so is actually quite readable. (There is a movie by Volker Schlondorff based on the book, it is also quite good.)

  2. Fermat’s last theorem has been proven.

    Maths 1 - Impossibility 0


    I, for one, sympathise with the agnostic dilemma and all that but linking uncertainty with math is blasphemy.Stop staring at me, I'm an engineer!Ok?!

  3. Alok, Lalbadshah: okay okay, my deductive logic is clearly off! But I'm still having a bit of trouble understanding why the black-crow example is self-evidently different from the "n" example. I think that second passage suggests (perhaps in a non-serious, imp-of-the-perverse sort of way) a magical side to Mathematics that would naturally extend beyond the clinical world of formulae and "proofs". That's where I thought it tied in with reason-vs-belief.

  4. Arjun,

    A very interesting way of presenting the agnostic dilemma using a mathematical metaphor! Lot of things in life don’t need refutation. For instance, you are under no obligation to disprove my assumption that goblins hide under my bed at night, or you are actually an alien who got lost while hitchhiking across Milky Way. The big question is whether you want to place the issue of God’s existence in the same category!

    There is probably only one solution to the agnostic dilemma. You have got to make up your mind :-)

  5. I thought the example of Fermat's last theorem was appropriate. For the longest time, mathematicians could not find a proof but they couldn't disprove it either. Hardy died much before it was finally proven.

  6. Jai,-
    Fermat's last theorem demands a direct proof. You have to prove such a number CANNOT exist.

    The "black crow" statement is an exact illustration of fermat - it is true for every crow that we have so far seen just as fermat was true for the first million "n" seen. That doesn't mean it is always true - inductive logic suggests that it may be so but we cannot directly deduce it.

    Incidentally read the etymology of "The Black Swan" (Nassim Nich Taleb) - that is a real-life example. It was assumed that there were no black swans until people hit Australia. You could even try reading the book - you may actually enjoy it.

    Since Wiley solved FLT before Leavitts was written, he could have used the Riemann Zeta Hypothesis.

    That stands in the same place - it's known to be true for every "n" so far investigated but nobody has proved that it MUST be true.