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Archive for the ‘Mathematics’ Category

Infinity and Beyond

In Mathematics on September 3, 2012 at 2:17 PM

I noticed a cute story on the BBC website where a four year old wanted to know the number before infinity. Johnny Ball, a television presenter who specialises in explaining mathematical ideas to children, does his best to explain it. But the question reminds me of Hilbert’s paradox of the Grand Hotel:

Imagine a hotel with an infinite number of rooms, and all the rooms are occupied. To this hotel…comes a new guest and asks for a room. “But of course!” – explains the proprietor, and he moves the person previously occupying room N1 into room N2, the person from room N2 into room N3, the person from room N3 into N4, and so on…. And the next customer receives room N1, which becomes free as the result of these transpositions.

Source: Patrick Hughes and George Brecht, Vicious Circles and Infinity: An Anthology of Paradoxes, (Penguin Books, 1978) p.29.


On comprehending numbers and the gleeful dismissal of mathematics

In Finance, Mathematics, Twitter on July 11, 2012 at 12:00 PM

This morning I started a fight on Twitter. I didn’t need to but I wrote something deliberately provocative. Nick Cohen quoted a line from Barbara Gunnell’s article “Staying alive in Britain: can the poor afford it?” The selected sentence quoted was this:

Like most financial figures, the estimate of personal debt has too many zeros to comprehend.

The personal debt figure that Ms. Gunnell refers to is £1,460,000,000,000. Her sentence annoyed me. I interpreted it as saying that the number 1.46 trillion is incomprehensible not just to herself, but to anyone. Why it annoyed me is that not only do I think I can comprehend a figure of that magnitude, but that I think many people with a reasonable aptitude for numbers can do so too. I responded as follows:

People without a mathematical background often say things like that. What would arts people think of those who might say “Charles Dickens’ novels have too many long words to comprehend.”  Howls of laughter and ridicule.

I meant it. Earlier this year I read Dickens’ Great Expectations. I used the dictionary to look up words that the celebrated author had used in that novel including, but not limited to, “betimes,” “chary,” “clew,” “contumaciously,” “epergne,” “equipage,” “fain,” “fetters,” “gewgaws,” “indite,” “jorum,” “prolix,” “sconces,” “slue,” “stolidity,” and “victualling.” I am not proud that I had to use a dictionary to find out the exact meanings of these words. It is an embarrassing fact. Woe betide anyone who said to a journalist, who by job definition should have an excellent command of language and a wide vocabulary, that Dickens is incomprehensible.

Just because I had to look up “gewgaws” as I had no idea what the word meant, it does not mean that Ms. Gunnell did not know either. She might well do. And just because she cannot comprehend a number such as 1.46 trillion, it does not mean that I can’t, as I can.

In her defence, Ms. Gunnell did not write that sentence without what she believed was a good reason to do so. In fact, she had an impeccable source. The Nobel Prize winning psychologist Daniel Kahneman was recently interviewed for an article published in this week’s Observer. He stated:

Human beings cannot comprehend very large or very small numbers. It would be useful for us to acknowledge that fact.

Statements by Nobel Prize winners should not be dismissed lightly and it is perfectly reasonable for a journalist such as Ms. Gunnell to rely upon Kahneman. Thinking about this, I do not believe that Kahneman was referring to a number as small as 1.46 trillion when he said humans cannot comprehend very large numbers. The clue is that he also says that humans cannot comprehend very small numbers. Most people might think that, for example, the number 3 is a very small number. But if I said that humans cannot comprehend the number 3, I would not be taken seriously. Kahneman must therefore be referring to a number such as 0.00000000000000000000000000000000000001. This I cannot really comprehend and nor do I suspect many, if any, people can. 1,460,000,000,000 might sound like a very large number, but in mathematical term it is hardly a googolplex. Now that is a number that I cannot comprehend.

I have no desire to personally attack Ms. Gunnell, indeed I think her article is well worth reading; my concern is to defend finance and mathematics as important subjects and ones that should be encouraged to be studied rather than gleefully dismissed as incomprehensible.