Women’s brains are 10% smaller than men’s, yet the sexes are equally intelligent. It might not have turned out like this.  It’s easy to imagine a world in which natural selection had made men brighter than women. The image is an awful one, a dystopia in which the relationship between the sexes would surely have been that of master and slave.  Thank goodness it didn’t turn out like that.

But actually, it did turn out like that: not with brains, but with brawn. We’re so used to men being bigger and stronger that we don’t regard this disparity with horror. Yet most men can dominate most women physically, and I believe this is the main reason why in most societies throughout history, the men have been in control. Why do men dominate women? Because they can.

In recent generations, thanks to the cognitive demands of advancing civilization, we are beginning to rise above the physical differences and approach equality. It might not have turned out like this. If the dimorphism had been in brains rather than brawn, the advance of civilization might have increased the inequality between the sexes rather than diminishing it.

[22 April 2011]

In Madrid this week, I spent a lot of time looking at Las Meninas.  What a friend it became!

A salient feature of art and literature, and a recurring theme in these notes, is the power of ambiguity.  The experience of a great work may be due half to the artist, who brings so much to the canvas, and half to the viewer, in whose eye and mind the image resonates.  Las Meninas has become my archetypical example of this synergy.  Standing in front of it in that great hall of the Prado, I felt that Velazquez and I were working together to create this viewing experience.

It is key that he is looking at us. He’s daring us with the question, what do you think of this?  What do you make of the dog, and the dwarf, and the cavernous dark upper half of my painting?  Can you imagine the secrets I know of this crazy decadent court of Philip IV?

Yet there is so much Velazquez did not know!  He did not know that Britain and its colonies in America would build a new world as Spain declined ever further. He did not know that the infanta Margarita would die at age 21.  It may be just my fancy, but I like to think that Velazquez had a sense of the uncertainty of the future and the genius to craft a work that would take strength from that uncertainty.

[3 April 2016]

What’s the smallest nonprime that is not obviously nonprime? For Ramanujan the answer would have been in the thousands, but for the rest of us, I think it’s 91. The sequence of nonobvious nonprimes runs 91, 119, 133, 143, 161, 187,….

Here’s my reasoning. If a number is divisible by 2, 3, or 5, that’s obvious, and if it is 11 times a single digit, that’s obvious too. Also, perfect squares are pretty familiar. This leaves us with 7•13, 7•17, 7•19, 11•13, 7•23, 11•17,….

[14 May 2016]

Oxford dons are expected to have glittering conversations at high table, and we had a good one today.

One of the fellows mentioned a psychologists’ theory that an infant first understands the number 2, then later the number 3, then later the number 4, before finally figuring out the general notion of number.  However, she said, she and her husband hadn’t been able to detect stages 3 or 4 in their own little boy.

I mentioned the physicists’ theory of the period-doubling transition to chaos.  This involves an infinite succession of orbits, each one 4.669… times shorter than the last (Feigenbaum’s constant).  Maybe the little one went through all the number stages, but too fast to observe?

It was David Wallace who wrapped up the interchange with a philosophers’ twist.  If an infant masters the number 2 in one week, the number 3 in the next half-week, the number 4 in the next quarter-week, and so on, why then, in a fortnight he’ll have mastered all the numbers, an infinite collection of specific notions in finite time!  Who needs the general concept?

[14 April 2016]

I got my first pair of varifocals a few months ago. When I put them on I felt I was swimming in a sea of blur and I was tempted to abandon the experiment.  Nathaniel said, don’t worry, in a few days you’ll be used to them and the world will look good again.

He was right.  My brain adjusted comprehensively and my new eyesight quickly became the new normal.  On a glorious walk along the South Downs Way yesterday, for example, I almost never noticed that the focussed fraction of my field of vision is half what it was a year ago.

Your first thought may be pleasure at how easily our brains adjust, but your second should be one of horror.  How diabolically our brains disguise our declining powers!  My visual input is half what it was a year ago, and I don’t notice.  It’s probably a tenth what it was when I as a teenager, and I don’t notice that either.  It’s a safe bet the power of my thought has diminished too.  As the engine slows down, occasionally there are puzzling artifacts out there in the periphery, but for the most part, we don’t notice.

[11 April 2016]

Every adult knows that the matter of God is a serious one, quite unrelated to the case of Santa Claus, who is a harmless invention for children.

Kate and I saw a millennium-scale version of this disparity on display this week in hall after hall of the Prado.  Half the paintings of the Renaissance seem to depict stories of the Christian holy family and the saints, and the other half, stories of Greek and Roman gods and mortals.  In the first category, we have for example Fra Angelico’s beautiful “Annunciation”, which shows Mary being dazzled by a sunbeam of gold, symbolic of God impregnating her.  This led to the birth of Jesus, who was both god and man.  In the second category, we have Titian’s beautiful “Danaë and the shower of gold”, which shows Danaë being dazzled by a cascade of gold coins, symbolic of Zeus impregnating her.  This led to the birth of Perseus, who was half god and half man.

I doubt one Prado visitor in a hundred notices how indistinguishable the Christian and Greek/Roman stories would be to the proverbial Martian.  Even then, to make the parallel feels boorish and simplistic. One is tempted to think, well, that’s very clever, but of course, the two cases are completely different.  Everyone knows those Greek and Roman stories are just stories!  When in doubt of your doubt: remember Fra Angelico’s Mary and Titian’s Danaë.

[3 April 2016]

We are machines, of course, but with the curious feature that we can see ourselves from the inside.

It was strange to see myself from the inside just now here in the guest rooms of St. John’s College, above the candy shop.  There are two toilets, side by side, interchangeable. Yesterday evening I needed the toilet and arbitrarily chose the one on the right. Today I needed the toilet again and realized there was no choice in the matter at all — my steps were quite autonomically taking me back to the same one I had used yesterday.

Why?  Why in the world must I use the same toilet on day 2 as on day 1? Presumably because, for some excellent reason of natural selection, we are built like this.

As a matter of principle, just to show I could, I overrode the machine and used the one on the left.

[7 January 2016]

The sun is 100,000,000 miles away, 1,000,000 miles across, whereas the moon is 100,000 miles away, 1000 miles across.  (All numbers are rounded to the nearest power of 10.) Our eyes miss these vast scales entirely and see both objects as of comparable size and distance, much closer than they really are. They seem, what, maybe 10 miles away and 1/10 mile across?

I’ve long been interested in what I call “the other moon illusion”: if you consider the sun and moon together in the sky, you will judge, the moon should look fuller!  This illusion results from failure to perceive that the sun is further away than the moon.  It amuses me to note that to eliminate it, we wouldn’t have to perceive the truth, that the sun is a thousand times further than the moon, let alone that it is ten million times further than it looks.  If we could only perceive it as 20 miles away instead of 10, that would be enough.

[7 January 2016]

Mathematicians are known for their electric writing style, but even in this tough competition, I think the opening sentence of the 1985 monograph by T. B. Burton stands out:

“To a reader who is unacquainted with differential equations, an overview such as this can be quite meaningless.”

[27 December 2015]

I just spent six weeks in an apartment at NYU, and in the corner was a table with the television and telephone on it.  There were five remote controllers supplied to control these devices in their numerous capabilities, with a total of 188 buttons.

Teenagers find this stuff obvious.  Me, I can just about figure out what I need when I need it.  Older people?  I think a lot of them are shut out, just as my aunt Jane lost the ability to communicate with me and Gwyned when her keyboard was upgraded to a tablet.  Phones and TVs were so much simpler in the old days!  It cannot be beyond our ability to make them simple again for the tens of millions of people who would like that.  I predict this will happen within a few years.

188 buttons.

[12 October 2015]