A Disanalogy of Disanalogies


by Roland Bolz

The following is ascribed to the 20th Century Polish mathematician Stefan Banach:

A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.

Let us imagine this ultimate mathematician at work. What would their thought process be like? What connections would they entertain, given their obsession with spotting analogies?

There is a famous phrase often (mis)attributed to Groucho Marx, which starts with the analogy ‘time flies like an arrow.’ Surely our mathematician would recognize therein an analogy between the passing of time and the flight of an arrow – the type of comparison one entertains when meditating in an armchair on the sorry finitude of all life on earth. The mood is philosophical.

But the phrase goes on with another analogy. ‘Time flies like an arrow. Fruit flies like a banana.’ Perhaps our gifted mathematician now believes that they are being presented with one of those rare and delicious analogies between analogies! So they read the second analogy analogously to the first. The flight of a piece of fruit is like the flight of a banana.

Certainly the conjured image of flying fruit alone is enough to put into question whether our first analogy, concerning the swift passing of time, is so sophisticated after all. Is the author of our analogy of analogies conveying to us that the philosophical mood we bought into just seconds ago is laughable? Why else juxtapose deep and shallow, philosophy and slapstick?

But the second analogy as parsed by the ultimate mathematician does not read right. ‘The flight of any piece of fruit is like the flight of a banana’? Are we being told that the flight of apples, blueberries and pineapples can all be studied by studying the flight of bananas? The banana as the perfect stand-in fruit? The objects in question (flights of bananas, flights of apples) are too similar to be analogous at all. Analogy requires deep differences, such as between the passing of time and the flight of arrows.

Luckily, the ultimate mathematician always sees several possible perspectives on the same material. The second sentence, which we first took as an analogy, can be parsed with ‘fruit flies’ as subject, ‘like’ as verb, and ‘a banana’ as object. Now the banana no longer appears as an object of comparison but as an object of desire. A very understandable desire for a fruit fly, no less. What does a fruit fly want? They’ll take a banana, thank you.

But what has happened to our sublime analogy of analogies? It seems to have disappeared like an arrow. Perhaps our author is conveying yet again that the time-arrow analogy can only be uttered by a philosophical poseur – the type of person who likes deep thoughts for the sake of being deep. Fruit flies, to the contrary, much prefer a simple banana over deep thoughts. And what would be a deep thought for a fruit fly anyway? Perhaps ‘time flies like a banana?’

The ultimate mathematician, however, will not give up at this impasse. They simply cannot yield. What if the second sentence (‘fruit flies like a banana’) has now been parsed correctly but the first one (‘time flies like an arrow’) has not? … And this is how our ultimate mathematician, seeker of analogies of analogies, ends up pondering what the f*** time flies are, and why they like arrows – just like fruit flies like bananas.



Image by Jean-Pierre Dalbéra via Flickr (cc).