Descartes’ Flies

There are a healthy number of legends about René Descartes. Some of them may be true. I know the one I like is the story that this superlative mathematician, philosopher, and theologian (fields not so sharply differentiated in his time as they are today; for that matter, fields still not perfectly sharply differentiated) was so insistent on sleeping late and sufficiently ingenious in forming arguments that while a student at the Jesuit Collè Royal Henry-Le-Grand he convinced his schoolmasters to let him sleep until 11 am. Supposedly he kept to this rather civilized rising hour until he last months of his life, when he needed to tutor Queen Christina of Sweden in the earliest hours of the winter morning.

I suppose this may be true; it’s certainly repeated often enough, and comes to mind often when I do have to wake to the alarm clock. I haven’t studied Descartes’ biography well enough to know whether to believe it, although as it makes for a charming and humanizing touch probably the whole idea is bunk and we’re fools to believe it. I’m comfortable being a little foolish. (I’ve read just the one book which might be described as even loosely biographic of Descartes — Russell Shorto’s Descartes’ Bones — and so, though I have no particular reason to doubt Shorto’s research and no question with his narrative style, suppose I am marginally worse-informed than if I were completely ignorant. It takes a cluster of books on a subject to know it.)

Place the name “Descartes” into the conversation and a few things pop immediately into mind. Those things are mostly “I think, therefore I am”, and some attempts to compose a joke about being “before the horse”. Running up sometime after that is something called “Cartesian coordinates”, which are about the most famous kind of coordinates and the easiest way to get into the problem of describing just where something is in two- or three-dimensional space.

We already know that we can put numbers on a straight line, or on a curved line which does not cross itself, so that each point on the line corresponds to just one number and each number within the range corresponds to just one point. The problem is whether we can talk about points over an area or within a space using some sort of number. In principle, we can, thanks to the wondrous space-filling curves, which will let us trace out a single curve that touches (or, to be precise, comes arbitrarily close to touching) every point in the area. In practice, that’s too difficult a system to work with.

It’s not enough to have a number for every point in an area. We want the numbers to have some extra properties. For example, starting from our original reference point — let me call it P — we move a little bit to a new point — let me call it R — then we want the number describing where R is to be close to the number describing where P is. This may start as an appeal to intuition, and intuition can and often does fail us, but it also lets us start describing the idea of continuity. We know in common-language terms what continuous things are, and that they’re nice to have. We can create a mathematical idea of continuity, and it’s quite close to the common English idea, and this mathematical continuity is also quite nice to have. It gives us nice luxuries like the ability to round off calculations, or let computers do the calculating for us.

One of the stories about Descartes, and I do not assert that this is true, is that he lay in bed looking at the tiles of the ceiling, and at the flies which sometimes landed on it. And in one of those great insights he realized he could say where a fly was by counting the number of tiles between the corner and the position. One number wouldn’t do for this; there would be one number for how far the fly was horizontally, and a separate number for vertically. (It’s not really vertically, but you know what I mean, and if someone has a good word for that second direction I would appreciate learning it. ‘Good’ here must include the quality that the meaning of the word is clear in context without explanations like this parenthesis to back it up.)

And that is his idea, as wonderful and simple as that: that we can describe locations by using a set of numbers. If it’s a location within a two-dimensional area, we rely on two numbers, given in a sequence and for which the order is significant: (2, 6) is some place within an area. This grouping is called an ordered pair of coordinates, or just an ordered pair. And while we need two numbers to represent a place, we have that nice continuity idea: (1.9, 6.01) is a point pretty close to (2, 6), and it’s plausible for a point near (2, 6) to have the coordinates (2.003, 5.86).

As with Arabic numerals, we can’t expect to swap digits around with impunity; “two to the right and six up” is not the same as “six to the right and two up”. That’s the “ordered” part of the ordered pair and hearing it written out as in the previous sentence shows why the ordering is so important.

Is the pair important? Could it be three numbers instead, or more? Well, to describe just a position somewhere in an area, yes, we could use three or more numbers. It’s just hard to see why we would want to: the two numbers are enough, and mathematicians generally like to make do with as few things as possible, particularly if those things are numbers. It’s so much more convenient using two numbers rather than one for positions in an area, but it’s not any more convenient to use three numbers.

However, if we wanted to pick a place in a volume, somewhere in three-dimensional space, then three numbers does turn out to be convenient. We could, in principle, use two or even one number and appropriate space-filling curves to squeeze down how many numbers are demanded, but three fits so much more nicely in general. Then we need an ordered triple, something like (2, 6, 5). Once again the ordering is important; that it’s three numbers, less so.

It’s harder to visualize, and harder to talk about without weakly defined science fiction concepts meaning “big strange stuff going on” intruding, but we can imagine higher-dimensional spaces, places with four or five or more dimensions. Then we need four or five or more numbers, again in order. Around this point we stop trying to figure out what the terms after ‘pair’ and ‘triple’ are, and settle for calling something an ‘ordered N-tuple’, where N is however many things there are: 4-tuple, 5-tuple, 6-tuple, and so on, and we let the people who know how to speak Latin or Greek or whichever that ending is wince. I’ve generally heard the word pronounced with a long vowel, as if someone tried to say ‘four tulip’ and got the last syllables jumbled, but there are some who use a short ‘u’, as if not quite saying ‘four topple’ right. Both seem to be accepted.

I go into this detail because when I first saw the mysterious N-tuples in a textbook I was stuck on how to say them, and I was stuck imagining ‘entipple’ until the instructor got around to saying the name. I remember the textbook also not being precisely clear on just what N was, and how it related to how it connected to the pairs and triples shown up to that point. This must have set my entire career back dozens of minutes, and I don’t want to hold anyone else up for the same cause.