Daily Archives: June 21, 2016


There are many roads that have entirely different spatial narratives for me depending on which direction I travel them in. The same buildings, trees, hills, curves, and power lines seem and feel not the opposite but entirely elsewhere when seen from a diametrically opposed perspective. Sometimes I do not even think of the places as the same places, the route as the same route; it takes an act of mental abstraction to unify them if I can at all. But only when a route is walked over both ways can it be fully understood.

This has a parallel, or anyway an analogue, in relations between people. Any two people may see the same things, the same moments, the same interactions, as parts of two quite different schemata, two quite different narratives, two quite different ideas of what is happening and how and why and what its significance is. Each may feel sure that his or her view of the situation is accurate, and yet what is happening for one is not the opposite of what is happening for the other, but not the same. At best, it may be the reciprocal.

What is the reciprocal? Reciprocal has different uses in different places. Between persons it is used to imply mutuality or at least an even balance sheet. In mechanics it (or its related form reciprocating) can refer to something that comes and goes repeatedly, like a piston (the word reciprocal seem to me a natural fit for such an engine, sounding like a cycling cylinder). In math, it is – to quote the Oxford English Dictionary – “A function, expression, etc., so related to another that their product is unity.”

Unity? There are two ways of looking at mathematical reciprocals. One is to see all numbers as fractions, and the reciprocal of each as being the reversal of it – topsy-turvy, upside-down, arse over teakettle. The reciprocal of 3/4 is 4/3; the reciprocal of 2 (which can be written as 2/1) is 1/2.The other way is to see a reciprocal as 1 divided by the number. The reciprocal of 3/4 is 1/(3/4), which is 4/3; the reciprocal of 2 is 1/2.

If you multiply a number by its reciprocal you get 1: for example, 3/4×4/3 = 12/12 = 1 and 2/1×1/2 = 2/2 = 1. This might seem a magical fact the first time you see it if you have thought of a reciprocal as being the flipped version of a number, but if you think of it as 1 divided by the number then you see it is necessarily true. Mutual reciprocals must multiply to unity. The road out and the road back combine to make the whole route.

It is tempting to add that this mutuality leading to unity can only work if both sides keep “I” out of it. You may know that i is a mathematical constant equal to the square root of –1. Thus if you have reciprocals such as 3/4 and 4/3, but you multiply i into each, you have 3i/4 and 4i/3 (which are no longer reciprocal), and if you multiply them together the result is a negative one: 3i/4×4i/3 = 12i2/12 = (12×–1)/12 = –1. But, you know, that’s just convenient, coincidental. You can’t have a perspective without a percipient; you can’t have an eye without an I. And in real life, the result of squaring up an I with another I is not a negative one. Any time you bring two roots or routes together you gain something.

Where does this word reciprocal come from? Apparently from Latin recus ‘backward’ (from re– ‘back’) and procus ‘forward’ (from pro– ‘for’). Something that is reciprocal is complementary, or mutual, or mutually dependent, or back and forth, or alternating, or contrary, or opposing. You see: even the meaning covers a 180-degree arc from sense to countersense.

And yet it still all makes sense. I see you: you see me. We see different things. Perhaps our views are in concord, perhaps in discord; it may be that we don’t even realize that they are as different as they are. But if we are on the same road, or sitting at the same table, the product of our fractions can put us at one.