Please excuse me for borrowing a couple of terms from elementary mathematics:
— A relation is said to be reflexive if it works in both directions. For example, "can see" is reflexive in the sense that, if John can see Mary, then normally Mary can see John. But "loves" is not necessarily reflexive, because John can love Mary whereas the sentiment might not be reciprocal.
— A relation is said to be transitive when its effects are, as it were, cumulative. For example, "is greater than" is transitive. If X is greater than Y, and Y is greater than Z, then X is necessarily greater than Z.
A few days ago, while watching my daughter and my dog scampering over the slopes at Gamone, I found myself wondering, for a few instants, whether "can see" might be, not only a reflexive relation, but transitive too. For example, if I can see simultaneously both my daughter and my dog up on the hillside, does this mean that they can see one another? This, of course, is a stupid question. Clearly, the answer is no. For example, you might be able to see two individuals in adjacent rooms, whereas their mutual vision is blocked by a wall. In other words, "can see" is not a transitive relation.
Here's a view of the circus of Choranche, as seen from my house:
Of an evening, I often see a bright electric lamp at the far end of the valley, at the spot where I've put a red dot. This lamp has always intrigued me, for three reasons. First, it's the unique source of light in this entire direction. (In other words, as soon as the Sun goes down, when the lamp is unlit, the entire scene of the photo is plunged in darkness.) Second, I've never been able to determine with certainty the precise place where this light is located. Third, the lamp is only lit at certain short periods of the year, which don't necessarily seem to coincide with holiday dates.
Behind the red dot in the photo, the massive rock wall that fills in the horizon between the cliffs of Presles and the slopes of the Bournillon is called Chalimont. On the far side of their crest, the vast forest of Herbouilly stretches out over the Vercors plateau in the direction of Villard-de-Lans.
Just beneath the red dot in the photo, you can see a curved line of clifftops, lying above the River Bourne, which tumbles down from a break in the Chalimont (hidden, in the photo, by the cliffs of Presles). In the middle of this curved line, some seven kilometers from my house (as the crow flies), the lowest point is a pass (for experienced rock-climbers) known as the Devil's Doorway. A nearby hole in the cliff is referred to as the Gaul's Cave. Besides, it's perfectly possible that human Bronze-Age cavemen might have used this place, two or three thousand years ago, as a base camp for their summer hunting season. And somewhere between the red dot and the curved line of clifftops, there's a sizable village, St-Julien-en-Vercors, lying alongside a major road that runs from the Bourne across to the village of La-Chapelle-en-Vercors (located behind the Bournillon plateau).
At the end of my article of 26 December 2009 entitled More fallen rocks [display], I explained that, to escape from Choranche, I have to choose a route up over the surrounding mountains. The other day, I left early and headed up towards St-Julien-en-Vercors, while saying to myself that I might find time to finally elucidate the puzzle (which arose for the first time in May 2004) of the lamp at the end of the valley. By chance, the first villager I encountered happened to be (I learned later) the most informed person in existence concerning St-Julien and its surroundings. As soon as I told him I came from Choranche, he said "I've never liked that village. No charm whatsoever." I found this frankness reassuring. There was no chance that this fellow would tell me bullshit. In fact, within a few minutes, we had become firm friends, and he told me everything I needed to know about the mysterious lamp. So, here's a summary of the affair.
The light comes from a forestry hut, high up on the slopes of the Chalimont, several hundreds meters above the village of St-Julien. The hut and a surrounding forest zone belong to a retired member of the French merchant navy, who lives down at Cassis, near Marseille. He and his wife drive up to the Vercors and stay up in the hut (accessible only on foot, and surrounded by snow at present) whenever the owner has to handle various aspects of the management of his trees. Since I left my name and address, the fellow phoned me up yesterday, introducing himself with humor as my "next-door neighbor". This afternoon, I used a telephoto lens to take a photo of what I believe to be his log cabin:
It sure looks icy up there. It seems to be so far away, and yet this Siberian scene lies just at the end of my long-focal lens.
Now, let me return to the definitions of mathematical relations at the start of this article. I said that the "can see" relation is reflexive. So, since I can see the lamp of this log cabin up on the slopes of the Chalimont, then the occupants should be able to see the lights of my house at Gamone. When I asked the owner what he could actually see from his log cabin when he looked in the direction of Choranche, I was surprised to learn that he can apparently see many interesting places. If I understand correctly, of an evening, he can see so many lights that he's not at all sure which one is my house at Gamone. Usually, he has a clear view of the autoroute that leads south in the direction of the Mediterranean. Furthermore, on clear days, he can often detect a celebrated mountain range in the south of France: the Cévennes.
That last detail set me thinking. If I can see my neighbor up on the slopes of the Chalimont, and he can see straight down to the south of France, then what a pity that the "can see" relationship is not transitive... otherwise I too should be able to gaze down at the south of France. This, of course, is totally unthinkable. From Gamone, I can't even see as far south as the first village in the Drôme, Saint-Eulalie, which is no more than a kilometer away.
During the first half of the 19th century, the French engineer Claude Chappe invented and installed a vast semaphore system throughout France, which concretized the transitive nature of the "communicate with" relation. During the Napoleonic Wars, for example, a series of Chappe towers could receive and retransmit information so rapidly that a message could be sent from one side of France to the other in a quarter of an hour. Now, that approach would in fact enable my Chalimont neighbor to inform me visually, every evening, what the weather had been like down in the south of France during the afternoon. He could use his powerful lamp to send me messages in Morse code. In fact, this won't be necessary, because I've already given him, not only my phone number, but my email address. Still, I get a thrill out of thinking that, at Gamone, I'm a mere hair's breadth away from being able to gaze down upon Provence.