The Net Advance of Physics RETRO:

2013 June 9
PART 5: Poincaré's words are in boldface.

The hypersphere, Poincaré's favourite four-dimensional object. [Illustration ©2006 Eugene Antipov]

So the characteristic property of space, that of having three dimensions, is only a property of our table of distribution, an internal property of the human intelligence, so to speak. It would suffice to destroy certain of these connections, that is to say of these associations of ideas, to give a different table of distribution, and that might be enough for space to acquire a fourth dimension.

Some persons will be astonished at such a result. The external world, they will think, should count for something. If the number of dimensions comes from the way we are made, there might be thinking beings living in our world, but who might be made differently from us and who would believe space has more or less than three dimensions.

Has not M. de Cyon said that the Japanese mice, having only two pair of semi-circular canals, believe that space is two-dimensional? And then would not this thinking being, if he is capable of constructing a physics, make a physics of two or of four dimensions, and which in a sense would still be the same as ours, since it would be the description of the same world in another language ?

It seems in fact that it would be possible to translate our physics into the language of geometry of four dimensions; to attempt this translation would be to take great pains for little profit, and I shall confine myself to citing the mechanics of Hertz where we have something analogous. However it seems that the translation would always be less simple than the text, and that it would always have the air of a translation, that the language of three dimensions seems the better fitted to the description of our world, although this description can be rigorously made in another idiom.

Besides, our table of distribution was not made at random. There is connection between the warning A1 and the parry B1, this is an internal property of our intelligence; but why this connection? It is because the parry B1 affords means effectively to guard against the danger A1 ; this is a fact exterior to us, it is a property of the exterior world. Our table of distribution is therefore only the translation of an aggregate of exterior facts; if it has three dimensions, this is because it has adapted itself to a world having certain properties; and the chief of these properties is that there exist natural solids whose displacements follow sensibly the laws we call laws of motion of rigid solids. If therefore the language of three dimensions is that which permits us most easily to describe our world, we should not be astonished ; this language is copied from our table of distribution, and it is in order to be able to live in this world that this table has been established.

I have said we could conceive, living in our world, thinking beings whose table of distribution would be four-dimensional and who consequently would think in hyperspace. It is not certain however that such beings, admitting they were born there, could live there and defend themselves against the thousand dangers by which they would be assailed.

Japanese dancing-mice [1 min 38 sec, all pretty much the same]


Several historians have remarked that Poincaré, like Max Planck, was a most reluctant revolutionary, that he had the soul of a conservative. The truth of this observation is evident in the present section.

At the beginning of the section we find a speculation as radical as any in the whole corpus of Nineteenth Century science and philosophy. Having argued that geometry is simply a collection of algorithms turning arbitrary inputs into outputs favoured by Darwinian natural selection, he proposes that these algorithms need not be unique. The same survival-enhancing outputs could perhaps be generated from the input data in more than one way! For example, besides the recipe known as "three-dimensional geometry" there might be other recipes -- "two-dimensional geometry", "four-dimensional geometry" -- that produce exactly the same relationship between stimulus and response, just as several completely different computer programs might be able perform the same task. He even cites what he believes to be an example of creatures with a different perception of the world's dimensionality -- Japanese dancing-mice.

Jerkers, as Japanese dancing-mice are usually called today, are otherwise ordinary mice who spend nearly all of their spare time -- that is, the time when they are not eating, sleeping, grooming themselves, &c. -- running in small circles, sometimes jerking their heads as they do so. They rarely move long distances in a straight line without zig-zagging. They are usually deaf, which immediately suggests a connexion between their constant circular motion and the organs of balance in the inner ear.

Modern biologists explain jerkers' strange behaviour as the result of a spontaneous point-mutation on Mouse Chromosome 4; whenever two copies of the mutation are inherited, the protein espin is suppressed, and the hairs of the inner ear, reponsible for both balance and hearing, are unstable and degenerate. Jerkers, in this view, believe themselves to be almost constantly falling to one side, and are attempting to correct for this by turning the other way. (This of course does not seem to explain why the jerkers periodically reverse the direction in which they spin, nor why they are able to stop spinning when the necessity of performing a task requires them to stand still.)

Nineteenth Century biologists also thought the mice's "waltzing" was at least partly an inherited trait, and "mouse fanciers" considered the Japanese dancer a distinct breed of Asian origin. The physiologists Bernhard Rawitz and Élie de Cyon studied dancers, paying special attention to the gross anatomy of their ears, and claimed to have found that they have only two, rather than the normal three, semi-circular canals. Since the canals detect motion in three-dimensional space, Cyon wrote in his L'oreille [Paris: Alcan, 1911] that Japanese dancers cannot perceive the third dimension (except visually -- he said it gives them vertigo, because they cannot understand it!) and came up with rather vague arguments relating the mice's behavioural peculiarities to their two-dimensional world-view. These findings -- even the alleged absence of a third canal -- were not universally accepted, and controversy raged among mouse experts in the early 1900s. Robert Yerkes, one of the founding fathers of Behaviourism (later also famous or notorious as a primatologist and an advocate of eugenics) devoted his first book-length publication to the matter: The Dancing Mouse: A Study in Animal Behavior [New York: Macmillan, 1907]. Then, as so often happens, the subject fell into obscurity for decades; it re-emerged briefly in the 1940s when raising laboratory mice had become an industry, and again quite recently when the connexion to espin production was suggested.

To Twentieth and Twenty-first Century observers, dancing mice appear to be sick, the victims of a genetic disease which produces a kind of tic or chorea. Even the now-preferred name -- "jerker" -- suggests an unhealthy animal undergoing spasms. In the Long Nineteenth-Century, an age when formal ritualistic behaviour was socially approved, they just seemed to be "waltzing". Cyon wrote of "the voluntary character of their dancing movements ... these movements are by no means forced movements, but rather the mice execute them with a certain pleasure." Even Cyon's opponents seem to have generally accepted that dancing was more of an eccentricity than an undesirable or maladaptive trait.

For Poincaré, the dancing mouse proved that space dimensionality is a mere convention. The mouse functions successfully in the world, but does not believe in the existence of a vertical direction obvious to all other animals. Still ... and here Poincaré's conservative persona comes to the fore.

Still ... dancing in the mousey sense may not be harmful, but who would want to spin around all the time? Surely if there were an advantage to the two-dimensional point-of-view, all mice would adopt it, not just those of (in Yerkes's characteristic terminology) a single "race"? Does this not suggest that, of the many collections of algorithms producing the same output from given input data, some are more efficient than others? (Any modern computer-scientist will instantly nod in agreement.) Therefore, might there not be a sense in which the three-dimensional view is better than any other?

Having begun thus to retreat, Poincaré continues with evident relief. Space may not be "out there", but at least the world is. The "threats" are real; input data received by the senses comes from sources which have objective existence. Moreover, this objective world is organised in a way more nearly resembling three-dimensional (and presumably Euclidean) geometry than any other.

While the recovery of a real world outside the self is undoubtedly necessary both for doing science and for mental health, the insistence that the real world must be tridimensional is a bit of a surprise coming from Poincaré. Not only one of the founders of special relativity but also an expert on four-dimensional geometry in the abstract, he here rejects the fourth dimension as a practically useful feature for science's description of the universe. "The translation [into 4D language] would always be less than the text, and ... would always have the air of a translation." Beings who "think in hyperspace" would be like jerkers only worse off, incapable of "defend[ing] themselves against the thousand dangers by which they would be assailed."