A Thought Experiment in the Development of the Formal Sciences
Friday 08 December 2023
Last Monday I stayed up all night writing, working on several different projects, among them some remarks on formalism from 1998 that I had disinterred exactly a year ago and was turning into a paper on formalism in December 2022. I didn’t go to bed until after 7:00 am, but that’s okay as my head was quite full of ideas, and I could have just kept going. There is probably nothing else that gives me as much pleasure as a thought experiment, and hitting upon a new thought experiment is always a source of enjoyment, especially when new aspects of the thought experiment unfold the more one thinks about it. During my sleepless night an odd counterfactual occurred to me, which suggested a thought experiment.
Non-Euclidean geometry was probably independently formulated by at least four different men, first Gauss (in about 1813), who said nothing of his discovery, and then, almost at the same time, János Bolyai (a Hungarian living in what is now Romania, who published a pamphlet in 1832), Lobachevsky (a Russian in Kazan, who published his findings as course notes in 1829–1830), and later, in a different form, Bernard Riemann (a German and a student of Gauss, communicating his findings in a lecture in 1854). Simultaneous scientific discoveries are not unknown; Leibniz and Newton independently discovered the calculus, probably Newton first, but Leibniz published first, and this led to a long conflict over priority that eventually came to have nationalistic overtones. In any case, Bolyai and Lobachevsky came to non-Euclidean geometry through the axiomatic method, and both formulated a version of hyperbolic geometry, in which there are an infinitude of parallels and space has a negative curvature. Riemann came to non-Euclidean geometry through a metrical method, and he formulated a version of elliptic geometry, in which there are no parallels, and space has a positive curvature. Of course, the history of non-Euclidean geometry is more complicated than this, but the above gets some of the essentials, though it is a long way from exhausting the details — and the details often matter.
The thought experiment, then, is this: could Bolyai or Lobachevsky have arrived at elliptic geometry by their methods as readily as they arrived at hyperbolic geometry, or could Riemann have arrived at hyperbolic geometry as ready as his methods led him to elliptic geometry? Asked another way, is there anything intrinsic to the axiomatic method that suggests hyperbolic rather than elliptic geometry? Is there anything intrinsic to Riemann’s metrical method that suggests elliptic rather than hyperbolic geometry? Is it just an historical coincidence that Bolyai and Lobachevsky both arrived at hyperbolic geometry by way of axiomatic reasoning, and that Riemann arrived at elliptic geometry by following through ideas about measuring space? The axiomatic method exemplifies a kind of formal rigor that is closer to logic, while Riemann’s metrical method is closer to arithmetical thinking. Summer before last I wrote, “There is a fundamental intellectual difference between those who get their rigor primarily from mathematics, and those who get their rigor primarily from logic.” Part of the interest of this thought experiment for me is the light that is sheds on these different modes of thought.
The axiom of parallels had been a sore spot in mathematics even since Euclid formulated it. Few have considered it self-evident, which axioms are (or were) supposed to be, and now that axioms are no longer considered self-evident, but rather the necessary assumptions to get started with mathematical reasoning (i.e., hypothetico-deductivism), today we would say that Euclid’s axiom doesn’t possess the requisite appeal to (geometrical) intuition that would make it a good axiom. Such objections were partially overcome when Euclid’s clunky formulation of the parallel postulate was replaced by Playfair’s axiom, which makes the parallel postulate seem much more intuitive: given a line, and a point outside the line, only one line can be drawn parallel to the line through the point. In hyperbolic geometry an infinitude of lines can be drawn parallel through the point, and in elliptical geometry no lines can be drawn parallel through the point. In an axiomatic context, the procedure is highly formal, and it seems just as likely that one could swap out Euclid’s parallel axiom with an axiom consistent with hyperbolic space or with an axiom consistent with elliptical space. Probably because it was all so unfamiliar, none of this seemed obvious at the time. And, as with the substitution of Playfair’s axiom for Euclid’s parallel axiom, the more mathematicians thought about elliptical and hyperbolic spaces, the more intuitive examples they were able to find. Today, we can find these intuitive examples in any number of books on non-Euclidean geometry, but the hard journey to converging on those intuitions took decades, and, in some cases centuries.
Here is one intuitive way to think about elliptic, Euclidean, and hyperbolic surfaces: if you take a circle, cut a pie shape out of the circle, and then tape the circle back together without the pie shape, you get an elliptic surface — elliptic because it “falls short” of a flat plane (“elliptic” comes from the same root as “ellipsis”). And if you press this elliptic surface down onto a flat surface, something will crumple or break in order to make the elliptic surface conform to the flat surface. If you take another circle, cut from the edge to the center, and then tape in the pie-shaped piece you’ve cut out of the elliptic circle, you now have a hyperbolic surface — hyperbolic because it is in excess of a flat plane, and if you press the hyperbolic surface down onto a flat plane it will fold over itself in some region so that parts of the flat plane are doubly covered by the hyperbolic surface.
It is often pointed out the “geometry” gets its name from earth measurement, as it seems to have been derived from surveying. By the time we get to the great Greek mathematicians, geometry had become a “pure” science, and the Greeks pursued it from sheer love of theory rather than in an effort to become better surveyors. Yet the measurement of Earth, if conducted with sufficient care and precision, can demonstrate to us that we do not live on a perfectly (ideally) flat Euclidean plane. As we scale up our measurements and refine our precision, we eventually find that Earth is roughly spherical, and that a triangle inscribed from the pole down to the equator, then a quarter of the way around the equator, and then back up to the pole, is a triangle that has three right angles, which works fine on a sphere — indeed, we could even say that it is intuitive on the surface of a sphere — but makes no sense on a flat plane, where the angles of a triangle add up to 180 degrees. The surface of a sphere is elliptical (it is a surface of constant positive curvature, and no longitude lines are parallel; all intersect), so that a sufficiently precise measure of the Earth could lead us to elliptical geometry, even if we have not thought about geometry axiomatically.
Spheres form naturally due to the force of gravity. Perhaps in a differently constituted universe pseudospheres would form instead of spheres, and then measuring the pseudosphere upon which one’s species evolved one could, by a practical, metrical method, arrive at hyperbolic geometry. However, a planet in a shape of a pseudosphere seems rather unlikely, though I doubt it could be ruled out on the scale at which we consider the possibility of different physical laws for different universes. Certainly a pseudospherical planet seems unlikely in our universe. It has been argued that a torroidal planet is physically possible, even if unlikely (Anders Sandberg has written about this), but as strange as a torroidal planet would be, it would still be an elliptical surface, and measurement of its surface by beings who evolved on a torroidal planet would lead them to elliptical and not hyperbolic geometry. However — since I’m getting pretty deep into counterfactuals here — one might suspect that intelligent beings on a torroidal planet might have an incentive to develop topology earlier than topology was developed in terrestrial history.
So there seems to be an intuitive connection between elliptical geometry and the geometry of planets, and, given the planetary endemism of human beings, our minds may have evolved with some (limited) degree of elliptical geometrical intuition, although we have supposed our geometrical intuitions to be primarily Euclidean. We can put the equivocation down to appearance and reality: Earth’s surface has the appearance of a flat plane, but the reality of the surface of a sphere, and that reality has conditioned our responses on a subconscious level, so that our minds are adapted to an elliptic surface whether or not we know this on a conscious level. And, of course, we had already discovered in classical antiquity that we live on the surface of a sphere. Eratosthenes demonstrated this, but anyone who watched a ship disappear over the horizon would have known this too, even if they couldn’t describe it. Given our long knowledge of being resident on the surface of a sphere, why did elliptic geometry take so long to be made explicit? And why, given that this is the case, would the first non-Euclidean geometries have been hyperbolic? One can speculate that, by the time one comes to question Euclid’s axioms, it becomes a matter of simply throwing out the parallel axiom altogether, and, proceeding as if there were no axiom at all to govern parallels, that parallels would be unlimited. This seems rather weak, and I’m sure that someone who knows mathematics could come up with a better explanation.
In a thought experiment I have discussed on several occasions (e.g., Technological Civilization: Second Addendum to Part III), I have suggested the swapping of Euclid and Darwin, which is a symbolic way of suggesting that we might have had an advanced biology in classical antiquity, and not have had an advanced mathematics until the 19th century. This thought experiment again begs the question of why scientific thought developed in the way that it did. Is there an intrinsic reason that geometry developed before biology, or is this sequence of scientific knowledge an historical accident that might have been different? Here I think we are in firmer, if still speculative ground, to observe that evolutionary biology touches the human condition in a way that geometry does not, and it required another couple of thousand more years for Western civilization to get to the point at which this bridge could be crossed — but while I have weaving counterfactuals, I might as well also suggest a counterfactual classical antiquity in which human beings would have been ready to accept evolutionary biology.
But this leads us to further thought experiments: it would be historically lazy to suggest that, after ancient Greek mathematics, intellectual creativity ceased and so mathematical discovery more-or-less came to a halt. This is an oversimplification, but in some respects it is true. With the collapse of the Western Roman Empire, literacy fell off dramatically and abstract studies other than theology got short shrift. It is true that mathematics came to be widely studied in Islamic civilization, and many advances were made (including the invention of algebra), but this was a thousand years later. Another counterfactual would be the continuing development of advanced mathematics, continuous from classical antiquity.
How would intellectual history have differed if non-Euclidean geometries had been developed two thousand years earlier? Here we have to imagine not only the practical applications of greatly expanded mathematical knowledge — which, to be honest, are few in a non-technological civilization — but also the intellectual consequences of developing axiomatic thought and hypothetico-deductivism at this level much earlier in history than was the case in fact. It is possible to imagine this in a specifically Greek civilization had continued to flourish and expand, but this was a history that did not happen, and arguably could not have happened.
Both Euclid and Aristotle belonged to Hellenistic civilization rather than to the Golden Age of Greek civilization before the Peloponnesian War. In Spenglerian terms, Greek culture grew up to the point of the Peloponnesian War, and then, with decline setting in, Greek civilization appears, and this is the Hellenistic Age, dominated first by Alexander the Great and his successors, and then by Rome. Greek civilization, by this interpretation, had undergone a natural evolution, and the geometry that we got from the later efflorescence of Greek thought was all that Greek thought had to give. And, in Huizingan terms, the Hellenistic twilight of Greek thought was not the promise of a new age to come, but the afterglow and a fulfillment of a great age that had passed.
