More Counterfactual Histories of Formal Thought
Friday 15 March 2024
In newsletter 266 I formulated a thought experiment in the history of the formal sciences, asking in the context of the simultaneous discovery of non-Euclidean geometry by at least four mathematicians, whether it would have been possible for those using an axiomatic method to have discovered elliptic geometry and for those using a metrical method to have discovered hyperbolic geometry. This poses the question of whether there is anything intrinsic in these methods of research that suggest the outcomes that occurred, or whether this merely an historical accident that had nothing to do with methodology. Another thought experiment I have suggested is swapping the positions of Darwin and Euclid in history, which is a symbolic way to suggest the counterfactual history of a biology of natural selection in classical antiquity, with advanced mathematics not appearing until the nineteenth century.
Another thought experiment in the history of the formal sciences that combines the above two exercises would be another symbolic swap, so that non-Euclidean geometry appeared in antiquity, or even that geometry originated as non-Euclidean geometry instead of as Euclidean geometry. Given that non-Euclidean geometry appeared independently several times at about the same point in history implies that the time was “ripe” for this development. The same can be said of the theory of diminishing marginal utility in economics, which, like non-Euclidean geometry, appeared in the late nineteenth century independently at least four times — in the work of Jevons, Walras, Menger, and Clark. We now call this the “marginal revolution.”
When a number of closely related discoveries occur at about the same time, as I noted above, history seems to be ripe for the discovery. This, in turn, implies a thesis in the philosophy of history, and this thesis (I imagine) would be that the boundary conditions for the discovery in question had been met, so the discovery became more-or-less inevitable, and we only need to wait and watch for it to appear (retrospectively, as historians, since we can’t foresee a scientific discovery before it happens). However, if an idea appears in the work of only a single thinker, then presumably the boundary conditions for this have also been met, but the discovery was only made by one man. There are several ways to account for this. One could argue that an individual working on a research program a bit out of the mainstream (like Einstein in 1905 or Gödel in 1931) will themselves construct some of the missing boundary conditions themselves, with their earlier work in constructing the boundary conditions being the condition of the possibility of their later work. Another obvious explanation would be that the crucial intuitive leap necessary to the discovery is simply more difficult in some cases, so that leap is made by one man, while the intuitive leap in another case is far less vertiginous, and so is made by a number of individuals more rapidly than it can be communicated and so becomes common knowledge.
This latter observation points to the social conditions of a scientific discovery, which also play a role. In the premodern world, a discovery might be made within one civilization, and circulated only with that civilization, but not be communicated to any other geographically isolated civilization. Since the advent of planetary scale communications and commerce networks, a discovery is far less likely to remain within a single civilization. Also, the organization of educational institutions and the publication of scientific journals contributes to the rapid dissemination of an idea. Prior to regularized institutions and publishing, an isolated scholar might make a great discovery, and it could lie hidden in his papers for decades or even centuries before it is discovered, and in all that time someone else might discover the idea under changed conditions and have it rapidly disseminated even while an earlier discovery remains unknown. Gregor Mendel is a rare example of lost work that got credited ex post facto when others who later duplicated his work unaware of his priority found his work in a scientific literature search.
The convergence of a society upon the intellectual boundary conditions that make a major scientific discovery possible points to a network of ideas that exist in a relationship of mutual implication, or perhaps partial mutual implication. Once part of the puzzle is before us, figuring out the rest of the puzzle is a matter of systematically working through the implications of the boundary conditions. This is particularly the case for ideas that are highly dependent on their context. While giving the details would be difficult, I think the case can be made the some ideas are reasonably simple and could be grasped by any intelligent individual if explained clearly, and this simplicity of an idea makes it, to a certain extent, context-independent. (This is a naturalistic way of formulating the view of knowledge in Plato’s Meno, in which a slave boy is led to seeing for himself how to double the area of a square, which involves an irrational.) Other ideas are far more context-dependent, so that explaining them requires the explanation of a large amount of subsidiary results. Another way to formulate this would be to say that some ideas have a quantifiably greater number of boundary conditions, and some ideas have far fewer boundary conditions. The amazing thing is that an idea with few boundary conditions can remain undiscovered for thousands of years simply because no one thought of it.
I think this is part of the intuitive appeal of my thought experiment of swapping Darwin and Euclid, though the two, given the above discussion, are asymmetrical cases. The idea of natural selection is, I maintain, a reasonably simple idea that can be explained to any intelligent individual. Because of this simplicity, it could have been introduced in classical antiquity, but no one thought of it. One could argue that the whole of ancient thought was counter to natural selection, but I could just as well argue that the whole of nineteenth century thought was no less teleological and design-centered. Darwin’s intuitive leap in conceptualizing natural selection, I argue, was equally possible in antiquity. However, in the case of natural selection, Alfred Russel Wallace simultaneously discovered the idea, so this points to something else taking place in the intellectual context that made the idea of natural selection conceivable in a way that it was not earlier conceivable. Euclid’s work, on the other hand, is far more context dependent. Euclid was a systematizer, and he gathered together the results of geometry up to his time, and supplied then with what Kant called architectonic. There were any number of other ancient mathematicians who might have done this if Euclid had not done it. Whether Euclid has a special place in framing the formal concepts of an axiom system, or whether this was also present earlier in antiquity, is not known to me.
I think that the ideas of non-Euclidean geometry are sufficiently simple that many of them could have been introduced in classical antiquity, and, if they had, this would have had a significant impact on the development of the Western intellectual tradition, as Euclid long served as a model and a exemplar of rational thought. But one of the reasons that Gauss didn’t publish any of his results on non-Euclidean geometry was its apparently radical departure from all mathematics heretofore, and we can easily imagine discoveries being ignored or forgotten or neglected just because they constitute too radical of a departure from what everyone “knows” to be true. There is an implicit limitation on the expansion of knowledge that follows from unstated condition of proximity and similarity to current practice. Darwin faced exactly this problem.
It is possible to formulate a counterfactual history in which, not long after Euclid completed the Elements, other ancient geometers, bothered by the parallel postulate (which proved to be a bother for about two thousand years), introduce alternative axiomatizations, thereby introducing non-Euclidean geometry in antiquity. However, Riemann’s metrical method presupposes analytic geometry, and analytic geometry presupposes algebra, so we can see how the subsequent centuries of mathematical development slowly built up the boundary conditions that made Riemann’s work possible.
In the discussion of the history of philosophy after Plato and Aristotle, many historians have observed that the subsequent period saw fewer fundamental philosophical ideas introduced, concluding that antiquity was entering a more stagnant intellectual period — a stagnation that gave way to the Dark Ages, in which knowledge was actually lost, rather than merely not being increased, as is implied by a stagnant period. This period of late-classical intellectual stagnation also corresponds to when Euclid’s Elements was written, so if this is a valid historical observation about stagnation, then it would make sense that Euclid’s work would be treated as canonical (or, if you prefer, treated as a kind of orthodoxy), and no one under these conditions would have thought of fundamentally altering Euclid by changing his axioms or postulates. Others were to extend and build upon Euclid, as with Apollonius of Perga’s work on conic sections, but it was not subjected to the kind critique that could have resulted in the discovery of non-Euclidean geometry in antiquity.
The ancient work on conic sections is also of interest because it was done as a purely theoretical exercise, for the love of mathematics, and without any thought of utility. After more than a thousand years elapsed, it was discovered that the trajectory of a cannon ball can be described by a conic section, and suddenly the work of Apollonius was relevant to military science. One might say this is much less common today, i.e., for pure mathematical work to be without practical application, but while this may be less common, it is not unknown. Starting with Cantor in the nineteenth century, set theory and transfinite numbers have been developed in a similarly purely theoretical vein, and while set theory has proved to have a multiplicity of applications, there have been virtually no applications for transfinite number theory. Now, a great deal of foundational work on mathematics derived from the same intellectual milieu — not least Gödel’s limitative theorems — so one could plausibly argue that the theory of transfinite numbers entailed research into the foundations of mathematics, which in turn had enormous practical consequences.
Is set theory sufficiently simple that it could have been introduced much earlier in history — in classical antiquity, for example, as in the other thought experiments I have suggested — or does it involve concepts that we take for granted but which are the result of a long period of development? Supposed we perform another thought experiment based on another symbolic swap — Euclid for Cantor — so we get transfinite numbers in antiquity, but we don’t get systematic geometry until the nineteenth century. In this thought experiment, the theory of transfinite numbers might be built up as an elaborate edifice, like conic sections, beautiful and fascinating for mathematicians, but with little utility other than as an intellectual exercise. Whereas if civilization were deprived of Euclidean geometry until much later (like the nineteenth century), there is so much of practical utility that would not have been possible that civilization could have not developed as it did in the absence of Euclid (again, I am using Euclid here as a symbol representing an architectonically organized geometry, including the ideas of a formal system).
Here’s another thought experiment: suppose that, instead of the idea of a formal system being introduced with geometry, it was introduced instead with arithmetic (the application of formal systems to arithmetic didn’t occur until the work of Peano and his successors; this is a large part of the story of twentieth century foundations of mathematics research). Here the obvious stumbling block is the absence of Hindu-Arabic numerals in classical antiquity, which are a boundary condition for any degree of sophistication in arithmetic. Yet another thought experiment, then, would be the introduction of Hindu-Arabic numerals in classical antiquity. This could have happened (if anyone had thought of it) independently of the architectonic organization of arithmetic as a formal system, so that an axiomatic arithmetic in antiquity would first require Hindu-Arabic numbers and then the desire to organize the knowledge of arithmetic into a formal system.
Here the questions I would ask are similar to the questions I asked in newsletter 266 about the different methods of arriving at non-Euclidean geometry: is there anything intrinsic to geometry that particularly lends itself to axiomatic exposition, and is there anything intrinsic to arithmetic that particular lends itself to an intuitive exposition and not an axiomatic one? Frege at the beginning of Foundations of Arithmetic argues that geometry was axiomatized because the Greeks reasoned more rigorously, and arithmetic had not been subject to this intellectual milieu in its introduction.
Frege’s innovation of quantification in logic is another idea of which we could ask if it is sufficiently simple that it could have been introduced earlier in history, but was not. The ancient logical tradition was reasonably sophisticated, so it seems like it could have been possible to explain quantification to someone who had studied ancient logic, but this is probably a misleading way to think about it. Aristotle’s logic was a logic of classes. A quantifiable predicate calculus presupposes a range of ideas that were foreign of classical antiquity. The terminist logic of late Scholasticism — again, another sophisticated logical milieu — maybe possessed more of the boundary conditions necessary to make sense of quantification, but even here I would hesitate. Both set theory and quantification theory seem to require the concept of a variable, which comes to us from algebra, and so presupposes a long historical development and contact among civilizations, since it was Islamic mathematicians who developed algebra. We take variables as a matter of course, and view them as being as intuitive as the concept of zero, but zero was another achievement in mathematical knowledge, an intuitive breakthrough, which is a presupposition of most advanced mathematics, which cannot make do without zero.
I can imagine a reader shouting into my ear that the real difference would have been the introduction of calculus in classical antiquity, as this would have facilitated advanced engineering only possible with the calculus. But calculus, even more than Riemann’s metrical method for arriving at non-Euclidean geometry, has pretty substantial boundary conditions. It is another classic case of simultaneous discovery, and I would argue that the time was ripe for calculus when discovered by Newton and Leibniz, but it is definitely not a largely context-independent idea like natural selection or set theory that could be lifted out of one intellectual milieu and inserted into another. Simple intuitions can be very powerful, and they can go a long way, but they require some formal framework within which to work out their implications.
One more thought experiment before I take my leave: again in newsletter 266 I quoted an aphorism of mine, “There is a fundamental intellectual difference between those who get their rigor primarily from mathematics, and those who get their rigor primarily from logic.” Both logic and mathematics were developed in antiquity, but mathematics was the dominant tradition of formal thought, with a far greater number of practitioners than logic. Suppose the roles were reversed: suppose that logic had been the dominant tradition in antiquity, with a great many more practitioners, and it continued to dominate Western intellectual history thereafter, while mathematics was the subordinate tradition, with far fewer practitioners. Given the conditions of this thought experiment, Western civilization would be changed from one in which rigorous thought was primarily mathematical to one in which rigorous thought was primarily logical. I think it is fair to say that the outcome would be unrecognizable, as the tradition of civilization would be radically altered by this change in the primary emphasis in formal thought.
