Elucidating Ontological Intuitions
Friday 22 March 2024
In newsletter 266, and again in newsletter 280 last week, I formulated a number of thought experiments about the history of formal thought, which were essentially variations on the theme of known history, but do any of these variations have any substantive bearing on formal thought? If we vary the historical order in which branches of mathematics appeared, or when important individual ideas appeared, does that ultimately make any substantive difference? This question needs to be cleaned up; we need to make a distinction between whether the order of discovery makes a difference to formal thought itself, or whether the order of discovery makes a difference to us, to our historical development, to civilization.
Hans Reichenbach made a distinction between the order of discovery and the order of justification. The history of Greek geometry prior to Euclid gives us the order of discovery; Euclid gives us the canonical order or justification, or, if you prefer, the order of exposition. As new discoveries are always being made, and new ways of derivation are being worked out, the order of discovery and the order of justification are parallel histories that interact over historical time, but which are not identical. The appearance of constructivism (of many varieties) in the twentieth century entailed a distinctive order of exposition, and one which pruned away all discoveries that could not be justified by constructive methods. Thus novel orders of justification are possible with the formulation of novel methods of derivation, and a canonical order of justification as in Euclid has the weight of tradition in its favor, but is not necessarily an optimal order, and indeed refinements to our deductive apparatus may well show us better orders of justification.
It doesn’t end there. In addition to the orders of discovery and justification, there are other histories and weave in and out of these histories, as, for example, the order of practical application of some ideas from the formal sciences, which may lie unused for centuries, or be taken up immediately. This was an important theme of the previous newsletter: a discovery might be made, but be of little immediate utility. However, we can make one broad generalization, and that is that the scientific revolution began the process of formal ideas bearing immediately upon scientific discovery (mathematics is the language and unifies the sciences) and the industrial revolution began (or, at least, greatly accelerated) the process of bringing formal ideas immediately to bear upon technological innovation and their engineering application.
From this macroscopic perspective, the formal sciences could appear in any number of variant orders of discovery, as long as the necessary concepts are available for the scientific revolution, and then for the industrial revolution. Thus another tranche of thought experiments could be formulated on the basis of the scientific revolution appearing earlier, or appearing in the context of some other civilization, and similarly with the industrial revolution. This is probably a more familiar theme than my counterfactual thought experiments in the formal sciences. In Cosmos, Carl Sagan imagined a Greek civilization that continued to innovate from classical antiquity and never experienced a dark age to retard its progress. One of my most viewed answers on Quora is Did any ancient civilizations come close to industrializing?
A complex historical event like the industrial revolution involves a great many boundary conditions, and probably many more boundary conditions that the appearance of an innovation of formal thought, so we need to see these boundary conditions as sequential, or nested, because we couldn’t have the kind of counterfactual Greek civilization that Carl Sagan imagined unless that Greek civilization had available to it the intellectual resources that make science, technology, and industry possible. And even if the boundary conditions are met, there’s no guarantee that the elements will come together to constitute the boundary conditions for the next development.
The resources of our formal thought may or may not be used to build a technological civilization, but none of this gets us closer to whether these counterfactual thought experiments in the formal sciences have any substantive relevance to formal thought. We have to take a different tack.
One distinction that comes up constantly in philosophy of history is that between the use of “history” to refer to past actuality or to refer to a body of knowledge. This is alleged to be ambiguous, but, as I have pointed out many times, it is an ambiguity that potentially dogs all the sciences, but the other sciences don’t seem to labor under this ambiguity as a particular burden. But we can gain some insight into our ontological intuitions when we apply this distinction to the formal sciences: “mathematics” can apply to mathematical actuality, or to a body of knowledge. If the former, we are dealing with Platonism, and it is understood that the body of knowledge we call mathematics is a discovery of Platonic truths independent of the body of knowledge. Here the ontological priority is clear: Platonic entities precede their formalization as knowledge. If the latter, the question immediately becomes that of the status of the body of knowledge, and here there are many different positions that one might hold. There are varieties of anti-realism, nominalism, fictionalism, and some forms of constructivism that hold mathematics is “a subject with no object” (as in the title of the book by John P. Burgess and Gideon Rosen).
If one holds that mathematics (and the other formal sciences) is just a body of knowledge (with the implication that it is a “mere” human construction, not inherent in the nature of things) that corresponds to no ontological domain, then we can say that my counterfactual thought experiments in the formal sciences are substantively relevant to these formal sciences. In other words, given fictionalism or some equally strong form of nominalism, there is no ontological order of concepts that could be distinct from an order of discovery or an order of justification. These disciplines are nothing but their human discovery and human exposition. If, on the other hand, one has strong Platonic intuitions and insists that the truths of mathematics are revelatory of intrinsic properties of the world, then these intrinsic properties might be reflected in the order of discovery or the order of justification… or they might not. My question about the substantive relevance of counterfactual thought experiments is only a live option given the Platonic scenario, so here I have already betrayed by intuitions.
Is there an ontological order of the formal sciences that ought to be reflected by human knowledge of this ontological order? Given the distinction between order of discovery and order of justification, there are at least four permutations for their relationship to an independent ontological order:
- neither discovery or justification do, or neither should, reflect the underlying ontological order,
- discovery reflects but justification does not,
- justification reflects but discovery does not,
- both discovery and justification do in fact reflect (or should reflect) the underlying ontological order.
Given the ambiguity of these formulations, it would be better to further break this down into eight permutations, distinguishing between relations that do obtain in fact and relations that ought to obtain. It is possible that the order of justification reflects some deep ontological structure of formal truth, but the order of discover is more-or-less accidental. That, at least, is my intuition, but I can’t rule out the possibility that the order of discovery reflects this deep ontological structure, and the order of justification is a concession to the limited epistemic horizons of human beings.
Bertrand Russell often emphasized that what we come to know first is not necessarily that which is logically simple, and it often requires considerable effort to converge upon the logically simple, from which all our non-simple intuitions can be derived. This is one way to account for the distinction between the order of discovery and the order of justification. It is also a tacit distinction between appearance and reality, and, as I have noted in many contexts, the distinction between appearance and reality is the fundamental metaphysical idea. For Russell, intuitions of logical simplicity are mere appearance, and the deeper reality is the absolutely logical simple that is determined as the indispensable assumption necessary for the derivation of our knowledge. There is also a circularity here: we are assuming a certain body of knowledge is normative, and the correct logical simples are those which allow us to derive the normative body of knowledge most economically. There is also, then, an assumption of parsimony: that the simplest derivation is to be preferred, as though we had some a priori assurance that the ontological architecture of formal truth is simple.
My intuition that justification reflects ontological structure, but discovery is an accident of history, is of a piece with Russell’s assumptions that intuitions are mere appearance, and absolute logical simples are the underlying reality. If initial intuitions grasp what is easiest to comprehend of underlying ontological reality, there might be any number of such easily grasped ideas (two obvious classes would be arithmetical ideas and geometrical ideas), and any number of circumstances of place, time, and culture that would make some intuitions easier to grasp than others. Thus the first formal ideas one grasps are accidents of history. But as we deepen our knowledge of the formal realm and respond to the desire to unify this knowledge in a way that makes it hang together systematically, we will probably find that the ideas we grasped first are not the most fundamental ideas, so the order of justification upon which we ultimately converge will diverge from our initial intuitions and will come to correspond to the ontological structure, and not that which one merely happened to first grasp.
We can distinguish the degree to which an idea is accidental (in the sense above). It is an accident of history that we find base ten counting to be most intuitive, and this is an accident that traces back to biological origins — back, in fact, to the first five-digited bilateral human ancestors that crawled onto the land hundreds of millions of years ago. Thus the “idea” of base ten numeration goes deep into our history. On the other hand, the thought experiment that I suggested last week, that arithmetic might have been formulated axiomatically before geometry was so formulated, doesn’t seem to have such deep roots. There may be some deep time biological reason for our early axiomatization of geometry, but I haven’t seen an argument for this, and I have no argument of my own. For something like this, that goes back about two thousand years, as opposed to base ten numeration, which goes back hundreds of millions of years, I assume that there is a great deal more freedom in how the order of discovery manifests itself in history.
Though my intuitions are in part consonant with Russell’s, I have to admit that my intuitions are not univocal on this point. I had an experience some years ago that sounds utterly trivial, but it really struck me at the time. Having read a lot of philosophy of logic and philosophy of mathematics, and so having a reasonably clear idea of where my views were to be located within the defined positions that are routinely argued (tending toward Platonism in my case), I happened to see a news report on a crop circle in the shape of a Mandelbrot set. I kid you not, my heart sunk in my chest at that instant because I immediately saw that aliens drawing a Mandelbrot set in someone’s wheat field was a very strong expression of Platonism, but my instinct was that the Mandelbrot set was a human artifact, so for the crop circle version of a Mandelbrot set to show up after the Mandelbrot set had been discovered by a human mathematician showed that those making crop circles were simply picking up on bits of mathematics available in popular culture.
I would like to emphasize to you that I have zero interest in crop circles or UFOs or anything like that; the instinctive response I felt was no expression of disappointment that space aliens aren’t real. I have no stake whatsoever in this pop culture debate; in fact, I find the whole thing distasteful. It was the fact that my instinct was to assume idea diffusion rather than independent discovery as the basis of the image that struck me. I wasn’t the Platonist that I thought I was; I was deceived about my own intuitions. Can there be a greater shame for a philosopher than to experience self-deception in regard to one’s intellectual intuitions?
There are, of course, many ways to analyze the Mandelbrot crop circle and my response to it. If one doesn’t mind flaunting Ockham’s razor, one can posit space aliens under a strict epistemic embargo regime according to which they cannot use any symbol to communicate with human beings before it is already available in channels of human communication.
