A Natural History of Intuition
Friday 16 August 2024
In last week’s newsletter I argued that all conceptual frameworks are kluges, and that I had come to this conclusion after reading Weyl’s The Open World. I don’t think I made it clear that the character of Weyl’s lectures that brought me to this point of view was the admixture of ideas that he brought to his analysis of the continuum and the infinite. I did point out that the obvious alternative would be to go all-in on a new approach and sweep all past ideas aside. Weyl didn’t do this. He drew on a wide range of influences, as though to synthesize and rationalize them. If so, it is an incomplete synthesis and an incomplete rationalization, hence a kluge.
One problem that has occupied me for many years is being able to express the nature of intellectual intuition and its relation to more formalized thought that exists at a distance from intuition, and this was one of the problems I was struggling with reading Weyl. We make use of both intuition and formalization in our attempts at scientific reasoning. We know that even to begin reasoning, to make a start at science, we must make certain assumptions. There is probably a fundamental temperamental difference between those who view the assumptions necessary to begin reasoning as a learning of the rules of the game, and those who view these assumptions as intuitions that are imperfectly captured by a given formalism, but subject to improvement, with the ultimate goal being the convergence upon a formally perfect capture of intuition.
The same division of attitudes can be found in distinct approaches to empirical science, viz. whether it’s just about getting the predictions of observable phenomena right, or whether it’s really about some fundamental process of the actual world that is imperfectly captured by a theory. This dichotomy was expressed in a quote attributed to Enrico Fermi in The Pope of Physics: Enrico Fermi and the Birth of the Atomic Age by Gino Segrè. The context was a conversation with Dyson, which dashed Dyson’s hopes but also saved him time that would have otherwise have been spent on a blind alley:
“There are two ways of doing calculations in theoretical physics. One way, and this is the way I prefer, is to have a clear physical picture of the process you are calculating. The other way is to have a precise and self-consistent mathematical formalism. You have neither.”
I have run into this quote in more than one source, and I may well have quoted it in an earlier newsletter. The distinction Fermi made in this quote isn’t precisely equivalent to the distinction I was making, which was more like Machian phenomenalism vs. intuitive understanding, but having a clear physical picture of what one is calculating is one way of having an intuitive understanding, while a precise and self-consistent mathematical formalism is what is needed for making precise and consistent predictions.
The quantum revolution in physics has dealt a body blow to intuition, because there are no intuitive models of quantum theory; there is no clear physical picture of quantum theory, which leaves us only with Fermi’s other alternative: a precise and self-consistent mathematical formalism. I believe this to be a problem, though I can easily imagine that there are those among us who are content with accurate predictions without any intuitive basis. I see it as a failure of intuition and imagination. Of course, we have no a priori guarantee that the world will be intelligible to the human mind, although this is a presupposition of most science, but even when the world has proved recalcitrant to yield up its secrets in the past, eventually human persistence overcame the obstacle. If science from here on out follows the quantum model, this will constitute a discontinuity in the history of human thought: up the quantum revolution, scientific knowledge reflected intuitive norms, but after the quantum revolution it ceased to do so.
I suspect that we will, at some time, return to a more intuitive science, but not necessarily any time soon. The key is that intuition is not fixed, but is adaptable. The way we talk about intuition in formal thought often implies that intuition is based on access to a Platonic realm of the eternally true, but formal thought is perhaps the best evidence we have of the flexibility of intuition. Hindu-Arabic numerals, zero, the number line, and Playfair’s axiom were all were all the result countless generations of human thought converging on better intuitions (sometimes slowly, sometimes in a sudden insight). The human brain has not appreciably evolved since these concepts were introduced, but now that we have these concepts available to us, they can be taught to any intelligent child, who is then capable of entertaining mathematical concepts that would have challenged mathematicians who never knew them. Of course, the same is true of language: no individual would be able to create any of the modern languages in isolation, but with the tool of language at our disposal, we can conceptualize that which would be intellectually inaccessible to us without language.
Probably it is human neuroplasticity that gives us our adaptable intuition. While we are stuck with the size of our brain, the large-scale structure of the central nervous system, and our natural sensory endowment, the brain can reconfigure itself over time. This is obvious when we learn to ride a bike or to juggle — initially difficult tasks which eventually come naturally (one might even say intuitively) to us. There’s a reason it is often said that you never forget how to ride a bike, even though it is an acquired skill. I have no doubt that there are bizarre mental conditions in which one forgets how to ride a bike or how to use zero.
The many books of Oliver Sacks take a keen delight in presenting us with unlikely and unusual neurological disorders that one would never have even guessed at before having read an account of them. That is one of the charms of Sacks’ writing. But this is part of the larger swing of the pendulum toward idiographic science, which also includes the habit of anthropologists to cite virtually unknown societies possessing some unlikely social convention, as a scientific “gotcha” moment that is supposed to confound those with the temerity to form general concepts. This swing toward idiographic science (previously discussed in newsletter 287) has occurred at the same time as the quantum revolution, and it makes me wonder whether the two are connected in some way.
Intuitions shade off at their lower end into instincts, and at their upper end into concepts. Instincts themselves are sometimes the function of the autonomous nervous system, and reactions to stimuli that involve the autonomous nervous system don’t even involve the brain: it would take too much time, and probably too much processing power, to judge and respond to every stimulus that is a potential threat to life, so the spinal column responds before the brain even knows what happened; awareness lags considerably behind such autonomous functions. Darwin described this in The Expression of the Emotions in Man and Animals:
“I put my face close to the thick glass-plate in front of a puff-adder in the Zoological Gardens, with the firm determination of not starting back if the snake struck at me; but, as soon as the blow was struck, my resolution went for nothing, and I jumped a yard or two backwards with astonishing rapidity. My will and reason were powerless against the imagination of a danger which had never been experienced.”
Today we know why this is the case, and we can explain the anatomy behind this reaction in great detail. But not all instincts are responses of the autonomous nervous system. Those instincts mediated by the brain, and not confined to the spinal column, are subject to the complexity of the human brain and the adaptability of neuroplasticity. Some are little within our control, but others can be altered, managed, and repurposed. As instincts depart ever so gradually from the immediate needs of survival, and are less subject to the harsh dictates of natural selection, where to fail is to die, they become biologically-based intuitions, deeply embedded in out evolutionary psychology, to be sure, but not beyond revision or replacement.
At their upper end, as I noted previously, intuitions fade into concepts. Concepts are explicitly formulated and known to be concepts. They are, in principle, entirely subject to human cognition, but since they are an outgrowth of a long evolutionary continuum, some concepts feel more natural that others, and some are rooted more deeply in our biology and our evolutionary psychology than others. A concept is an abstraction, and sometimes this is obvious to us, while at other times we can forget that we are employing an abstraction when we are employing a concept. Some abstractions are so perfect that it requires a conscious act of will to realize that we are thinking conceptually when we use them. The natural numbers, especially when represented by Hindu-Arabic numerals, are like this. The natural numbers are supremely abstract but, at the same time, perfectly intuitive. Once you learn to count with the natural numbers, it would be difficult to count in any other way. I call such concepts intuitive formalisms — a formally perfect capture of intuition.
For much of Western history, geometry was also treated as an intuitive formalism, and it had the additional benefit of having been formalized in an axiomatic system, which gave it special credibility as a representation of pure rationality. Thus the discovery of non-Euclidean geometry was an intellectual crisis on a level with the crisis in the foundations of mathematics in the early twentieth century. It was a body blow to intuition, something like the body blow to intuition delivered by quantum theory. However, as mathematicians began to think about non-Euclidean geometry, we learned that we could form intuitive concepts about different geometries, and we started to see space in new ways, from many perspectives, some of which reflected our intuitions about plane geometry captured by Euclidean geometry, but some of which were startlingly novel. Our total inventory of intuitions expanded, and human thought became more comprehensive and more flexible. Also, we became more critical and analytical, having learned that our intuitions about the nature of space enshrined in Euclidean geometry were not the final word on the structure of space.
With Weyl’s exposition of the infinite in The Open World, I found a more comprehensive conception of the infinite than would have been found before Cantor or Brouwer, with both cultivated formal intuitions and more demanding formal concepts that couldn’t quite be called intuitive. But there was also the retention of old intuitions with new intuitions imposed upon them. There are, I think, at least two ways that a conceptual framework can grow and be absolutely larger than it was in the past. We can grow our formal conceptual framework as we have grown the number system, adding rationals, reals, transcendentals, imaginary numbers, and so on. Here the concept of number itself is subject to expansion and more detailed articulation.
We can also expand our conceptual framework by retaining past conceptions alongside contemporary conceptions, as when we continue to number chapters with Roman numerals when we have Hindu-Arabic numbers available to us. Experts in ancient languages might be able to count in Cuneiform: numbers were among the first symbols of ancient languages to be deciphered. Of course, we can do both at the same time if we have the mental capacity to master both the entire history of a discipline and its most complex contemporary concepts.
The concessions we make to older intuitions is probably due to their relative proximity to biological instincts and evolutionary psychology, but, as we have seen, this isn’t in any sense absolute. It may well be that there is considerable individual variation, so that some individuals are quite captive to intuitions with a biological basis, while others have such a degree of neuroplasticity that they can continually revise their intuitions and maintain only the most tenuous links to biologically-based cognition. This begs the question of whether we might, someday, employ pharmacological, surgical, or DNA therapies to enhance our neuroplasticity and further free up our intuitions to adapt to contemporary formal or scientific concepts. This in turn begs the further question of the limits to such interventions, which could risk so removing us from our biology that we could no longer survive. It might be like the final scene in Forbidden Planet when the doctor (if memory serves) hooks himself up to the alien machine and briefly experiences cognitive enhancement before dying.
I still haven’t adequately described the point I wanted to make about conceptual frameworks being kluges as a result of retaining former intuitions, now somewhat dated, alongside later intuitions. These newsletters are also always a kluge, as I struggle to express problems that I am currently working on, rather than offering an exposition of matters that have been fully thought through. Old ideas exist cheek-by-jowl with elusive, not-yet-clarified ideas, and the result is probably jarring and confusing by turns. Such is the nature of philosophical research.
