This is really just me playing with ideas. Could schoolkid maths have got Dunne out of some of the corners he painted himself into, or at least have enabled a few more people to understand what he was getting at?
For his theory of Serial Time, first put forward in An Experiment with Time, J W Dunne tried hard to provide a sound mathematical basis in accordance with the physical theories of the day, especially with Einstein's newly-developed theory of relativity. But, sadly, he was no mathematician and the task was hopelessly beyond him. He had no idea what he was talking about, gleaning what little understanding he could from popular accounts; he did not even understand vectors or the differential calculus, let alone the vector calculus which underpins relativity. His constant appeals to relativity, with its imaginary coordinates of time and Lorenz-Fitzgerald contraction, were wholly naive and spurious.
Nevertheless his aims were laudable. Can anything be salvaged from the wreckage? Had he been familiar with even the simplest differential calculus, I suspect that he would have presented his ideas in a very different way, far more succinctly and in like measure far more clearly. The standard schoolkid notation I have used here would have meant nothing to him - but, if you are or once were a standard schoolkid struggling with your maths, maybe this is for you.
In modern physics, and specifically in relativity, the passage of time is represented as a distance along the time dimension t in some four-dimensional block spacetime. For example in passing from some moment t0 to a later moment t1, the elapsed time Δt = t1−t0.
But what about the rate at which time passes, the rate at which it changes? Mathematically we write the rate at which say x changes as dx/dt, the differential of x with respect to time. Asking about the rate of change of time is asking to evaluate dt/dt which is, trivially, 0. This result at least confirms that, in models of block spacetime, there is no passage of time; all of time is mapped out and the physicist may descend upon the map where and when they wish.
Yet we all experience the passage of time, and it is hard to make any sense of quantum physics or thermodynamics, those other pillars of modern physics, without such a concept. dt/dt does not shed any light on this problem.
Dunne's solution was simple enough. He proposed two time dimensions; the familiar physical t1 and a second, perceived or conscious dimension of time t2. We may then express the passage of physical time with respect to conscious time, as dt1/dt2, which does allow some sensible meaning.
But then it got complicated. He observed that conscious time also passes and that the same problem recurred; as I put it here, dt2/dt2 = 0, which is as unhelpful as before. So he proposed yet another time dimension, as t3, with which to express dt2/dt3 (There was also another level of consciousness to go with it). And so on in an infinite or serial regress.
Given this model, we might ask, how does physical time look from say the third level? How do the relative rates of passage of the various times work out? We are asking what is the rate of change, with respect to t3, of the rate of change of t1 with respect to t2. That is, (dt1/dt2)/dt3. Assuming that the passage of physical time is a constant, say k, that is easy to calculate. By definition we have dt1/dt2 = k. Since k is a constant, we can immediately say that dt2/dt3 = dk/dt3 = 0. Thus, physical time appears as a block time to any higher level of consciousness. This accords well with Dunne's theory that, in dreams and altered states of consciousness, the higher mind is liberated from the attention focus of full consciousness and can wander freely along its physical timeline in both directions.
But is the regress in itself justifiable? Few commentators have ever thought so.
One critique of Dunne's argument is to suggest that, given the idea that perceived time is a function of brain activity and that Dunne accepted this mind-brain parallelism, it might be more sensible to ask how psychological time t2 varies with the physical time t1 of the brain's biology. In other words, to treat dt2/dt1 as the relevant differential. This is the opposite of Dunne's argument. We have long known that psychological time can speed up or slow down quite substantially, so that in general, dt2/dt1 is not a constant but varies according to the brain's state of activity. This does not entirely resolve the problem of the passage of time, but it does break the direct link between the physical and the experiential. And it was this link which Dunne used to create his circular paradox.
But one may wish to allow a less constrained view of human consciousness, perhaps to provide more elbow room for free will or a soul. In that case, on what basis can one argue that the passage of t3 time has any meaning? Meditation adepts report that, in their highest meditative states, they cease to experience any flow of time. One might suggest that, at worst, we can happily stop here. t3 can be understood to represent Eternity, a dimension of time in which there is no flow and indeed dt3/dt3 = 0 is where the arguments conclude.
This gives us a manageable framework for asking whether there is really a need for t2 to flow. Given the exact parallelism we nowadays assume between conscious experiences and brain signals, each of t1 and t2 must have its parallel in the other realm; the physical in the mental, the experiential in the physical.
The objective, physical t1 provides the basis on which the brain signals evolve. Our flow of consciousness must therefore, at the objective level of the electroencephalograph, follow this in strict lockstep.
However our experience of that time evolution is a subjective model. This model is known to incorporate many fictional adjustments, such as erasing time delays associated with the differences in processing times of visual and aural information so that we experience sight and sound as synchronised (exactly the same adjustment is made by digital video and TV systems, only the exact lengths of the time delays differ between neuron and transistor). Then again, there are the well-known subjective effects of emotional and mental states on our perceptions of time passing. What emerges from all this is t2, a far less strictly clock-driven t2 than Dunne would have us consider.
It seems reasonable therefore to suggest that Dunne's t2 is a delusion, cobbled together by the brain in order to reconcile t1 with the other things going on in our minds, not least memories. It needs no further explanation and may be safely dismissed. The problem of t1 passing may be directly referenced to Eternity which, for a brief moment in my argument, drops a level to become our new t2.
In somewhat philosophical vein, one might suggest that Eternity is not a kind of Time at all, since it does not flow or pass. Perhaps it would be clearer to label it, say, E. This leaves only t1, and we may conveniently describe the rate of time passing as dt/dE, the rate of change of time with respect to Eternity.
Oddly enough, this is roughly where Dunne ended up towards the end of his life. He originally declared an "observer at Infinity", thus going further than the logic of stepwise regression could take him. His highest level of time was indeed Eternity, but he was forced to fudge over how he eventually got off the ladder of regressions. Then, many years later, his friend J B Priestley argued forcefully to him that the endless regress was logically flawed and that two or three dimensions were quite enough. Dunne eventually agreed, though he still clung to the idea that our perceptions of time only made sense within his arguments; that is, the flaw lay in our human limitations and not in his reasoning.
What then of this E, this timeless Eternity? Does it have any genuine philosophical value or have I been playing fatuously trivial games? It evidently has a mathematical utility, in allowing the short and clear exposure of certain flaws in bad models of time.
Philosophically, does such an Eternity have a sensible meaning in any material sense, or is it solely the domain of the metaphysician and the theologian? Some contemporary philosophers did take Dunne's second time dimension seriously, though they all differed in their criticisms and ideas about it, and none gave their wholehearted support. But does dt/dE have any deeper meaning beyond a mathematical trick? Is it in any way testable, verifiable or falsfiable (choose your pedant to taste) as an aspect of reality? This question remains wide open.
Updated 9 April 2021