The passage of time and the mathematical universe

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If, in this era of science, you ask what philosophy can do that physics cannot do, the answer might be: make the unimaginable imaginable. The true philosopher hovers between poetry and mathematics.

As the church father Augustine famously wrote, time is a word which every one of us perfectly knows how and when to use, but no one knows precisely what time is. And yet time, together with consciousness, ranks among our most fundamental intuitions. We can close our eyes and imagine a world without space, but we cannot think and be conscious without the intuition of time. Not everything is spatial but everything is in time, including space, for we cannot cover even the shortest distance in space without spending time.

Plato defined time as the moving image of an immutable eternity. Plato’s love of mathematics undoubtedly played a role here: the quiet and stable realm of numbers and geometrical figures, undisturbed by the perpetual change that pervades the world around us.

Isaac Newton said: time is what keeps moving even if nothing else moves. Newton’s argument was theological rather than physical, for Newtonian physics does not need absolute time. But, as Newton argued, if the universe were empty, there would still have to be time and space, for else God would not have a place to live in.

Newton’s great rival in the field of mathematics, the German philosopher Gottfried Leibniz, astutely remarked that time is the order of all events that occur in the same place. In other words: time ensures that not everything happens all at once.

Speaking about time we could discuss its topology: whether time has a beginning and an end, which Aristotle and Stephen Hawking denied, albeit with very different arguments; or whether time could be circular, as the ancient Stoics and later Friedrich Nietzsche believed, so that we shall live this exact same life over and over again.

We could speak about the microstructure of time. Are there smallest and indivisible units of time? If so, these time atoms cannot have duration, for if they had, they could be further divided. It is difficult to see, however, how any duration can arise from time atoms that have no duration themselves.

If, on the other hand, time is like the mathematical continuum, infinitely divisible, we must solve the paradoxes of Zeno. The athlete Achilles cannot overtake a tortoise because over an infinite number of moments the lead of the tortoise may become infinitely small but it will never reach zero. An arrow appears to fly through the air but must nevertheless stand still at each infinitely short moment of its flight, for else it would never be anywhere. Although clearly refuted by observation, these paradoxes have to this day not been fully resolved from a purely mathematical point of view.

We could discuss the direction of time: the steady increase of entropy in the universe, which, in simple terms, means that differences in the distribution of energy are gradually levelled out. Only while the universe is still in its youth this trend is locally halted by the creative power of gravity which forms the stars and through the stars the chemical elements of which we are composed. Eggs break, electromagnetic waves disperse from your mobile phone outwards, and living organisms die and rot, but the reverse never happens, although the laws of physics allow it. Is this simply a matter of statistics, or does time, mysteriously, have a direction in and of itself?

We could touch upon the psychology of time, the ‘specious present’, which is not an indivisible moment but which comprises three different actualities: the presence of things gone, which is memory, the presence of things before us, which is perception, and the presence of things to come, which is expectation. The enjoyment of musical melody rests on this extended present, connecting the note that we hear to the one we just heard and the one we expect immediately to follow.

The most intriguing question, however, one that lies at the heart of our intuition of time, is the question whether time passes, whether nothing remains but everything becomes and passes away.

In this form, time is the great destroyer, the enemy of existence, relentless drift of all and everything towards nothingness and the grave.

But the passage of time is also the great creator, source of all movement, of life and thought, of science itself, for science is the transition from a state of ignorance to a state of knowledge through observation and experiment.

This time is the greatest problem of philosophy and the greatest obstacle to a unifying theory in modern physics.

Time is, first of all, the greatest problem in philosophy, for if nothing is and everything becomes, how then is metaphysics as the science of being possible? How is true knowledge possible, if the thing that we think we know has already changed the moment we know it? As Heraclitus wrote in antiquity: ‘In the same rivers we step and do not step, we are and are not.’

Time is also a fundamental problem in modern physics, as it pits against each other the two great theories of the early twentieth century. In Einstein’s relativity, time coalesces with space into spacetime, where moments exist next to one another, not after one another. There, the past still exists and the future already exists and is fixed. On the quantum level, however, the old time reigns. There, the future is open and things happen that are not merely practically but essentially unpredictable. As the physicist Freeman Dyson pointed out, relativity fits the past while quantum physics agrees with our idea of the future.

It may be remarked, by the way, that in both theories, the idea of free will, hotly debated by philosophers, not so much by physicists, is problematic. It is evidently questionable when the future is fixed and already determined. But it is equally problematic if, as the British physicist and Nobel laureate Roger Penrose has suggested, our decisions ultimately rest on an unpredictable quantum event in the brain. As Arthur Schopenhauer said, we may sometimes be free to do wat we want, but it is not our own decision to want what we want.

There seems to be an obvious way out of these conundrums, a strategy shared by both metaphysicians and physicists, which is simply to deny the reality of time, the reality of the passage of time in particular. Since Galilei and Newton, physics is mathematical physics. Its astounding success in describing the workings of the world rests upon the construction of mathematical models. But although time can be easily introduced in a mathematical formula as a variable, mathematics does not capture the essence of what we experience as the passage of time. In fact, to imagine time, we are forced to imagine it as something static and spatial, a line, for example, where the points, lying next to one another, represent the moments that according to our intuition follow upon one another.

Why is this so? The Enlightenment philosopher Immanuel Kant was the first to tackle this question in depth. Every single mental image we have of the world contains space, Kant remarked, but a single mental image of the world cannot contain time, for time is not an aspect of a single image but the order of subsequent images in the mind. While space is in each of our mental images, time is outside them as their frame or container. For this reason, the human intellect can imagine subsequent states of the world in time but it cannot imagine time itself.

We shall elaborate a little more on the relation between time and mathematics. Logic and mathematics are timeless, it seems, and their truths hold forever. But take the basic law of logic, formulated by Aristotle and later by Ludwig Wittgenstein: either something is the case or it is not the case; there is no third way, or in Latin, tertium non datur. This timeless law of logic holds only in time, for it is only through the passage of time that your coffee is either hot or cold, that each of us is either still alive or already dead. Without time, the coffee shall be both hot and cold, and we shall be both dead and alive.

Indeed, as Leibniz suggested, it is time which ensures that not everything occurs all at once. Time is the great differentiator, unfolding space and unwinding the history of the universe, from its early hot and dense beginning to the cold and dark infinity that awaits it in the far future.

It has been argued that time cannot be real because time itself suffers from a fatal contradiction. For, as the argument goes, how can one and the same event, say this lecture, appear to be both past and present and future: future from yesterday’s point of view, present now, and past tomorrow — as if the same building were the Eiffel Tower when approached from the east but the Colosseum for visitors from the west?

Now, the argument holds if we assume that logic and mathematics hold sway over time, that time, like everything else in the world, must bend to their timeless laws. Yet as the law of tertium non datur demonstrates, it may well be the other way around. In this case, we must perhaps reconcile ourselves to the fact that there cannot be a single and complete description of the world without contradiction or ambiguity.

The advocates of the so-called mathematical universe take their argument a step further. They suggest that not only can the world be completely described through mathematical models, but the world is itself a mathematical model, the world is mathematics and mathematics is all there is, a proposition which would most certainly have appealed to Plato.

In philosophical terms this might be an instance of the fallacy of the maker or transcendental fallacy. More than once in the history of science, when scientists found a construct or a model which seemed adequately to reflect essential properties of the world, they tended to equate the world with the construct or the model. In the seventeenth century, the universe was conceived of as a gigantic clockwork, fine-tuned by God. More recently, everything is said to be information, or we are supposed to exist as parts of a software program run on a mysterious device outside our time and space.

Against this idea of the mathematical universe goes the fact that mathematics does not satisfactorily explain our intuition of the passage of time. Nor does it explain that other basic intuition closely connected with time, which is consciousness. Now the proponents of the mathematical universe will retort that the intuition of passing time is an illusion. They will also argue that self-consiousness, the unique and inexpressible awareness of being me, is a feeling, not a scientific concept, and can therefore be ignored.

And yet, if we may believe the Dutch mathematician Bertus Brouwer, the creator of mathematical intuitionism, it is not the perception of static space that gives us mathematics, but the close intertwinement of consciousness and time. For it is time that splits every single mind-state in two, thereby making it possible for this state to reflect upon itself, and for us to be self-conscious. We thus have the transition from one to two, which gives rise to the notion of multiplicity and sequence and order. And this is indeed what mathematics is in the broadest sense: the study of multiplicity and sequence and order.

But the idea of the mathematical universe also suffers from an inadequate understanding of reality. ‘Real’, from Latin, literally means ‘like a thing’. Now saying that the world and the things in it are real does not teach us much, for obviously things are ‘like things’. In Dutch, however, one speaks of werkelijkheid, and the German word is Wirklichkeit, which suggests that the world and the things in it work, that they have an effect on other things and on us.

This is the only meaningful question that physicists can ask, and the only thing they are interested in: not what a photon is, or what space is, or what this chair is, but what it does, how it works, and how it interacts with other physical objects and systems. And working is by its very nature a temporal process. It implies movement and change.

Reality is working; the world is dynamic. If a thing does not in any way interact with other things, it is shut off from the universe and we cannot meaningfully say that it exists. Yes, one can model this dynamic to a certain extent with mathematics, but a mathematical model as such is static, does not change and does not work. Mathematics is an extremely useful bridge between our intellect and the world, an extremely succesful tool for understanding the world, but mathematics is not the world, is not the dynamic of the world.

In one of his later works, Plato introduces a unnamed foreigner who asks young Socrates if he indeed believes that in the purest form of reality, the true world, there is no time but only eternity. Yes indeed, Socrates replies, upon which the foreigner asks: Are you so easily convinced that in the true reality there is no movement, no life, no thought, that the true world does not live and think but supremely remains the same, without thought, without change?

Whatever counter-intuitive shape the world may take at the deepest level, as quantum foam, or as the tiny multi-curled strings according to the theory named after them, or as a structure best described by non-commutative geometry, the world is dynamic on each and every level, and therefore there is time, therefore the universe evolves, therefore we exist, not frozen in an eternal present, but living and thinking and constructing our mathematical models of the world. At the deepest level there may not be time quite as we experience it, but there will be dynamic, proto-time if you will, from which our familiar time springs.

Finally, if the passage of time is real, does that mean that we too must inexorably pass, that we are doomed to sink into nothingness? Yes and no. For time is in the universe, is a property of the universe, but the universe is not in time. Or we could say that time is the dynamic of the world which makes everything pass, but time itself, as the dynamic of the world, does not begin nor end nor pass. If this paradox defies logic, then remember that time is the father of logic, not the other way around.

Everything on this earth, everything in the universe shall pass, but the world as a whole, all that is, with all its dynamic, shall not pass. We are mortal and every moment of our consciousness is fleeting, but as part of all that is, embedded in the dynamic of time, we are eternal, albeit eternal in our temporality.