Essay/Mathematics

Future tension

Facts about the past and present are either true or false. Can knowledge of the future offer the same degree of certainty?

Anthony Sudbery

Will the sun rise? Photo taken from the ISS on 1 March 2016. Photo courtesy NASA

Anthony Sudbery

is professor emeritus of mathematics at University of York. His research focuses on quantum theory. He lives in York.

4,700 words

Edited by Sam Dresser

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Que sera sera
Whatever will be will be
The future’s not ours to see
Que sera sera.

So sang Doris Day in 1956, expressing a near-universal belief of humankind: you can’t know the future. Even if this is not quite a universal belief, then the universal experience of humankind is that we don’t know the future. We don’t know it, that is, in the immediate way that we know parts of the present and the past. We see some things happening in the present, we remember some things in the past, but we don’t see or remember the future. 

But perception can be deceptive, and memory can be unreliable; even this kind of direct knowledge is not certain. And there are kinds of indirect knowledge of the future that can be as certain as anything we know by direct perception or memory. I reckon I know that the sun will rise tomorrow; if I throw a stone hard at my kitchen window, I know that it will break the window. On the other hand, I did not know on Christmas Eve last year that my hometown of York was going to be hit by heavy rain on Christmas Day and nearly isolated by floods on Boxing Day.

In the ancient world and, I think, to our childhood selves, it is events such as the York floods that make us believe that we cannot know the future. I might know some things about the future, but I cannot know everything; I am sure that some things will happen tomorrow that I have no inkling of, and that I could not possibly have known about, today. In the past, such events might have been attributed to the unknowable will of the gods. York was flooded because the rain god was in a bad mood, or felt like playing with us. My insurance policy refers to such catastrophes as ‘acts of God’. When we feel that there is no knowing who will win an election, we say that the result is ‘in the lap of the gods’.

Aristotle formulated the openness of the future in the language of logic. Living in Athens at a time when invasion from the sea was always a possibility, he made his argument using the following sentence: ‘There will be a sea-battle tomorrow.’ One of the classical laws of logic is the ‘law of the excluded middle’ which states that every sentence is either true or false: either the sentence is true or its negation is true. But Aristotle argued that neither ‘There will be a sea-battle tomorrow’ nor ‘There will not be a sea-battle tomorrow’ is definitely true, for both possibilities lead to fatalism; if the first statement is true, for example, there would be nothing anybody could do to avert the sea-battle. Therefore, these statements belong to a third logical category, neither true nor false. In modern times, this conclusion has been realised in the development of many-valued logic.

But some statements in the future tense do seem to be true; I have given the examples ‘The sun will rise tomorrow’ and, after I have thrown the stone, ‘That window is going to break.’ Let’s look at these more closely. In fact, no such future statement is 100 per cent certain. The sun might not rise tomorrow; there might be a galactic star-trawler heading for the solar system, ready to scoop up the sun tonight and make off with it at nearly the speed of light. When I throw the stone at the window, my big brother, who is a responsible member of the family and a superb cricketer, might be coming round the corner of the house; he might see me throw the stone and catch it so as to save the window.

We did not know that the sun would fail to make its scheduled appearance tomorrow morning; I did not know that my naughtiness would be foiled. But this lack of knowledge is not a specific consequence of the fact that we are talking about the future. If the Spaceguard programme had had a wider remit, we might have seen the star-trawler coming, and then we would have known that we had seen our last sunrise; if I had known my brother’s whereabouts, I could have predicted his window-saving catch. In both these scenarios, the lack of knowledge of the future reduces to lack of knowledge about the present.

The success of modern science gave rise to the idea that this is always true: not knowing the future can always be traced back to not knowing something about the present. As more and more phenomena came under the sway of the laws of physics, so that more and more events could be explained as being caused by previous events, so confidence grew that every future event could be predicted with certainty, given enough knowledge of the present. The most famous statement of this confidence was made by the French mathematician Pierre-Simon Laplace in 1814:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

This idea goes back to Isaac Newton, who in 1687 had a dream:

I wish we could derive the rest of the phenomena of Nature by the same kind of reasoning from mechanical principles, for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards one another, and cohere in regular figures, or are repelled and recede from one another.

In this view, everything in the world is made up of point particles, and their behaviour is explained by the action of forces that make the particles move according to Newton’s equations of motion. These completely determine the future motion of the particles if their positions and velocities are given at any one instant; the theory is deterministic. So if we fail to know the future, that is purely because we do not know enough about the present.

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For a couple of centuries, Newton’s dream seemed to be coming true. More and more of the physical world came under the domain of physics, as matter was analysed into molecules and atoms, and the behaviour of matter, whether chemical, biological, geological or astronomical, was explained in terms of Newtonian forces. The particles of matter that Newton dreamed of had to be supplemented by electromagnetic fields to give the full picture of what the world was made of, but the basic idea remained that they all followed deterministic laws. Capricious events such as storms and floods, formerly seen as unpredictable and attributed to the whims of the gods, became susceptible to weather forecasts; and if some such events, like earthquakes, remain unpredictable, we feel sure that advancing knowledge will make them also subject to being forecast.

This scientific programme has been so successful that we have forgotten there was ever any other way to think about the future. Mark G Alford, a physicist at Washington University, writes:

In ordinary life, and in science up until the advent of quantum mechanics, all the uncertainty that we encounter is presumed to be … uncertainty arising from ignorance.

We have completely forgotten what an uncertain world was inhabited by the human race before the 17th century, and we take Newton’s dream as a natural view of waking reality.

Well, it was a nice dream. But it didn’t work out that way. In the early years of the 20th century, Ernest Rutherford, investigating the recently discovered phenomenon of radioactivity, realised that it showed random events happening at a fundamental level of matter, in the atom and its nucleus. This did not necessarily mean that Newton’s dream had to be abandoned: the nucleus is not the most fundamental level of matter, but is a complicated object made up of protons and neutrons, and – maybe – if we knew exactly how these particles were situated and how they were moving, we would be able to predict when the radioactive decay of the nucleus would happen. But other, stranger discoveries at around the same time led to the radical departure from Newtonian physics represented by quantum mechanics, which strongly reinforced the view that events at the smallest scale are indeed random, and there is no possibility of precisely knowing the future.

Quantum theory is so puzzling, it’s not clear it should be described as an ‘explanation’ of the puzzling facts it subsumes

The discoveries that had to be confronted by the new physics of the 1920s were two-fold. On the one hand, Max Planck’s explanation of the distribution of wavelengths in the radiation emitted by hot matter, and Albert Einstein’s explanation of the photoelectric effect, showed that energy comes in discrete packets, instead of varying continuously as it must do in Newton’s mechanics and James Clerk Maxwell’s electromagnetic theory. On the other hand, experiments on electrons by George Paget Thomson, Clinton Davisson and Lester Germer showed that electrons, which had been firmly established to be particles, also sometimes behaved like waves.

These puzzling facts found a systematic, coherent, unified mathematical description in the theory of quantum mechanics which emerged from the work of theorists after 1926. This theory is itself so puzzling that it is not clear that it should be described as an ‘explanation’ of the puzzling facts it subsumes; but an essential feature of it, which seems inescapable, is that, when applied to give predictions of physical effects, it yields probabilities rather than precise numbers.

This is still not universally accepted. Some people believe that there are finer details to be discovered in the make-up of matter, which, if we knew them, would once again make it possible to predict their future behaviour precisely. This is indeed logically possible, but there would necessarily be aspects of such a theory that would lead most physicists to think it highly unlikely.

The format of quantum theory is quite different from previous physical theories such as Newtonian mechanics or electromagnetism (or both combined). These theories work with a mathematical description of the state of the world, or any part of the world; they have an equation of motion that takes such a mathematical description and tells you what it will change into after a given time. Quantum mechanics also works with a mathematical object that describes a state of the world; it is called a state vector (though it is not a vector in three dimensions like velocity), and is often denoted by the Greek letter Ψ or some similar symbol.

But this is a different kind of mathematical description from that in mechanics or electromagnetism. Each of those theories uses a set of numbers that measure physical quantities such as the velocity of a specified particle, or the electric field at a specified point of space. The quantum state vector, on the other hand, is a more abstruse object whose relation to physical quantities is indirect. From the state vector, you can obtain the values of physical quantities, but only some physical quantities: you can choose which quantities you would like to know, but you are not allowed to choose all of them.

Moreover, once you have chosen which ones you would like to know, the state vector will not give you a definite answer; it will give you only probabilities for the different possible answers. This is where quantum mechanics departs from determinism. Strangely enough, in its treatment of change, quantum mechanics looks like the old deterministic theories. Like them, it has an equation of motion, the Schrödinger equation, which will tell you what a given state vector of the world will become after a given time; but because you can get only probabilities from this state vector, it cannot tell you what you will see after this time.

State vectors, in general, are puzzling things, and it is not at all clear how they describe physical objects. Some of them, however, do correspond (if you don’t look too closely) to descriptions that we can understand. Among the state vectors of a cat, for example, is one describing a cat sitting and contentedly purring; there is another one describing it lying dead, having been poisoned in a diabolical contraption devised by the physicist Erwin Schrödinger.

But there are others, obtained mathematically by ‘superposing’ these two state vectors; such a superposed state vector could be made up of a part describing the cat as alive and a part describing it as dead. These are not two cats; the point of Schrödinger’s story was that one and the same cat seems to be described as both alive and dead, and we do not understand how such states could describe anything that could arise in the real world. How can we believe this theory, generations of physicists have asked, when we never see such alive-and-dead cats?

it follows from quantum mechanics that although cats have states in which they seem to be both alive and dead, we will never see a cat in such a state

There is an answer to this puzzle. If I were to open the box in which Schrödinger has prepared this poor cat, then the ordinary laws of everyday physics would ensure that, if the cat was alive, I would have the image of a living cat on my retina and in my visual cortex, and the system consisting of me and the cat would end up in an understandable state in which the cat is alive and I see a living cat. If the cat was dead, I would have the image of a dead cat, and the system consisting of me and the cat would end up in a state in which the cat is dead and I see a dead cat.

It now follows, according to the laws of quantum mechanics, that if the cat is in a superposition of being alive and being dead, then the system consisting of me and the cat ends up in a superposition of the two final states described above. This superposition does not contain a state of my brain seeing a peculiar alive-and-dead state of a cat; the only states of my brain that occur are the familiar ones of seeing a live cat and seeing a dead cat. This is the answer to the question at the end of the earlier paragraph; it follows from quantum mechanics itself that although cats have states in which they seem to be both alive and dead, we will never see a cat in such a state.

But now the combined system of me and the cat is in one of the strange superposition states introduced by quantum mechanics. It is represented mathematically by the familiar sign +, and called an entangled state of me and the cat. How are we to understand it? Maybe the mathematical sign + just means ‘or’; that would make sense. But unfortunately this meaning, if applied to the states of an electron, is not compatible with the facts of interference observed in the experiments that show the electron behaving like a wave. Some people think that this + should be understood as ‘and’: when the cat and I are in the superposition state, there is a world in which the cat has died and I see a dead cat, and another world in which the cat is still alive and I see a living cat. Others do not find this a helpful picture. Perhaps we should just take it as (in some sense) a true description of the cat and me, whose meaning is beyond us.

Now let us broaden our horizon and consider the whole universe, which contains each one of us considered as a sentient, observing physical system. According to quantum mechanics, this has a description by a state vector in which the sentient system is entangled with the rest of the universe, and several different experiences of the sentient system are involved in this entanglement. The same overall state vector of the whole universe can be seen as such an entangled state for every sentient system inside the universe; these are simply different views of the same universal truth.

But saying that this is the truth about the universe seems to conflict with my knowledge of what I see. To illustrate this, let us again consider a little universe containing just me and a cat. Let us suppose that the cat survived when I did Schrödinger’s experiment. Then I know what my state is: I see a living cat. From this I know what the state of the cat is: it is alive. The entangled state of my little universe that was produced by my experiment also contains a part with a dead cat and my brain full of remorse.

But seeing a live cat, as I do, I reckon that this other picture is not part of the truth; it describes something that might have happened but didn’t. In general, considering the whole universe, I know that I have just one definite experience. But this contradicts what was asserted in the previous paragraph. Which of these is the truth?

This contradiction is of the same type as many familiar contradictions between objective and subjective statements. In The View from Nowhere (1986), Thomas Nagel shows how some of these contradictions can be resolved: we must recognise that there are two positions from which we can make statements of fact or value, and statements made in these two contexts are not commensurable. This applies to the puzzle presented by quantum mechanics as follows. In the external context (the God’s-eye view, or the ‘view from nowhere’) we step outside our own particular situation and talk about the whole universe. In the internal context (the view from now, here), we make statements as physical objects inside the universe.

Thus, in the external view, the entangled universal state vector is the whole truth about the universe; the components describing my different possible experiences, and the corresponding states of the rest of the universe, are (unequal) parts of this truth. But in the internal view, from the perspective of some particular experience that I know I am having, this experience, together with the corresponding state of the rest of the universe, is the actual truth. I might know what the other components are, because I can calculate the universal state vector using the equations of quantum mechanics; but these other components, for me, represent things that might have happened but didn’t.

Since I cannot see the future, none of the worlds of the future are singled out for me

We can now look at what quantum mechanics tells us about the future. As we should now expect, there are two answers, one for each of the two perspectives. From the external perspective, the universe at any one time is described by a universal state vector, and state vectors at different times are related by the Schrödinger equation. Given the state vector at the present time, the Schrödinger equation delivers a unique state vector at any future time: the theory is deterministic, in complete accord with Laplace’s world-view (in a quantum version).

From the internal perspective, however, things are quite different. We now have to specify a particular observer (who has been me in the above discussion, but it could have been you or anyone else, or indeed the whole human race taken together), with respect to which we can carve up the universal state vector as described above; and we have to specify a particular experience state of that observer. From that perspective, it is by definition true that the observer has that definite experience, and that the rest of the universe is in a corresponding definite state.

So quantum mechanics tells us that at this moment there are a number of different worlds, but I know that one of them is singled out, for me, as being the world that I see and whose finer details are revealed to me by experiment. But when we turn to the future the situation is different. Since I cannot see the future, none of the worlds of the future are singled out for me. Even if there is only one world now, and what I see agrees with the universal state vector of quantum mechanics, it might happen that the laws of quantum mechanics produce a superposition of worlds at a future time. For example, if I start with the experience of setting up Schrödinger’s experiment with the cat, then at the end of the experiment the universal state vector will be the superposition that we have already encountered, with one part containing me seeing a living cat and another part containing me seeing a dead cat. Then what can I say about what I will see at that future time?

I found this rather startling when I first encountered it. I was used to thinking that there is something awaiting me in the future, even if I cannot know what it is, and even if there is no law of nature that determines what it is. Whatever will be will be, indeed. But Aristotle already saw that this is wrong. Statements in the future tense do not obey the same logic as present-tense statements: they do not have to be either true or false. Logicians following Aristotle have allowed the possibility of a third truth value, ‘undetermined’ or ‘undecided’, in addition to ‘true’ and ‘false’.

However, Aristotle also pointed out that, although no one statement about the future is actually true, some of them are more likely than others. Similarly, the universal state vector at a future time contains more information, for me, than simply what experiences I might have at that time. These experiences, occurring as components of the universal state vector, contribute to it in different amounts, measured by coefficients that are usually used in quantum mechanics to calculate probabilities. So we can understand the future universal state as giving information, not only about what experiences are possible for me at that future time, but also about how probable each experience is.

Now, truth and falsity can be expressed numerically: a true statement has truth value 1, a false one has truth value 0. If a future event X is very likely to happen, so that the probability of X is close to 1, then the statement ‘X will happen’ is very nearly true; if it is very unlikely to happen, so that its probability is close to 0, then the statement ‘X will happen’ is very nearly false. This suggests that the truth value of a future-tense statement should be a number between 0 and 1. A true statement has truth value 1; a false statement has truth value 0; and if a future-tense statement ‘X will happen’ has a truth value between 0 and 1, that number is the probability that X will happen.

The nature of probability is a long-standing philosophical problem, to which scientists also need an answer. Many scientists take the view that the probability of an event makes sense only when there are many repetitions of the circumstances in which the event might occur, and we work out the proportion of times that it does occur; they hold that the probability of a single, unrepeated event does not make sense. But what we have just outlined does seem to be a calculation of a single event at a time that will come only once. In everyday life, we often talk about the probability that something will happen on just one occasion: that it will rain tomorrow, or that a particular horse will win a race, or that there will be a sea-battle. A standard view of such single-event probability is that it refers to the strength of the belief of the person who is asserting the probability, and can be measured by the betting odds they are prepared to offer on the event happening.

But the probability described above is an objective fact about the universe. It has nothing to do with the beliefs of an individual, not even the individual whose experiences are in question; that individual is being told a fact about his future experiences, whether he believes it or not. The logical theory gives an objective meaning to the probability of a single event: the probability of a future event is the truth value of the future-tense proposition that that event will happen. I explore this view of probability, and the way that quantum mechanics supports the associated many-valued logic of tensed propositions, in ‘The Logic of the Future in Quantum Theory’ (2016).

It has now become clear that the description of the physical world given by quantum mechanics, namely the universal state vector, plays very different roles in the two perspectives, external and internal. From the external perspective, it is a full description of reality; it tells how the universe is constituted at a particular time. This complete reality can be analysed with respect to any given sentient system, yielding a number of components, attached to different experiences of the chosen sentient system, which are all parts of the universal reality.

Those things that might have happened, but didn’t, some of which we don’t even know about, might still affect the future

From the internal perspective of this system, however, reality consists of just one of these experiences; the component attached to this experience is the complete truth about the universe for the sentient system. All the other non-zero components are things that might have happened, but didn’t. The role of the universal state vector at a later time, in this perspective, is not to describe how the universe will be at that time, but to specify how the present state of the universe might change between now and then. It gives a list of possibilities at that later time, with a probability for each of them that it will become the truth.

It might seem that we can at least know these probabilities for the future, being able to calculate them from our certain knowledge of our present experience, using the Schrödinger equation. But even this is uncertain. Our present experience could well be only part of the universal state, and it is the whole universal state vector that must be put into the calculation of future probabilities. Those things that might have happened, but didn’t, some of which we don’t even know about, might still affect the future. However, if those things are sufficiently different from our actual experience on a macroscopic scale, then quantum theory assures us that the effect they might have on the future is so small as to be utterly negligible. This consequence of the theory is known as decoherence.

Knowledge of the future, therefore, is limited in a fundamental way. It is not that there are true facts about the future, but the knowledge of them is not accessible to us; there are no facts out there, and there is simply no certain knowledge to be had. Nevertheless, there are facts about the future with partial degrees of truth. We can attain knowledge of the future, but that knowledge will always be uncertain.

An expanded version of this article will appear in Space, Time and the Limits of Human Understanding, ed. Shyam Wuppuluri and Giancarlo Ghirardi, to be published by Springer

 

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