On Real Numbers, Free Will and Open Future
By: Michele Diodati
Tags: Philosophy, Real Numbers, Science, Physics, Philosophy Of Science

”La liberté est un mystère” — Nicholas Malebranche

The great capacity demonstrated by Newtonian mechanics to predict the motion of celestial bodies provided 18th-century philosophers and scientists the theoretical foothold on which to build the vision of a mechanical, material Universe, governed exclusively by cause and effect relationships between physical objects. In such a view of the cosmos, every event is the necessary consequence of pre-existing conditions. Everything can only happen in the way it happens, given all the conditions that precede it.

Determinism and necessity

Determinism is the philosophical position that expresses this conception of the Universe and the phenomena that occur in it. One of his major theorists was the French philosopher and mathematician Pierre-Simon de Laplace. In his work Essai philosophique sur les probabilites, dated 1814, we find a thought that admirably sums up the essence of determinism:

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.

Pierre-Simon, marquis de Laplace (1749–1827)

For Laplace, and for all determinists, the Universe is an immense, very complicated mechanism governed by natural forces. The all-human uncertainty about the course of future events is nothing but ignorance. If we could conceive the totality of all existing bodies, their motions, and the forces that act on them at all times, no event could ever surprise us. Everything would seem to unfold according to necessity, following a pre-established order, like the sequence of the frames in an already seen movie (a comparison that Laplace could not yet have done, but that would certainly come to his mind if he had been born a century and a half later).

But determinism can assert itself as a dominant philosophy only if two illustrious victims are sacrificed on its altar.

The first is time as a creative agent. For the determinist, time is that ticked by clocks — merely a way to mark chains of events flowing under the laws of nature. Conceived in this way, the passing of time does not bring with it any inherently unpredictable novelty. What appears to us as chance is only the result of imperfect knowledge of the conditions that determine its occurrence.

The second victim is free will. In a world where every event is the necessary consequence of previous conditions, there is no possibility of exercising genuinely free choices. Human beings are mortal, material beings, subject to the same set of forces and causes that regulate the events of the physical world. Therefore, the idea of ​​exercising free will in making a decision is a pure illusion. We have to choose what we choose, given the set of conditions that determine our choices.

The greatest eighteenth-century philosopher, Immanuel Kant, correctly understood the significance of this loss for the moral action of human beings. In a clockwork universe, where events occur according to necessity, there is no room for the principle of responsibility. If our actions are the forced outcome of causal chains of physical, material events (such as the biological mechanisms regulating the functioning of the brain), then we cannot be held morally responsible for what we do. Moral responsibility presupposes free will.

Conditional freedom

In the third antinomy of pure reason, Kant lays bare the essence of the conflict between free will and determinism. The thesis reads:

Causality according to the laws of nature, is not the only causality operating to originate the phenomena of the world. A causality of freedom is also necessary to account fully for these phenomena.

But the antithesis, which expresses the essence of determinism, immediately after denies that thesis:

There is no such thing as freedom, but everything in the world happens solely according to the laws of nature.

Immanuel Kant (1724–1804)

How to reconcile two positions that could not be more opposite? Kant resorted to a sort of philosophical sleight of hand, with which he “restored” free will to human beings without denying determinism. Conciliation requires accepting a scientifically unprovable idea, that is, the possibility of a double causality.

Without going into the details of Kantian reasoning, the principle postulated by the German philosopher states that, alongside the causal series of physical events, it is possible to conceive another causality born directly from the inner spontaneity of the thinking subject, free from the constraints of the material world. Exercising this freedom allows one to alter the course of physical events, initiating new and independent chains of causes and effects. In the whole Critique of Pure Reason, it is one of the rare cases in which Kant uses an example to clarify the concept:

When, for example, I, completely of my own free will, and independently of the necessarily determinative influence of natural causes, rise from my chair, there commences with this event, …, an absolutely new series; although, in relation to time, this event is merely the continuation of a preceding series.

If you look at the scene from the outside with the eye of the determinist scientist, the rise from the chair is the result of a mechanical chain of causes and effects, based on the integrated functioning of the muscles, the nervous system and the human body as a whole. At no time the natural order of phenomena is violated. But, for the subject who performs the action, it is instead a deliberate act of will, an exercise of free will.

For Kant, the existence of this inner freedom does not contradict the causal order of the deterministic Universe. According to his theory, the world can be viewed from two different points of view, which coexist side by side — one purely mechanistic, deterministic, founded on the observation and reconstruction of mere physical facts; the other based on the direct inner experience of one’s capability to make free choices.

Albert Einstein (1879–1955)

But Kant’s solution was too philosophical to elicit full adherence among scientists. The great success of what we now call classical physics led physicists, in particular, to stick to a properly deterministic language and mentality.

All this stands out sharply, for example, in Einstein’s thought. Amid the crisis created by quantum mechanics, which had forcefully brought uncertainty and randomness back into the fabric of physical reality [1], Einstein was still firmly anchored in the idea of a world dominated by deterministic natural laws. Such laws, once fully understood, would prove that there is no place in the Universe for blind randomness. Einstein returned to the subject several times during his life, for example in a letter of September 7, 1944, addressed to his friend Max Born, in which he wrote:

You believe in the God who plays dice, and I in complete law and order in a world which objectively exists, and which I, in a wildly speculative way, am trying to capture. … Even the great initial success of the quantum theory does not make me believe in the fundamental dice-game, although I am well aware that our younger colleagues interpret this as a consequence of senility.

The idea of a rigid causal order, which connects events deterministically in an endless chain of causes and effects, also leads Einstein to deny the profound reality of time as the root of change and novelty. This idea is clearly expressed in a letter dated March 1955, sent to the family of the lifelong friend Michele Besso, who died a few days before (the letter precedes by a few weeks the death of the same Einstein):

Now he has again preceded me a little in parting from this strange world. This has no importance. For people like us who believe in physics, the separation between past, present and future has only the importance of an admittedly tenacious illusion.

But the determinists, although stuck to their faith in the existence of a natural order ruled by necessity, could not give up entirely the idea of freedom. A whole philosophical current, compatibilism, was born to try to somehow save free will, or at least a semblance of it, even within a deterministic framework, in which necessary laws govern the course of events.

Arthur Schopenhauer (1788–1860)

The essence of compatibilism lies in the idea that we as human beings are free, but within the limits in which our innate character and the circumstances in which we act allow us to be so. We can do whatever we want, but we can’t decide what we want. Will drives us to action, but the drive to act comes from interests that make that action a necessary consequence, as Schopenhauer explains in his 1839 Essay on the Freedom of the Will:

… a human being can no more get up from his chair before a motive pulls or pushes him than a billiard ball can be set in motion before it is struck; but then his getting up is as necessary and inevitable as is the rolling of the ball after it is struck. And to expect someone to do something without his being urged to do so by any interest at all is like expecting a piece of wood moving toward me without a cord that draws it.

Real numbers, math monsters

Given the radical consequences of determinism, it is legitimate to raise doubts and ask questions. Is free will possible, or is it just an illusion? Does determinism have a real foundation, or is it only a kind of metaphysical faith? Is there a necessary natural order that connects all the phenomena of the Universe in a series of causal chains, so that the future is already determined in detail by the conditions that precede it? Does time really flow, or is it just a tenacious illusion, as Einstein thought?

All these issues are at the center of a series of articles published between 2019 and 2020 in some of the major scientific journals. Their author is Nicolas Gisin, a Swiss physicist, professor at the University of Geneva.

The Swiss physicist Nicolas Gisin

Gisin, who is almost 68 years old (he was born on May 29, 1952), is a successful scientist and entrepreneur. As a theoretical physicist, he is the author of hundreds of scientific articles, mainly devoted to topics concerning quantum information, cryptography and teleportation. As an experimental physicist, he led pioneering experiments of historical importance, which proved beyond doubt the non-local nature of quantum phenomena.

The ideas that Gisin has disseminated in his recent articles, summarized in brief, are the following:

determinism is based on a metaphysical illusion; time really flows giving rise to unpredictable events, so the future is open, and free will is not an illusion but an actual reality that underlies our capacity to know and act in the world.

Let’s start from the beginning. What is the illusion on which determinism is based? For Gisin, that real numbers [2] are physically real.

In classical physics, determinism stems from the idea, well summarized by Laplace’s thought reported at the beginning, that, if the initial conditions of a physical system could be known with an arbitrary degree of precision, all the future evolution of that system could be derived from those conditions.

The existence of such a chain of necessary causes and effects rests on the assumption that the initial conditions, even those of the whole Universe since the Big Bang, can be encoded through real numbers, with all the precision necessary to derive the most minute and distant consequences.

In the case of physical systems of little complexity, classical mechanics, using dynamical equations based on the use of real numbers, has shown that it can effectively predict the future evolution of those systems with a high degree of precision. Tables with dates of coming eclipses of the Sun and Moon or oppositions of Mars are just one of many possible examples. But it is all to prove, instead, that there are real numbers that contain the complete information that, being able to access their content, would allow deriving in detail, and for as long as you want, the future evolution of a chaotic dynamical system, such as the evolution of weather over a thousand years.

It is not a problem of capacity, that is, the possibility of incurring a limit on the ability to store data by a real number. Such numbers are infinite by definition. The infinite decimal development of a real number could be used, for example, to memorize the set of all the questions, and the relative answers, which it is possible to formulate in any human language (it is one of the examples proposed by Gisin to show what kind of “monsters” are these numbers).

An example of how one can use the infinite decimals of a real number to encode all possible questions (rendered in binary language, that is, in series of 0 and 1) and the related answers that can be formulated in any human language. The questions are coded by sequences of Sn symbols. Each sequence of Sn is followed by two control bits. If the previous series does not represent a question, the value of the two bits is 00. If it represents a question and the answer is yes, the two bits assume value 01; if the answer is no, the value is 10

The problem strongly emphasized by the Swiss physicist is that these numbers are mathematical entities, very useful for performing calculations, but still abstract. They are entities of thought, like Platonic ideas. They have no real physical existence. The argument that Gisin brings to support his thesis is straightforward [3]:

… a finite volume of space can’t contain more than a finite amount of information, hence … the mathematical real numbers are not physically relevant.

Let’s try to understand. Information, to be usable in physics, must have a material existence. For example, it must be contained in a physical medium such as a computer hard drive. But storing information requires space and energy. Even if a real number existed that held the whole set of initial conditions of the Universe (for example information on the position and motion of all the existing particles), such that it could be used to derive the entire cosmic evolution in a deterministic way from the Big Bang onwards, not all the atoms in the Universe would be enough to memorize that number. Starting from similar considerations, Gisin concludes that [3]:

All numbers containing an infinite amount of information, cannot represent physical entities; specifically, they cannot be used, and in fact are not used, to describe initial conditions.

There is no need for the excluded middle

Even classical physics — Gisin notes — uses only numbers that contain a finite amount of information. Yet, despite this, most physicists still attribute a deterministic character to classical physics. They are prompted by adherence to a metaphysical principle [4], which Gisin calls of infinite precision. This principle comes in two forms, the first concerning reality, the second concerning knowledge of reality [5]:

there exists an actual value of every physical quantity, with its infinite determined digits (in any arbitrary numerical base). despite it might not be possible to know all the digits of a physical quantity (through measurements), it is possible to know an arbitrarily large number of digits.

Both postulates are illusory because they assume that real numbers are real in the true sense of the word, which is impossible if it is true that a finite space cannot contain an infinite amount of information.

Denying the physical reality of real numbers has extraordinarily important consequences. First, it allows us to recognize that classical physics, as well as quantum mechanics, is also indeterministic. It means that the future state of a chaotic dynamical system cannot be known a priori in the smallest details, not because we lack knowledge of its initial conditions, but because its future has not yet been written, it is not predetermined. In short, there is no real number stored somewhere in the Universe that contains all the information that determines its evolution over time.

To understand the difference, let’s think about the following forecast: “It will be raining in exactly one year time from now at Piccadilly Circus”. Gisin writes [6]:

If one believes in determinism, then this proposition is either true or false, though it might be impossible in practice to know which alternative holds. But if one believes that the future is open, then it is not predetermined that it will rain, hence the proposition is not true, and it is not predetermined that it will not rain, thus the proposition is also not false.

The difference between the two conceptions of time is enormous. For the determinist, the future is already written in the conditions of the past. We do not know if in precisely one year it will rain in Piccadilly Circus because we do not know the complete initial conditions that would allow us to calculate it. Still, those conditions must exist, so one of the two possibilities must be true, and the other must be false: In a year’s time at Piccadilly Circus, it’ll either rain or it won’t. Tertium non datur (“There is no third possibility”), as the logical principle of the excluded middle establishes.

Luitzen Egbertus Jan Brouwer (1881–1966)

Instead, in the concept proposed by Gisin, which explicitly refers to intuitionistic mathematics founded by the Dutch Luitzen Egbertus Jan Brouwer at the beginning of the 20th century, the law of the excluded middle does not apply. For intuitionist logic, time is creative; it brings unpredictable novelties. It follows that the evolution over time of a chaotic system, such as that of meteorological weather, is not predetermined a priori by real numbers that contain the totality of the initial conditions of the system, but materializes moment by moment.

In intuitionistic mathematics, the decimal digits that describe the state of a physical system are created over time and are not infinite. In this sense, real numbers are for all practical purposes random numbers, whose decimal digits have the same random values ​​that are obtained by measuring the state of a quantum system. It is the exact opposite of what determinism postulates.

This change of perspective makes the future unpredictable and open. It justifies the possibility that a proposition P that speaks of the future (“It will be raining in exactly one year time from now at Piccadilly Circus”) may be not true and not false at the same time.

It is a change of perspective that affects not only the future but also the present. For the determinist, the present does not have a measurable dimension. It merely is the continuously shifting dividing line that separates the past from the future. The intuitionist’s point of view is entirely different. Gisin writes about it [6]:

Physicists often have the intuition that the present is thick — that is, the present is not of measure zero. This corresponds naturally to the intuitionistic concept of the viscous continuum: the continuum cannot be sharply cut in two. When trying to cut it, something sticks to the knife.

Free will and open future

The philosophical view put forward by Gisin in the series of articles published over the past two years has its foundation in the idea that, as thinking beings, we are endowed with free will. For the Swiss physicist, the freedom to choose between different alternatives and different futures is not an illusion, contrary to what the determinists believe. This sort of philosophical genealogy is clearly and concisely set out in six points, reported at the beginning of a short essay, contained in a volume of collected writings published in 2017 [7]:

Free Will comes first in the logical order. Indeed, without free will there is no way to make sense of anything, no way to decide which arguments to buy and which to reject. Hence, there would be no rational thinking and no science. In particular, there would be no understanding. Since free will is the possibility to choose between several possible futures, point 1 implies that the world is not entirely deterministic. Non-determinism implies that time really exists and really passes: today there are facts that were not necessary yesterday, i.e. the future is open. In addition to the geometrical time, there is also creative time. … The tension between free-will and creative time on one side and scientific determinism on the other side dissolves once one realizes that … the real numbers that theories use as initial conditions and parameters are not physically real. Hence, neither Newtonian, nor relativity, nor quantum physics are ultimately deterministic. Consequently, neither philosophy nor science nor any rational argument can ever disprove the existence of free will, hence of the passage of time.

In conclusion, this series of philosophically-based articles published by Gisin, somewhat unusual for a physicist, can be regarded as a coherent attempt to free physicists from the metaphysical illusion of infinite precision, from the dogma that it is possible, at least in theory, to derive in a compelling way all the events occurring over time from a set of initial conditions defined once and for all. No, says Gisin, this is not possible even in theory. As a good realist, the Swiss physicist is willing to believe only what empirical facts say, as bizarre as they may seem and far from the metaphysical idea of a clockwork Universe ruled by necessity [8]: