Indeterminism in Physics and Intuitionistic Mathematics

Home , Quantum indeterminacy, Time in physics

Nicolas Gisin Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland (Dated: November 9, 2020) Most physics theories are deterministic, with the notable exception of quantum mechanics which, however, comes plagued by the so-called measurement problem. This state of aﬀairs might well be due to the inability of standard mathematics to “speak” of indeterminism, its inability to present us a worldview in which new information is created as time passes. In such a case, scientiﬁc determinism would only be an illusion due to the timeless mathematical language scientists use. To investigate this possibility it is necessary to develop an alternative mathematical language that is both powerful enough to allow scientists to compute predictions and compatible with indeterminism and the passage of time. We argue that intuitionistic mathematics provides such a language and we illustrate it in simple terms.

I. INTRODUCTION I have always been amazed by the huge difficulties that my fellow physicists seem to encounter when con- templating indeterminism and the highly sophisticated Physicists are not used to thinking of the world as in- circumlocutions they are ready to swallow to avoid the determinate and its evolution as indeterministic. New- straightforward conclusion that physics does not neces- ton’s equations, like Maxwell’s and Schrödinger’sequa- sarily present us a deterministic worldview. But even tions, are (partial) differential equations describing the more surprising to me is the attitude of most philoso- continuous evolution of the initial condition as a func- phers, especially philosophers of science. Indeed, most tion of a parameter identified with time. Some view this philosophers who care about quantum physics adopt ei- state of affairs as the paradigmatic signature of scientific ther Bohmian mechanics [2–4] (become a Bohmian, as rigor. Accordingly, indeterminism could only be a weak- one says) or the many-worlds interpretation [10]. Appar- ness due, for instance, to an incomplete description of ently, to most of them adding inaccessible particle posi- the situation. tions or inaccessible parallel worlds to their ontology is a There are essentially two kinds of common objections reasonable price to pay in order to avoid indeterminism. to the above determinsitic worldview. First, not all evo- The central claim of Bohmians is that they circumvent lution equations one encounters in physics have a unique the quantum measurement problem. The latter has two solution for all initial conditions. Even the classical sides. First, quantum theory is silent about when poten- Hamiltonian equations may fail to satisfy the Lipschitz tialities become actual. Second, quantum randomness conditions and thus allow for several solutions, possibly seems to emerge from nowhere (from outside space-time, even a continuous infinity of solutions, as is the case as I sometimes wrote) and does thus not satisfy Leibniz’s of Norton’s dome [1]. These cases are fairly contrived principle of sufficient reason. Admittedly, the first point and exceptional, but still worth remembering when col- is serious. However, the second one is unfair: in all fun- leagues claim that determinism is obvious. Second, there damentally indeterministic theories there must be events is quantum physics, which is generally presented as in- that happen - and thus information that gets created trinsically indeterministic. Actually, one can even prove - although their happenings were not necessary. Spon- the existence of quantum randomness from two highly taneous quantum collapse models are good examples of 2 plausible assumptions: physical distances exist (noth- such consistent indeterministic theories [11–14]. Con- ing jumps arbitrarily fast from here to there1), and no sequently, since indeterminism is precisely the view that super-determinism (no combination of determinism and time passes and creates new information, one can’t ar- conspiracy, i.e. there are de facto independent processes) gue against indeterminism by merely asserting that the [5–8]. But this quantum indeterminism is immediately creation of new information is a priori impossible. associated with difficulties. These come either under the A very different reaction to these difficulties is illus- name of the measurement problem [9], or claims of the in- trated by Yuval Dolev’s claim that tense and passage are completeness of quantum theory that should be comple- not, never were, and probably cannot be part of physics mented by additional variables, as in Bohmian mechanics and its language [15]. [2–4], or by the radical many-worlds view [10]. I address But is indeterminism truly that difficult to conceptual- quantum indeterminacy in appendix A. But allow me to ize and contemplate [16]? Isn’t indeterminism pervasive straightforwardly continue with my main motivation. in our lives, and shouldn’t physics not only lead to fasci-

1 In other words, no influence “propagates” at infinite speed; this 2 Moreover, several of such collapse models also answer the first excludes, among others, Bohmian mechanics [2–4]. side of the measurement problem. 2 nating technologies and sophisticated theories, but also numbers, i.e. numbers x for which there is an algorithm 5 tell stories of how nature does it in a language that al- that generates a series of approximations xn such that −n lows humans to gain intuitive understanding? If not, how |x−xn| ≤ 2 for all positive integers n. Although essen- could our intuition develop? And, is there deep under- tially all numbers one encounters are exceptional (they standing without intuition? Shouldn’t one reply to Dolev have names and there are only countably many names), by adapting the mathematical language used by physics the vast majority of real numbers are not computable; to make it compatible with indeterminism? Shouldn’t their bits (and digits) are truly random, as random as one paraphrase Rabelais and state that “Science without the outcomes of quantum measurements, and are of max- time is but the ruin of intelligibility” [17]? imal Kolmogorov complexity. This randomness is crucial It is worth trying to accept indeterminism at face value for Bohmian mechanics to reproduce quantum statistics 6 and see where this leads us. Accordingly, let’s assume and for classical dynamical systems to produce chaos . that nature has the power to continually and sponta- The latter is often termed “deterministic chaos”, though neously produce randomness in the form of entirely new is it based on real numbers whose decimals are random information (though information without meaning). For [26, 27]. simplicity, and because we need to start somewhere, let’s In [24] I argued that one should avoid infinite infor- assume this power expresses itself by a natural random mation densities, i.e. that a finite volume of space can process that continually produces random bits at discrete contain at most a finite amount of information (see also instants of time. Continuous random processes and larger [28–30]). Then I concluded that the mathematical real alphabets would also work, but let’s start simple, i.e. with numbers should be replaced by finite-information num- binary information in the form of bits produced at dis- bers and that the usual real numbers would be better crete time steps, which I name “time-instants”. Is it named “random numbers”. In this way, whenever the truly impossible to think of such a fundamental Natural equations have an analytic solution and the initial con- Random Process (NRP)? Not a human-made one, but - ditions and the time t are given by finite-information again - as a power of nature. Isn’t that very natural if one numbers, then the state at time t is also given by finite- wants to contemplate the possibility that our world is in- information numbers. And if the solution is numerical, trinsically indeterministic? Can one start anywhere else then obviously it is given by finite-information numbers. than by assuming that our world (nature) has the power If, on the other side, the dynamical system is chaotic, to produce random bits? Let me emphasize that I am then the finite-information number encoding the initial not claiming that physics is indeterministic; more mod- condition must be complemented by fresh random bits estly, I am interested in showing that our best physics is as time passes, which is equivalent in practice to the ran- 3 compatible with indeterminism . domness hidden in the real/random numbers that usually Let us contrast the above assumption of the existence describes the initial conditions. of a natural random process with the common assump- Consequently, intrinsic randomness is already hidden tion that real numbers faithfully describe our world, in in our archetype of a deterministic physics theory, i.e. in particular that the positions of elementary particles (or classical mechanics, and also in Bohmian mechanics. But their centers of mass) in classical and in Bohmian me- then, why not make the assumption of fundamental ran- chanics are faithfully described by mathematical real domness, as a power of nature, an explicit assumption? numbers [24]. Typical real numbers contain infinite Admittedly, for practical reasons it might be useful to (Shannon) information. Hence, the “real number as- keep standard mathematics for our computations. For, sumption” is far from cheap. It assumes that ungras- in any case, both classical mathematics and the mathe- pable infinite quantities are at the basis of our physics. matics adapted to finite information always agree on the This is certainly not an obvious and clean assumption4. outcomes of computations. But fundamentally, we have The infinite information necessary to describe a typ- to choose a perspective. Either all digits of the initial ical real number is clearly seen in its binary (or deci- conditions are assumed to be determined from the first mal) expansion: the series of bits (digits) never ends and moment, leading to timeless physics, or these digits are - typically - has no structure whatsoever. Admittedly, initially truly indeterminate and physics includes events there are exceptional numbers for which the bits have that truly happen as time passes [17]. Notice that in a great deal of structure, like rational and computable

5 The bits of computable numbers may look random, but are fully 3 Rietdijk and Putnam have argued that (special) relativity is in- determined by a (finite) algorithm. Interestingly, there are rel- compatible with indeterministim [18–20]. However, I believe that atively simple formulas that allow one to compute any bit of π Stein and Savitt’s responses are convincing [21, 22]. Einstein is without the need to first compute all previous bits (see subsec- often quoted for his claim that “God doesn’t play dice”, but his tion III B and [25]). This clearly illustrates that these bits are position was actually quite more subtle [23]. not random but determined. 4 6 For example, when one says “let x0 be the initial condition”, one Recall that almost all classical dynamical systems are non- is effectively saying “let x0 denote an inaccessible infinite amount integrable, hence their solutions are hyper-sensitive to the initial of information about the initial condition”. condition, i.e. are chaotic. 3 both perspectives chaotic systems would exhibit random- physics, from classical Platonistic to intuitionistic mathe- ness. In the first case, from the point of view of classical matics, could well make it easier to express some concepts mathematics, all the randomness is encoded in the ini- and to rebut Dolev’s claim that passage cannot be part tial condition. In the second case, randomness happens of physics [15]. as time passes. This is the point of view of intuitionistic mathematics, as developed by L.E.J. Brouwer, where the dependence on time is essential [31]. II. INTUITIONISTIC MATHEMATICS Quite independently of the above motivations, L.E.J. Brouwer, a 20th century Dutch mathematician, refused Mathematics is the language of nature, as famously to admit ungraspable actual infinities in the foundations claimed by Galileo and most scientists. But which math- of mathematics [32]. Accordingly, for Brouwer and his ematics? Classical mathematics, i.e. God’s mathematics, followers, the mathematical continuum can’t be an infi- in which every number is a completed individual number, nite collection of completed individual points, each repre- although most of them are not computable, a mathemat- sented by a real number and each being 3 times infinite: ics that assumes omniscience, i.e. that every proposition an infinite equivalence class of infinite series of rational is either true or false. Or a form of constructive mathe- numbers, each rational being itself an equivalence class matics, i.e. a human’s mathematics concerned with finite of infinitely many pairs of natural numbers. Brouwer de- beings, a mathematics that doesn’t postulate the law of nied that the continuum is built up from independently the excluded middle because humans can’t prove every existing points: there must be more to the continuum proposition true or false. Obviously, the sort of math- than the collection of (lawlike, that is, algorithmically ematics physicists use greatly influences, almost deter- generated) real numbers [33]. Brouwer - the mathemati- mines, what physics says about nature. It is not that cian - also insisted that in some cases the future values these different mathematical languages change the ev- (of numbers) must be indeterminate, and mathematical eryday working of physics; the practical predictions are experience is a deeply non-deterministic temporality [33]. the same. However, the mathematical language strongly Reading Brouwer’s intuition about the continuum with suggests the worldview offered by physics. From God’s the eye of today’s scientists makes it hard to understand viewpoint, everything is already there, fully determined, (though see the very accessible book by Carl Posy [33]). waiting for us to discover, to sort out into categories Indeed, today’s scientists have been selected by their abil- and sets. But for humans, things are always unfinished, ity, among others, to digest modern mathematics. How- continually evolving, progressing; only the computable is ever, Brouwer was able to develop an alternative math- fully determined and only the finite can be grasped. ematics which avoids ungraspable infinities, specifically Erret Bishop [35] wrote beautifully about the construc- to avoid the standard description of the continuum as an tive mathematics side of our story (though without men- infinite collection of individual completed real numbers. tioning physics): The classicist wishes to describe God’s Moreover, as we shall see in the next section, Brouwer de- mathematics; the constructivist, to describe the math- scribes the continuum as a “viscous” medium [34], as he - ematics of finite beings, man’s mathematics for short; and I - desired. Brouwer’s construction of the continuum how can there be numbers that are not computable (...)? is the heart of intuitionistic mathematics. Does that not contradict the very essence of the con- Intuitionistic mathematics is rarely presented, and cept of numbers, which is concerned with computations? when it is, it is presented in all its complexity, includ- and Constructive mathematics does not postulate a pre- ing intuitionistic logic and axiomatics (about which not existent universe, with objects lying around waiting to be all intuitionists agree). I believe this is not necessary, collected and grouped into sets, like shells on a beach. It at least not for physics. Moreover, there is an intimate is worth also quoting Carl Posy [33]: We humans have fi- connection between intuitionistic mathematics and inde- nite memories, finite attention spans and finite lives. So terminism in physics. One may guess that intuitionistic we can fully grasp only finitely many finite sized pieces of mathematics is poorly known also because of Brouwer’s a compound thing; There’s no infinite helicopter allowing complex personality, quite an extreme idealist, while ap- us to survey the whole terrain or to tell how things will plying his mathematics to physics should be the task of look at the end of time. a solid realist, aiming at bringing physics closer to our In this section I would like to present a succinct in- experience. troduction to a specific form of constructive mathemat- The objective of this article is to allow physicists and ics: intuitionistic mathematics. I’ll emphasize the as- philosophers of science (and anyone interested) to make pect of intuitionistic mathematics which is most rele- their first steps in intuitionism and to understand why vant to physics, in particular to an indeterminsitic phys- this allows one to re-enchant physics, introducing a model ical worldview. Accordingly, I emphasize in which sense of an objective “creative” time, i.e. a dynamical time that numbers in intuitionism are processes that develop in allows for an open future and the passage of time. Much time. However, contrary to Brouwer and Bishop, I do remains to be done; I don’t claim completion of a pro- not think of these processes as driven by some ideal- gram, but rather, more modestly, to open a new door. ized mathematician [32] nor constructing intelligence [35], Indeed, changing the mathematical language used by but merely driven by randomness, a randomness con- 4 sidered as a power of nature, like other laws of nature. adapt the intuitionistic concept of number to the com- Hence, Brouwer, the father of intuitionistic mathematics, mon usage of physicists. Indeed, physicists often use would probably not have liked my presentation, because intuitionistic concepts and reasoning without realizing in my view nature (physics) plays an essential role, while it. Common sense leads physicists to realize that some Brouwer was a strong idealist. Admittedly, here math- classical mathematical facts are physically meaningless, ematics is presented as a tool to study nature, i.e. for like the infinitely precise initial conditions necessary for doing physics. deterministic chaos in classical mechanics and the finite In my presentation I limit myself to the basic tool - probability that a billiard ball tunnels through a wall in sequences of computable numbers determined by random quantutm mechanics. For example, Born, one of the gi- bits - and the two main consequences - the continuum is ants of quantum theory, wrote: Statements like ‘a quan- viscous and the non-validity of the law of the excluded tity x has a completely definite value’ (expressed by a real middle. I believe that all this is quite easy to grasp intu- number and represented by a point in the mathematical itively. continuum) seem to me to have no physical meaning [39]; The most common case found in the intuitionistic lit- see also Drossel’s view on the role of real numbers in sta- erature defines real numbers as provably converging se- tistical physics [28]. Also, scientists working on weather quences of rational numbers. This is very similar to the and climate physics explicitly use finite-truncated num- standard view of real numbers. The main difference is bers and stochastic remainders [40]. In a nutshell, for that, in intuitionistic mathematics, at every instant only physicists, “real numbers are not really real” [24]. It a finite initial sequence is determined, but the future of is only when real numbers and some other mathemati- the sequence may still be open, i.e. undetermined. As cal objects are taken literally that some odd conclusions time passes, that initial sequence develops with the addi- follow, like - again - that classical mechanics imposes a 9 tion of new rational numbers that get determined by fresh deterministic worldview . information. Hence, from the very essence of intuitionist numbers, time is essential in intuitionistic mathematics [31]. This strongly contrasts with numbers in classical III. INTUITIONISM: A FIRST ENCOUNTER mathematics where all objects, including real numbers, are considered to exist outside time, in some Platonistic world, i.e. are considered from a God’s eye point of view. Intuitionists reject ungraspable infinities. Hence, they Accordingly, it should not come as a surprise that clas- reject the typical real numbers of classical mathemat- 10 sical mathematics makes it difficult to describe the pas- ics . For an intuitionist at any time-instant, every num- sage of time in physics, while intuitionistic mathematics ber is determined by only finite information, for example allows for such a description. Somehow, time is expelled by a finite series of digits, perhaps generated according from classical mathematics; hence also the sense of the to some principle, but still only a finite number. How- flow of time is expelled from physics when it “talks” the ever, this series of digits is not frozen, but evolves as language of classical Platonistic mathematics. time passes: new digits can be added, though only a finite number at a time. More precisely, new information Here however, we don’t start with sequences of ratio- is added - again perhaps according to some principle - but nal numbers, but use sequences of computable numbers, only a finite amount of information. Hence, let me stress i.e. numbers defined by a finite deterministic algorithm, that numbers are processes that develop as time passes. a concept that was under development at Brouwer’s The new information can be entirely fresh. Brouwer, as a time, but which is much better suited for applications in proper idealist, thought of a sort of idealized mathemati- physics. Indeed, in physics one often uses computable cian who would produce this new information, e.g. by functions, like exponentials and sines, to describe the solving mathematical problems [33]. This is not very ap- evolution of simple dynamical systems. Such functions pealing to physicists, nor more generally to realists. Nev- are expressed by algorithms that map rational and com- ertheless, the idea that the passage of time, the creation putable numbers to computable numbers7. Hence, if one of new information and an indeterminate future may en- would like to keep these simple descriptions, one had better at such a basic level as numbers is highly attractive to ter concentrate on computable numbers. those who believe that time, passage and an open future We shall also relax some constraints, requiring only are essential features of reality. convergence of the sequences of computable numbers with probability one, not with certainty8. In this way we

gence with probability one, see subsection III E. Here I leave this as an open question for future work, however see [37, 38] 7 Note that there are some contrived counter-examples that build 9 Note that classical physics is compatible with a deterministic on discontinuity, see [36] worldview; we merely stress that this is not the only possibility. 8 Admittedly, introducing probabilities into these sequences may 10 Emile Borel nicely illustrated the inﬁnite amount of information re-introduce the classical real numbers into intuitionism, some- contained in typical real numbers by noticing that the digits of thing one would like to avoid. Actually, among all the examples one single real number could contain the answers to all questions presented in the next section, only the last one assumes conver- one could formulate in any human language [41]. 5

Hence, let us posit that nature has the power to contin- α(n − 1), as α(n − 1) is determined by the available ually produce truly random bits. More precisely, at every information α(0), n−1, r(1), ..., r(n−1); but it is discrete instant of time n, where n denotes a positive in- simpler to deﬁne fct as in (3). teger, a fresh new random bit, denoted r(n), comes into existence. It is totally independent of all the past, in par- 4. The sequence α(n) converges with unit probability (where the probability is over the random bits). For ticular of all previous random bits r(1), r(2), ..., r(n − 1). −n The most straightforward way to use these random bits example one may impose |α(n) − α(n − 1)| ≤ 2 to construct an intuitionistic real number α in the unit which guarantees convergence. interval [0..1] is to use binary format and to add this fresh Brouwer named the series α(n) choice sequences. Here bit to the already existing series of bits: we use equivalently the terminology of sequence or series α(n). α(n) = 0.r(1)r(2)r(3)...r(n) (1)

The above expression of α should be seen as a never end- ing process, a process that develops in time. Intuitively, as time passes the series α(n) converges to the real num- ber11 α. Crucially, at any finite time, only finitely many bits r(n) are determined and thus accessible. This simple first example of intuitionistic real numbers cleanly emphasizes the intrinsic randomness of the bits in FIG. 1: We model an indeterministic world by a Natural Ran- the binary expansion of all typical real numbers [26, 27]. dom Process (NRP) and by intuitionist numbers α that are The above expression for α is only the simplest and processes fed by the NRP. At every time-instant n, the output most straightforward intuitionistic element of the contin- of the NRP, denoted r(n), is used to determine the computable uum. A first slight generalization assumes that the series number α(n). The sequence of α(n) converges, but, generally, of random bits starts at any position; that is, one could at any finite time the sequence is still ongoing, hence the intuitionist number α is never fully determined. This illustrates add any integer α0 to α and some initial finite series of bits λ : the fact that at any finite time, α(n) contains only a finite 0 amount of information, it is only the entire sequence of α(n) α(n) = α .λ r(1)r(2)r(3)...r(n) (2) that may contain an unlimited amount of information. The 0 0 collection of all α’s recovers the continuum, though a sticky More generally, intuitionist numbers allow one to use all or viscous continuum, because at every time there are some the existing random bits in any (finite) computable way. sequences α(n) still ongoing. Hence, at any finite time, one can’t pick out a point α out of the continuum, as it is not yet Accordingly, as illustrated in Fig. 1, an intuitionist real determined where exactly that sequence will converge. Some- number α is given by a sequence of computable numbers how, it sticks to all the other α’s that had, so far, the same α(n) for all positive integers n satisfying the following sequence of α(n). This is in sharp contrast to the classical conditions: description of the continuum where every point is represented by a completed real number. Somehow, the classical contin- 1. α(0) is a given computable number. uum is analogous to the intuitionistic continuum, but viewed from the “end of time”, i.e. a God’s eye view. 2. For every integer n ≥ 1 there is a random bit r(n) and all r(n)’s are independent of each other, i.e. the random bits are assumed to be i.i.d.12 with uniform Let me emphasize several important points: probability. 1. Notice how general this definition is. Indeed, the 3. There is a computable function13, denoted fct, s.t. function fct is only required to be computable, but beyond that can be whatever. As it depends on the α(n) = fctα(n − 1), n, r(1), ..., r(n), (3) time-instant n, it can be tailored to adapt to each instant n; it may fully depend on the values of the hence, α(n) is a computable number for all n. Note random numbers r(n). Note that one does not need that the function fct doesn’t need to depend on to restrict the r(n)’s to bits; they could as well des- ignate any finite fresh information, e.g. any finite series of random bits, as illustrated in subsection III D. 11 Admittedly, we use the same notation α for the sequence and for the real number it converges to, despite the fact that these are 2. I like to think of the random bits as produced by a two different things, hoping the reader will not get confused by Natural Random Process: NRP. The basic idea is this. that if indeterminism exists, then, as time passes, 12 Independent and identically distributed. nature is able to produce true random bits r(n), 13 My definition is close to what modern intuitionists call “projec- tions of lawless sequences”, see p.68 of [33]. The function fct acts i.e. true little acts of creation which definitively dif- in a way similar to what Brouwer called a “spread”, see p.30-31 ferentiate the past, the present and the future: at of [33]. time-instant n, past random bits r(p), p < n are 6

determined, r(n) has just been created, while fu- 1...n. And there are countably infinitely many com- ture r(f), f > n are indeterminate14. These ran- putable functions. dom bits are used in a process that eventually determines a real number. Hence, numbers are pro- 8. We define autonomous numbers as those sequences cesses. Note that at any finite time, the process where the function fct does not depend on n. is still ongoing, i.e. the number is never fully de- The following sub-sections present examples of intu- termined, except for a countable subset of all real itionist numbers defined as choice sequences based on the numbers. From the end of time point of view, Natural Random Process NRP. These examples assume the set of all intuitionist numbers α cover all the binary notations and numbers in the unit interval [0..1], continuum. But that’s only the end of time view- but can easily be extended to arbitrary numbers in any point. More precisely, it is impossible to describe basis. classically a real number such that one knows (in- tutionistically or classically) that no intuitionistic real number will ever coincide with it. A. Totally Random Numbers 3. Each sequence α(n) uses one NRP. A priori, two sequences α(n) and β(n) use two independent NRP, Assume the NRP outputs at each instant, n, one bit although it would be interesting to also consider r(n), see Fig. 2. correlated NRP. Define α(n) = α(n − 1) + r(n) · 2−n. Hence:

4. Obviously, intuitionistic mathematics requires time α(n) = 0.r(1)r(2)...r(n) (4) [31], just as counting requires time. This is not surprising; new information means that there was a Here we assumed α(0) = 0, but one can easily allow any time when this information did not exist, so inde- initial α(0) and merely add it to the above. terminism implies an open future. Time passes, we all know that, and intuitionistic mathematics inte- grates that fact in its heart and builds on this fact. This is in strong contrast to classical mathematics, which assumes numbers and other objects to be given all at once, somehow from outside time, or from the end of time, existing in some idealized Platonic world. Hence, unsurprisingly, classical mathematics is at odds with time and is a poor FIG. 2: Assume the NRP outputs independent random bits, r(n), all uniformly distributed. At each time-instant n, the tool for theories that like to incorporate a descrip- bit r(n) is merely added to the bit series of α, i.e. α(n) = tion of time, or at least a description of stuff that 0.r(1)r(2)...r(n). Such choice sequences define typical real evolves in time. numbers, i.e. numbers with no structure at all. We name such intuitionist numbers Totally Random Numbers 5. At every time-instant n, all future random bits r(f), for f > n, are totally indeterminate. The conditions defining intuitionist numbers are 6. At every time-instant the process α(n) might still clearly satisfied. We name such α’s Totally Random be ongoing, possibly forever. However, it is also Numbers. They correspond to typical real numbers, and possible that at some instant nd the processes ter- all real numbers can be seen as the result of such a minates (dies) and all future α(f) equal α(nd): process, including the typical non-computable real num- 15 α(f) = α(nd) for all f ≥ nd, see examples in sub- bers . However, intuitionist numbers are much richer, section III D. as illustrated in the following sub-sections. 7. At any finite time-instant n, the theretofore determined sequences α(n) constitute only a count- B. Computable Numbers ably infinite set. Indeed, for a given computable function fct there are only finitely many sequences The computable function fct could be independent of α(n) determined by the finitely many r(k), k = all the random bits r(n) and determine one computable number. A well-known example is the number π, the ratio of the circumference of a circle to its diameter.

14 If we want to be able to conceptualize and contemplate “indeterminism” - and not stay stuck with determinism - we have to admit as a basic fact of nature that entirely new events happen, that new information gets created, for example in the form 15 Note that one may somewhat artiﬁcially construct “mixed” num- of new random bits, that the NRP considered here continually bers, e.g. every second bit is random while every other bit is given outputs. by the corresponding bit of a computable numbers like, e.g., π. 7

Computable numbers are thus clear examples of in- the function fct depends only on the first k random bits tuitionist numbers, though not the typical ones. Here, r(1), r(2), ..., r(k). In practice, our computers emulate there is nothing creative; there is no creative dynamical the NRP with movements of the mouse or the coinci- time, though computing the next digit, or next approxi- dences between a key stroke and the computer’s internal mation of a computable number, necessarily takes some clock, or similar event arising from outside the computer. time. Clearly, after time-instant k, the random bits that define Note that the bits (and digits) of computable numbers the seed are determined and the pseudo-random sequence may look random, even if one knows the function fct, is fully determined, as any computable number. i.e. one knows the algorithm that allows one to compute Accordingly, computable numbers contain finite infor- it up to any arbitrary precision. However, this appar- mation even in the limit n going to infinity, i.e. the in- ent randomness differs profoundly from true randomness, formation defining the algorithm plus possibly a finite randomness that involves the creation of new informa- set of random numbers that may determine the seed of tion. One way to illustrate this is a remarkable algorithm pseudo-random numbers. Hence, no α(n) ever contains [25] that allows one to compute the number π: more than that finite information. Consequently, even intuitionistically one can think of computable numbers as X 1 4 2 1 1 given all at once, contrary to all non-computable num- π = − − − (5) 16k 8k + 1 8k + 4 8k + 5 8k + 6 bers that all contain an infinite amount of information. k≥0 Note however, that, except for rational numbers, not all the infinite set of bits is given (determined) at once; only There are many different algorithms (i.e. different fct’s) the information needed to compute any finite set of bits that compute π, but this one is of special interest is given18 (more precisely, only the information needed here because it allows one to compute the bits of π at to compute any approximation ±2−n is given at once). any position without the need to first compute all the previous bits16. This clearly illustrates that all bits of π already exist and can be accessed, without the need to wait a time corresponding to their positions C. Finite Information Quantities - FIQs in the series of bits. This is in total opposition to the totally random numbers presented in the previous A particularly physically relevant example of intuition- subsection. Consequently, the randomness of the bits of ist number is the following. Let k ≥ 3 be a fixed positive computable numbers is only apparent, as they are all odd integer. At each time-instant n we keep the k last fully determined by a finite algorithm17. random bits: r(n − k + 1)...r(n) and forget about all “older” (previous) random bits. Define α(n) as α(n − 1) Another interesting example of computable numbers to which one adds as nth bit the majority vote of these are so called pseudo-random series of bits, as produced by last k random bits. Formally: all modern computers and heavily used in today’s cryp- Pk . If j=1 r(n − j + 1) > k/2, then α(n) = α(n − 1) + tography. There are many families of pseudo-random 2−n, else numbers, each defined by a computable function fct. if Pk r(n − j + 1) < k/2, then α(n) = α(n − 1). Each function uses a finite sequence of bits, r(1)...r(k), . j=1 for a fixed integer k, as a seed to generate highly complex sequences of bits, complex enough that without the This corresponds to what we named “Finite Informa- knowledge of the seed it is extremely difficult, possibly tion Quantities” (FIQs) in [16]. At time-instant n, the impossible in practice, to guess the next bit from only n first bits in the binary expansion of α are determined the knowledge of the function fct and of the previous and equal either to 0 or to 1, and the far down the series bits. For an intuitionist these pseudo-random series of of bits are totally indeterminate. However, interestingly, bits are just an example of a choice sequences α(n) where the k bits in the intermediate positions, n+1 to n+k−1, may have a non-trivial “propensity” to end up equal to Pk 0 or to 1. For example, if the sum j=1 r(n − j + 1) is k+1 larger than 2 , then the (n + 1)th bit is already deter- 16 The formula (5) is tailored for hexadigits, i.e. digits in bases 16, mined whatever r(n + 1) will turn out to be. Or, if the but also applies to base two, i.e. to bits. In a nutshell, the nth k−1 sum is smaller than 2 , then the (n+1)th bit is already hexadecimal is given by bπ16nc modulo 16 (where bxc denotes determined and equals 0. The next bit, i.e. bit number the closer integer smaller than x). Apply that to (5) and split the sum in one sum over k ≤ n and a sum over k > n. Realize that n + 2, might also be already determined, but there is the second sum doesn’t contribute, hence a finite computation a larger possibility that it is still undetermined, though suffices to compute the nth hexadecimal. not fully random: if the sum is rather large, then there 17 In an indeterministic world the weather in both one and two is a similarly large “propensity” that it will eventually years’ time is, today, undetermined. In two years time it will be determined. However, first the weather in one year from now will be determined. This is in strong contrast to the bits of π that can be accessed - and are thus determined - without first accessing the previous ones. 18 For a variety of indeterminacy in intuitionism see [42]. 8 get determinated to the value 1. The propensities of the further bits tend to move away from the extremal values 1 and 0 and from bit number n + k all “propensities” 1 are fully random, i.e. equal 2 . Below and in Fig. 3 an example is presented. It is useful to introduce some notations. Denote qj the propensity of the jth bit to equal 1. Hence qj = 1 means that the jth bit equals 1 and similarly qj = 0 means that the jth bit equals 0. With this notation, at each time-instant n, α(n) can be expressed as a series of propensities, reminiscent of the usual series of bits, but where each qj is a rational number: 1 1 FIG. 3: Example of a Finite Information Quantity (or FIQ). α(n) = 0.q ...q q q ...... (6) 1 n n+1 n+k−1 2 2 At each time-instant n the NRP outputs a bit r(n). The nth bits of α is determined by the majority of the last 5 random where the first n qj’s necessarily equal 0 (if bit number j is bits r(n − 4)...r(n). At step n the bits n + 1 to n + 4 of α are 0) or equals 1 (if bit number j is 1), while qn+1 to qn+k −1 already biased either towards 1 or towards 0, indicated here equal rational numbers between 0 and 1 corresponding in parenthesis by the propensities to eventually be determined to the propensity that they eventually, after k new time- (at later time-instants) by future outputs of the NRP to the instants, acquire the value 1 or 0 depending on future bit value 1. NRP outputs. Here is an example of the first steps of a FIQ (finite information quantity). Assume k = 5 and the first 4 bits may ask what the value of k is for the dynamical sys- are given and all equal to 0, hence we set α(0) = α(1) = tem one is considering. Such questions are usually hid- ... = α(4) = 0. Assume the first 4 outputs of the NRP den in classical statistical mechanics, where one assumes equal 1101. Below we illustrate a possible growth of the that all initial conditions that satisfy some constraints, FIQ sequence α using the propensity notations: like fixed energy, are equi-probable. Here also, one could ask whether real-valued initial conditions with correlated r(1)...r(5) = 11011 (7) bits, with correlation length k, are not more likely than 3 7 11 1 others. α(5) = 0.000011 (8) 4 8 16 2 r(2)...r(6) = 10110 (9) 1 3 1 5 1 D. Mortal Numbers α(6) = 0.000011 (10) 2 4 2 16 2 r(3)...r(7) = 01100 (11) Here is another example of intuitionist numbers de- 1 1 1 5 1 fined by a sequence much inspired by Posy’s example α(7) = 0.0000110 (12) based on the Goldbach conjecture, see [32, 42]. In this 2 4 8 16 2 example the sequence may terminate (die) after a fi- In this example, at time-instant n = 5, the NRP out- nite time or may continue forever, depending on chance, puts r(5) = 1 which determines the 5th bit of α(5): i.e. depending on the outputs of the NRP. We name such q5 = 1. Furthermore, the correlation present in this kind numbers mortal, because the corresponding process may of random number is such that the 6th bit of α is al- suddenly end. 19 ready determined: whatever r(6), one has q6 = 1. Still Define α(1) = 0.1 and set a flag f = 1. Assume that at the time-instant n = 5, the 7th bit is already biased at time-instant n the NRP outputs not just one bit, but towards 1: 3 out of the 4 possible random bits r(6) and outputs n (independent) random bits r(n)1, r(n)2..r(n)n. r(7) determine that 7th bit to the value 1, hence, at that For all n ≥ 2: time-instant n = 5, the propensity q = 3 , as indicated 7 4 . If the flag is set to 1 (f = 1) and at least one of the in (8). And so on, here assuming r(6) = r(7) = 0. At n random bits {r(n)j}j=1..n equals 0, then α(n)n = 1, time-instant n, at least n bits of α are determined and i.e. one adds a bit 1 at the end of the bits defining α(n) at most k − 1 bits are biased towards 0 or 1, i.e. have a and the flag remains at 1. propensity different from 1 . 2 . If the flag is set to 0 or if all random bits are equal It seems that all indeterministic physics can be mod- to 1, then the series terminates, i.e. α(n) = α(n − 1) and elled using only FIQs [16]. Note that the FIQs presented the flag is set to 0. here slightly differ from those of [16] because here we may have correlations between the bits of α. This allows one to define arithmetic for FIQs, see subsection III F. Let us conclude this sub-section with a comment on 19 Note that the value of this flag f(n) can be computed from the the parameter k that enters the definition of a FIQ. One series r(1)...r(n), but it is simpler to keep track of it. 9

Formally: value of the function at the point(s) of discontinuity. f(n) = f(n − 1) · 1 − r(n)1 · r(n)2 · ... · r(n)n (13) Yet another mortal number is defined as follows. Let 1 −n f(n) = 0 ⇒ α(n) = α(n − 1) (14) r(n) = ±1 and define α(n) = 2 + r(n) · 10 . The series terminates at time-instant n when, by chance, the f(n) = 1 ⇒ α(n) = α(n − 1) + 2−n (15) previous n/2 random bits r(j) happen to have the same Interestingly, since the probability that the flag is value, all +1 or all -1, and n is even and larger or equal switched to 0 decreases exponentially with the time- to 4. Clearly, for n = 4 the series terminates whenever r(3) = r(4), i.e. with a probability of 1 : P (n = 4) = instant n, there is a finite chance that the flag remains 2 stop 1 at 1 and the α(n)’s increase continuously, approaching 2 . For n = 6 the series terminates whenever r(6) = 1. The probabilities that the series of α(n) terminates r(5) = r(4) 6= r(3), where the last constraint implies that at time-instant n is 2−n. Hence, the probability that the the series did not already terminate at instant n = 4: series never dies equals: Pstop(n = 6) = 1/8. However, computing Pstop(n) for arbitrary even n is non-trivial because for n ≥ 8 the 4 P rob(endless) = P rob(α = 1) last random bits do not overlap with the early bits. The −n general formula reads, for all n ≥ 3: = Πn≥2(1 − 2 ) ≈ 0.5776 (16) bn/2c X Consequently, a priori one doesn’t know whether the P (2n) = 1 − P (2j) · 2−n (17) series of α(n) will terminate and settle to a rational stop stop j=2 number 0.11..1 with k 1’s, where k is a priori undetermined, or whether the series converges to 1 (recall that where bn/2c denotes the largest integer smaller or equal 1 = 0.11..1.., with an unending series of 1’s). Hence, to n/2. The first term in (17) is the probability that the as long as the sequence α(n) hasn’t terminated, it is series did not stop before instant 2bn/2c and the factor undetermined whether α = 1 or α < 1, i.e. α = 1 is not 2−n the probability that the last half of bits are all equal true and α 6= 1 is not true, an example of intuitionistic and the series did not stop after instant 2bn/2c but before logic in which the law of the excluded middle doesn’t instant 2n. hold. This should not surprise us, because the question Accordingly, the probability that the series terminates whether α equals 1 or not is a question about the reads: future, the open future! If one asks a question about X the weather in a year’s time, no one is surprised that Pstop = Pstop(4) + Pstop(2n) (18) the answer is not yet determined, i.e. that the state- n≥3 ment “it will rain on the fieldhockey pitch in Geneva bn/2c in precisely one year from now” is presently neither 1 X X X = + 2−n − P (2j)2−n (19) true nor false. At least, that’s the case in an indeter- 2 stop ministic world. As Posy put it [33]: The law of the n≥3 n≥3 j=2 bn/2c excluded middle fails because objects and the world are 3 X X = − P (2j)2−n (20) not-determinate, so truths about them are indeterminate. 4 stop n≥3 j=2 Here is another example of a mortal number which 3 1 < (21) might oscillate forever between below and above 2 . 4 The rule for the flag is as above and if f(n) = 0, P −n 1−k then α(n) = α(n − 1), again as above. However, where we used n≥k 2 = 2 . Accordingly, the 1 −n probability that the series never terminates is larger if f(n) = 1, define α(n) = 2 + (−2) , for all n = 2, 3, 4, .... Accordingly, as long as the sequence than a quarter. Numerically I found Pendless ≈ 0.31668. doesn’t terminate, α oscillates between below and above Importantly, this number is strictly positive. 1 1 1 1 1 1 1 1 1 1 2 : 2 , 2 + 4 , 2 − 8 , 2 + 16 , 2 − 32 , .... This mortal number illustrates the concept of viscosity: there is no way to It is fun to invent further mortal numbers. Appendix cut the continuum in two, as there always are intuitionist B presents some useful general formulas to evaluate the numbers which are never definitively on one side of the dying probabilities of mortal numbers. cut, nor definitively on the other side, as here illustrated Finally, no autonomous number can die with certainty. 1 However, the next subsection shows that autonomous with a tentative cut at 2 . Note that a straightforward consequence is the absence of step functions and more numbers can die with probability one. generally of discontinuous total functions of the unit interval20, as it would be impossible to determine the E. Autonomous numbers

So far all non-computable numbers we presented use 20 A function is total if it is deﬁned everywhere. a function that depends explicitly on the time-instant n, 10

i.e. are not autonomous. It is difficult to see how to guar- 1. First, the integers k and n have to be constructable, antee convergence without such a dependence. However, not merely exist in some Platonistic sense. it is natural to ask for autonomous numbers, i.e. numbers defined by a function independent of n. Here is such an 2. Second, these constructions should be possible at example: the time-instant nnow when the assertion is made, i.e. the constructions cannot access any of the fu- α(n + 1) = α(n) + r(n) · η · α(n) · 1 − α(n) (22) ture outputs r(f) of the NRP, f > nnow, because these future r(f) are not yet determined. where r(n) = ±1 denotes the outputs of the NRP and η Consequently, the above defines only a partial order, as is any given computable number between 0 and 1 which the following does not hold for all α and β: determines the rate of convergence. This sequence has two fixed points at 0 and 1. Inter- (α = β) ∨ (α < β) ∨ (α > β) (24) estingly, for any value of α(n) the probability (over the r(k) for k > n) that the sequence converges to 1 equals where α = β means that both sequences converge to α(n). This sequence converges only with probability one, the same real number whatever future bits are produced as some sequences of random numbers r(n) do not lead by the NRPs. A first example is provided by the mortal 21 number α that oscillates between below and above β = 1 to convergence, like an infinite√ sequence of alternating 2 η+2− η2+4 as long as it didn’t die, see subsection III D. As another +1 and −1 and α(0) = . 2η example, consider two FIQs, α and β, based on two inde- This example is inspired by the old gambler’s ruin pendent NRPs, and assume that up to the present time- problem [43]. Interestingly, this has inspired (some- instant nnow all the bits determined by then coincide: times unknowingly) early attempts to solve the quantum α(n) = β(n) for all n ≤ nnow. That does not guarantee measurement problem by adding stochastic terms to the that at the next time-instant the next bits still coincide. Schrödingerequation [44, 45]. This led to spontaneous But even if they do not coincide, e.g. α(nnow + 1) = 1 collapse models, quite reminiscent of intuitionist num- and β(nnow + 1) = 0, it could still be the case that the bers. two sequences converge to the same value if, by chance, all future bits of α are 0’s and all next bits of β are 1’s. All this may seem surprisingly, but only if one forgets F. Arithmetic and logic of intuitionist numbers that intuitionist numbers are processes that develop as time passes22. Numbers are there to count and more generally to Here, again, it would be only at the end of time – compute and to compare. Intuitionist numbers can if there were such a thing– that (24) could hold: at straightforwardly be used in any algorithmic computa- any finite time we have no “infinite helicopter”, using tions. It suffices to apply the algorithm to each term of Posy’s illustration, to see how the sequences will develop. the sequence α(n). For example, the addition α + β is given by the sequence α(n) + β(n), and the exponential Let us consider one more definition. We say that α of α is given by the sequence exp{α(n)}. Since we and β are apart, denoted α#β, iff one can construct two defined intuitionist numbers as sequences of computable integers k and n such that for all m ≥ 0 numbers (and not merely rational numbers), every |α(n + m) − β(n + m)| ≥ 2−k (25) computable function (i.e. algorithmic function) can be applied to any intuitionist number, though care has Intuitively, α and β are apart if one can put one’s finger to be paid to the convergence as, e.g., in the sequence in between α and β. Interestingly, one can prove that 1/α(n) that diverges if α = 0. if two intuitionist numbers are not apart, i.e. if one can constructively prove that one can’t put one’s finger in The set of intuitionist numbers can be ordered (see between, then they are equal [33]: section 2.2.2 of [33]): α > β iff one can construct two integers k and n such ¬(α#β) ⇒ (α = β) (26) that for all m ≥ 0 From the above one is temped to deduce (α 6= β) ⇒ α(n + m) − β(n + m) ≥ 2−k (23) (α#β). But the latter is wrong! The assumption α 6= β

At first sight this may look very similar to well-known definitions in classical mathematics. However, there are serious caveats: 22 Of course, at any time-instant, n, a version of (24) (where the ordering is over the computable numbers), holds for the computable approximations, α(n) and β(n), of α and β that are determined at n. However, for all n, there are (α, β) such that (24) does not hold at time-instant n where the ordering is defined 21 1 More generally, let r(n) = sign[ 2 − α(n − 1)]. by (23). 11 does not allow one to construct the two integers k and quite different proofs. An example is Gleason’s theorem, n needed to prove “apartness”, a nice illustration of the which plays a central role in the foundations of quan- non-validity of the law of the excluded middle in intu- tum theory. Hellman noticed that the original proof by itionistic mathematics. Gleason is not constructive, hence in particular not valid The fact that the law of the excluded middle fails in intuitionistically [46]. However, a few years later, Rich- intuitionism is surprising to classical eyes. But, if one man and Bridges gave a very different constructive proof thinks more about this, it is very natural and even nec- of the original Gleason theorem [47]. essary in an indeterministic world. Indeed, at any finite In summary, there are deep differences between classi- time, there are propositions that can’t be proven in a fi- cal and intuitionistic mathematics. These difference are nite number of steps using only the information existing precisely those needed to describe indeterminacy and in- at that time. As previously stated, the excluded mid- determinism, both in the physical world and in mathe- dle fails because the world is not-determinate, so truths matics. What is important for the practitioner is that about it are indeterminate. And - my emphasize - this all of physics that can be simulated on a (classical23) holds also for mathematical objects. computer can also be derived using only intuitionistic A basic logical consequence runs as follows. Assume mathematics. that a proposition P is true, i.e. there is a finite proof of P . Then, it is impossible to prove P false, hence P ⇒ ¬¬P . However, because of the lack of the excluded IV. INDETERMINISTIC PHYSICS AND middle, a proof that it is impossible to prove P false is INTUITIONISTIC MATHEMATICS not a proof of P : ¬¬P 6⇒ P . In physical terms: the impossibility to prove that it will rain in a year’s time is no evidence that it will be sunny. Indeed, the weather in If one wants to seriously consider the possibility of a year’s time could merely be undetermined. However, indeterminism, i.e. to negate the necessity of determin- a proof of the impossibility to prove that it will not be ism, it makes plenty of sense to postulate that nature not rainy in a year time implies that it will not be rainy: is able to continually produce new information, here ¬¬¬P ⇔ ¬P . modeled by true random numbers. Indeed, the very Because of the non-validity of the law of the excluded meaning of indeterminism is that nature is able of true middle, some classical theorems are not valid intuition- little acts of pure creation: the nth output of the NRP, istically. However, there are also some new theorems, r(n), was not necessary before it happens, hence was invalid classically but valid intuitionistically, and some totally impossible to predict, but after time-instant n it theorems valid both classically and intuitionistically, but is a fact of nature. These random numbers allow one to that require very different proofs. Let me illustrate these define choice sequences (in Brouwer’s terminology) that 3 cases. represent the continuum, but with numbers that are First, let’s consider a theorem that holds in intuition- not all given at once, contrary to Platonistic/standard ism, but not classically: all total functions (i.e. functions mathematics, numbers that are processes that develop defined everywhere) are continuous. This is known as as time passes. Table I illustrates the close connection Brouwer’s theorem. This excludes, among others, step between concepts in indeterministic physics and in functions from the collection of total functions. Indeed, intuitionistic mathematics. according to intuitionism, at no finite time could one define the value of the function at the point(s) of disconti- Below I briefly comment each line of Table I. nuity, because, at that time, some choice sequences are still fluctuating above and below that point. Note, how- 1. In indeterministic physics the past, present and fu- ever, that intuitionism accepts arbitrarily close approx- ture are not all given at once, contrary to the block imations to discontinuous functions, i.e. functions with universe view. Analogously, in intuitionistic mathe- arbitrarily fast transitions from one value to another. matics the continuum is described by numbers that Second, here is an example of a classical theorem not are not all given at once; for most of them their se- valid intuitionistically. The classical intermediate value ries of digits is still an ongoing process, contrary to theorem is not valid in intuitionism: for a continuous classical real numbers, whose infinite series of deci- function f(x) for which there exist real numbers a < b mals is assumed to be completed since ever and for such that f(a) < 0 < f(b), one can’t construct a point ever. x0 s.t. a < x0 < b and f(x0) = 0. At first sight this may look shocking to classical eyes. However, what physicists really need in practice is a weaker form of the intermediate value theorem, a form that holds intuitionistically: 23 Some quantum information processing - on so-called quantum under the same assumptions and for all > 0 one can computers - cannot be efficiently simulated on classical comput- construct an x s.t. a < x < b and |f(x )| < . ers. However, all quantum information processing can be sim- 0 0 0 ulated in a finite time on a classical computer. Accordingly, Finally, there are also theorems that hold in both clas- everything that can be simulated by a “quantum computer” also sical and intuitionistic mathematics, but which require holds intuitionistically. 12

Indeterministic Intuitionistic 6. Some things merely are, but most things are chang- Physics Mathematics ing, events are becoming. Becoming is central in in- 1 Past, present and future Real numbers tuitionistic choice sequences, in the way bits come are not all given at once are not all given at once into existence one after the other. 2 Time passes Numbers are processes 7. In indeterministic physics the future is open, again 3 Indeterminacy Numbers can contain only is sharp contrast to the block universe view. Con- ﬁnite information sequently, statements and propositions about the 4 Experiencing Intuitionism rests on future need not be either true or false. For example, the proposition “it will rain in exactly one grasping objects year time from now at Piccadilly Circus” is nei- 5 The present is thick The continuum is viscous ther true (because it is not predetermined that it 6 Becoming Choice sequences will rain), nor is it false (because it is not prede- 7 The future is open No law of the excluded middle termined that it will not rain). Hence, in a world (a proposition about the future with an open future, the law of the excluded middle can be neither true nor false) does not always hold. Analogously, in intuitionistic mathematics, as long as a number is not completed, TABLE I: This table illustrates the close connections between there are statements about it that are neither true the physicist’s intuition about indeterminism in nature and nor false, and in intuitionistic logic the law of the the mathematics of intuitionism. excluded middle doesn’t hold.

2. In indeterministic physics, time is modeled as pass- V. INTUITIONISTIC AND/OR CLASSICAL ing, contrary to the block universe in which every- MATHEMATICS? thing is fixed and frozen. Analogously, in intuitionistic mathematics the numbers that fill the contin- There are several mathematical languages. It is not uum are processes that develop as time passes. that one is correct and the other one wrong. Hence, the question is not intuitionistic or classical mathemat- 3. In intuitionistic mathematics, at any time, num- ics. Both exist on their own, independently of physics; bers, like all mathematical objects, are finite, in both have their beauties and roles. However, the differ- particular they contain finite information. Hence, if ent languages make it clear that some conclusions one is the complexity of the evolution requires unbounded tempted to infer from physics are, actually, inspired by information, as in classical chaotic dynamical sys- the language, not by the facts. A central claim of this pa- tems, then the evolution is necessarily indetermin- per is that intuitionistic mathematics is better suited to istic. describe a world full of indeterminacy, a world in which time passes and the future is open. However, admittedly, 4. Physics is not only about fascinating technolo- intuitionism does not prove that our world is indetermin- gies and highly abstract and sophisticated theories. istic, it only proves that physics is equally compatible Physics should tell stories about how the world is with an indeterministic worldview as it is with a deter- and functions. Physics should help us to develop ministic one. our intuition, like, e.g., how the kangaroos manage Intuitionistic mathematics is a form of constructive not to fall off Earth, how the moon drives tides, mathematics, i.e. all objects are defined by finite how transistors allow our computers to operate and information at all times. Additionally, intuitionism how time passes. In a nutshell, physics should not incorporates a dynamical time. As we have seen, in be too far from our experiences. Similarly, mathe- intuitionistic mathematics, at every time instant, there matics should not be too far from our intuition and is only finite information and there are ever ongoing in particular should avoid the ungraspable infinities processes; this is the mentioned dynamical time. I like that plague classical Platonistic mathematics. to call this dynamical time “creative time”, as truly new information is continually created [55]. This new 5. Many physicists have the intuition that the present information feeds into the mathematical objects and is thick, that it can’t be of measure zero, infinitely is necessary to provide a mathematical framework to squeezed between the past and the future. The describe indeterminism in physics and the passage of present and passage are necessary ingredients to time: there is a time before and a time after the creation tell stories. As Yuval Dolev nicely put it, “To think of the new bits of information. Contrary to Bergson, I of an event is to think of something in time” [15]. do see this dynamical/creative time as entirely objective, In intuitionistic mathematics the continuum is vis- it is a purely natural process. One consequence is that cous; it can’t be neatly cut in two. This might well the law of the excluded middle and the principle of provide the thickness that our model of the present sufficient reason don’t hold in intuitionism, as, I believe, needs for a faithful description. it has to be the case in any theory that faithfully 13 describes the world as indeterminate and its evolution numbers, the numbers that are at the basis of the scien- as indeterministic. tific language. In classical mathematics, these numbers are called “real”, for historical reasons. For intuitionists, those “real” numbers are never completed, at least never in a finite time, which are the only times that there VI. CONCLUSION are. Their infinite number of digits, coding infinite information, are, again, only the view from the end of time. Physicists produce models of reality. The models Accordingly, “real” numbers hide all the future far down should be as faithful as possible, in particular produce in their series of digits: “real” numbers are the hidden correct empirical predictions. This is the first criterion variables of classical physics [50]. It is the common usage to judge physical models. However, it is not the only one. of real numbers in physics that produces the illusion that Physical models should also allow humans to tell stories the future is already fixed. In a nutshell, “real numbers about how nature does it, e.g. how the moon drives the are not really real” [24], a fact deeply incorporated in tides, how white bears and kangaroos remain on Earth, intuitionistic mathematics. how lasers operate and how time passes. One should not Finally, replacing the real numbers physicists use with confuse the model with reality, hence our models can at “random” numbers, i.e. intuitionist numbers based on best help us to gain understanding and develop our in- natural random bits, as presented here, might turn out to tuitions of how nature does it. From this point of view, help overcoming the conundrum in which today’s physics physics should model the passage of time. In this article is locked, between a quantum theory full of potentialities we argued that this can be done by modeling dynamical and indeterminacy and the block universe view provided “creative” time at the level of numbers, by the contin- by general relativity. Indeed, it is the Platonistic mathe- ual creation of new information, modeled by new inde- matics that physicists use unconsciously that leads them pendent random bits. This should go down all the way to trust the block universe view. into the mathematical language we use to formulate our physical models. Surprisingly, such a language has already existed for a century, with Brouwer’s intuitionism and his choice sequences, and especially with Kreisel’s Acknowledgment lawless choice sequences [48] In intuitionism, the law of the excluded middle holds only if one assumes a look from the “end of time”, that Useful critics and comments by Carl Posy, Yuval Dovel, Gilles Brassard, Ben Feddersen, Barbara Drossel, Valerio is, a God’s eye view [17]. But at finite times, intuitionism Scarani, Christian Wüthrich, Flavio Del Santo, Jon Lench- states that the law of the excluded middle is not neces- ner, Tein Van der Lugt, Stefan Wolf and Michael Esfeld are sary, that there are propositions that are neither true nor ackowledged, as well as the many colleagues who send me false, but merely undetermined. Such propositions might comments on my Nature Physics contribution [17]. Financial be about the future, the open future, as already empha- support by the Swiss NCCR-SwissMAP is greatfully acknowl- sized by Aristotle [49]. But they might also be about edged.

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Quantum physics is “officially” non-deterministic, fol- [34] C.J. Posy, Epistemology, Ontology and the Continuum, lowing the standard Copenhagen interpretation. How- in E. Grosholz and H Breger (eds.) The Growth of Math- ever, most physicists seem not to entirely admit that ematical Knowledge, Kluwer, 2000. state of affairs. In particular, high-energy physicists [35] E. Bishop, Aspects of Constructivism, Notes on the lec- tures delivered at the Tenth Holiday Mathematical Sym- mostly swear only by relativity and the (beautiful in their posium held at the New Mexico State University, Las eyes) block universe, in which everything is frozen since Cruces, December 27-3l, l972. ever and for ever. Physicists and philosophers working on [36] M.B. Pour-El and I. Richards, The wave equation with the foundations of quantum physics also mostly adhere computable initial data such that its unique solution is to alternative interpretations. Hence, in this appendix, I not computable, Advances in Mathematics 39, 215-239 summarize my understanding of quantum indeterminacy. (1981). Surprisingly, the official Copenhagen interpretation, [37] B. van Fraassen, Probabilistic Semantics Objectified: with its top-down indeterministic causation from large Postulates and Logics, J. Phil. Logic, 10, 371-391 (1981). classical measurement apparata down to microscopic [38] C. Morgan and H. Leblanc, Probabilistic Semantics for Intuitionistic Logic, Notre Dame J. of Formal Logic, 24, quantum systems, is rarely seriously defended by physi- 161-180 (1984). cists because of its vagueness, a weakness that admittedly [39] Born, Max (1969). Physics in my Generation, New York: plagues this standard interpretation. Hence, only spon- Springer-Verlag. taneous dynamical collapse models aim at providing a [40] T.N. Palmer, Stochastic Weather and Climate Models, serious physical description of the indeterminism present Nature Reviews— Physics 1,463-471 (2019). in quantum physics [11–14]. These models assume a fun- 15 damental random process that presents the Schrödinger circumventing quantum indeterminism [10]. Here, the equation as an approximation to a more fundamental basic idea is that all possibilities, however small their stochastic evolution equation. The stochastic term in quantum probabilities, are realized in parallel. However, the equation is always present and active, though its ef- physicists don’t realize this directly, because they also are fect is small for systems with a few particles and huge for in “superposition” of experiencing all possibilities. And system composed of many (Avogadro number) particles. the physicists’ friends also “see” their friends with dif- The random process, typically a Wiener process, is pre- ferent experience, i.e. the friends and eventually the en- sented as fundamental, not decomposable into more basic tire universe evolve into an enormous superposition state elements. It is not a field; it is merely a power of nature in which everything that could happen actually happens (a disposition) that affects all quantum evolutions. [53, 54]. The evolution of this multiverse is described The alleged incompleteness of quantum theory goes by the deterministic Schrödingerequation. Hence, inde- back to the hugely influential (and beautiful) paper by terminism is expelled from this view, at the cost of an Einstein, Podolsky and Rosen in 1935 [51]. It sparked enormoulsy complex multiverse. Apparent indetermin- large research programs, which, on one side, definitively ism is merely due to our “splitting” (superpositions) into excluded the possibility of a completion in terms of local several persons with different experiences. variables [8, 52], and on the other side led to Bohmian mechanics [2–4]. Interestingly, the first side brought along the understanding that quantum indeterminism APPENDIX B: GENERAL FORMULAS FOR can be mathematically proven from two natural assump- MORTAL NUMBERS tions: no arbitrarily fast influences and no superdeter- minism, plus the experimental fact that Bell inequalities Denote q(n) the probability that a mortal series termi- can be violated [5–8]. nates at instant n, given that it did not stop before: The second consequence of the EPR paper, i.e. Bohm’s model, can be intuitively understand as follows. In q(n) = P rob(stop @ n|did not stop before) (B1) Bohmian mechanics one adds positions of every particle to the usual quantum state and adds evolution equations and p(n) the probability that the series stops at instant of those particles’ positions to the Schrödinger equation. n and did not stop before: All these evolution equations are deterministic and the apparent indeterminism is all due to the unknown initial p(n) = P rob(stop @ n & did not stop before) (B2) particles’ positions. As Bohmian mechanics’ predictions are equivalent to standard quantum theory, it has to in- We get, for all n ≥ 2: corporate some non-locality. And indeed, the particles’ n−1 positions evolutions depend not only on the quantum X state nearby, but on the entire quantum state, including p(n) = q(n) · 1 − p(m) (B3) arbitrarily distant parts of the quantum wave-function. m=1 n−1 Bohmian mechanics is a perfect example that all indeter- = q(n) · Πm=1(1 − q(m)) (B4) ministic theoretical models can be complemented such that the complemented model is deterministic. In full The equality between (B3) and (B4) is easy to prove generality, it suffices to assume as supplementary vari- by induction, starting from p(1) = q(1). Hence: ables all results of all possible future measurements, while X making sure that these remain inaccessible until the mea- Pendless = 1 − p(n) = Πm≥1(1 − q(m)) (B5) surements take place [24]. In the case of Bohmian me- n≥1 chanics, this is done by assuming that the particles’ positions remain hidden, i.e. that one can’t gain more infor- Note that if q(n) = 2−n−1, then p(n) ≈ 30%, and if −n−1 1 mation about them than what is provided by the usual p(n) = 2 then q(n) = 2+2n . quantum state and measurement results. This “trick” There are not many infinite products with simple for- guarantees also that the non-locality intrinsic to Bohmian mulas. Consider the case q(n) = 1/n2, for all n ≥ 2, with mechanics can’t be used for faster than light (actually, for q(1) = 0. For example, at each time instant r(n) consists arbitrarily fast) communication. of 2 n-dits, r(n) = r1(n), r2(n) with rj(n) ∈ {0, 1, ..., n − The many-world interpretation of quantum theory is 1} and the series terminates iff r1(n) = r2(n) = 0. Then 1 1 another example of quite an extreme view that aims at Pendless = Πn≥2 1 − n2 = 2 .