arXiv:1909.03697v2 [quant-ph] 26 Nov 2019 s tt opt h usqeteouin eetdevel- Recent evolution. subsequent the compute the and to universe it the of use state to the required about is information future, s complete the In predict (demon), perfectly intelligence” to [8]. able “superior posedly thermodynamics hypothetical of law the second fact, the with of incompatible be derivation even the might determinism that and argum [5–7], information-theoretic on based con severe difﬁculties to ceptual leads determinism phys of classical advocacy the in that determinism showing of critique the into breathed 4]. developed [3, systematically understood fully was implications systems its and chaotic af century, of twentieth the theory of the half second the and in [2] fostered determi were classical (see about doubts physics These classical thereof). references in indeterminism argu M. all genuine and Schr¨odinger standpoints– for E. different with Exner, physicists –admittedly F. preeminent who Boltzmann, Born, of L. works of likes the determinis the in this epis- remarkably to exceptions view, merely notable tic as ﬁnd indeterminacy could introduce one temic), to taken is (which t [1]. precision” and inﬁnite world with physical states the past of its states retrodict and future the nature predict of int to superior laws gence a enable the will of conditions boundary knowledge the appropriate sufﬁcient be a to that assumed usually “view is determinism demon), Laplace’s istic is electrodynamics) Maxwell’s and mechanics ton’s oee,i sol nrcn er htnwlf a been has life new that years recent in only is it However, physics statistical of development the by stimulated Yet, New- (i.e., physics classical that accepted generally is It H SCASCLPYIS(IN)DETERMINISTIC? PHYSICS CLASSICAL IS WHY ettn aosagmn u oLpae(nw as (known Laplace to due argument famous a Restating . nata au feeypyia uniy ihisinﬁnite its with quantity, physical on every relies of physics value in classical actual indeterminism of an genuine determinism introduced alleged have the that to theory ﬁrst the nnt precision inﬁnite h lsia n h unu esrmn rbes n pro and causation”). problems, “top-down and measurement pro models” quantum measurement the “quantum and the classical d of) the become (part values as indeterminate mechanics the quantum i how indeterministic explaining any e of that strong evident problem as make (such we Finally, open could issues. physics indeterministic an that quantum on of based interpretations indeterminism well-known classical to those link also We nhsclcneune,w osdrpsil alternativ physi possible a consider of we attribution consequences, the unphysical into translates (which precision hsc ihu eemns:AtraieInterpretation Alternative Determinism: without Physics lsia hsc sgnrlyrgre sdeterministic, as regarded generally is physics Classical aut fPyis nvriyo ina otmngse5 Boltzmanngasse Vienna, of University Physics, of Faculty ru fApidPyis nvriyo eea 21Geneva 1211 Geneva, of University Physics, Applied of Group utinAaeyo cecs otmngse3 -00Vie A-1090 3, Boltzmanngasse Sciences, of Academy Austrian .Bidn nrcn nomto-hoei ruet sho arguments information-theoretic recent on Building ). nttt o unu pisadQatmIfrain(IQOQ Information Quantum and Optics Quantum for Institute nt nomto quantities information ﬁnite Dtd oebr2,2019) 27, November (Dated: determin- lvoDlSanto Del Flavio ioa Gisin Nicolas nism have elli- ents up- ics, ter ed n o - - neemnsi nepeain fcasclphysics. classical of interpretations indeterministic e a enn omteaia elnmes ed to leads numbers) real mathematical to meaning cal essaesbett h aelmttosipsdb h law the ( by physics imposed pro- limitations of these same the such, to physical subject As are a cesses (computa- (communication). manipulated into transmitted be embodied and to tion) is it sci- allowing information natural (encoding), the system the if in only value ences meaningful a concep acquires well-formalized information conclusion mathematically the abstract, to the led that thermodynamics) infor- quantum quantum and of physic mation frameworks to the application within its blossomed and (mainly theory information of opments alc’ eo,adhnedtriim ed otocat- two of to problem the leads problems: determinism, of egories hence and demon, Laplace’s of lo o h prto faLpaindmnal oanalyze to able demon Laplacian a of operation Univer the entire for the allow in resource He physical insufﬁcient [...] is Universe. observable there by the above in from particles bounded of contain, number be the could it would atoms limit it of a number include world, the would physical which constraints our physical hypothet- in to (even subject constructed were be demon to this, a ically) About such “If evolution. concludes: subsequent ma- Blundell to the be to supposed compute then are to Universe, whole nipulated that the systems of information physical the encode the of capability ory trrtto fpyiswudhv oda ihthe with deal to have would physics of nterpretation 1 sopsdt unu ehnc hti considered is that mechanics quantum to opposed as h ai,mtpyia supinta hr exists there that assumption metaphysical tacit, the rdtrie iis(hc ename we (which digits predetermined n ihu eott nomto-hoei ruet nR gene in more arguments information-theoretic a to in resort discussed without recently and been have problems Similar egne,a ela oeptnilproblematic potential some as well as mergence), h omro hs rbesi ietyrltdt h mem- the to related directly is problems these of former The inﬁnitesimals ehnc.I atclr epooeamdlof model a propose we particular, In mechanics. opyis ecalneti iwb arguing by view this challenge We physics. to oeiesfrpsil ouin eg,“collapse (e.g., solutions possible for ideas pose FQ) oevr edsusteperspectives the discuss we Moreover, (FIQs). triae rbe nw ntecnetof context the in known problem a eterminate, lm.W ics oesmlrte between similarities some discuss We blem”. adursprinciple Landauer’s -00Ven,Austria Vienna, A-1090 , . 1 igta h rnil finﬁnite of principle the that wing n,Asraand Austria nna, ,Switzerland 4, fCasclPhysics Classical of s I), 9) ntefc fthis, of face the In [9]). inﬁnities rnil of principle n h problem the and f [10]. ef. a context ral eto se of t nce on s s 2 even a relatively limited macroscopic physical system” [7].2 as a fundamentally indeterministic theory, contrarily to its The problem of inﬁnitesimals, instead, is related to the standard formulation. The latter can be considered, in this question of whether it would be possible to know, even in view, a deterministic completion in terms of (tacitly) posited principle, the necessary boundary conditions with inﬁnite pre- hidden variables. This situation resembles (without however cision.3 Moreover, do these inﬁnite-precision conditions (i.e., being completely analogous) the contraposition between the with inﬁnite predetermined digits) exist at all? As Drossel standard formulation of quantum mechanics –which consid- pointed out, this is related to the problem of determinism in so ers indeterminism an irrefutable part of quantum theory– and far as the “idea of a deterministic time evolution represented Bohm’s [13] or Gudder’s [14] hidden variable models, which by a trajectory in phase space can only be upheld within the provide a deterministic description of quantum phenomena – framework of classical mechanicsif a point in phase space has adding in principle inaccessible supplementary variables. inﬁnite precision” [8]. To address the problem of inﬁnites- imals, and in doing so challenging determinism both at the epistemic and ontic levels, one can again use an argument that relates information and physics, namely the fact that ﬁ- nite volumes can contain only a ﬁnite amount of information Before further discussing, in what follows, the arguments (Bekenstein bound, see [6]). for alternative interpretations of classical physics without real In this respect, one should realize that classical physics is numbers –and the issues that this can cause– some general not inherently deterministic just because its formalism is a set remarks on indeterminism seem due. It appears, in fact, that of deterministic functions (differential equations), but rather a common misconception concerning physical indeterminism, its alleged deterministic character is based on the metaphys- is that –in the eyes of some physicists and philosophers–this is ical, unwarranted assumption of “inﬁnite precision”. Such a taken to imply that any kind of regularity or predictive power hidden assumption can be formulated as a principle –tacitly looks unwarranted –the supporter of determinism would ask: assumed in classical physics– which consists of two different how can you explain the incredible predictive success of our aspects: laws of physics without causal determinism? Yet, an indeter- Principle of inﬁnite precision: ministic description of the world does not (at least necessar- 1. (Ontological) – there exists an actual value of every phys- ily) entail a “lawless indeterminism”, namely a complete dis- ical quantity, with its inﬁnite determined digits (in any arbi- order devoid of any laws or regularities (quantum mechanics trary numerical base). with its probabilistic predictions provides a prime example of 2. (Epistemological) – despite it might not be possible to these indeterministic regularities). We would like to deﬁne in- know all the digits of a physical quantity (through measure- determinism trough the sufﬁcient condition of the existenceof ments), it is possible to know an arbitrarily large number of some events that are not fully determined by their past states; digits. in the words of K. Popper, “indeterminism merely asserts that In this paper, we further develop the argument put forward there exists at least one event (or perhaps, one kind of events in Ref. [6], wherein it has been outlined a concrete possibil- [...]) which is not predetermined” [15]. Such a remark is im- ity to replace the principle of inﬁnite precision. According portant because, historically, classical mechanics allowed to to this view, the limits of this principle rely on the faulty as- predict with a tremendous precision the motion, for instance, sumption of granting a physical signiﬁcance to mathematical of the planets in the Solar System and this led Laplace to for- real numbers. We would like to stress that such an assump- mulate his ideas on determinism, which became the standard tion cannot be whatsoever justiﬁed at the operational level, as view among physicists and remained for a long time unchal- already stressed by Born, as early as 1955: “Statements like lenged. In fact, thinking that physics is deterministic seems ‘a quantity x has a completely deﬁnite value’ (expressed by completely legit, in so far as certain physical systems exhibit a real number and represented by a point in the mathematical extremely stable dynamics –e.g. an harmonic oscillator (pen- continuum) seem to me to have no physical meaning” [12]. dulum), or the (Newtonian) gravitational two-body problem, Relaxing the assumption of the physical signiﬁcance of math- and in general any integrable systems. However, this justiﬁ- ematical real numbers, allows one to regard classical physics cation of determinism can be challenged on the basis of two considerations. On the one hand, as already remarked, the ex- istence of some very stable systems (for all practical purposes 2 The problem with (physical) inﬁnities is connected to the so-called treated as deterministic) does not undermine the possibility of Hilbert’s Hotel paradox, proposed by D. Hilbert in 1924 [11]. This para- indeterminism in the natural world. On the other hand, in the dox illustrates the possibility for a hypothetical hotel with (countably) inﬁ- nite rooms, all of which already occupied, to allocate (countably) inﬁnitely last one century a systematic study of chaotic systems –which many more guests. are not integrable– has been carried out, giving us good rea- 3 In what follows, boundary and initial conditions will be used interchange- sons to doubt determinism. Indeed, chaotic systems are not ably, for they are conceptually the same in our discussion. In fact, they both stable under perturbations, meaning that an arbitrarily small serve as necessary inputs to the (differential) dynamical equations in order to give predictions. Thus, if either of the initial or the boundary condi- change in the initial conditions would lead to a signiﬁcantly tions are not determined with inﬁnite precision, they affect the subsequent different future behavior, thus making the principle of inﬁnite dynamics in the same way. precision even more operationally unjustiﬁed, and therefore 3 representing a concrete challenge to determinism.4 free parameters of these equations. Thus, if one aims at elimi- Incidentally, it is interesting to stress that, in the context of nating determinism as an unfounded interpretational element, quantum physics, Bell’s inequalities [16] have given us good there seems to be different possibilities, either involving the reasons to believe, if not directly in indeterminism –which in- laws of physics or the characterization of the IC: deed cannot be conﬁrmed or disproved within the domain of 1. The laws of motions are fundamentally stochastic. In science (see further)– that having at least one not predeter- this case, however, we cannot speak of an interpretation of mined event in the history of the Universe could have tremen- the theory, but an actual modiﬁcation of the formalism is re- dous consequences on the following evolution. In fact, an quired. In fact, in this case not only chaotic systems but also experimental violation of Bells inequalities guarantees that integrable ones would exhibit noisy outcomes, leading to ex- if the inputs (measurement settings) were independent of the perimentally inequivalent predictions. This case is the ana- physical state shared by two distant parties, then the outcomes logue of spontaneous collapse theories in quantum mechanics would be genuinely random (i.e., they cannot have predeter- [22–25], which modify the Schr¨odinger’s equation with addi- mined values). Yet, the amount of random numbers gener- tional non-unitary terms. ated in a Bell’s test can be greater than the number of the 2. The IC cannot, in principle, be fully known. Without corresponding inputs (in terms of bits of information): Bell’s any ontological commitments, one can take seriously the epis- tests performed on quantum entangled states can be thought temological statements of the principle of inﬁnite precision of as machines to increase the amount of randomness in the and push it to the extreme, namely asserting that there are in Universe (see also [17]). Surprisingly enough, recent results principle limits to the possibility of knowing (or measuring) [18, 19] showed that it is not even necessary to have a sin- certain quantities. This is what is entailed by the standard gle genuinely random bit from the outset, but it is sufﬁcient interpretation of the Heisenberg uncertainty principle that is to introduce an arbitrarily small amount of initial random- usually stated as: there is in quantum physics a fundamental ness (i.e., of measurement independence) to generate virtu- limit to the precision with which canonically conjugated vari- ally unbounded randomness. Hence, if one single event in the ables can be known. Such an uncertainty can be introduced past history of the Universe was not fully causally determined in classical physics as well, and would be characterized by a beforehand, there is an operationally well deﬁned procedure new natural constant ε (e.g., the standard deviation of a Gaus- that allows to arbitrarily multiply the amount of indetermin- sian function centered in the considered point). This would set istic events in the future. Namely, it would be enough to use an epistemic (yet fundamental) limit of precision, with which the randomness of the one indeterminate event as input in a physical variables could be determined, as a classical ana- Bell’s test to extract more randomness (through the violation logue of Heisenberg’s uncertainty relations in quantum the- of a Bell’s inequality). The random outputs can be used as ory. This viewpoint appears similar to the one proposed in new inputs for more Bell’s experiments, and the process can Ref. [8]. Notice, however, that since this approach is agnostic be repeated arbitrarily many times [20]. However, the initial with respect to the underlying ontology, it is fully compatible arbitrarily small amount of randomness (or of indeterministic with a realist position that takes this uncertainty as being an events) cannot be demonstrated by physics and its justiﬁcation ontological indeterminacy. 5 (such as in weather forecasts). can only come from metaphysical arguments (see [21]). So more general arguments than theoretical reduction would be desirable. 3. The IC are not fully determined. One can think that the FORMS OF INDETERMINISM IN CLASSICAL AND fundamental limit of precision in determining physical quan- QUANTUM PHYSICS tities is not merely epistemic, but actually is an objective, on- tologic indeterminacy that depends on the system and its in- Before proposing some possible models of indeterministic teractions at a certain time. This view is the one that will be classical physics, in this section we shortly discuss some gen- pursued in what follows, when we will propose ways to re- eral features of deterministic and indeterministic theories. In move real numbers R from the domain of physics. Although doing so we aim at clarifying possible similarities between this case does not seem to be the analogue of any speciﬁc in- classical and quantum physics. Both classical and quantum terpretation of quantum theory, it clearly goes in the direction mechanics are, in fact, formalizedby a set of differential equa- tions (laws of motion) that govern the dynamics of systems, together with appropriate initial conditions (IC) that ﬁx the 5 As a supporting argument for this fundamental epistemic limit, one can even think in terms of theoretical reduction (i.e. when a theory is super- vened by a more fundamental one, of which it represent an approxima- 4 Sometimes a distinction is made between strong and weak determinism. tion). In fact, determining positions in classical physics with higher and The former can be intuitively deﬁned as “similar initial conditions lead higher precision, means to access digits that are relevant at the microscopic to similar trajectories and it is fulﬁlled by any integrable system. On the scale, and thus the Heisenberg’s indeterminacy relations need to be applied other hand, weak determinism can be deﬁned as “identical conditions lead (leading to the identiﬁcation ε = ~). However, it ought to be remarked that to identical trajectories. This holds for classical chaotic systems, however it in certain classical chaotic systems the digits that are to become relevant is not empirically testable, for it would require the knowledge of the initial are not necessarily the ones in the microscopic domain, but could be those conditions with inﬁnite precision. which will become relevant only after a longer time 4 of realistic interpretations. It ought to be stressed, however, 2. Rational numbers. that if the initial conditions are not fully determined, this even makes (quantum) Bohmian mechanics becoming indetermin- Another possibility is to consider that physical quantities istic. take value in the rational numbers, Q. Even if this sounds somewhat strange, one can argue that, in practice, physical measurements are in fact only described by rational numbers. For instance, a measurement of length is obtained by compar- ing a rod that has been carefully divided into equal parts (i.e. INDETERMINISTIC CLASSICAL PHYSICS WITHOUT a ruler) with the object to be measured and determining the REAL NUMBERS best ﬁt within its (rational) divisions. And even probabilities are obtained as limits of frequencies of events’ occurrences In Ref. [6], a model of indeterministic classical mechanics (i.e. ratios of counts). However, while rational numbers do has been sketched, which, while leaving the dynamical equa- eliminate the unwanted inﬁnite information density, they do tions unchanged, proposes a critical revision of the assump- not seem to remove determinism in so far as all the digits are tions on the initial condition. In this view, the standard inter- fully predetermined. Moreover, the use of rational numbers pretation of classical mechanics has always tacitly assumed leads to those that can be named “Pitagora’s no-go theorems”. as hidden variables the predetermined values taken by physi- Indeed, positing a physics based on rational numbers, would cal quantities in the domain of mathematical real numbers, R. rule out the possibility of constructing a physical object with The physical existence of an inﬁnite amount of predetermined the shape of a perfect square with unit edge or a perfect circle digits could lead to unphysical situations, such as the afore- with unit diameter. In fact, by means of elementary mathemat- mentioned inﬁnite information density, as explained in detail ical theorems, their diagonal and circumference, respectively, in Refs. [5] and [6]. would measure √2 and π, hence resulting to be physically un- acceptable. Additionally, if one plugs in the equations of mo- In this section, we discuss some possible solutions to elim- tion initial conditions and time both taking values in the ratio- inate these unwanted features, namely possible ways of car- nal numbers, the solutions are not in general rational numbers. rying out the relaxation of the postulate according to which These problematic issues lead to consider yet another possible physical quantities take values in the real numbers. These so- solution. lutions are however intended to be merely different interpreta- tions of classical mechanics, i.e. they ought to be empirically indistinguishable from the standard predictions (in the same 3. “Computable real numbers”. way as interpretations of quantum mechanics are) [26]. A further alternative is to substitute the domain of phys- ically meaningful numbers from (mathematically) real num- bers to the proper subset thereof of “computable real num- bers”; that is, to keep all real numbers deprived of the ir- 1. “Truncated real numbers”. rational, uncomputable ones. In fact, even irrational, com- putable real numbers can encode at most the same amount A ﬁrst possibility is to consider physical variables as taking of non-trivial information (in bits) as the length of the short- values in a set of “truncated real numbers”. This, as already est algorithm used to output their bits (i.e., the Kolmogorov noted by Born, would ensure the empirical indistinguishabil- complexity). Uncomputable real numbers are in this model ity from the standard classical physics: “a statement like x = π instead substituted by genuinely random numbers, thus in- cm would have a physical meaning only if one could distin- troducing fundamental randomness also in classical physics. guish between it and x = πn cm for every n, where πn is the These numbers, together with chaotic systems, lay the founda- approximation of π by the ﬁrst n decimals. This, however, tions of an alternative classical indeterministic physics, which is impossible; and even if we suppose that the accuracy of removes the paradox of inﬁnite information density. How- measurement will be increased in the future, n can always be ever, this proposal could be considered an ad hoc solution, chosen so large that no experimental distinction is possible” since it maintains a ﬁeld of mathematical numbers as physi- [12]. If one, however, wishes to attribute an ontological value cally signiﬁcant, but removes “by hand” those that are prob- to such an interpretation has to identify n with a new univer- lematic (which admittedly are almost all). sal constant, that, independent of how big it could be, sets a limitation to the length of physically signiﬁcant numbers, ul- timately including the life of the Universe, if time too is to be 4. “Finite information quantities” (FIQs). considered a physical quantity. Another problematic issue is that n would be dependenton the units in which one expresses Developing further the proposal in [6], we put forward an the considered physical variables. This leads to consider a sec- alternative class of random numbers which are for all practi- ond possible solution. cal purposes (in terms of empirical predictions) equivalent to 5 real numbers, but that have actually zero overlap with them extreme, if a bit has an associated propensity of 1/2, it means (they are not a mathematical number ﬁeld, nor a proper subset that the bit is totally random. Namely, if one were to mea- thereof). We refer to them as “ﬁnite-information quantities” sure the value of this bit, there would be an intrinsic property (FIQs). In order to illustrate this possible alternative solu- that makes it taking the value 0 or 1 with equal likelihood (we tion to overcome the problems with the principle of inﬁnite don’t use “probability” to avoid formal issues, see footnote precision, let us consider again the standard interpretation in 6). All the intermediate cases can then be constructed. For in- greater formal detail. A physical quantity γ (which may be the stance, a propensity qk = 0.3 means that there is an objective scalar parameter time, a universal constant, as well as a one- tendency of the kth digit to take the value 1, quantiﬁed by 0.3, dimensional component of the position or of the momentum, and thus the complementary propensity of taking the value 0 etc.) is assumed to take values in the domain of real numbers, would be 0.7 (how this actualization occurs is an open issue, i.e., γ R. Without loss of generality, but as a matter of sim- as we discuss in the next section). We would like to stress that plicity,∈ let us consider γ to be between 0 and 1, and that its while we assume propensities to be an (ontic) objective prop- digits (bits) are expressed in binary base: erty, at the operational level they lead to the measured (epis- temic) frequencies, but they supervene frequencies insofar as γ = 0.γ1γ2 γ j , propensities can describe single-time events. ··· ··· + red We posit that propensities take values in the domain of where each γ j 0,1 , j N . This means that, being γ ∈{ } ∀ ∈ ∈ rational numbers such that they contain only a ﬁnite amount R, its inﬁnite bits are all given at once and each one of them of information. Hence, postulating them as an element of re- takes as a value either 0 or 1. ality, does not lead to the same information paradoxes of real In an indeterministic world, however, not all the digits numbers. It also follows from the deﬁnition, that the propen- should be determined at all times, yet we require this model to sities q j for the ﬁrst N(t) digits of a quantity γ(N(t)) at time t give the same empirical predictions of the standard one. We are all either 0 or 1, i.e. q j 0,1 , j [1,N(t)]. therefore require a physical quantity to have the ﬁrst (more As a function of time, propensities∈{ } ∀ must∈ undergo a dynami- signiﬁcant) N digits fully determined –and to be the same as cal evolution. We envision more than a way to evolve propen- those that give the standard deterministic predictions– at time sities in time, although we do not propose an explicit model t, and we write γ(N(t)), whereas the following inﬁnite digits to describe this. On the one hand, one can think of a dynam- are not yet determined. This reads: ical process similar to spontaneous collapse models of quan- tum mechanics. Admittedly, spontaneous collapse models re- γ(N(t)) = 0.γ1γ2 γ ? ?k , ··· N(t) N(t)+1 ··· ··· quire to modify the fundamental dynamical equation of quan- tum physics, the Schr¨odinger’s equation. Hence these mod- where each γ j 0,1 , j N(t), and the symbol ?k here means that the∈{kth digit} ∀ is a≤ not yet actualized binary digit els are not merely interpretations, but testable different theo- (see further). ries. Nevertheless, for propensity this is not necessarily the Despite the element of randomness introduced, the transi- case because they are a postulated element of reality which is tion between the actualized values and the random values still however not observable. Thus, even if it would be desirable to be realized does not need to be a sharp one. In fact, one to have an explicit form for the equations governing the dy- can conceive an objective property that quantiﬁes the (possi- namics of propensities, the measured values of physical (ob- bly unbalanced) disposition of every digit to take one of the servable) quantities would evolve in the usual way. Thus we two possible values, 0 or 1. This property is reminiscent of maintain that our proposed new interpretation is indeed an in- Popper’s propensities [27],6 and it can be seen as the element terpretation and not a different testable theory. of objective reality of this alternative interpretation: On the other hand, intuitionistic mathematics could be the Deﬁnition - propensities tool to solve the issue of the evolution of propensities (see There exist (in the sense of being ontologically real) physical “choice sequences” below). In fact, one can start from an inﬁ- nite sequence of completely random bits (or digits), and then properties that we call propensities q j [0,1] Q, for each digit j of a physical quantity γ(N(t)). A∈ propensity∩ quantiﬁes the number representing a physical quantity evolves according the tendency or disposition of the jth binary digit to take the to a law (a function of these random bits). However, despite value 1. this law would describe the evolution of propensities, it is dif- The interpretation of propensities can be understood start- ferent from a standard a physical law, for it is a different way ing from the limit cases. If the propensities are 0 or 1 the to construct mathematical numbers –which in turn describe physical quantities– in time. meaning is straightforward. For example, q j = 1 means that the jth digit will take value 1 with certainty. On the opposite Making use of propensities, we can now reﬁne our deﬁni- tion of physical quantities: Deﬁnition - FIQs A ﬁnite-information quantity (FIQ) is an ordered list of 6 propensities q1,q2, ,q j, , that satisﬁes: While propensities were for Popper an interpretation of mathematical prob- { ··· ···} abilities proper, we are not here necessarily requiring them to satisfy Kol- 1. (necessary condition): The information content is ﬁnite, mogorov’s axioms, as discussed in Ref. [28]. i.e. ∑ Ij < ∞, where Ij = 1 H(q j) is the information content j − 6 of the propensity, and H is the binary entropy function of its 5. “Choice sequences”. argument. This ensures that the information content of FIQs is bounded from above; Finite Information Quantities are not numbers in the usual 2. (sufﬁcient condition): After a certain threshold, all sense, because their digits are not all given at once. On the the bits are completely random, i.e. M(t) N such that 1 ∃ ∈ contrary, the “bits” of FIQs evolve as time passes, they start q j = 2 , j > M(t) 1 ∀ from the value 2 and evolve until they acquire a bit value of either 0 or 1. In a nutshell, FIQs are processes that develop in It ought to be stressed that this view grants a prior fun- time. Interestingly, in intuitionistic mathematics, the contin- damentality to the potential property of becoming actual (a uum is ﬁlled by “choice sequences”, as Brouwer, the father of list of propensities, FIQ), more that to the already actualized intuitionism, and followers named them [29]. This is not the number (a list of determined bits). In fact, the analogue of a place to present intuitionistic mathematics (see, e.g., [30]), but pure state in this alternative interpretation of classical physics let us emphasize that this alternative to classical (Platonistic) would be a collection of all the FIQs associated with the dy- mathematics allows one to formalize “dynamical numbers” namical variable (i.e., the list of the propensities of each digit). that resemble much our FIQs [31]. Interestingly, using the Namely, this represents the maximal piece of information re- language of intuitionistic mathematics makes it much easier garding a physical system. Yet, even having access to this to talk of indeterminism [32]. knowledge (which is admittedly not possible due to the fact that propensities are not measurable) would lead to in princi- ple unpredictable different evolutions. Thus, two systems that are identical at a certain instant of time (in the sense that they STRONG EMERGENCE are in the same pure state, i.e. the propensities associated to their variables are all the same) will have, in general, different observable behaviors at later times. However, the merit of this The argument for determinism seems to rely, to a certain view is that the bits are realized univocally and irreversibly as extent, on the tacit assumption of reductionism in its stronger time passes, but the information content of a FIQ is always form of microphysicalism, i.e. the view that every entity and bounded, contrarily to that of a real number. A physical quan- phenomenon are ultimately reducible to fundamental interac- tity γ reads in this interpretation as follows: tions between elementary building blocs of physics (e.g., par- ticles). In fact, in a completely deterministic picture, every ? , with q (0,1) particular phenomenon can be traced back to the interactions z k }|k∈ { between its primitive components, along a (ﬁnite) chain of γ(N(t),M(t)) = 0. γ1γ2 γN(t) ?N(t)+1 ?M(t) ?M(t)+1 . | ···{z } ··· | {z ···} causally predetermined events. In this way any form of strong determined γ j 0,1 ? , with q = 1 ∈{ } l l 2 emergence seems to be ruled out, and it becomes only appar- ent (i.e. a weak or epistemic emergence). On the other hand, Notice that none of the FIQs is a mathematical number, but admitting genuine randomness in the universe, allows in our they capturethe tendency (propensity)of each bit of a physical opinion the possibility of strong emergence.8 quantity to take the value 0 or 1 at the following instant in As a concrete example, consider the kinetic theory of gases. If time. This admittedly leads to problematic issues, such as the one starts from a molecular description of the ideal gas, from problem of how and when the actualization of the digits from the perspective of standard, deterministic classical mechan- their propensity take place: it thus introduces the analogue of ics, the stochasticity is only epistemic (i.e. only an appar- the quantum measurement problem also in classical physics ent effect due to the lack of complete information regarding (see further). positions and momenta of every single molecule). Thus, the Moreover, FIQs partly even out the fundamental differ- deterministic behavior of the law of the ideal gas is not ex- ences between classical and quantum physics, making both pected to be a strong emergent feature, but solely a retrieving of them indeterministic (and making so even Bohmian inter- at the macroscopic scale of the fundamental determinism of pretation).7 Table I compares some possible combinations of the microscopic components. In the perspective of the alter- deterministic and indeterministic interpretations of quantum native indeterministic interpretation (based on FIQs), instead, and classical physics. the deterministic law of the ideal gas, ruling the behavior at the macroscopic level, emerges as a novel and not reducible

7 There is yet another deterministic interpretation of quantum mechanics, the so-called many-worlds interpretation that grants physical reality to the 8 For a deﬁnition of strong emergence, see, for instance, Ref. [33]: “We wave function of the Universe, which always evolve unitarily. If FIQs are can say that a high-level phenomenon is strongly emergent with respect to introduced in that interpretation, then the realization of all the values of a low-level domain when the high-level phenomenon arises from the low- the bits would actually take place, each of which being real in a different level domain, but truths concerning that phenomenon are not deducible “world”. even in principle from truths in the low-level domain”. 7

Classical Quantum ψ L2(RN ) IC FIQs | i ∈ ∈ Measurement postulate Newton’s equation IC (position) FIQs and ψ L2(RN ) ∈ | i ∈

Indeterministic Bohm’s guidance equation admitted IC (position) R and ψ L2(RN ) ∈ | i ∈ IC R Bohm’s guidance equation ∈ Newton’s equation Schr¨odinger’s equation Many worlds interpretation (?) Deterministic

TABLE I: A table comparing deterministic and indeterministic interpretations of classical and quantum physics. Note that the substitution of FIQs in the place of real numbers makes not only classical physics indeterministic, but also Bohm’s interpretation of quantum physics (which is usually taken to restore determinism). feature, from fundamental randomness.9 TOP-DOWN CAUSATION Notice that the historical debate on the apparent incom- patibility between Poincar´e’s recurrence theorem and Boltz- The idea of strong emergence, including emergent deter- mann’s kinetic theory of gases does not arise in the frame- minism, is related to the concept of “top-down causation” work of FIQs. Poincar´e’s recurrence theorem, in fact, states [34, 37]. In this view, microphysicalism is not necessarily that continuous-state systems (i.e., in which the state variables rejected ontologically (i.e., it admits that complex structures change continuously in time) return to an arbitrarily small are hierarchical modular compositions of simpler ones), but neighborhood of the initial state in phase space. However, the fact that the behavior of macroscopic events is fully de- Poincar´e’s theorem relies on the fact that the initial state is termined by the interactions of the macroscopic entities is perfectly determined (i.e., it is a mathematical point identiﬁed revised. Top-down causation maintains that the interactions by a set of coordinates which take values in the real numbers) between microscopic entities do not causally supervene the in phase space. Thus in a FIQ-based alternative physics the macroscopic phenomena, but rather it posits a mutual inter- theorem simply cannot be derived. In fact, FIQs interpreta- action where also the macroscopic (strongly emergent) laws tion features genuinely irreversible physical processes. impose constraints on the behavior of their constituents. Note Similarly, Drossel has recently pointed out that, in a physics that top-down causation requires indeterminism (at least at where it is impossible to determine points of phase space with the lower level of the constituents) to be in principle conceiv- inﬁnite precision, “the time evolution of thermodynamics is able [35]; this was already remarked by Popper, when stating: undetermined by classical mechanics [...]. Thus, the second “[A] higher level may exert a dominantinﬂuence upon a lower law of thermodynamicsis an emergentlaw in the strong sense; level. For it seems that, were the universe per impossibile a it is not contained in the microscopic laws of classical me- perfect determinist clockwork, there would be no heat produc- chanics” [8]. tion and no layers and therefore no such dominating inﬂuence At this point one should ask oneself whether there are ex- would occur. This suggests that the emergence of hierarchical amples of emergence that possibly go beyond some form of levels or layers, and of an interaction between them, depends the law of large numbers. Admittedly, we are unsure about upon a fundamental indeterminism of the physical universe. this. Clearly, in all indeterministic physical theories, the law Each level is open to causal inﬂuences coming from lower of large numbers will play an important role and lead to some and from higher levels” [36]. stability and hence to some form of determinism at the larger scale (or higher-level description). It seems that this question Concerning indeterministic interpretations of physical the- is closely related to possible top-down causation, the topic of ories, top-down causation could help to understand how the the next section. determination of dynamical variables (i.e., the actualization of their values) occurs in the context of indeterministic theories. Namely, the reason why –and under what circumstances– a single deﬁnite value is realized among all the possible ones. 9 It is true that also the law of the ideal gas would not be perfectly determinis- tic in a FIQ-based physics, however its stability makes it almost determin- In the FIQ-based indeterministic interpretation of classical istic for all practical purposes, whereas at the microscopic level, chaotic physics here introduced, this translates into the understand- behaviors multiply the fundamental uncertainty of the single molecules. ing of how the bits of physical variables becomes fully deter- 8 mined, namely how their propensities become either 0 or 1. sity of taking that (or another) possible value. Consider for We envision two possible mechanisms that could explain the example the chaotic systems analyzed in [6] (a simpliﬁed ver- actualization of the variables: sion of the baker’s map). One can then think of a “faster” dy- 1. The actualizations happens spontaneously as time namics that, at every time step, shifts the bits by not only one passes. This view is compatible with reductionism and it digit toward the more signiﬁcant position, but, say, it shifts does not necessarily require any effects of top-down causa- them by 1000 digits (or any other arbitrarily large ﬁnite num- tion. Note that this mechanism resembles, in the context of ber). This clearly entails that the rate of change of propen- quantum mechanics, objective collapse models such as the sities depends on the dynamical system under consideration, “continuous spontaneous localization” (CSL) [23, 25]. and cannot be thought of as a universal constant of “spon- 2. The actualization happens when a higher level requires taneous” actualization. A possible solution is to introducea it. This means that when a higher level of description (e.g., the model of measurement that makes the digits becoming actual macroscopic measurement apparatus) requires some physical (and therefore stable) up to the corresponding precision. This quantity pertaining to the lower-level description to acquire a clearly resembles the solution to the quantum measurement determined value, then the lower level must get determined. In problem provided by the objective collapse models, such as quantum mechanics a similar explanation is provided by the the CSL [23, 25] or the GRW [22, 24]. The latter model, Copenhagen interpretation and, more explicitly, by the model indeed, posits a modiﬁcation of the standard Schr¨odinger’s in Ref. [37]. equation which accounts for a spontaneous random “collapse” In fact, the latter mechanisms are strongly related to what of the wave function, occurring with a certain natural rate. has been discussed at length in the context of quantum theory, Under certain assumptions, this model leads to a mechanism namely the long-standing “quantum measurement problem”. that changes the rate of spontaneous collapse, which increases This comprises the problem of “explaining why a certain out- linearly with the number of componentsof a system (thus dur- come –as opposed to its alternatives– occurs in a particular ing a measurement the wave function of a microscopic system run of an experiment” [38]. In fact, some of the most com- in contact with a macroscopic apparatus collapses extremely monly accepted interpretations of quantum mechanics (e.g. fast). An analogous solution can be in principle proposed the Copenhagen interpretation) uphold the view that it is the for the rate of actualization of the propensities that deﬁne the act of measurement to impose to microscopic (quantum) ob- FIQs. However, this seems to mean that the dynamical equa- jects to actualize one determined value, out of the possible tions need to be modiﬁed (in the same fashion as the GWR ones. model modiﬁes the Schr¨odinger’s equation), thus leading toa Note that, despite it has been already remarked in the lit- different formalism and not only an interpretation. erature that every indeterministic theory has to deal with a Coming back to top-downcausation, this could explain why “measurement problem” (see e.g. [38]), it seems that there every time one performs a measurement the determined dig- has hardly been any consideration of this issue in the context its remain stable. In fact, the act of a measurement can be of other indeterministic theories than quantum mechanics. In regarded as the direct action performed at the higher level the next subsection we will discuss what we call, by analogy, which imposes to the lower level to get determinate. This is the “classical measurement problem’. We will then draw a very similar to what is taken to be the solution to the quantum connection with top-down causation, and show how this could measurement problem within the Copenhagen interpretation, help to shed light on this problem. wherein the higher level is the macroscopic measurement ap- paratus, whereas the lower level is the measured microscopic system. However, this kind of solutions lacks a clear deﬁni- The “classical measurement problem” tion of what is to be considered a measurement and how to identify higher and lower levels of description. It is a very corroborated experimental fact that if a quantity As a matter of fact, the “classical measurement problem” is measured twice with the same instrument, we expect a cer- here introducedremains so far unresolved,as well as the quan- tain amount of digits to remain unchanged, and it is essential tum measurement problem and, more in general, the problem to scientiﬁc investigation that such a knowledge is intersubjec- of the actualization of physical variables in any indeterminis- tively available (up to the digit corresponding to the measure- tic theory. Yet, it is desirable that the topic of the measure- ment accuracy). Moreover, if a more accurate measurement ment problem should ﬁnd room in the debate on foundations instrument is utilized, we expect not only the previous digits of physics, in more general discussions than those centered on to remain unchanged, but also to determine some new digits quantum mechanics only. that then become intersubjectively available. How to recon- cile this stability of the measured digits with a fundamental uncertainty in the determination of a physical quantity? How CONCLUSIONS does potentiality become actuality? In the proposed FIQ-based indeterministic interpretation of We have discussed arguments–primarily based on the mod- classical physics, too, one has to carefully deﬁne how the dig- ern application of information theory to the foundations of its of physical quantities realize themselves from the propen- physics– against the standard view that classical physics is 9 necessarily deterministic. We have also discussed concrete [12] Born, Max. (1969). Physics in my Generation. New York: perspectives to reinterpret classical physics in an indetermin- Springer-Verlag. istic fashion. We have then compared our indeterministic pro- [13] Bohm, D. (1952). A suggested interpretation of the quantum posals with some interpretations of quantum physics. How- theory in terms of “hidden” variables. I. Physical review, 85(2), 166. ever, it seems clear that the empirical results of both classical [14] Gudder, S.P. (1970). On HiddenVariable Theories. Journal of and quantum mechanics can ﬁt in either a deterministic or in- Mathematical Physics, 11(2), 431-436. deterministic framework. Furthermore, there are compelling [15] Popper, K. R. (1950). Indeterminism in quantum physics and in arguments (see e.g., [6, 21, 39]) to support the view that the classical physics. Part I. The British Journal for the Philosophy same conclusion can be reached for any given physical theory of Science, 1(2), 117-133. –a trivial way to make an indeterministic theory fully deter- [16] Bell, J. S. (1964). On the einstein podolsky rosen paradox. mined is to “complete” the theory with all the results of every Physics Physique Fizika, 1(3), 195. [17] Gisin, N. (2010). Is realism compatible with true randomness?. possible experiments that can be performed. arXiv preprint arXiv:1012.2536. In conclusion, although the problem of determinism versus [18] Colbeck, R. and Renner, R., (2012). Free randomness can be indeterminism is in our opinion central to science, the hope ampliﬁed. Nature Physics, 8(6), 450. to resolve this problem within science itself has faded, and [19] P¨utz, G., Rosset, D., Barnea, T.J., Liang, Y.C. and Gisin, N. this is ultimately to be decided on the basis of metaphysical (2014). Arbitrarily small amount of measurement independence arguments. is sufﬁcient to manifest quantum nonlocality. Physical Review Letters, 113(19), 190402. [20] Gallego, R., Masanes, L., De La Torre, G., Dhara, C., Aolita, Acknowledgments L. and Ac´ın, A. (2013). Full randomness from arbitrarily deter- ministic events. Nature communications, 4, 2654. [21] Suppes, P. (1993). The transcendental character of determinism. We would like to thank the participants to the Workshop Midwest Studies in Philosophy, 18(1), pp.242-257. “Experiencing Reality Directly: Philosophy and Science”, [22] Ghirardi, G. C., Rimini, A., Weber, T. (1986). Uniﬁed dynamics held in Jerusalem on May 20-22, 2019, for many discussions for microscopic and macroscopic systems. Physical Review D, that provided interesting inputs. We also thank George Ellis, 34(2), 470. Veronica Baumann, Arne Hansen and Stefan Wolf for useful [23] Gisin, N. (1989). Stochastic Quantum Dynamics and Relativity. Helvetica Physica Acta, 62(4), 363371 comments. FDS acknowledges the ﬁnancial support through [24] Bell, J. S. (1987). Are there quantum jumps?, in Schr¨odinger. a DOC Fellowship of the Austrian Academy of Sciences. Centenary of a polymath, Cambridge University Press. [25] Ghirardi, G. C., Pearle P., Rimini A. (1990). Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles. Physical Review A, 42: 78. [26] Baumann, V. and Wolf, S. (2018). On formalisms and interpre- [1] Falkenburg, B., Weinert, F. (2009). Indeterminism and deter- tations. Quantum, 2, p.99. minism in quantum mechanics. In Compendium of Quantum [27] Popper, K. R. (1959). The propensity interpretation of proba- Physics. Springer, Berlin, Heidelberg. bility. The British journal for the philosophy of science, 10(37), [2] Del Santo, F. (2019). Striving for Realism, not for Determinism: 25-42. Historical Misconceptions on Einstein and Bohm. APS News, [28] Gisin, N. (1991). Propensities in a non-deterministic physics. 18(5). Synthese, 89(2), 287-297. [3] Ornstein, D. S. (1989). Ergodic Theory, Randomness, and [29] Brouwer, L. E. J. (1949). Consciousness, Philosophy and Math- Chaos. Science, 243(4888), pp.182-187. ematics, in Proceedings of the Tenth International Congress of [4] Prigogine, I. and Stengers, I. (1997). The end of certainty. Si- Philosophy, vol. III, North Holland, Amsterdam 1949. mon and Schuster. [30] Posy, C. J. (1976). Varieties of Indeterminism in the Theory of [5] Dowek, G. (2013). Real numbers, chaos, and the principle of General Choice Sequences. Journal of Philosophical Logic, 5, a bounded density of information. In International Computer 91-132 . Science Symposium in Russia. Springer, Berlin, Heidelberg. [31] Troelstra, A. S. (1977) Choice Sequences, a chapter of intu- [6] Gisin, N. (2019). Indeterminism in Physics, classical chaos and itionistic mathematics, Oxford University Press. Bohmian mechanics. Are real numbers really real?. Erkenntnis, [32] Gisin, N. (2019) Real Numbers as the Hidden Variables of Clas- https://doi.org/10.1007/s10670-019-00165-8. sical Mechanics. In Quantum Studies: Mathematics and Foun- [7] Blundell, S. J. (2017). Emergence, causation and storytelling: dations, in press; see also Gisin, N. Indeterministic Physics and condensed matter physics and the limitations of the human Intuitionistic Mathematics, in preparation. mind. Philosophica, 92, 139-164. [33] Chalmers, D. J. (2006). Strong and weak emergence. The [8] Drossel, B. (2015). On the relation between the second law of reemergence of emergence. Oxford University Press. thermodynamics and classical and quantum mechanics. In Why [34] Ellis, G. F. (2011). Top-down causation and emergence: some More Is Different. Springer, Berlin, Heidelberg. comments on mechanisms. Interface Focus, 2(1), 126-140. [9] Landauer, R. (1996). The physical nature of information. [35] Drossel, B. (2019). Strong emergence in condensed matter Physics letters A, 217(4-5), pp.188-193. physics. Preprint arXiv:1909.01134v1. [10] Ellis, G.F., Meissner, K.A. and Nicolai, H. (2018). The physics [36] Popper, K. R., Eccles, J. C. (1977). The Self and Its Brain: An of inﬁnity. Nature Physics, 14(8), p.770. Argument for Interactionism. Springer Heidelberg Berlinclock. [11] Kragh, H. (2014). The true (?) story of Hilbert’s inﬁnite hotel. [37] Drossel, B. and Ellis, G. (2018). Contextual wavefunction col- arXiv preprint arXiv:1403.0059. lapse: An integrated theory of quantum measurement. New 10

Journal of Physics, 20(11), p.113025. [39] Werndl, C. (2009). Are deterministic descriptions and inde- [38] Brukner, C.ˇ (2017). On the quantum measurement problem. In terministic descriptions observationally equivalent?. Studies in Quantum [Un] Speakables II. Springer. History and Philosophy of Modern Physics, 40(3), 232-242.