Indeterminism in Physics and Intuitionistic Mathematics 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 Schr¨odinger'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 jx−xnj ≤ 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.