<<

comment Corrected: Publisher Correction Mathematical languages shape our understanding of in Physics is formulated in terms of timeless, axiomatic . A formulation on the basis of intuitionist mathematics, built on time-evolving processes, would ofer a perspective that is closer to our experience of physical reality. Nicolas Gisin

n 1922 , the , met in Henri Bergson, the philosopher. IThe two giants debated publicly about time and Einstein concluded with his famous statement: “There is no such thing as the time of the philosopher”. Around the same time, and equally dramatically, mathematicians were debating how to describe the continuum (Fig. 1). The famous German mathematician was promoting formalized mathematics, in which every real number with its infinite series of digits is a completed individual object. On the other side the Dutch mathematician, Luitzen Egbertus Jan Brouwer, was defending the view that each point on the line should be represented as a never-ending process that develops in time, a view known as intuitionistic mathematics (Box 1). Although Brouwer was backed-up by a few well-known figures, like Hermann Weyl 1 and Kurt Gödel2, Hilbert and his supporters clearly won that debate. Hence, time was expulsed from mathematics and mathematical objects Fig. 1 | Debating mathematicians. David Hilbert (left), supporter of axiomatic mathematics. L. E. J. came to be seen as existing in some Brouwer (right), proposer of intuitionist mathematics. Credit: Left: INTERFOTO / Alamy Stock Photo; idealized Platonistic world. right: reprinted with permission from ref. 18, Springer These two debates had a huge impact on physics. Mathematics is the language of physics and Platonistic mathematics makes it difficult to talk about time. Hence, the So, it may seem that physics should give than a finite amount of information and sense of flow of time was also expulsed up telling stories and concentrate on more concluded that physically relevant numbers from physics: all events are the ineluctable and more abstract theories. But is this really can’t contain infinite information. I’ve consequences of some ‘quantum fluctuations’ the only alternative? Physics is likely to be since discovered, thanks to Carl Posy, that happened at the origin of time — the in danger of coming to a halt when faced that intuitionistic mathematics comes . Accordingly, in today’s physics with claims like “time is an illusion”4. Just surprisingly close to my — and I bet there is no ‘creative time’ and no ‘now’. as François Rabelais stated that “ many ’ — intuition about the This had dramatic consequences, in without consciousness is only ruin of the continuum6,7. In this Comment, I would like particular when one remembers that physics soul”, wouldn’t it be appropriate, then, to share my excitement about this finding, is not only about technologies and abstract to say that “Science without time is only and argue that the debates between Einstein theories, but also about stories on the ruin of intelligibility”? and Bergson, and Hilbert and Brouwer, workings of nature. Time is an indispensable The mathematical language that ought to be revisited. ingredient in all narratives. As Yuval physicists use makes it easy or difficult to Bergson never agreed with Einstein’s Dolev emphasized, “To think of an is formulate some concepts, like the passage statement, and Einstein himself felt to think of something in time. […] Tense of time. The same holds for our notion uncomfortable with his beloved physics and passage are not removable from how we of continuum. In ref. 5 I argued that a lacking the concept of ‘now’ — although, think and speak of events”3. finite volume of can’t contain more admittedly, he didn’t see any way to

114 Nature Physics | VOL 16 | February 2020 | 114–119 | www.nature.com/naturephysics comment

physicists reject the faithfulness of such Box 1 | Choice sequences as real numbers in intuitionist mathematics a representation, but admit — often with regret — that they don’t see how one could A way to understand the continuum in α(n) converges to a computable number. do otherwise. For example, Max Born, one intuitionistic mathematics, well suited In other examples, the digits of α(n) are of the fathers of quantum theory, stressed to physicists, assumes that nature has correlated, for example, new digits of α(n) that “Statements like ‘a quantity x has a the power to produce true randomness, depend on the previous k random numbers completely definite value’ (expressed by a here illustrated as a true random number r. By choosing suitable functions f, an real number and represented by a point in generator (RNG) that outputs a digit r(n) infnity of classes of series can be defned. the mathematical continuum) seem to me to at each time step n. Historically, Brouwer, the father of have no physical meaning”10 (see also ref. 11 At each time step n, a rational , did not use random for the view of a computer scientist). This is number α(n) is computed by a function number generators, but mathematical where intuitionistic mathematics can help. f: α(n) = f(α(n−1), n, r(1), …, r(n)), as objects he named choice sequences, In intuitionistic mathematics, numbers shown in the fgure. Diferent functions where the choices were made are processes that develop in time; at f defne diferent classes of series α(n). by an idealized mathematician19, and each of time, there is only finite Te series α(n) is assumed to converge; what I name ‘classes’ were termed information. One way to understand this however, at any time, only fnite ‘spreads’ by Brouwer. unusual claim goes as follows. Assume information about the series exists, in nature has the power to produce random accordance with the basic idea that the True RNG numbers. One may think of a quantum random number generator is a genuine random number generator. That would do, n = 1 f(1, r(1)) endless process that develops in time. r(1) (1) but here it is preferable not to think of a A frst simple example assumes that α human-made randomness source, but rather the function f merely adds the random n = 2 as a power of nature: nature is intrinsically r(2) (2) digit r(n) as the nth digit of α, that is, α and fundamentally indeterministic. f( (1), 2, r(1), r(2)) α(n) = α(n−1) + r(n) × 10−n = 0 . r(1) α Now, this source of randomness feeds r(2) r(3) … r(n). Usually, it is assumed ... the digits of typical real numbers, as that α(0) is a given initial rational number, illustrated in Box 1. Let me emphasize that but that is not essential. Note that if the n the digits of all typical real numbers are random number generator is actually a r(n) α(n) truly random — as random as the outcomes pseudo-random number generator, then f(α(n–1), n, r(1), r(2), ..., r(n)) of quantum , as has been nicely emphasized for instance by Gregory Chaitin12,13. Moreover, typical real numbers contain infinite information. This makes incorporate it. He thus concluded that one space reduces to the unit interval and it possible, for example, to code in a single has to live with this state of affairs. Hilbert consider discrete time steps. The state X is number the answers to all questions one also felt uncomfortable by the infinities thus described by a number formed by an may formulate in any human language, as 14 that his beloved axiomatized mathematics infinity of digits bn that follow zero. At each noticed by Émile Borel . introduced, firmly stating that physics time step, going from t to t+1, the digits are So, we have to choose a perspective. should never incorporate actual infinities8,9. shifted by one place to the left, with the first Either all digits of the initial conditions are In order to illustrate the tension physics digit dropping out: assumed to be determined from the first faces, let us consider classical mechanics. moment, leading to timeless physics; or This is a very well understood with Xt 0 : b1 b2 b3 b4 ¼ bn ¼ these digits are initially truly indeterminate known limiting cases — relativity and ¼ and physics includes events that truly quantum theory. One may argue that happen as time passes. quantum theory is more fundamental and Xt 1 0 : b2 b3 b4 ¼ bn ¼ Notice that in both perspectives chaotic is intrinsically indeterministic. However, þ ¼ would exhibit randomness. In the the interpretation of quantum theory is After n time steps, the most relevant digit first case, from the point of view of classical still heavily debated and , is bn — the nth digit of the initial condition. Platonistic mathematics, all randomness with its enormous explanatory power, is This digit could, for example, represent is encoded in the initial condition. In the often presented as the ideal of a physical today’s weather — imagine 0 for pouring second case, randomness happens as time description of nature. Hence, for the rain and 9 for bright . passes — as described by intuitionistic following discussion we will consider only If one assumes all digits of the initial mathematics, where the dependence on classical mechanics. condition faithfully represent physical reality, time is essential15. One may object that Although classical mechanics is usually then, according to our simple model, the intuitionism doesn’t derive indeterminism, illustrated in terms of and other weather of the entire is already fully but assumes it from the start. That is correct. cog wheels, almost all classical dynamical determined. This illustrates the absence of Likewise, classical mathematics assumes systems are of a very different nature: any ‘creative time’; nothing really happens, actual infinite information from the start. they are chaotic. Because of the as everything is determined by the initial These two views cannot be distinguished hypersensitivity to initial conditions, the condition and the deterministic empirically. One can always claim that future of typical classical systems depends equations. At least, this is the consequence of instead of God playing dice every time a on far down the series of digits of the initial representing the points on the line between 0 random outcome happens, God played all conditions. A simple but relevant example and 1 by classical Platonistic mathematics. the dice at once before the Big Bang and of typical classical dynamical systems can But is such a representation truly faithful encoded all results in the ’s initial be modelled as follows. Assume the state and necessary? My experience is that most condition. Despite the empirical equivalence

Nature Physics | VOL 16 | February 2020 | 114–119 | www.nature.com/naturephysics 115 comment

Table 1 | Intuitionist mathematics as a language for indeterministic physics Scientists working on weather and climate physics explicitly use finite-truncated Indeterministic physics Intuitionistic mathematics numbers and stochastic remainders17. , and future are not all given at once Digits of real numbers are not all given at once All those who speak more than one language know that some concepts are easier Time passes Numbers are processes to express in one language than in another. Indeterminism Numbers contains finite information I believe the same is true of the language of The present is thick The continuum is viscous physics. Which mathematics we adopt when The future is open No law of the excluded middle: a proposition ‘talking physics’ will influence the possibility about the future can be neither true nor false of indeterminism in physics (Table 1). I believe that the notion of a deterministic Becoming Choice sequences and timeless world does not arise from Experiencing Intuitionism the huge empirical success of physics, but Concepts that appear natural in indeterministic physics are easily expressed in terms of intuitionist mathematics. from considering Platonistic mathematics as the only language for physics. Physics can be as successful if built on intuitionistic mathematics, even if this breaks its of the two views, they present us with very (and n is even and larger than 2), after which marriage to determinism. Contrary to usual different pictures of our world. Somehow, the series terminates and all future α(n) expectations, I bet that the next physical the real numbers are the hidden variables of remain constant. Since at each time step theory will not be even more abstract than classical mechanics16. the probability that the series terminates quantum field theory, but might well be But intuitionism goes way beyond the decreases exponentially, there is a finite closer to human experience. ❐ description of the continuum. Physicists probability that α(n) oscillates forever often have the intuition that the present between below and above 1/2. Accordingly, Nicolas Gisin is thick — that is, the present is not of as long as the series did not terminate, the , Geneva, . measure zero. This corresponds naturally proposition α < 1/2 is neither true nor false. e-mail: [email protected] to the intuitionistic concept of the viscous Hence the continuum is viscous6,7: it can’t be continuum: the continuum cannot be sharply cut in two, above and below 1/2. Published online: 6 January 2020 sharply cut in two. When trying to cut it, In these two examples, the law of the https://doi.org/10.1038/s41567-019-0748-5 something sticks to the knife. excluded middle holds only if one assumes This aspect of intuitionistic mathematics a look from the ‘end of time’, that is, a God’s- References is closely related to the law of the excluded eye view. But at finite , intuitionism 1. Weyl, H. Te Continuum (Dover, 1994). middle, which does not hold in intuitionistic states that the law of the excluded middle 2. Gödel, K. Collected Works Vol. IV (eds Feferman, S. et al.) p. 269 (Oxford Univ. Press, 1995). logic. In intuitionist logic, a proposition P is not necessary, that time truly passes and 3. Dolev, Y. Eur. J. Philos. Sci. 8, 455–469 (2018). could be neither true nor false. This is very the future is open. Looking at the law of the 4. Darbour, J. Te End of Time (Oxford Univ. Press, 2001). difficult to swallow for today’s scientists, excluded middle in this way makes its absence 5. Gisin, N. Erkenntnis https://doi.org/10.1007/s10670-019-00165-8 who have been educated within the remit in intuitionistic mathematics easily acceptable (2019). 6. Posy, C. J. J. Philos. Logic 5, 91–132 (1976). of classical mathematics. But think of a and makes the present naturally thick. 7. Posy, C. J. Mathematical Intuitionism (Cambridge Univ. Press, in proposition about the future, for example, “It Let me emphasize that classical the press). will be raining in exactly one time from formal mathematics and real numbers 8. Hilbert, D. in Philosophy of Mathematics (eds Benacerraf, P. & Putnam, H.) 183–201 (Cambridge Univ. Press, 1984). now at Piccadilly Circus”. If one believes in are marvellous tools that should not be 9. Ellis, G. F. R., Meissner, K. A. & Nicolai, H. Nat. Phys. 14, determinism, then this proposition is either abandoned. However, their practical use 770–772 (2018). true or false, though it might be impossible should not blind the physicists; after all, 10. Born, M. Physics in My Generation (Springer, 1969). 11. Dowek, G. in Computer Science – Teory and Applications in practice to know which alternative holds. their use does not force us to believe (eds Bulatov, A. A. & Shur, A. M.) 347–353 (Springer, 2013). 5 But if one believes that the future is open, that “real numbers are really real” . In 12. Chaitin, G. in Meta Math! Ch. 5 (Vintage, 2008). then it is not predetermined that it will rain, other words, one should not confuse the 13. Chatin, G. J. Preprint at https://arxiv.org/abs/math/0411418 (2004). 14. Borel, E. in From Brouwer to Hilbert (ed. Mancosu, P.) 296–300 hence the proposition is not true, and it is epistemological usefulness of classical (Oxford Univ. Press, 1998). not predetermined that it will not rain, thus mathematics with the ontology, which 15. Iemhof, R. Intuitionism in the philosophy of mathematics. the proposition is also not false. might well be better described by Stanford Encyclopedia of Philosophy (Summer 2019 Edition) https://go.nature.com/2E99Hqe (2019). There are similar examples using choice intuitionistic mathematics. 16. Gisin, N. Quantum Stud.: Math. Found. https://doi.org/10.1007/ sequences α(n) (Box 1). Assume that Furthermore, in practice one never uses s40509-019-00211-8 (2019). random numbers are not digits, but binary real numbers with all their infinitely many 17. Palmer, T. N. Nat. Rev. Phys. 1, 463–471 (2019). r(n) 1, and let (n) 1/2 r(n) 10–n. digits. Think, for example, of computer 18. van Dalen, D. L.E.J. Brouwer – Topologist, Intuitionalist, = ± α = + × Philosopher (Springer, 2013). This goes on until, by chance, the last simulations that necessarily at each time 19. Brouwer, L. E. J. in Proc. 10th International Congress of Philosophy, n/2 consecutive r take the same value only consider finite information numbers. vol. III (North-Holland, 1949).

116 Nature Physics | VOL 16 | February 2020 | 114–119 | www.nature.com/naturephysics