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How Future Depends on Past and Rare Events in Systems of Life


The dependence on history of both present and future dynamics of life is a common intuition in biology and in humanities. Historicity will be understood in terms of changes of the space of possibilities (or of “phase space”) as well as by the role of diversity in life’s structural stability and of rare events in history formation. We hint to a rigorous analysis of “path dependence” in terms of invariants and invariance preserving transformations, as it may be found also in physics, while departing from the physico-mathematical analyses. The idea is that the (relative or historicized) invariant traces of the past under organismal or ecosystemic transformations contribute to the understanding (or the “theoretical determination”) of present and future states of affairs. This yields a peculiar form of unpredictability (or randomness) in biology, at the core of novelty formation: the changes of observables and pertinent parameters may depend also on past events. In particular, in relation to the properties of synchronic measurement in physics, the relevance of diachronic measurement in biology is highlighted. This analysis may a fortiori apply to cognitive and historical human dynamics, while allowing to investigate some general properties of historicity in biology.

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  1. 1.

    This is the space of all pertinent parameters and observables, as measurable quantities (see below for more on this notion, which, in physics, is a precise, mathematical frame for the “space of all possibles dynamics”).

  2. 2.

    Cf., in physics, kinetic and potential energy are different observables in the same dimension, energy.

  3. 3.

    An “invariant” is characterized by the transformations that preserve it—typically, transformations in space and time as reference systems, like in Galileo's or Einstein's relativity, or, in mathematical terms, by groups or category-theoretic isomorphisms.

  4. 4.

    Pierre Musso suggested the adjective “historicized” in order to express our concern for a conceptual transfer of the notion of invariance from a physico-mathematical frame to an historical science.

  5. 5.

    On unpredictability and randomness. We stress that randomness, for us, means unpredictability in the intended theory (Calude and Longo 2015). This relativizes randomness to the theory and its symmetries (Longo and Montévil 2017). Without an at least tentative theoretical frame, one cannot talk of unpredictability: in order to “(un-)pre-dict” one needs to try first to “say” something (“dicere”, in latin). In this understanding, classical and quantum randomness differ, first since the two theories give non-classical probability values to entangled events (Einstein et al. 1935; Aspect et al. 1982), second, because, by measurement and Schrödinger's equation, randomness is “integrated” (is intrinsic) to Quantum Mechanics. Moreover, but this is a further issue, classically any event has a cause, may it be a non-measurable fluctuation or perturbation, while in the current interpretation, some quantum random events may be a-causal, such as the spin up or down of an electron. Hidden variable theories claim that there is always a hidden cause; but then they get into different troubles: they need non-local variables in order to handle entanglement, a physico-mathematical inconsistency. In short, randomness is diverse and theory dependent in physics; a further notion of randomness is proposed here for biology (comparative analyses are presented in Buiatti and Longo 2013; Bravi and Longo 2015; Calude and Longo 2015).

  6. 6.

    The aspects of history and philosophy of science in this analysis are the core issue of the project directed by the author at the IEA of Nantes, “Lois des dieux, des hommes et de la nature” (2014–2020), in collaboration with historians and jurists:

  7. 7.

    The changing phase spaces of biological evolution was explicitly mentioned, in different frames and languages, independently by Kauffman (2002) and Bailly and Longo (2011; in French: 2006). Yet, we were all preceded by R. Thom, in his papers in Amsterdamski et al. (1990). For Thom, in scientific analyses, the mathematical phase space pre-exists to the randomness (“noise”) affecting the system (p. 70); thus, “it is the lack of the definition [of the virtual possible] that affects—very seriously—the scientific nature of Darwin's Theory of Evolution.” (p. 271). Following Darwin, instead, we work at a science where changes occur at the very level of the phase space, of the virtual possible, in Thom's terminology, and where random events may modify it. Thom's remark resembles the insight in Einstein et al. (1935): “Quantum Mechanics is incoherent or incomplete, since it implies particles' entanglement”, a property that they formally derive in QM and that contradicts the separability of distinct and measurable events—an absurdity in Einstein's view, as much as Darwin's theory is not scientific for Thom. Great minds may see the key point, even when they are wrong: Darwin's Theory of Evolution does not allow to pre-define the “virtual possible”, yet it is scientific; Quantum Mechanics allows to derive entanglement, against Relativity Theory, yet entanglement has been corroborated, it is not absurd (Aspect et al. 1982).

  8. 8.

    Also H. Weyl considers the mathematical framing of natural sciences as based on the preliminary construction of a pre-given mathematical space of possibilities, to be a priori set in the background of any analysis. As this must be grounded on fundamental pre-listed symmetries and their possible breaking, he hints to the difficulties this may present as for a mathematical foundations of biology (Weyl 1949).

  9. 9.

    Globally, astrophysics is more a science of some fundamental, largely invariant, processes, such as star or planets' formation (Longair 2006). Yet, the comparatively unbalanced amount of chemical elements is often explained by invoking a particular history of the universe; this is analogous, though, to the understanding of the particular position of a ball downhill by knowledge of its path: an original symmetry breaking or the iteration of a few of them fully describes the state of affairs (see the following examples on “path dependence”). A more interesting example is given by alloys. The precise course of annealing can cause the same alloy to end up with drastically different properties: traces of the past (or “shape memory”) play a role in some critical transitions or bifurcations in the intended transformation. However, the entire process and its time are analyzed in a pre-defined phase space. Even a recent radical stress on historicity in cosmology, the changes of value of the fundamental physical constants, refers to changes of numerical values within pre-given dimensions, that of G, c, h (or of a-dimensional constants, such as alpha), see Uzan (2011).

  10. 10.

    This is the case also for equilibrium thermodynamics, where path dependent observables (often called variables) are entropy, enthalpy, pressure ….

  11. 11.

    In statistical mechanics, as recalled in Longo and Montévil (2014), one may have a randomly varying number of particles. Thus, the dimension of the state space, stricto sensu, is not pre-defined. However, the range of possibilities is known: the particles have a known nature, that is the relevant observables, the equational determination and the probabilities of each particle's phase space are given. In other terms, even if the number of dimensions of the space may be unknown, it has a known nature and probabilities—we know the probability it will grow by 1, 2 or more dimensions, and, most importantly, they are formally symmetric. Then the number of (possibly extra) particles just becomes a new parameter. Thus, the situation is delicate, but mathematically fully mastered (see Sethna 2006, for an introduction).

  12. 12.

    A classical example of protention is the predator's eye jerk preceding the prey's trajectory (Berthoz 2000).

  13. 13.

    There exist relevant neurophysiological and neuro-imaging evidence for (pre-conscious) protention and the dependence of protention on retention (Szpunar et al. 2007; Botzung et al. 2008).

  14. 14.

    An elementary example: a straight line may be defined as an axis of rotations, that is as the invariant of a group of symmetries, which are transformations of the tridimensional space in itself.

  15. 15.

    As for the difficult notion of “interpretation” by life forms, we do not mean here conscious “meaning as reference”, but en-action in an “umwelt” in the sense of von Uexküll (Brentari 2015), or meaning as encored on action, in a broad, (bio-)semiotic, sense (see also Deacon's reference to Peirce, e.g. in Deacon 2015). This further relates to the rest of the paper, which is on biological evolution.

  16. 16.

    Prochiantz (1997) observes that a pianist thinks also with the hands, a dancer with the feet. Neural structures and organs (muscles) are jointly modified and retention is distributed (see also the analysis of vestibular memory in Berthoz 2000).

  17. 17.

    Planarians are shown to recover some memory after their cut head regenerates, as if the bodily traces of a past and memorized activity could influence the newly formed brain (Shomrat and Levin 2013). As observed in the previous note, it is known that pianists and violinist have visible neural synaptic reinforcement somehow corresponding to local muscles increase (yet, it is not known whether the experiment with planarians would work with pianists or violinists, for lack of volunteers).

  18. 18.

    The co-constitutive history of physics and mathematics, the latter being a limit construction, is grounded, in particular, on the genericity both of mathematical concepts and of physical objects: they are invariant of the theory and experiments, e.g. any right triangle works for a fully general proof of Pythagora's Theorem, any stone or electron would do in a pertinent experiment, from Galileo to Planck. In other words, they are symmetric or invariant under permutations with other right triangles, stones or electrons. Moreover, physical trajectories are specific, that is, they are geodetics in suitable phase spaces. These issues are discussed at length in Bailly and Longo (2011) and Longo and Montévil (2014). In our approach, the mathematical tools for physics cannot be transferred as such to biology, except for some local applications (some aspects of morphogenesis, for example), since biological objects (the organisms) are specific (historical, individuated) and their trajectories are generic (they are possible evolutionary and ontogenetic paths), a crucial duality with physical theories, see Longo and Montévil (2014).

  19. 19.

    Galileo started the dance of modern physics by observing momentum: inertial movement is momentum conservation.

  20. 20.

    This is explained at length in the introduction to Longo and Montévil (2014). For example, Boltzmann asymptotic unification of classical trajectories and conservation laws with thermodynamics is discussed as a counterexample to prevailing reductionists modes. Boltzmann's approach lead to a new unifying theory and its mathematics: statistical mechanics, see Chibbaro et al. (2014) for this and more on the anti-reductionist history of physics.

  21. 21.

    Independently of life, it is even difficult to assign probability values to the Earth or the Planets positions in the Solar system at that or at a slightly larger time scale (Laskar 1994). Yet, the space of possible positions is known: the phase space of the classical dynamics of the Solar System. Thus assigning probabilities is legitimate, though technically difficult.

  22. 22.

    Technically, planets are in resonance when they are aligned with the Sun. This produces major instabilities in their orbits, very well expressed by non-linearity, and massively contributes to chaos, even in the Solar System (Laskar 1994). Yet, this happens within one level of determination, the equations for the planetary system. Bio-resonance, instead, refers to the interferences, by regulation and integration, between different levels of organization in an organism, possibly given by different forms of (mathematical) determination (e.g. statistical networks as for cells' interactions and classical non-linear dynamics as for organogenesis (Buiatti and Longo 2013).

  23. 23.

    This flat application of classical geodetics to phylogenetic trajectories is nicely criticized also by physicists working in biology, see Goldenfeld and Woese (2011), which “highlights the importance of collective interactions and the interplay between environmental fluctuations and evolution, which are neglected in the Modern Synthesis” as well as in geodetic/optimality approaches to evolution based on it. The best is to quote their text: “… horizontal gene transfer is now known to be present in multicellular Eukaryotes as well, as a result of genome-wide surveys published in the past year or so … Inheritance of acquired characters … not only through horizontal gene transfer (e.g. in microbes), but also through so-called epigenetic mechanisms that bypass the usual modes of inheritance, especially in ciliates. Not only is the Modern Synthesis afflicted by strong interactions, but its very foundation is questionable. The evident tautology embodied by “survival of the fittest” serves to highlight the backward-looking character of the fitness landscape: Not only is it unmeasurable a priori, but it carries with it no means of expressing the growth of open-ended complexity and the generation of genetic novelty. Thus, the Modern Synthesis is, at best, a partial representation of population genetics; but, this on its own is a limited subset of the evolutionary process itself, and arguably the least interesting one” (p. 383). The “a priori unmeasurability” for us is due on the co-constitution of phylogenetic trajectories and their phase space; the open-ended random complexification of organisms is formalized in Bailly and Longo (2009) and Longo and Montévil (2014).

  24. 24.

    See the footnote above on Goldenfeld and Woese (2011) and West-Eberhard (2003) for more extensive references.

  25. 25.

    This is needed for the application of the renormalization methods to this phenomena (Binney et al. 1992).

  26. 26.

    For a comprehensive view in this perspective, see the journal special issue (Soto and Longo 2016).

  27. 27.

    In the immense literature on mutations as well as on different uses of DNA in evolution, some recent striking advances are in Harms and Thornton (2014).

  28. 28.

    In todays' mathematics, Grothendieck's approach is the finest existing form of relational mathematics and its application to contemporary physics is being organized in this style, see Zalamea (2012). The definition of new concepts and structures, in Grothendieck's relational approach, develops but goes beyond the approach based on invariants and transformations, which range from Klein to Weyl and MacLane (the founder of Category Theory): definitions a la Grothendieck are given in the “purest” yet meaningful mathematical frame, so that their invariance and the intended transformations are “intrinsic” to the new notion, see Longo (2015).

  29. 29.

    An organism is an ecosystem, with crucial phenomena of symbiosis of different species. Yet, an ecosystem is not an organism: its coherence structure is more weakly correlated, at least by the metrics (but also by the functions). For example, two cells in the same tissue will never depart too far and maintain their organismal functions, while, even within a colony, say, two … workers ants may go from 0 to tenths of meters apart and still be functional. But, of course, there is more than this as for the greater coherence and autonomy of an organism (Moreno and Mossio 2015; Montévil and Mossio 2015).

  30. 30.


  31. 31.

    Typically, in octopuses, the nerve fibers pass behind the retina, while, in vertebrates, they route before it and disrupt it by a “blind spot”. The different phylo- and onto-genetic histories of the two neural systems help to (partially) understand this difference and show that there is no optimality, since the vertebrates' eye may be considered '”better” as for the presence of a cornea, but not as for the nerves' routing (the rumor has it that Helmholtz, an early profound analyst of the eye structure, proposed to fire the Designer of the vertebrates' absurdly connected eye …).

  32. 32.

    As for the liver and in contrast to the immune systems and even lungs, in Bravi and Longo (2015), we naively accepted the claims of “noise biology”, an approach that we criticize (randomness should be considered “noise”, in biology, to be analyzed by averaging out, central limits etc): in liver, we wrote, what only matters is the average enzymatic production. Yet, even in this case, the statistical, “averaging out” approach is wrong. That is, even such an apparently “dull” organ contributes to the organism's and to its own stability by an extensive aneuploidy and polyploidy of its cells (almost 50%). By this form of cells' diversity, the liver better responds to toxic injuries and damages: “subsets of aneuploid hepatocytes that are differentially resistant to the injury remain healthy, regenerate the liver and restore function” (Duncan 2013).

  33. 33.

    Rarity, though, may be historical or relative. In experimental evolution, in Echerichia coli, extremely rare mutations unexpectedly turned out to be relatively frequent after a particular history of the population. In particular, their phenotypic expression may contingently depend on prior mutations in that population, along a 30,000 generations' history (Blount et al. 2008). Yet, according to these authors, it does not seem to be the result of gradual, cumulative changes: rare events may become frequent, after a long history in a rare environment [for E.c., an artificial glucose-limited medium that also contains citrate (Blount et al. 2008)].


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Claus Halberg and the referees made several insightful and constructive comments.

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Correspondence to Giuseppe Longo.

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This work is part of the project “Lois des dieux, des hommes et de la nature” at IEA-Nantes. A very preliminary version in Italian appeared in Paradigmi, XXXIII, Mag-Agosto, 2015.

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Longo, G. How Future Depends on Past and Rare Events in Systems of Life. Found Sci 23, 443–474 (2018).

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  • Biological Evolution
  • Space of Possible Phenotypes
  • Invariance
  • Processual Time
  • Historical Time