Bailly F., Longo G. Mathematics and the Natural Sciences: The Physical Singularity of Life

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Extended Criticality 253

In contrast, it appears that for biolo gy, the cells of an organism, the

organisms of a species, the species of an environment (that is, the biolons),

are concerned with “generic trajectories”: all those which remain compatible

with their perduration, even their transformations (mutations). It would

be the falling back upon the speciﬁcity of this generic which would cause it

to los e its biological character each time a physical reduction is attempted.

From then on, it is p ossible to understand the extended critical situa-

tion as the expressio n o f this genericity in contrast with the localized critical

transition of physics (within the space of parameters and within time, due

to their incess ant ﬂuctuations). To put it in other words, there would be a

sort of duality between physics and biology: speciﬁcity of the trajectories

and “locality” of the criticality for physics, genericity of trajecto ries and

extension of criticality for biology. At the same time that would enable us

to “naturally” relate the biological to the consider able variability of which

it is the locus given that this generic compatibility would allow for the ex-

istence of a great number of possible “trajectories.” Invariants should no

longer then be deﬁned within a given phase space but over the set of phase

spaces that are compatible with this genericity, e ach spe c iﬁcity being able to

modify the local structure of this set without, howe ver, disappearing. This

is maybe what would explain the great invariants (approximated), which we

have mentioned, almost always conce rning considerable sets of organisms

(mammals, oxygen metabolisms, the animal kingdom itself, e tc.), because

they would constitute some of the ra re invariants with regard to this global

plasticity. These invariants endure, within the variability of living phenom-

ena, a nd contribute to its stabilizatio n; moreover, this biological variability

diﬀers from physical variability, because the latter does not normally con-

tribute to stability (at least in non-linear cases) and ﬁnds itself within a

framework of determination, while the former may go beyond it.

To conclude, we believe that, for biology, it is the collection itself of rel-

evant objects and of parameters (phase space) which will follow the current

situation, which is indeterminate and that this indeterminatio n should be

intrinsic to the theory, as the co-constitution of living individual/species

(biolon) with and within an ecosystem is a major theoretical challenge in

biology. Within this changing frame, the trajectories themselves (a notion

that is preserved although it becomes problematic also in q u antum physics),

diﬀer from those of classical physics bec ause they are not speciﬁc (geodesics

within g iven spaces), but generic (p ossibilities within changing spaces). In

a sense, there would therefore be two levels of inter-c orrelated indetermi-

nation: that of the passing into the new ecosystem and that of the “choice”

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254 MATHEMATICS AND THE NATURAL SCIENCES

of trajecto ry for each biolon, among all compatible trajectories. Over the

course of phylogenesis, typically, the formation of a new species, which par-

ticipates in the ecosystem, e ven the emergence of a new organ, modiﬁes

the evolutionary space, even the space of ontogenesis, and modiﬁes the set

of possible trajectories. Once a gain, the genericity of the trajectory ought

not to be analyzed only within the same phase space, but also in terms of

passage to a new possible space.

For this reason, in our opinion, the analysis of the dynamics of living

phenomena cannot be reduced to the terms of cla ssical deterministic unpre-

dictability (with its geodes ics, of which the choice is more or less sensitive

to bounda ry conditions), but to those of an intrinsic indetermination of the

evolution of living organisms, to be understood in terms of the genericity

of the trajectories and of changes of phase spaces. As a result, similarly as

classical dynamics and quantum mechanics propose two causally diﬀerent

notions of randomness (a dice and spin-up/spin-down are random in a very

diﬀerent sense), so our approach to biological phenomena suggests a third

form of randomness: the indeterminatio n loca tes itself at the level of the

very space of phases or evolutions and coexists with a relative structural

stability of individual biolons.

The extension of critical situations, together with the intrication of the

levels of organization and their eﬀects of re c iprocal “resonance,” proposes an

intelligibility of the physical type, inasmuch as physics succeeds in provid-

ing us with adequate metaphor s. As quantum physics has succeeded with

regard to classical mechanics, it would be necessary to provide ourselves

with autonomous concepts and, if ever po ssible, ma thema tical structures ,

in order to better g rasp the ﬁelds of living phenomena and their dynam-

ics.

one phase space. It is there, we believe, that we must shift from a

determined trajectory to an intrinsic indetermination, comparable to the

possible and indeterminate paths of quantum physics, but concerning the

phase space itself. Moreover, as we have stres sed many times , within each

space, it is the generic trajectories that play an important role, not just the

geodesics.

The theory of viability (see Aubin, 1991) proposes a mathematical analysis which is

close, but diﬀerent, to the approach which we propose: we replace the evolution equation

(dx/dt = . . . ) membership within a s et of possible evolutions (dx /dt ∈. . . ). However,

the function x (t) and the set, which determines this passage, are given beforehand and

contain, according to our interpretation, the lis t of all possible future spaces, with their

geodesics; x (t), particularly, represents a “trajectory” of phase spaces, instead of a tra-

jectory within

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Extended Criticality 255

6.5 Another View on Stability and Variability

We have mentioned earlier the stabilizing role of reg ulation and integration

for the intertwining and the intrication of levels of organization. Our anal-

ysis, in terms of the genericity of the trajectories and dynamics o f phase

spaces should contribute to making intelligible the fact that the phenome-

nal life is characterized by the preservation of a few key inva riants not only

due to stability, but also to variability. For this reason, we have consid-

ered the intertw ining and the interactions of organization levels on the one

hand as one of the components of stabilization, and on the other as fo rming

part of biological indeter mination (as resonances destabilizing each level of

organization). Following varia bility, typically, biolons all diﬀer from one

another: the individuals of a spec ies are not and must not be identical.

Even cells have a certain form of “individuality.” Indeed, va riability is at

the ce nter of evolution and, in conjunction with individuation, it is also at

the basis of ontogenesis. Clearly, here lies another singularity of living phe-

nomena in refere nce to the physical descriptions, where individual entities

of the “same type” are all identical and their description can be, conceptu-

ally as well as mathematically, perfectly stable (the speciﬁcity of physical

trajectories , which we have a ddressed).

6.5.1 Biolons as attractors and individual trajectories

Let’s now try to understand this approach, described by indeterministic

dynamics within globally s table frameworks, and this stability/instability

dialectic, in diﬀerent terms, which are, nevertheless, compatible with the

former.

On the one hand, a species presents a global structural s tability, whereas

its members may display diﬀerent variations in their nature and behavior.

Likewise, in its own time scale, an individual is (relatively) stable, while its

cells evolve and die, each according to a diﬀerent path. In one or the o ther

case, the global structure is essentially (but not entirely) preserved while

local variations occur. If we take a physical analogy, the greater biolon

(a species, a metaz oan) can be desc ribed by means of the geometry of an

attractor, of which the globa l dynamic is essentially stable a s long as it

remains within the same phase spac e . The relative instability of individual

trajectories within the attra c tor, those of the smaller biolons in it (the cells

in a metazoan, the individuals in a species), contributes to variability o f

the global structure. Yet, the local biolons, a s part of orgons, are stabilized

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256 MATHEMATICS AND THE NATURAL SCIENCES

by the function (of the orgon they may belong to – a population in the

case of a species ). Thus, the functionality contributes to the local and

global stability, by maintaining the function. While their minor instabilities

constitute adaptability, by (minor) changes in the individual trajectories

(or, say, variations in the fractal structure of lungs yield changes of survival

in changing environments).

We can understand the change of phase spaces as the toppling of the

attractor into another space, with the characteristics o f indetermination,

which we have discussed. The diﬃculty would then consis t in the search

for the correct classes of universality, or of the great biological invariants,

such as, for instance, absolute numbers (the clocks of living matter, etc.),

which concern the species, even entire phyla, and which would allow us to

sp e ak of a toppling of the “same” attractor.

This dynamic stability is as compatible with the variations and even the

instability of the individual trajectories, which may fall within the same at-

tractor (the smaller biolons, included within the largest – a metazoan, a

cell, respectively). In this sense, a biolon of the intermediate level (a meta-

zoan) co nstitutes a trajectory for a species, the latter being considered as

an attractor; a t the same time, this biolon behaves as a stabilizing attra c tor

for its own component, its cells.

As we have observed, in physics, the dynamic of the trajectory is given

by the geodesic within a predeﬁned phase spac e ; thus, it is selected accord-

ing to the highest stability and ge nerally in a unique way. In other words,

it is speciﬁc and optimal in the sense given by the metric space, a measure.

In contrast, individual biolons are, in our opinion, represented by diﬀer-

ent and g e neric simultaneous paths, and simply submitted to compatibility

constraints with conditions at the boundaries (which, in turn, do not have

to be stable and can be modiﬁed by continuous processes). This approach

would enable us to simultaneo usly capture the properties associated w ith:

• the interaction of relative global stability and individual variability, as

attractor vs trajector y;

• compatibility instead of “optimality” of geodesics, as recalled above,

given that trajectories must re main within the evolutionary boundaries

of the attractor.

Each individual trajectory would thus ﬁnd its origin in an unsta ble

situation and would actualize various contextual potentialities (typically

the ﬁrst mitosis in embryogenesis, within the same genotype) and would

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Extended Criticality 257

evolve according to the potentialities accessible within the framework of the

global structural stability.

Natural selectio n may reduce variability at the spec ies level, all the while

increasing it at the individual level. The more or less tolerable pathologies

would rather concern the cells within an individual. Naturally, this con-

ceptual mo de perfectly adapts to the above-proposed notion of extended

critical situation, as a paradigmatic state of biolons. Besides, it may also

be seen as a generalization of Waddington’s notion of chreode (Wadding-

ton, 1977), by the aspe c ts of genericity and indetermination upon which we

have insisted.

Notice that this schema refers solely to biolons, at various levels, given

that only biolons are concerned with identities (identities preserved by

changes). Orgons would rather constitute, through the energetic exchanges

and the functional activities to which they ar e the locus, the mater ial sup-

port to stability a nd to varia tion: an orgon is no t a trajectory,

but may

be at the origin of the continuo us variation of the individual represented

by the trajectory.

Physical paradigms have aided us in formulating these notions, which

are not of physical nature proper. We would like to re c all, once more,

that all the while working within a monist framework, we face va rious

phenomenalities, such at least as they are co -constituted with our living

and historical being. Over the course of history, these phenomenalities

have been made (partially) intelligible, by their organization using various

conceptual structures (and if possible, mathematical ones) – especially over

the course of the XXth century, so rich in science, before aim ing toward

“uniﬁcation.” A s ynthesis, in fact, is far from obvious; it must be constructed

by means of a new (and mathematical) conceptual uniﬁcation as its long-

term objective. As has occurred in physics, the c onceptual and technical

uniﬁcation with the biological sciences will be possible only after having

sp e c iﬁed, we believe, the causal structure and the dynamics speciﬁc to each

phenomenal level, in its theoretical and experimental autonomy and after

having established the conceptual bridges co nnecting one intelligibility to

another, without nece ssarily, or immediately, reducing o ne to the other.

Let’s remind ourselves that a faltering orgon may be replaced by a “prosthesis,” an

artifact, likely to serve its purpose without decisively aﬀecting the biolon to whi ch it

participates (organs within an organism, for instance). This is generally not possible for

biolons: in no way can we replace an individual cell.

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Chapter 7

Randomness and Determination in the

Interplay between the Continuum and

the Discrete

Introduction

The idea hinted at in Chapter 5, is that the mathematical structures, con-

structed for the intelligibility of physical phenomena, according to their

continuo us (mostly in physics) or discrete nature (generally in computing),

may propose diﬀerent conceptions of nature. We will futher develop this

issue her e

As a matter of fact, the causal re lations, as structures of intelligibility

(we “understand nature” by them), are mathematically related to the use of

the continuum or the disc rete. In particular, as the myth that the world “is”

(in the end) discrete or tha t the universe is, such as the brain or the mecha-

nisms of biological heredity a re “discrete algo rithms” (the brain as a digital

computer, the genetic program), is still largely common, this chapter will

challenge these views, in biology and even cognition, from within physics.

The fo c us on randomness will explain our approach to determination by a

conceptual duality.

But what discr e te (mathematical) structures are we talking about? We

believe that there is one clear ma thema tical deﬁnition of “discr e te,” that we

will use in this paper: a structure is discrete w hen the discrete topology on

it is “natural.” Of course, this is not a formal deﬁnition, but in mathema tics

we all know what “ natural” means. For example, one can endow Cantor’s

real line with a discrete topology (all points are isolated), but this is not

“natural” (you do not do much with it, nor do you better understand the

reals nor the notion of “continuous functions”). On the other hand, the

integer numbers o r a digital database are naturally endowed with a discre te

A preliminary version of this chapter is in Math. Struct. in Comp. Science, vol 17, n.

2, pp. 289-307, 2007.

259

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260 MATHEMATICS AND THE NATURAL SCIENCES

top ology (even though one may have good reasons to work with them also

under a diﬀerent topological structuring).

In the sequel, the randomness/unpredictability issue in quantum me-

chanics is going to be discussed. Our approach will then stress that, in the

space-time of modern microphysics, in no way one may consider the discrete

top ology as “natural.” Our re ﬂec tion will be based exactly on quantum non-

locality and non-separability results that, in our view , propose the exact

opposite of an underlying discrete space-time. Indeed, the dis c rete topology

“separates ” and “localizes” the elements of mathematical structures, this is

its job. Of course, quantum mechanics started exactly by the discovery of a

fundamental (and unexp e c ted) discretization of light absorption or emission

sp e c tra of atoms (spec iﬁcally the atom of hydrogen). Then, a few dared to

propose a discrete lower bound to the mea sure of action, that is, the prod-

uct energ y×time. It is this physical dimension that has a discrete structure.

Clearly, o ne can then compute, by assuming the relativistic maximum for

the spe e d of light, a Planck length and time. But in no way are space

and time thus o rganized in small “quantum boxes.” And this is one of the

most striking and crucial fea tures of quantum mechanics: the global and

entanglement e ﬀects (Bell, 1964; Bo hm, 1951; Aspect et al., 1982). These

eﬀects suggest the opposite of a discrete, separated organization of s pace

and time and is at the c ore of its scientiﬁc originality.

In particular, entan-

glement motivates q uantum computing (as well as our analysis of quantum

randomness).

As for the continuum, its role will be stressed in understanding classical

determination, as mathematized in the geometry of dynamical systems. As

a matter of fact, the notion of randomness lies at the center of dy namic

unpredictability: a deterministic system is unpredictable, precisely when

it presents random evolutions. In quantum mechanics, this notion is also

evoked but in a very diﬀerent and intrinsic way (we will give a more precise

meaning to this term). And we will attempt to examine the following issues

closely: how does randomness present itself in to day’s natural sciences? Is

there a randomness speciﬁc to the various ﬁelds of physics? What impact

does this eventual diﬀerentiation/unity of the notion of physical uncertain-

ties have on the common/scientiﬁc conce pt of randomness? Is it correlated

to the various mathematical tools established (continuous vs disc rete, for

example)?

In ongoing work with T . Paul, a quantum physicist (see the papers downloadable from

http://www.di.ens.fr/users/longo/), we show how the idea of the topological separability

of quanta is the hypothesis underlying the (beautiful) but wrong argument in Einstein

et al. (1935).

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Randomness and Determination 261

We will not return to the terrain of spec iﬁcally biological randomness

(see Chapter 6) which rema ins , in our opinion, unexplo red. We will ﬁrstly

reﬂect upon class ical physics, in order to consider the problem of the mean-

ing of randomness in the context of diﬀerent theoretical fra meworks. We

will see that dynamical s ystems and quantum physics independently pro-

pose very important, but not identical, notions of randomness, giving rise to

a diﬀerent role for probabilities in their various contexts. Our distinguishing

criterion will refer to the notion of “epistemicity” as being in c ontraposition

to that of “objectivity,” the diﬀerence being related to the role of the know-

ing subject in the construction of scientiﬁc objectivity. This is an e ssential

role within modern physics, particularly w hen it is a question of probability

and randomness. In short, we will ﬁr st state the problem of knowing if a

disordered sequence (or more generally, a dis ordered state) is the eﬀect of

chaotic determinism or of pure rando m processes, of an epistemic or an

objective nature. In the absence of very general theorems on the matter

(which would contribute to a constitution of objectivity – as is the case for

“mixing” random sequences, or Bernoulli’s dynamics, that can be demon-

strated as being equivalent to – and thus interpreted as – “heads or tails”

type sequences; we will return to this), we can develop an argument, which

will have the eﬀect of somewhat displacing the problem all the while better

highlighting some of the constraints which it implies.

We will develop the thesis that in classical physics, randomness is of

an “epistemic” nature , particularly meaning that one will always be able

to interpre t an apparently random sequence as stemming from a chaotic

determinism or as stemming from “pure” randomness, which is analyzable in

statistical terms (independently on any possible determina tion). Thus two

diﬀerent approaches are possible, according to the viewpoint one assumes.

In contrast, we will c all “objective” the randomness speciﬁc to quantum

mechanics. O ur argument will base itself upon two elements justifying

this distinction. One is centered upon the role of theoretic al determinism,

of the “view” (constituted of the proposed theoretical frame work) and of

measurement. The other, stro nger in a c e rtain sense, is founded upon

the properties of quantum non-separability and of continuous mathematics,

which we will discuss in length.

The fundamental diﬀerence between classical and quantum random-

ness we will explore here may give an idea of one of the diﬃculties to be

found in any attempt to describe a proper “biological ra ndomness,” which

we mentioned in prev ious chapters. While waiting for uniﬁcation (classi-

cal/quantum), very expressive theories (classical and relativistic, quantum)

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262 MATHEMATICS AND THE NATURAL SCIENCES

have been built and randomness is separately analyzed and successfully ap-

plied in the diﬀere nt contexts. In systems biology, in evolution in particular,

it may be very ha rd to pr opose a suitable theory of randomness, also be-

cause this separation of theoretical context seems unsuitable: the least one

can say, is that classical interactions (at the phenotypic level typically) are

“entangled” with molecular or atomic interactions, and thus, most likely,

to quantum eﬀects (in mutations). Thus, r andomness in evolution may

require an understanding of the simultaneous action of both classical a nd

quantum ra ndomness. The discuss ion in this chapter of the di ﬀerences, in

the understanding of randomness within physics, is not devoid of interest

as for possible descriptions of the singularity of biology also concerning this

aspect of (in-)determination.

7.1 Determini stic Chaos and M athematical Randomness:

The Case of Classical Physics

Let’s therefore begin by analyzing classical randomness. To this end, we

assume the idea according to which the resu lt of a throw of dice, for ex -

ample, which we legitimately consider to be random, may be interpreted

as such given that the system of e quations and of constraints enabling us

to describ e it (and thus to determine it) is (very) sensitive to the initial

conditions. Note also that the reverse interpretation is quite diﬀerent: a

dynamic system deﬁned by its equations and presenting a chaotic behav-

ior has an objective character as sociated with these equations themselves

and to their intrinsic properties (it is described/given by physical o bjects,

with their properties, their invariants, their symmetries . . . deduced from

the equations, see Chapters 3 and 4). In short, from the modern (po st-

Laplacian) point of view, even this system, a paradigm of randomness, is

indeed deterministic, in the sense that a suﬃcient number (in fact a very

great numb e r) of equations could theoretically describe all the forces at play,

all of them being theoretically well-known (gravitation, various frictions).

However, this would tell us very little about its evolution: this classical

system is s o sensitive to the slightest variation in the boundary conditions,

which are, moreover, very numerous, that the mathematical eﬀo rt of de-

scribing the (many) equa tions determining it will not help us in practice,

even not qualitatively. And its evolution remains unpredictable: it is that

which leads us to consider it, within a class ical framework, as random all

the while being deterministic (and chaotic).