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

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Causes and Symmetri es 203

Turing has the huge merit of having invented the machines and of hav-

ing, in 1950, when he abandoned the myth of the great digital br ain, high-

lighted the diﬀerence between computational imitation, the game which

should demonstrate his machine to be indistinguis hable from a woman

(modulo the sole intermediary of a written interface, (Turing, 1950)), and

modeling. In fact, in his 1952 article o n morphogenes is, he presents a model

of chemical ac tions-reactions which generates for ms and which bases itself

upon a system of diﬀere ntial equations: tiny variations generate the variety

of forms in certain natur al phenomena (Turing calls this sensitivity “expo-

nential drift,” a quite relevant and origina l name! The notion of “sensitiv-

ity to the initial c onditions” dates back to the 1970s). T his ma thema tical

model, which he explicitly says could be false, nevertheless, attempts to

make the world intelligible; the 1950 imitation tends for its part to trick

the observer (and will thus be considered at the origin of cla ssical artiﬁcial

intelligence).

5.2.4 Rules and the algorithm

Computerized simulation transforms all physical evolution into an elabora-

tion of digital information. Particularly, the simulation of a geodesic in a

discrete universe s hould make a digital computation correspond to a tra-

jectory, and make the conservation of informa tion corr e sp ond to laws of

conservation (energy, movement, . . . ). Any state or property, in short, any

physical quantity, as a determination of objects (in the sense of D.1), are in

fact encoded by digital information; the quantity of movement is encoded

using 0s/1s, just as is the intensity of a ﬁeld or mass, and their evolution is

a calculus approximated by these 0s/1s. Is this encoding “conservational”

(do e s it preserve that which is important)? If a physical trajectory is a

geodesic, which geodesic do we associate with the c alculus in its digital

universe, w hich symmetry breaking do we associate with rounding?

Let’s begin by recalling the generality of the geodesic principles in

physics at the center of this science since Copernicus, Ke pler, and Galileo.

As we observed in Chapter 4 and as we recall again here: any fundamental

law of physics is the expression of a geodesic principle applied within the

appropria te space. The work of the physicist, who organizes and, by that,

makes the phenomena intelligible, consists in a good meas ure of the search

for this spa c e (conceptual, or mathematical) and for its relevant metric.

This approach leads us to understand the slide of meaning around the

concept of law, which intends to justify the identiﬁcation of mathematica l

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204 MATHEMATICS AND THE NATURAL SCIENCES

modeling w ith computational imitation. The notion of “law” has a socia l

origin: law is normative to human behavior. The tr ansferal of the concept

as it is to physics corresponds to an ordinary metaphysics: an a priori (di-

vine, if possible), which would dictate the world’s laws of evolution. Matter

would then conform itself to this pre-exis ting and normative ontology (as

mathematical laws of a Platonic universe, for instance). On the other hand,

the comprehension of the notio n of law as an explication of the regularities

and criticalities of a landscape, w ith its mountain passes, valleys, and peaks,

its geodesics, inverses this and highlights the transcendental constitution at

the ce nter of any construction of knowledge. Mathematics, the tool of for-

mal determination, then elabor ates itself upon the phenome nal veil at the

interface between us and reality, the reality which, of cour se, causes fr ic-

tion and ca nalizes the cognitive act, but which is also organized by this

same act. Laws are not “already there,” but are co-constituted in the in-

trication between ourselves and the world: the discernability of geodesics,

as formal determinations within the framework of a network of interac-

tions, is its main result. And their mathematical processing c oincides with

the beginning of modern sc ience . The normativity of physical law then

becomes only cognitive (in o rder to co nstruct knowledge), and not a n on-

tology. The diﬀerent forms of formal determination (laws) propose diﬀerent

causal regimes, the ulterior tools for intelligibility.

The identiﬁcation b e tween algorithm and law causes us to make a back-

wards s tep: the algorithm is normative for the machine, for its calculi,

exactly as God’s law governs any trajectory. The machine would not know

where to go; it would be static, without its primary motor, the program.

Once again, the myth of the computer-universe (the genome, evolution, the

brain . . . all governed by algor ithms) consists of a metaphysics and a notion

of determination which precedes the science of the XXth century and for

good historical reasons: the re-centering of the foundations of ma thema tics

upon a philosophy of the arithmetic absolute, at the fringes of the time’s

great scientiﬁc turning points, as we have mentioned a few times.

This way of understanding law should highlight the very ﬁrst diﬃculty

for the computational simulation of a physical trajecto ry by mea ns of a

calculus. Physical law and algorithm therefore do not c oincide: they do

not have the same epistemological status. Law is also not an algorithm for

another reason we have already evoked: the formal determination, as math-

ematical explanation of laws, does not imply the predictability of physical

evolution. On the other hand, any algorithm, implemented within a dis-

crete state machine, generates a predictable calculus, at least thanks to the

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Causes and Symmetri es 205

“symmetry” by repetition within time, which we mentioned e arlier (identi-

cal iteration always being p ossible for a sequential computer).

However, we absolutely need digital simulation, a tool which is indis-

pens able today for any sort of construction of scientiﬁc knowledge: by

highlighting the diﬀerences, outside o f any computational myth (the world

would be like a computer), we aim to better identify that which is doable,

and afterwards to do better, in terms of simulation-imitation. Let’s attempt

then to understand the evolution of a calculus in physical terms.

In the case of an is olated computer – a sequential machine – we remain

within a Newtonian framework: the absoluteness of the clock and of the

access to a database are its essential characteristics. The situation is more

complex with modern networks: the distribution of ma chines in physical

space and the ensuing relational time changes the situation. Ce rtain aspects

of the absoluteness of Turing ma chines are thrown into question (we discuss

this in detail in Longo (2007) and, more technically, in Aceto et al. (2003)).

However, the exactitude of the discrete database subsists, as well as the

issue o f rounding, of course.

In both cases, o f sequentiality and concurrence, we may nevertheless

understand calculus as a geodesic within the s pace (pre)determined by the

program (more spe c iﬁcally: by the programming environment, or all as-

pects of software – operating system, compilers and interpreters, programs,

. . . ). Shortly, while accepting the divine a priori of the programmer who

establishes, before hand, the rules of the game, the notion of “following a

geodesic” would be deﬁned by following the rule correctly. The hardware or

software bug would then be the ﬂuctuation or perturbation that cause s the

evolution to derail. However, this type of bug is not integral to the theory;

it is not inherent to it, contrarily to the theory of dynamic systems which

integrates the notion of sensitivity to the conditions at the edges as well

as measurement by interval. Moreover, hardware bugs and logical errors

are very rare (therefore, statistically very diﬀere nt to the variation due to

approximation within a dynamic); they are to be avoided and, in princi-

ple, are avoidable (or they may belong to another phenomenal level, which

is far from being integrated within the mathematics of eﬀective calculus:

quantum physics).

We are left with the issue of rounding, which is inherent to calculus. The

use of rounding, today, can be very dynamic a nd mobile: one c an aim to a

desired approximation a nd the end of a calculus and increase beforehand the

available decimals, up to hundreds, to end within the targeted interval, if

possible. The modern approach to analysis by intervals provides a powerful

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206 MATHEMATICS AND THE NATURAL SCIENCES

theoretical framework for these processes (Edalat, 1 997). Of course, the

sp e e d of the calculi is inversely proportional to the improvement of the

approximation. An excess of the latter may prevent us from following any

dynamic long enoug h.

This being said, this bound, the rounding, constitutive of the arithme-

tization of the world (a quite necessary arithmetization if one wants digital

machines to perform calculi and therefore to participate in science today),

modiﬁes the causal regime and the s ymmetries corr e lated to it, as we hinted

and keep demonstrating .

Firstly, let’s clear up a possible confusion: the interval inherent to classi-

cal physical measurement and quantum incertitude have nothing to do with

digital approximation. First, measurement as interval is a physical prin-

ciple, a classical one; it is not a “practical” issue: thermal ﬂuctuation, at

least, is a lways present above absolute zer o, by principle. And, as we have

already and often observed, the ﬂuctuation or perturbation below the ob-

serva ble amplitude participate in the evolution of a dynamic system, which

is somewhat unstable, be c ause it can break the symmetries of the evolution,

and thus be one of the causes of a speciﬁc trajectory. In computer science,

a bug which manifests below the ro unding is without eﬀect. Afterward, the

analogy sometimes made – naively – between digital discretization and that

of elements of “length” (time and space) induced by the Plank constant, h,

is not relevant. The non-separability, the non-locality, the essential indeter-

mination of which we speak in quantum physics are almos t the opposite of

the c e rtitude of the little boxes, well localized and stable, well separated by

predicates (the memory addresses), within which is distributed the digital

universe.

So we now fac e the principal issue: rounding entails a loss of information,

at each step of the calculus. It c an be associated with the irreversible growth

of a form of entropy, deﬁned as negative information (neg-information) So

if one enco des all determinations, forma l and objective, of a physical object

and process, all properties and states, in the form of digital information, the

elaboration of the latter, the dig ital calculus, will follow a geodesic which

is, normally, pe rturbed at each step by a loss of information. This pertur-

bation does not correspond to any phenomenon intrinsic to the process we

intend to simulate: the loss of information is not, generally, the encoding of

the change in objective determinatio n. It is a new type of symmetry br e ak-

ing. Will this inﬂuence the proximity of the virtual reality to the physical

phenomenon? Will it inﬂuence the quality of the imitation?

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Causes and Symmetri es 207

As we have alrea dy observed, arithmetic approximation does not aﬀect

the simulation of a linear or Laplacian proce ss: just as the a pproxima-

tion of the measurement, the rounding, does no t remove the computational

geodesic (the following of the rule) fr om the physical ge odesic. The initial

loss of informa tion is pres e rved, it remains o f the same order of magnitude

or it increa ses in a controlled way (technically: the e xtremes of the approx-

imation intervals a re pres e rved). This is not the case fo r non-linear cases.

Let’s consider, for illustrative purp oses, one of the simplest dynamics, one

that is well-known and one-dimensional: the discrete logistic equation,

n+1

= kx

(1 − x

)

For 2≤k≤4, this equation formally deﬁnes a sequence {x

} of rea l num-

bers, between 0 and 1 (a “time discrete” trajectory within a continuous

space). Particularly, for k = 4, it generates chaotic trajectories (sensitive

to the initial conditions, dense in [0, 1], with an inﬁnity of periodic points,

. . . ). Can we approximate any s e quence of real numbers thus generated

by a digital computer? Out of the question, at least in what concerns an

initial value of x

taken from a set o f measure 1 (that is, for almost any real

value in [0, 1]). Even if we choose a x

that can be represented exactly by

a computer, at the ﬁrst rounding in the course of the calculus, the digital

sequence and the c ontinuous sequence will begin to diverge. By improving

the approximation/rounding of 10

−14

to 10

−15

, for instance, after approx-

imately 40 iterations, the distance between the two seq uences will start to

oscillate between 0 and 1 (the gre atest possible distance). Likewise if, with

a rounding of 10

−15

, we begin using values that diﬀer from 10

−14

(of course,

if we want to, nothing would prevent us from restarting the digital machine

upon the exact s ame values and from calculating, with the same r ounding,

exactly the same discrete trajectory, . . . ). The technical problem ca n be

summed up by the observation that the dynamic is a “shuﬄing” one: the

boundaries of the interval a re no t preserved.

We therefore may not, in general, approximate, with the machine, a

continuo us trajectory; however, we may do the opposite. In fact, all that

can be proved, in dynamic (metric) contexts we will not specify here, is the

following “pursuit” lemma (see the Shadowing Lemma (Pilyugin, 1999):

notice the or der of the logical quantiﬁcation):

For any x

and δ there is a ε such as that for any trajectory f , ε-

approximated (or with a rounding-oﬀ not greater than ε at each step),

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208 MATHEMATICS AND THE NATURAL SCIENCES

there exists a continuous one, g, such as g approaches f by a diﬀerence of

at most δ at each step.

Even when considering the lucky case wher e we have δ = ε (this is pos-

sible in certain cases), it c omes to say that, globally, your digital sequences

are not so “wild”: they can be approximated by a c ontinuous sequence, or

. . . there are so many c ontinuous trajecto ries that, if we take a discrete

one, one can easily ﬁnd a continuous one, which is close to it. Thus, the

image of an attractor on the screen provides qualitatively correc t informa-

tion: the digital trajectories are a pproximated by trajectories of continuous

dynamics (determined by equations). But the reverse is not true: that is,

it is not true in general, for a trajecto ry given by analytical means, that

the computer may always approximate it. Diﬀerent versions of the pursuit

lemma apply to suﬃciently regula r chaotic systems. However, many dy-

namic systems do not even satisfy weak fo rms of this lemma (see Sauer,

2003). This signiﬁes the existence of initial values and intervals such that,

within these intervals, any rounding and any other initial value cause any

continuo us sequence to diverge from the given dis c rete sequence.

What happens, in the terms of our approach, which is geometric in

nature? To understand this in detail, it would be neces sary to refer to the

technical analysis which the authors are also developing. In this work o f

reﬂection, which nevertheless guides the mathematical and computational

analysis, let’s try to s e e it in a very informal manner. T he ﬁrst diﬃculty

resides in the necessity to place onese lf within the appropria te space, in

order to better understand. In short, it is necessar y to analyze the evolution

of a system, such as for instance the discrete logistic function, within a space

where the notion of neighbo rhood, between real numbers, corresponds to

digital a pproximation. A space which provides for such a metric is called a

“Cantor space.” In this space, which we will not deﬁne here, two real points

are close if and only if they have close binary or decimal representations (for

instance, 0.199999. . . to inﬁnity and 0.2 are very far from one another in

the Cantor space, whereas they are identical with regard to the usual real

line, thus posing numerous problems from the computational standpoint,

when we try to opera te upon their approximations).

We then see that at each of the digital calculus’ iterations, the round-

ing induces a loss of information corresponding to the deterioration of the

approximation around the point of the trajectory. If we measure the phe-

nomenon in terms of isotropy of spa c e (the points within this widening

neighborhood are “indistinguishable,” in a manner of speaking), this “gray”

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Causes and Symmetri es 209

zone, of isotropy, grows, thus augmenting the symmetries of space. A no-

tion of entropy as negative infor mation also enables us to grasp this change

in symmetries, as a loss of infor mation. Now, all tha t we have in the

machine is encoded information. Independently of what it enc odes, the

physical object’s formally determined properties or states, all is in the form

of digital information. So the objective determination, which is given by

the preservation of the theoretical symmetries, radically changes: we ar e

facing a change in symmetry tha t does not model a component of the evolu-

tion of the natural phenomenon, because it depends only upon the discrete

structure of the simulation universe and upon the imitation of the formal

physical determination by algor ithms (or, when we make a philosophy of

it, of the epistemological identiﬁcation of law with algorithm).

So there is, in terms of symmetry, the explanation o f the causal regime

of which we were speaking. The discretization, in fact the o rganiza tion of

the world by means of discrete mathematics proposes a causal regime (in

this case, an evolution o f symmetries), which is diﬀerent than that which is

proposed by continuous mathematics. It is not an issue of ﬁnitary transla-

tions of the same physica l world, but of scientiﬁc construction, b e c ause this

world is itself co-constituted by our formal and objective determinations.

When they change, its organization and its intelligibility also change. Once

more, this does not imply that the world is continuous “in itself”: we are

only observing that, since Newton, Leibniz, Riemann, Poincaré, . . . we

have organized and made intelligible some physical phenomena by means

of historical notions of continuity and of limit. If we want to do without

them, causal organization and intelligibility will be altered.

Another issue would also mer it to be detailed, but we will leave it fo r

latter work. Singularities in modern physics play an essential role. We

know for instance of shock situations, in no n-linear systems, wher e the

digital calculus does not come even remotely close to the critical situation.

We have the continuous description; the mathematics is clear, explica tive,

organizing for the physical phenomenon, we understand qualitatively, but

the numerical calculi chaotically revolve around the singularity, without

coming close to it. In fact, the current notions of limit and o f singular

point, which are absolutely necessary to the analysis of the phase changes,

the shocks, in order to e ven speak of renormalization processes in physics,

are not always co herently approximable. The loss of s ymmetries and the

change of a correlated causa l regime constitute our way of understanding

this problem, which is s peciﬁc to the digitalization of phenomena, without

referring to Laplacian myths and computational metaphysics. Computer

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210 MATHEMATICS AND THE NATURAL SCIENCES

science, a science which is now mature, deserves, from an epistemological

and mathematical standpoint, more attention and an internal view which

is able to assume the force and the limits of its own methods.

5.3 Causalities in Biology

While focusing now on biolog y, we will not address the discussions con-

cerning the biological levels of org anization, the intertwined hierarchies,

the crossed causalities, the ago-antagonistic eﬀects, the variabilities within

phenomena, the autopoietic processes, which we ﬁnd in biology. Of course,

all these properties will remain part of the backdrop of the approach, which

we propose here, but the approach will be mor e conceptual or, better ,

“schematic” (in the geometric sense of simple schemas or diagrams) than

thoroughly theoretical or descriptive: it will seek rather to contemplate a

framework of representation enabling us to extricate heuristic categories of

thought than to account for the e ﬀective phenomenality of life. Indeed, the

“relationships,” which we highlight, by means of very abstract little pat-

terns, do not necessarily correspond to “material relationships” or to phys-

ical conﬁgur ations; they are only organizing structures of thought, which

should aid the comprehension of phenomena, by proposing a conceptua l

framework. Moreover, is F = ma – at a much more elaborate and mathe-

matized level indeed – not a corre lation which organizes a phenomenon by

making it intelligible? Let’s also recall that this equation has been preceded

by the general concept of inertia, o r even, way befor e Galileo, by cosmo-

logical sp e c ulations and concepts as eminently philos ophical as profound

(see, for ins tance, the remarks of Giordano Bruno in “L'inﬁnito universo e

mondi,” 1584). T he physical intelligibility speciﬁc to this equa tion can be

the object of highly diﬀering conceptual “readings”: it may no longer be

primitive, but derived (from the Hamiltonian, from the Lagrangian, as we

have mentioned in the ﬁrst part), where it may be correlated to dis tinct

symmetry breaking, as we have also seen.

So, as we insist on emphasizing from the onse t, our approa ch rema ins

very specula tive here: it is for us the beginning of an attempt at a con-

ceptual categorization and schematization, which seeks to open new venues

without being sure of their outcomes and w hich will require, in order to

be continued, more discussion with biologists and the sanction of a certain

fecundity in the quest for a greater understanding of living phenomena. Of

course, we will remain within the framework, which we have de termined

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Causes and Symmetri es 211

for our selves, that is in the geometric terms of symmetries as an analysis

of physical causality; yet, we will also take into account here aspects which

are speciﬁc to biology, related to forms of teleonomies or of anticipation and

which we have already summed up with the concept of “c ontingent ﬁna lity”

(see the next section).

5.3.1 Basic representation

Let’s consider the dynamic functioning of a centrifugal governo r (or Watt

governor: two weights are lifted in rotation by pressure, making a valve open

so as to lower the pressure . . . ). This functioning is completely determined

by the data, obtained afterwards, concerning the initial conditions and the

physical laws. In this sense, its “b e havior,” though being well regulated and

leading to a dynamic equilibrium, is determined in a univocal and oriented

way (geodesics within a well – and pre-established phas e space).

In the case of living phenomena, the behavior (and functioning ) of an

organism does not appear to be determined in the same way. What ap-

pears to be determined like this in a more or less rigid way (within a given

domain, that is c ompatible with the or ganism’s surv ival), is what we could

call the aim of the functionings and behaviors, the functions to be fulﬁlled

in order to ensure homeostasis (or, better, homeorhesis, as any equilibrium

is dynamically, not statically preserved, in biology). Yet, what is not deter-

mined in such a manner, are on the one hand the possible ways to achieve

this and on the o ther hand the adaptations and modulations, which would

ensure the achieve ment of the functions and b e haviors. Speciﬁcally, to put

it in the language of physics, there are not only phase changes but also, as

we have seen in Chapter 3, changes in phase spaces, that is, observables

and relevant variables.

We know that the mathematical and equational formalization of this

situation (as took pla c e for physics and as we hope, in time, could be the

case for life sciences inasmuch as the adequate mathema tics would be elab-

orated) is confronted with profound diﬃculties, sometimes even diﬃculties

of principle, which the numerous and successive attempts at modeling have

encountered. Also, before any reitera ted attempt in this rega rd, it appears

to be necessar y to try to illustrate and to represent – in this case by means

of schematics, the ﬁrst abstract conceptual stage – that which appears to

characterize these modes of functioning and that which we could call the

“ﬁnalities” which interpret them, in the sense in w hich Monod could speak

of a teleonomy of life. These ﬁna lities, o f co urse, are neither necessary

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212 MATHEMATICS AND THE NATURAL SCIENCES

nor absolute; they rather participate in our view of living phenomena and,

most of all, they are contingent, as they are speciﬁc to living matter and

relative to its contexts. In short, they could not b e prese nt (no life, no spe-

ciﬁc specie or individual); they are relevant to various levels of organization

and to their corr e lations, or to their intertwining and looping, pa rticularly

in the form of integration and regulation (see Bailly and Longo, 2003 and

2009).

To make intelligible the notion of contingent ﬁnality, we will try to

organize into networks the interactions between the “material structure s”

and “functions” of living matter. It is at this level in fact that teleonomy

manifests: for instance, when an organic structure appears to be ﬁnalized

in relation to a certain function.

We therefore propose a conceptual framework, to organize knowledge

by means of our recourse – temporarily at least – to a description of this

sort (for a better, active, understanding, the reader may easily draw the

simple intended diagrams):

(1) We have a target set, co nstituted of several domains – the target do-

mains, which may or may not overlap – corresponding to the functions

to be ensured for the maintenance and the p e rdurance of the organism

and its species.

(2) We have a so urce set constituted of all the organic possibilities likely

to be mobilized to this end (biochemical reactions, transport agents,

etc.), also repres e nted by this set’s domains (source domains).

(3) We have a set of arrows, originating in the source domains to reach to

the target domains (these arrows correspond to the orientations and

functioning mo des aiming to ensure the functions) and which presents

the fo llowing particularities:

(i) Any target domain is reached by at least one arrow; usually, sev-

eral s ource domains are at the origin of the arrows reaching the same

target domain. An example of this situation is the conjunction of “oxy-

gen metabolism” and of “glucose metabolism” to ensure the mainte-

nance of muscular tissue such as the heart; in this case both source

domains are respectively deﬁned by the chemical reactions rela ted to

the speciﬁc energetic sources (availa bility of glucose) and by the cel-

lular assimila tion processes of the intake of oxygen obtained through

breathing (availability of oxygen), the a rrows corresponding for their

part to the var ious transport and transformation systems which enable