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

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Invariances, Symmetries, and Symmetry Breakings 163

respect to the reference systems.

More fundamentally still, it appears that the s ymmetry/symmetry

breaking pair is beginning to play, for the intelligibility of physics, a role

similar to the one alre ady played in other disciplinary ﬁelds by certain

foundational dichotomies such as, in mathematics, the dichotomies of ﬁ-

nite/inﬁnite, continuous/discrete, local/global (Lautman, 1977). As we

observed, the dialectic of the symmetry/breaking of symmetry pair thema-

tizes, on the one hand, invariance, conservation, regularity, and equivalence,

and on the other , criticality, instability, singularity, and ordering.

We have seen that through the pair ’s dialectic, it is an essential compo-

nent of the very identity of the scientiﬁc object that is presented and objec-

tivized. Could we go even further and consider that we have thus managed

to construct this identity at a level such that cognitive schemas conceived as

conditions of possibility for any construction of objectivity are henceforth

mobilized, thus reviving a form of transcendental approach (Petitot, 1991)?

In fact, we showed that ther e exists a clo se formal relationship between the

abstract properties o f sy mmetry captured by mathematical group struc-

tures and logical structures as fundamental as the equivalence relation. At

the same time, there may e xist a similar formal relationship between the

semi-group structure, of which an essential usage is made with regards to

the renormalized treatment of critical pheno mena, and the logical structure

of the order relation. Now, the theor e tical ana ly sis of the abstract notions

of space and of time demonstrates that, for their formal reconstruction,

these notions need to mobilize the mathematical structures of group and

of semi-group, respec tively. Indeed, regardless of the number of dimensions

considered, the displacement prop e rties, consubstantial to the concept of

space, refer to the determinations of the displac e ment group, whereas the

properties of irreversibility and of the passing of time refer to the character-

istics of the semi-group (generally, for one par ameter). We then witness the

constitution of a pair of abstract complexes which doubtlessly represents

one of the essential bases for any objective interpretation within the pro-

cesses of the construction of knowledge, as we have mentioned earlier: the

complexes of <space, group structure, equivalence relation> on the one

hand and of <time, se mi- group structure, order relation> on the other.

Let’s point out once more, in order to disp e l any possible confusion, that

the space and time evoked by these complexes no longer refer to physical

entities as such, but rather to the conceptual frameworks, which are meant

to enable any physics to manifest itself, that is, to abstract conditions of

possibility a nd not to eﬀective realizations, thus reactualizing a point of

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164 MATHEMATICS AND THE NATURAL SCIENCES

view (Kant, 1 986a). That is, space and time are no longer considered as

“objects” to be studied, but rather as the conditions of possibility for any

sensible experience. In this sense, the symmetries and breakings of sym-

metries associated with these gnosiological complexes appear not only as

elements of the intelligibility of physical re ality, but indeed as factors for

the scientiﬁc constitution of such reality.

So, could we say that we have clearly passed from a descriptive anal-

ysis of the theoretico-c onceptual situation currently dominant within the

epistemology of physics to that which may constitute the quasi prescriptive

foundations of a contemporary “natural philosophy,” where the constitutive

principles are to be found not only in the use of mathematical structures,

but also in the formal redetermination of these “forms of intuition,” which

are space and time? Not only would we simply operationalize them, but

by coupling them with the corresponding logical and mathematical deter-

minations (group structure, equivalence relation, etc.) we refer them to the

frameworks of invariance, which make them into reference structures that

are mathematically determined, rather than abstract and vague.

The concept of invariance, having already demonstrated its tremen-

dous technical and theor e tical fecundity in mathematics and physics, now

appears to be called to play an essential role in both the understanding

of cognitive pr ocesses themselves as well as in the formal reconstruction of

their conditions of possibility, doing so with and in contrast to the concept

of critical transition, by its invariance and symmetry breakings. Husserl

had demonstrated this on a philosophical level, following all the mathe-

matical advances of his time, a nd Thom semiotically remathematized it

with the application of his “catastro phe theory” (critical situations mani-

fest as singularities). And this is what we are trying to readdress when

we refer to the fact that mathematics constitutes the ﬁeld where concepts

and demonstrations are, epistemologically spea king, the most structurally

stable (invariant!) among all forms of k nowledge, all the while analyzing

the constitutive role of mathematics for a great number of scientiﬁc disci-

plines (it is by this a s ingularity within our symbolic c ulture ). But there

are still branches of the natural sciences, which, while con forming to sci-

entiﬁc method and r igor and having recourse to their own formalisms, are

still not mathematized in the way that physics can be. They continue,

however, to play a more descr iptive than prescriptive role, one that is more

classiﬁcator y than predictive, in that they are not very “calculatory.” Such

is the case with biology where keen questions relating to invariance and

variability are posed. It there fore seems interesting and instructive to try

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Invariances, Symmetries, and Symmetry Breakings 165

to test this general appro ach using this pa rticular case in order to analyze

its scope and limits, albeit summarily.

4.3 Invariance and Variability in Biology

4.3.1 A few abstract invariances i n biology: Homology,

analogy, allometry

In biology, there are numerous examples of symmetry properties and of se-

quences of symmetry breaking, be it an issue of morphology, of functionality,

or of embryogenesis and development. But rather than sticking to a dom-

inantly descr iptive approach to such properties that are lengthily exposed

in several texts (see for instance Vogel and Angermann, 1984), it appears

conceptually more interesting to try to fully enter into the very foundations

of modelizations and to consider more general principles, concerning both

the apparition and selection of se veral great levels of organization as well

as the transformation of organic structures and of physiological functions

(in this regard, see also the works of Bouligand, 1980, 198 9; Dellatre and

Thellier, 1980; Kauﬀman, 19 93; Thom, 1972, 1980). It is this a sp e c t which

we would attempt to brieﬂy explore here , in order to highlight some of its

impo rtant characteristics, namely within the framework of the theo ry of

evolution and of the interpretation that we may provide for some of its

realizations.

Indeed, if we consider not only s peciﬁc organs or organisms, or the

characteristics of speciﬁc species, but if from the onset we consider a more

global viewpoint such as the compar ison between diﬀerent species, be it at

a speciﬁc point in time – synchronously – or within an evolutive process –

diachronically –, we will be led to consider other types of inva riances and of

symmetries tha n those w hich we ﬁnd morphologically or functionally repre-

sented within speciﬁc biological structures. From an evolutive standpoint,

these are mainly of two types: those resulting fro m the evolution of the

same fundamental organ towards organs which may be quite diﬀerent but

which have a common origin, therefore entertaining relationships of homol-

ogy, and those which, by convergence, r e sult fro m the transformation, under

the pressure of selec tion, of fundamentally diﬀerent organs towards simi-

lar forms and identical str uctures that entertain therefore r e lationships of

analogy. From a more immediate point of view and, geometrically speaking ,

one which is mor e metrical, we would have for comparable biological struc-

tures relationships of allometry, that is, in the last analysis, scaling laws

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166 MATHEMATICS AND THE NATURAL SCIENCES

corresponding to the invariances of certain laws by dilatation (see Chapter

3 a nd below for the notion of scaling and allometry).

To begin, let’s consider homologies (same origins, but often having dif-

ferent functions) and analogies (same functions, but diﬀerent origins). They

are to be found throughout a ll living forms. In plants for instance, where tu-

bercles, tendrils, and stem thorns are homologous a nd where roots, stems,

and tendrilled leaves are analogous. In animals , the articulation of the

limbs of vertebrates can be found homologously in the ﬁns of the dolphin,

in the wings of ﬂying reptiles and of birds or bats: paleontology shows us a

common origin, despite their diﬀering functions. The mouthparts of insects

are also homolog ous, with their rather variegated functions, and so are the

cerebrums of vertebrates. While analogies are discernible b e tween the dif-

ferent types of excretory organs, between the wings of birds and of insects,

between burying and shovel-shaped legs of the mole and of the mole cricket,

or even between the tubercle of the potato and that o f the dahlia (in this

case the analogy is morphological).

These c omparisons and classiﬁcations are not limited to the relating

of organs and functions, they also extend to behaviors. Thus, there exist

homologo us movements among diﬀerent species of duck, and as well there

often exists for a number of other species a sequence of homologous behav-

iors which derive from an original specialized typical behavior. Likewise,

many gestures, stemming from situational convergences, are analogous be-

tween numerous species, s uch as acts of appeasement in the face of the

threat of aggression. With regard to the human species, the comparison

with animals highlights similar ities and diﬀerences: as for the other spe c ies,

there exist homologies as sociated with acquired phylogenetic traits stored

within the genes, as well as analogies associated with acquired o ntogenetic

characteristics stored within the central nervous system. However, there

also exist traditional homologies w here memo ry replaces the genotype as

the locus of information storage. Despite its exceptional character, it re-

mains necessary to note that this type of traditional homology, transmitted

via the memory of a social gro up, does not appear to be completely lack-

ing among other animal species, such as demo nstrated by the behavior of

the Japanese stumptail maca que or by the song of the African pin-ta iled

whydah, and many other animal populations, which have been shown to

have cultura lly transmitted behavior. All these characteristics are of in-

terest to us inasmuch as they manifest invariances which are interspeciﬁc,

organic, functiona l, and behavioral, beyond the varia bility associated with

the evolution a nd diﬀerentiation of life forms, a little bit in the way that

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Invariances, Symmetries, and Symmetry Breakings 167

geometrical or more abstract symmetries display and even pr e scribe invari-

ances in the variety and the apparent dispers ion of physical phenomena

(whereas in biology, we remain at a stage which is essentially descriptive).

With the properties of allometry (Thompson, 1961; Gould, 1966; West

et al., 1997) a nd of scaling (Peters, 1983; Schmidt-Nielsen, 1984), we are

no longer within a co ntext of invariances as sociated with an original form,

organ, or behavior type. Their homologous traces are to be found in the

evolutive variations o r in the permanence of functions which, by the selec-

tive pressure they exercise, govern the c onvergence of the forms or of the

analogous behavior. Now, instead, at a speciﬁc time in the history of evo-

lution, and still within the framework of interspeciﬁc comparisons, it is a

question of identifying and interpreting “scaling invariances,” expressed by

the scaling laws already mentioned in Chapter 3. These laws may be wit-

nessed between elements which are common to these species and relevant

to their characterization (metabolisms, lifespans, internal rhythms, . . . ,

but also relationships between physiological sys tems such as the nervous,

respiratory, and vascular systems). In a very general way, these allometric

relationships are expressed in power laws, as we have se e n, where the vari-

able is the characteris tic size o f the species under consideration, most often

measured by means of the a dult’s average weight, M , and is therefore of

the fo rm:

Q = AM

In this expression, Q represents the quantity under consideration, for

a species of which the average adult weight is M. A is an interspeciﬁc

constant which is characteristic of this quantity and k the scaling exponent,

which is also interspeciﬁc as well as being characteristic o f the quantity at

hand. Therefore, in principle, relative to such quantities, a who le set of

sp e c ies can ﬁnd themselves gro uped under a single class characterized by

the series of these invariable quantities which are the magnitudes A and k.

The distinctions only appear when the pa rameter M is taken into account,

distributing Q into diﬀerent sizes, each speciﬁc to a species.

Let’s consider a few particularly signiﬁcant examples. All metabolic

rates scale as the 3/4 power of mass (see Chapter 3), thus indicating a

strong structural stability for living organisms in the mobilization o f the

means of survival. Likewise, reg ardless of the spe c ies, all characteristic bi-

ological time spans (lifespan, ge station period, maturation period, inverses

of breathing frequencies, heart rates – when there is a heart ) appear all

to scale at the same power of mass, k = 1/4. T his characteristic has

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168 MATHEMATICS AND THE NATURAL SCIENCES

rather interesting consequences in the cases of periodic physiological phe-

nomena. Particularly, let’s recall that mammals (to which correspond, for

a given physiological function and their lifespan, a sing le consta nt A, with

the s ame value) are characterized by the same total number of breaths

(2.5 × 10

approximately) or of heart beats (10

approximately), whereas

the corresponding fre quencies or periods are obviously diﬀerent for each

sp e c ies. Thus, the passing of the singular ity from the speciﬁc to the gen-

eral is accompanied here by the passing fro m dimensional magnitudes to

pure numbers that are characteristic of the ge nerality thus constructed, a

manifest c onsequence of the invariance thus revealed.

Let’s just mention tha t there exis t scaling prop e rties, which lead to allo-

metric relationships other than the interspecies a nd mass related properties

we have just been considering. For instance, this occurs within the same

sp e c ies, as in the cas e of the rat for example, over the course of its develop-

ment, over its period of growth (Burri et al., 1977), for the ratio between

the total pulmonary or alveolar capillary surface and the lung’s volume,

or as in the case of the dog, fo r the p ost natal growth of its bronchial

tree (Horsﬁeld, 1977). But in these cases, and conversely to those which

we have examined precedently for the interspecies relationships, we cannot

identify manifest invariances. Therefore, the resulting symmetries by di-

lation remain conﬁned to a speciﬁc morphological description, even if we

can relate them to morphogenetic dynamics which enable us to account for

their development (Bailly et al. 199 4).

Let’s now return to the principles, which can eventually be derived from

these observations and analyses: even if we observe that redundancies and

degenerations are extremely frequent in biology, it very well appears that

certain symmetries conform to principles if not of economy, then at least of

optimization, as demonstrated by the analysis we c an make of the existence

of fractal g e ometrical structures among many organs engaged in vital phys-

iological functions and which must, let’s r e c all, ensure the compatibility

between three necessities which may appear contradictory: to ensure maxi-

mal eﬃciency and ﬂexibility of control (thus the forming of very lar ge active

interfaces or of arborescent networks of which the surface of the terminal

section is very high in order to involve a whole volume), while presenting

only an acceptable level of steric encumberment (if not minimal volumes,

then at lest very limited ones ), and behaving as truly diﬀerentiated organs.

These organs, typically the lungs or the vascular system, present within

this determined spatiality a homogeneity of structure and of functioning

which individuates them in a non-ambiguous fashion. One will note here

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Invariances, Symmetries, and Symmetry Breakings 169

the formal aﬃnity of this quest for optimization, which we have just men-

tioned, with the rules of optimization that are concomitant to the geodesic

principles of physics discussed in length earlier.

In other cases, it appears, fo r instance in what concerns the symmetries

of temporal periodicity, but also in spatial arrangements, that principles of

synchronization (spatiotemporal, then) are manifested, enabling the reg-

ulation and integration of diﬀerentiated structures and distinct levels of

organization within a whole which is suﬃciently perennial for survival and

reproduction. These eﬀects presumably derive from the evolutionary his -

tory of species and a number of them must be associated with the observed

homologies, which translate, as we have seen, an invariance of which the

origin is to be found within the existence of original forms to derived forms.

These pr inciples are clea rly not arbitrary, and must stem from evolutive

responses in terms of adaptation to environmental or functional selection

pressures. Therefore, they cannot generally be dissociated from the con-

ditions of apparition of the structures to which they apply and from the

transformations to which they have been submitted due to the historic mod-

iﬁcations of the environments to which they have been exposed. However,

in all likelihood, they also stem from internal tendencies that ar e more lo-

cal but nonetheless important, by the fact that living systems also present

at some levels physico-chemical aspects, which a re governed by the laws

and principles of physics, as local modelizations, by means of non-linear

dynamic systems clear ly show.

4.3.2 Comments regarding the relati onships between

invariances and the conditions of possibility for life

After these few discussions regarding certain symmetries and regularities

encountered in biology and the regulating principles to which they could

refer, we would like to orient the reﬂection towards the speciﬁcity of the

biological with regard to the physical in terms of invariances , s ymmetries,

and of the epistemological meanings we can derive from them.

If we attempt to outline a rough classiﬁcation of the set of determina-

tions involved in living phenomena, we would recall that any living organism

is characterized from a topological point of view by the distinction betwe e n

an inside and an outside, thus ena bling us to deﬁne a spatial individuation.

It is also characterized, from an energetic point of view, by the existence

of a metabolism regulating its exchanges by ﬂuxes between inside and out-

side, as well as, from an informational point of view, by a genetic expression

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170 MATHEMATICS AND THE NATURAL SCIENCES

morphologically and functionally translated by a structuration into levels of

organization

ensuring both regulation (mainly top-down functions) and in-

tegration (mainly bottom-up oriented activities). Finally, it is characterized

from an “identity” point of view by a reproductive capacity co rresponding

to the transmission of this g e netic and other heritages and of its aptitude

for evolutionary adaptation. We must also mention this extraordinary in-

variance, throughout time and between diﬀerent species, which constitutes

the stable universality of the genetic structure itself. These aspects may or

may no t need to be applied to the ﬁeld of physics: the inside/outside dis-

tinction is in no way necessary to the characterization of a physical object,

even if these determinations may prove to be re levant in deﬁning physical

systems. Conversely, the issue of energy exchanges is common although

strongly qualiﬁed by the fact that biologica l individuation has no bearing

on it, etc. If, furthermore, we are interested in the semi-empirical condi-

tions of possibility for the existence of such biological objects, we will ob-

serve, as we have already indicated in Chapter 3 and further specify below,

that from a spatial point of view, it very well appears that the embedding

space (or, explicitly, the space within which life manifests itself) must be

three-dimensional. Fr om a temporal point of view, it will be necessary to

articulate very distinct types of temporality and, from the intrinsic and

abstract structural point of view, a functional hierarchy of levels of o rgani-

zation will need to be constituted. We shall not return to this la st point,

which we considered earlier under a somewhat diﬀerent angle. Howeve r, we

shall continue here with a more detailed view upon an idea outlined in the

preceding chapter concerning the issue of spa tial dimensionality.

Some (meta)physi cs at few dimensions. L e t’s more expand on a dis-

cussion already mentioned. Could one conceive of a biological science which

would not constitute itself within a three-dimensional embedding space (in

the way that we have uni- or bi-dimensional physics, for insta nce )? This

question refers to the organizational constraints and the nec e ssity o f ar-

ticulating the local and the global, the whole and the parts. Indeed, it is

We use here the term “level of organization” i n a strong and non-metaphoric sense,

responding to a double determination: the change in level of organization must be ac-

companied on the one hand by the divergence (passing to the inﬁnite boundary) of an

intensive magnitude contributing to characterize the original level. On the other hand,

it must be accompanied by a change in the relevant object considered by the theory or

experiment (a way to signify that from this viewpoint the extensional composition of

elements at one level does not suﬃce to characterize both the nature and the behavior

of the object considered at a superior level). We will return to this in Chapter 6.

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Invariances, Symmetries, and Symmetry Breakings 171

technically necess ary to couple a globa l (individuating) homogeneity with

local distinctions, in a way that is not only compatible, but also coopera-

tive, and in o rder to ensure high levels of connexity all the while avoiding

confusions and mix-ups. This can lead to the following a post eriori reason-

ing:

1. If the number of spatial dimensions (d) is less than three, we will lack

homogeneity and connexity, because on the one hand, neighborhoods

between elements are too few (four closest neighbors within a square

network, for instance) and, moreover, the independent ro utes of com-

munication are too restricted or too complicated and have the risk of

intersec ting and mixing; the situation may be considered as non func-

tional, by excess of diﬀerentiation and by necessity of a too high level of

structural complexity. In short, crossing channels would intersect and

mix.

2. If, convers e ly, d > 3, then the situation will be the o pposite: the levels

of connexity (many channels are possible) and homogeneity are too

great to pr e serve a suﬃcient local diﬀerentiation having recourse to

elaborate structures governing transport and exchange, ensuring the

global functions and re gulating the local situations; for instance, in

physics, mean ﬁeld theories a re exact from d = 4 and up.

This may

be interpreted by saying that there arises no nec e ssity for distinction

and discrimination between structures in order to ensure a globality

from local interactions. The situa tion is this time non functional by an

excess of homogeneity and lack of structural diﬀerentiation.

3. Finally, if d = 3, we can then ho pe to combine a global connexity that is

just suﬃcient to coordinate an enormously diﬀerentiating heterogene-

ity, albeit by having recourse, for the articulation be tween structures

and functions, to these somewhat particular geometries, the fractal g e -

ometries we have just mentioned.

Physics thus provides good rea sons for mathematics to propose three di-

mensions for the realm of the biological. The reader may take this as he/she

likes and consider that, in a way, we have answered the metaphysical ques-

tion of why we live within a three- dimensional space.

The preceding remarks and investigations have thus led us to no te that

the terms of symmetry and of invariances take on meanings in b iology,

In short, according to these theories and their various contexts of applications, the

(observable) values at one point are determined, as averages, by neighborhood points’

values.

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172 MATHEMATICS AND THE NATURAL SCIENCES

which are akin to those prevailing in physics, without, nevertheless, be-

ing reduced to them. Indeed, if these properties are frequently referred to

and ex pressed within our spa c e -time (geometrical aspec ts, temporal pe ri-

odicities), they correspond just as often to much more abstract principles

inducing a recourse to internal s paces, which, in co ntrast to that which

occurs within the abstract internal spaces of physics, remain largely under-

determined. Such is the case, for instance , with the symmetry by inva riance

of the species’ genetic heritage from one generation to another despite its

individual variations, despite the concomitant invariance of great levels of

organization or of functional systems, or even the invariance of individual

or social behaviors as demonstrated by etholog ical studies. Complementar-

ily, we have noted s e veral times the importance of symmetry breaking for

biology, be it a question of contrasting the diﬀerentiated bio logical activity

(metabolic, enzymatic, . . . ) of molecules having left or right conﬁgura-

tions (sugar s, pr oteins, . . . ) or, at a very diﬀerent level of o rganization,

be it an issue of the irreversible character of the development and aging

of organisms. To a p oint where many have wondered about the intrinsic

relationship betwee n the exis tence o f life and that of c e rtain fundamental

symmetry breaking. These may b e sometimes of a spatial nature (in re-

lationship to diﬀerentiation and the organization into diﬀerent levels, for

example dur ing embryogenesis), but are mainly temporal (at the individual

scale – development then aging – as at the global scale – evolution).

Finally, and to conclude here with regard to the ﬁeld of biology, we

would like to retur n more sp e c iﬁcally to the po ssible relationships between

physics a nd biology, by taking a point of view which is completely formal

and beyond their res pective theoretical and conceptual co ntents, the latter

being our habitual standpoint. By attempting to place ourselves at a point

from which we would seek to mutually posit physics and biolog y, and by now

conceiving each of these disciplines as a model of natural science, almost

in the semantic sense of logical model theory. The logico-formal exercise

may propose an ulterior view upon our monism, which feeds on theoretical

diﬀerentiations all the while seeking possible uniﬁcations between theories.

The reader will be able to follow it even without knowing the terminology

of logic.

To do this, let’s suppose we have a ge neral axio matic framework for the

purpose of generally characterizing the sc ientiﬁc rationality of the natural

sciences and its formal and methodological requirements, even if they are

thematized diﬀere ntly according to discipline and sub-discipline. Now let’s

suppose that physics a nd biology co nstitute models o f realization, still in