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

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Invariances, Symmetries, and Symmetry Breakings 173

the sense of model theory, of this possible axiomatic. Let’s try to specify

from this point o f view their relationships, considering that through one of

their central problematic concepts, that of matter, they will meet, partially

at least, within the sa me semantic domain. In which sense could we say

that physics constitutes a standard model of this axiomatic whereas biolo gy

presents a non-standard model?

From the semantic point of view, it would be tempting to consider

biology as an “enlargement” of the physical model, that is, by adding

more “structure s” (more org anization, more observables . . . ). This may

be roughly assimilated with the technical sens e given to the notion o f en-

largement or extensio n in logic and model theory. This enlargement would

enable us to distinguish the concept of life within matter, a concept and a

reality that physics does not know how to “see” nor to “name.” Syntacti-

cally (cf. the IST approach of Nelson (1977)), we would then consider that

the introduction o f a new pr e dicate (“non-living”) enables us to formally

distinguish purely physical situations from biological ones. This predicate

would not be expressible within the theory, in the same way that formal

arithmetic cannot separate the “standard” number from the “non-s tandard”

by means of an internal predica te.

Such an appro ach, which has no thing of the absoluteness speciﬁc to

arithmetic or to formal set theory, only suggests a po ssible conceptual

framework for the unity and diversity we confer to natural, physical v s

living phenomena. From an epistemological o r even philosophical point of

view, it enables us to choose between the terms of an alternative:

• either, as it is often admitted, the axiomatic (virtual) presuppositions

which govern the constitution and development of physics and of bi-

ology are fundamentally diﬀerent, and these disciplines develop non

equivalent models from this point of view; we would then be fa c ing an

underlying dualism, within which the concept of “life” not only plays a

discriminating role, but also c orresponds to an axiomatic irreducibility

(we could even ﬁnd, theoretically, biological ﬁnalism, or even vitalism);

• or, acc ording to the proposition we have just presented, we would con-

sider a s ingle axiomatic for both disciplines, which would refer to a

sort of formal materialistic monism, which is nonetheless likely to lead

to the realization of semantically distinct models (physics and biology

being distributed be tween standard and non-s tandard, respectively);

syntactic analysis reestablishes a duality in this monism, but the irre-

ducibility is no longer axiomatic, it refers to the distinction between

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174 MATHEMATICS AND THE NATURAL SCIENCES

interna l and external sets. This would be compatible with our view of

life as a “physical” singularity, an external or non-standard bo undary

case, having a more spe c iﬁc counterpart the notion o f an “e xtended crit-

ical situation” to which we will return in Chapter 6. Strict theoretical

extensions of physical theories may then better gras p the singularity of

life.

Of course, this second term of the alternative is not necessarily asso-

ciated with the par ticula r and somewhat unilateral approach we have just

been considering: other “axiomatic” points of view would doubtlessly en-

able us to use it. But the important issue we would like to stress, by

means of something that is but an example of a possible approa ch, re-

sides in the capacity of neglecting in principle neither the unifying per-

sp e c tive of mathematical formalization, nor the variability of the empirical

phenomenality of that which makes the speciﬁcity and unity of life phe-

nomena. It is possible to elaborate a conceptualization, which, without,

however, falling into a physicalistic reductionism, does not renounce ﬁnd-

ing intrinsic formal relationships between the theory of physical (inert)

materiality and the theory of biological (living) materiality. It goes with-

out saying that, in accordance to what we have emphasized abundantly

above, such a physicalistic reduction does not exist, at least for the time

being. And this, not only for substantial reasons (even if matter – in-

ert or living – is unique) related to the fact that the separation/bridging

of the two disciplines involves the overstepping of a high number of o r-

ganization levels and a redeﬁnition of elementarity r e latively to s implic-

ity and to complexity, but also because there exists in fact many physi-

cal theo ries of biology – each of which can present very interesting illus-

trations for such or such a particula r aspect – and because an eventual

complete re ductio n would pose the question of knowing which would be-

come the most relevant and predominant. Finally, o ne should also con-

sider that, in spite of our eﬀort towards a uniﬁed understanding by the

geodesic principle, some key technical aspects of physics are still not uni-

ﬁed: the ﬁeld theories of (general) relativity and quantum mechanics, typ-

ically.

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Invariances, Symmetries, and Symmetry Breakings 175

4.4 About the Possible Recategorizations of the Notions of

Space and Time under the Current State of the Natural

Sciences

In all the preceding , it ha s appeared more or less clearly that the concepts

of space and time introduced and used in most contempor ary theories had

only remotely something to do with the spontaneous intuitions of these

notions and with their actual employment within the classical frameworks

of the natural sciences. Already, in order to account for this situation, we

have be e n led to introduce wha t we have called “gno siological complexes,”

referring to abstract structures that mobilize conce pts such a s those of the

group or semi-group in mathematics or, in mathematical lo gic, of equiv-

alence relations or of order relation. The most re c e nt developments in

physical theory, associated with more speciﬁcally biological considerations,

would incite us to attempt to possibly go even further, or at least in a new

direction, namely with regard to the notions of inside/outside and of the

conceptual couple of internal/external.

From a biological point of view, we know that this couple is concep-

tually structuring since the introduction, by Bernard, of the notion of

“milieu intérieur,” which we have already evoked in Chapter 3. In what

concerns space, that which pertains to an external environment, submit-

ted to heteronomic laws that are physically and biologically relevant, and

that which pertains to internal topologies (embryogenesis, development,

anatomo-physiological or ganization), referring to autonomy, are thus disso-

ciated.

These a re dissociated at least since one of the issues is precisely to

correctly describe their articulation.

In what co nce rns time, it is by no means less clear that two notions are

confronted depending on whether we ar e dealing with proc e sses of response

to “external” stimuli (presenting characteristic r e laxation times or periods,

being of a purely physical nature, expressed in dimensional terms – seconds,

hours . . . ) or, contrastingly, to internal regulation, of an iterative nature

(heart beats, breathing, . . . ), repr e sented in fact by these pure numbers

mentioned earlier and which only acquire such a tempor al dimensionality

by the fact of the scalings to which they obey relative to the “size ” (or

“weight”) parameter of organisms.

In this vein, see the s el f-organizing approaches, which aim to account for this situation

by introducing the concept of “organizational closure” (Varela, 1989), for instance, or

the use which is made in the same perspective of modelizations by means of dynamical

systems, as in Kaufmann (1993).

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176 MATHEMATICS AND THE NATURAL SCIENCES

The new epistemological fact, which we have already evoked but which

is now good to take again into consideration, is that physics, in turn, break-

ing with previous conceptions, is henceforth led to increasingly operate the

same kind of distinction between that which is an issue of an “exterior -

ity,” the space and time of relativity, and that which p e rtains to a sort

of interiority – ﬁbers, of which the geometrical and symmetrical proper-

ties contribute to characterize the quantum be haviors and identities. This

characteristic, already present in quantum mechanics as such, acquires con-

siderable scope in quantum ﬁeld theory and becomes truly dete rmining in

uniﬁcation theories (between quantum physics and general relativity) which

have recourse to string and superstring theories (introduc ing spaces of 10,

even 26 dimensions, of which the exceeding dimensions are compactiﬁed

These approa ches appear to be counter to usual intuition, but we very

well know tha t mathematization confers to reference universes an increas-

ingly counter- intuitive status. At the same time, it confers to “physical ob-

jects” themselves properties which see m increasingly strange, to the point

of putting into question the usual concepts of causality and locality. Once

more, we see the mathematical formalisms being developed (either diag ram-

matical or symbolic), while structural intuitions are elabo rated, these b e ing

new intuitions without much relationship to pre vious ones, and which are

both quasi inherent to the generativity of the mathematical principles of

construction as well as being les s and less empirical.

The problem w hich

then arises is that o f the rational articulation between these new types of

intuitions and the factual results which present themselves in the usual

universe of natura l phenomena (of physics or of biology ).

Conclusion In our brief inquiry concerning the mathematical princi-

ples currently appearing to be the mos t relevant in the attempt at rigorous

characterization and at rational reconstitution within the natural sciences,

it rather seems that both the theoretical results accumulated these last

years as well as the introduction of new concepts related to renewed ap-

proaches led us to put into perspective the epistemological interpretation of

In fact, and as we have already mentioned, the increase in dimensions of space-time

as a reference universe in view of a uniﬁcation between interactions is a method which

precedes its employment within the framework of quantum theories (see Chapter 3).

One will note that it is not the ﬁrst time in scientiﬁc history that cognitive phenomena

of this type occur. The passage from Ptolemy to Copernicus entailed a great loss of direct

intuition: the sun so clearly revolved around the earth! But even the Galilean theory

of movement and of dynamics clashed with the corresponding Aristotelian intuition and

according to several inquiries, still continues somewhat to do so.

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Invariances, Symmetries, and Symmetry Breakings 177

the status and of the deter mination of this objective knowledge. In contem-

porary physics, it is necessary to observe a radical rupture from previous

analyses: besides quantum and relativity theories, issues related to cr iti-

cal phenomena, to non-linear dynamics and to co mplexity have opened a

truly or iginal ﬁeld. The analyses of modes of construction of scientiﬁc ob-

jectivity ﬁnd themselves comforted at the expense of radical empiricisms

on the one hand, and of ontological realisms on the other. Manifestly, the

practice of contemporary rese arch as well as the presentation o f its results

and their conceptual interpretation articulates two moments which are in-

creasingly indissoc iable: the constructive moment and the transc e ndental

moment. The ﬁrs t is to be found in generalizing induction, the back and

forth movement betwe e n theory, e xperience, and observation, the search for

deducibility, the putting into form of hypotheses and conclusions. T he sec-

ond manifests itself as much in the intuitive movement which short-circuits

the various stages (though this may mean having to restitute them later),

by the passing to the limit which transforms facts into laws and laws into

principles. The abductive reasoning which is associated with this proc e ss,

the conceptual reversal w hich passes from the deductive to the prescriptive

and which ﬁnds in these laws and principles the expression of their abstract

conditions of possibility are part of this aspect.

Moreover and in this spirit, we wanted to show that it was possible to

consider the possibility of a sort of mathematical reconstruction of these

forms of intuition which are time and space, sta rting from the mathemat-

ical c oncepts of symmetries, of gro up, of equivalence on the one hand, of

symmetry breaking, of semi-group and of order on the other ha nd. And if

not acc omplishing a complete reconstruction of these structures, we wanted

in any case to open possibilities in this direction, using the results that the

most recent physical science may produce in terms of invariance. How-

ever, let’s reiterate that it is not a question of a c omplete and deﬁnitive

objective reconstruction: in the same way that continuous mathematics,

for instance, has and s till undergoes many changes in its most varieg ated

philosophico-mathematical determinations, from the soph istic paradoxes

to the recourse to large cardinals. From Aristotle to Leibniz, Cantor,

Dedekind, the continuous-discrete of the intuitionists (Bell, 1998) and of

non-standardists (Harthong, 1983), we can expect the mathematical deter-

minations of these fundamental structures which are space and time, and

which are prerequisite conditions to all objective knowledge in the sciences

of na ture, to continue to develop in step with the theore tical advances in

these disciplines such as the deepening of their mathematical foundations.

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178 MATHEMATICS AND THE NATURAL SCIENCES

Earlier, when addressing superstring theory, we also mentioned, a s ex am-

ples, some space s with high numbers of dimensions (10 or 26, of which most

cannot be observed since they are compactiﬁed), and which are endowed

with properties of supersymmetry (enabling the pas sage between fermionic

and bosonic ﬁelds). We mentioned this in view of unifying quantum ﬁeld

theory with r e lativistic gravitational theory (see Be nnequin, 1994).

It is, moreover, striking to observe in this regard that the iss ues raised

by the theory of elementary particles are akin to tho se which proved to be

the most relevant in terms of cosmology. Indeed, the states closest to the

Big Bang are also those where the various interactions are physically uni-

ﬁed and where the quantum constraints, most often associated with very

short time or magnitude scales, relate to those of a gravitational nature, the

latter being associated w ith the rather high densities on these very short

scales. And this back to the origin, as the B ig Bang itself is (hypotheti-

cally) attributed to a spontaneous symmetry breaking of the highly ener-

getic qua ntum void. It is such results that have led certain a utho rs (Weyl,

1918b; Cohen-Tannoudji et al., 1986) to introduce the matter-space-time

gnosiological complex, as mentioned earlier. T his doubtlessly conduces to

new reﬁnements in the mathematical characterization of that which consti-

tutes the physically relevant aspe c ts of the study of material phenomena.

It must be expected that pro gress in terms of a yet to be accomplished

mathematization in the ﬁeld of biology will lead to the introduction o f

a rigorous characterization of living phenomena , thus producing an even

more unifying gnosiological complex a nd thus enabling us to reinterpret

what signiﬁcances the associated spatialities and temporalities could elicit.

It is, however, necessary, at this stage, to note that we are thus pro-

gressively engaged in a somewhat para doxical situation from a conceptual

point of view: on the one hand a transcendental type approach conduces us

to consider space and time as primary data (forms of intuition, conditions

of possibility of all sensible experience ) and thus as strictly “empty” refer-

ential frameworks (pure for ms ). But on the other hand, and as with a ny

construction of scientiﬁc objectivity, we are constantly led to reconstruct,

according to theoretical advances, the mathematical structures which we

will also call space or time. These ar e less and less empirical and increas-

ingly determinant for the phenomena themselves, to a point where it is

virtually impossible to distinguish between events and the “locus” of their

occurrence. Each of these reciprocally determines the other, as is already

sp e c tacularly apparent in general relativity and in quantum ﬁeld theory.

Biology, as a science of processes, tends to increasingly fo ster such a repre-

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Invariances, Symmetries, and Symmetry Breakings 179

sentation. The distinction between forms of intuition and the mathema tical

intuitions which would c ome to “ﬁll” them, doubtlessly conceivable within

the framework of Newtonian physics and its abso lutiza tion of time and

space, becomes less operative in the sense, at least, that the unifying math-

ematical constitutive determina tions are increasingly so, even if this entails

going against the ﬁrst intuitions.

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Chapter 5

Causes and Symmetries: The

Continuum and the Discrete in

Mathematical Modeling

Introduction

How do we make sense of physical phenomena? The answer is far from

being univocal, particularly because the whole history of physics has set,

at the center of the intelligibility of phenomena, changing notions of cause,

from Aristotle’s rich classiﬁcation, to which we will return, to Galileo’s (too

strong?) simpliﬁcation and the modern understanding in terms of “struc-

tural relationships” or the replacement of these notions by structural rela-

tionships. It is then an is sue of the stability of the structures in question, of

their invariants and symmetries (Weyl, 192 7 and 1952; van Fraassen, 1989;

see the pre vious chapter). To the point of the attempt to completely dispel

the notion of cause, following a gre at and still open debate, in favor, for

instance, of probability correlations (in quantum physics, see, for e xample,

Anandan, 2002).

The situation is even more complex in biolog y, where the “reduction” to

one or another of the c urrent physico-mathematical theories is , as we have

seen, far fr om being accomplished, if ever possible. Our point of view should

henceforth be clear: the diﬃculties in doing this reside as much within the

sp e c iﬁcities of the causal regimes of physical theories – which, moreover,

diﬀer amongst themselves – as in the richness speciﬁc to the dynamics of liv-

ing phenomena. Our approach ha s attempted to highlight certain aspects,

such as the intertwining a nd coupling of levels of organization, which are

strongly related to the phenomena of autopoiesis, of ago-antagonistic eﬀects

(in short, a dynamic opposition of “forces” or causes), of the hybrid causal-

ities often mentioned in the theoretical reﬂections in biolo gy (see Varela,

1989; Rosen, 1991; Stewart, 2002; Bernard-Weil, 2002; Bailly and Longo,

2003).

181

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182 MATHEMATICS AND THE NATURAL SCIENCES

We will now return to some as pects of the construction of scientiﬁc ob-

jectivity, as explication of a theoretica l web of relationships. And we will

mostly spea k of caus al relationships, since causal links are fundamental

structures of intelligibility. Our approach will again be centered upon sym-

metries and invariances, because they e nable causes to mani fest themselves,

namely by the constraints they impose. In a strong sense, they thus present

themselves as conditions of possibility for the construction of mathematical

or physica l objectivity.

Now, if mathematics is constitutive of physical objectivity and if it

makes phenomena intelligible, its own “internal structure,” that of the con-

tinuum, for example, as opposed to the discrete, contributes to physical

and biological determination and structures their causal links. To put it in

other words, mathematical structures are, on the one hand, the result of

a historical formation of m eaning, w here history should be understood as

the constitutive process fr om our phylogenetic history to the construction

of intersubjectivity and of knowledge within our human communities. But,

on the other hand, mathematics is also constitutive of the m eaning of the

physical world, since we make reality intelligible via mathematics. Particu-

larly, it organizes regularities and correlates phenomena which, otherwise,

would make no sense to us. The thesis outlined in Longo (2007) and which

we further develop here, is that the mathematics of continua and discrete

mathematics, the latter characteristic o f computer modeling, propose dif-

ferent intelligibilities both for physical and living phenomena, particularly

for that which concerns causal determinations and relationships as well as

their associated sy mmetries/asymmetries.

In yhe ﬁnal section, we will a ttempt to addre ss the ﬁeld of bio logy by

questioning ourselves about the operational relevance and status of the

concepts thus under consideration. But in this text, we will ﬁrst propose to

illustrate, in the case of physics, the situatio n which we have just summarily

described, all the while returning to the reﬂection conducted in Chapter 4:

this will enable us to “enframe” physical causality and to compare it to

computational models and to biology.

5.1 Causal Structures and Symmetries, in Physi cs

The representation usually associated with physical causality is oriented

(asymmetric): an orig inary cause generates a cons e c utive eﬀect. Physical

theory is supposed to be able to express and measure this relationship.