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

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Space and Time from Physics to Biology 143

within a range of values of critica l parameters – an extended zone of criti-

cality. Exit from this zone implies the death of the organism: its underlying

physico-chemical structure can no longer sustain biolo gical functions. Any

biological system behaves in a manner characterized by a dense distr ibution

of critical p oints in the space o f control parameters, and not a discrete or

isolated one. Ho meostasis then, corresponds to a sort of s tructural stability

of the trajectories, relative to the attractor-basins of the dynamics.

3.3.5 Some philosophy, to conclude

Space a nd time, especially as they feature in the fra mework of modern

physics, are neither objects nor categories. To recall Kant’s formulation

(Kant, 1986a) they are “a priori forms o f sensible intuition,” and as such

the preconditions o f any possible experience. In the light of the most pro-

found analy sis our current physical theories allow us to make of them, they

seem to re ﬂec t the mathematical structure of a group and a semi-group,

respectively.

Indeed, the mathematical proper ties postulated for space, inas much as it

is the medium and suppo rt of displacements in general, necessarily connect

with and exemplify the group s tructure. Given the tight connection between

the group structure and its associated equivalence relations, an abstract,

epistemically basic frame of reference emerges which acts as a kind of pole of

attraction for any repre sentatio n of objectivity, namely the frame: <space,

group structur e , equivalence relation>.

Similarly in respect to time, the property of possessing an o rientation –

“time’s arrow” as the index of change – is reﬂected in the abstract structure

of a semi-group. That structure can be put in corr e spondence with an o rder

relation. This leads us to the recognition of a second epistemically basic

frame of reference: <time, semi-gro up, order relation>.

To repeat, for the avoidance of all confusion: the space and time which

feature in these frames are not so much an aspect of entities with an intrin-

sic nature, but rather of the conceptual grid presupposed by any natural

science: they are conditions of pos sibility r ather than of concrete actual-

ity. If this approach is correct, these two “poles” leave their mark in our

notions of permanence and change, stability and e volution, identity and

A semi-group is a set endowed with an internal operation which is associative; if the set

contains an identity with respect to the operation and each element possesses an inverse,

then it is a group. By this latter property, groups may be associated with symmetries,

see Chapter 4.

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144 MATHEMATICS AND THE NATURAL SCIENCES

diﬀerentiation. They delimit the “ﬁeld” of preconditions for any natural

science, to the extent that the phenomena studied therein are manifested

in a spatio temporal setting.

This Tableau General is still incomplete, notably in the life sciences.

But what clearly stands out is that contemporary theories across a whole

range of scientiﬁc domains involve conceptions of space and time that have

not yet been fully stabilized or clariﬁed (although in physics s upersym-

metric string theories are seen by many as leading to uniﬁca tion). Will

we continue to distinguish space radically from time despite the merging

of their status in the setting of relativity theory ? Shall we continue to

refer to a single notion of space in view of the variety of topological and

other structura l properties (compactiﬁcation, non-commutativity, internal-

ization, fractal dimension) envisaged for it in current physical theory? Or

in the face of the complexiﬁcation and fractalization of the forms of space

arising in biology?

Likewise shall we retain the representation of time as a unique param-

eter in physics or as intrinsically irr e versible? And how will the time of

physics turn out to be connected with that of bio logy? By jettisoning its

formalization in terms o f isolated bifurcations perhaps, in order to take into

account the synchronic and diachronic eﬀects in biological systems?

One reason for this rather confused state of aﬀairs (albeit one closer to

the often counter-intuitive nature of reality than our spontaneous percep-

tions can bring us) is that the very epistemo logical status of space and time

remains relatively problematic despite the formal categorization introduced

in the foregoing. One additional diﬃculty has been introduced by the fact

that since recent developments in quantum physics, surprisingly mirro ring

what has long been the situation in biology, we now have to take acco unt

of spaces external and internal to the systems under investigation.

We here encounter a distinction long made by philosophers – and no-

tably by Kant – but this time in a form lying on the side of the objects

themselves, whereas for Kant it was conceived as lying on the side of the

epistemic subject. It is the distinction between space as the form of exter-

nal a nd time as the form of internal sense (K ant, 1986b). Such a distinction

with respect to objects was unacceptable to the Kant of the Critique, for

having broken w ith an ontological characteriza tion of the objects of natural

sciences, any such internality was denied and location was considered some-

thing totally external. However, towards the end of his life, in re-examining

Newton’s Principia, Kant could not refrain, on the evidence of the Opus

Postumum, from thinking through this point afresh, in particular in relation

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Space and Time from Physics to Biology 145

to the questio n of energy. He renewed his investigation of certain aspects of

Leibniz’s thought on this subject which he had largely avoided beforehand:

he would perhaps have fo und in the latter-day evidence for a spatialization

(and temporalization?)internal to the objects of physics and biology, if not

the answer to his puzzle, at least a spur to new investigations.

Any attempt to make the internal/external dichotomy correspond in a

straightforward way to the distinction between space and time is infected

with artiﬁciality – the more so if the distinction is seen as lying within

reality itself r ather than the knowing subject. Nevertheless, taking into

account the Kantian view of s pace and time as the very conditions for the

constitution of objectivity, it does not seem too extreme to speak of them,

rather than of subjective “forms of sensible intuition,” of objective “forms

of sensible manifestation.” Such forms could themselves be connected with

the concepts of externality and internality. In the case of “the external,”

one essentially considers the phenomenal manifestation of relations between

objects (interactions and corresponding measurements). In the case of “the

interna l” one considers, rather, constraints concurrent with the phenomenal

manifestation of the continuing identity of a n object – or better, of its

identiﬁcation.

Notice that the distinction between internal and external aspe c ts is one

of the conceptually distinguishing features of biology. But it can be ex-

pected to be an element in our conception of the objects of theoretical

physics as well, since it now has an objective, mathematically expressed

counterpart in the distinction between the external space-time (the base

space) and internal spaces (the ﬁbers over the base) which is now a fully

developed aspect of the mathematical fo rmalism of key are as of quantum

physics. Brieﬂy (and ironically, in the light of certain epistemological ten-

dencies which have sought the reduction o f biology to physics), it is on the

side of biology that one now looks for the conceptual clariﬁcation allow-

ing the development of a more comprehensive abstract framework for the

understanding of physical phenomena.

In summary, we have thus arrived at a kind of conceptual “renormal-

ization” (if we may use, by a conceptual extension, this term from physics,

mentioned at the end of Chatper 2). The external space-time of physics can

be seen as a manifestation of the couple <s pace/relatio n> a nd as providing

a “base” relative to which the corresponding c ouple <time/identiﬁability>

can be viewed as the ﬁbration (the internal spaces). The further articulation

of these two elements (via the introduction of super symmetry and super-

space) gives ris e to what might be termed a space-time of a kind which

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146 MATHEMATICS AND THE NATURAL SCIENCES

allows us to take account of a ll the as pects in which an object becomes

determinate: b oth its r e lational properties and its internal structuring ac-

tivity which permits its continuing identity. In biology, it is the external

space-time which corresponds to the manifestation of the <s pace/relatio n>

couple, while morphogenesis and biorhythms correspond to the manifesta-

tion of the <time/identiﬁability> c ouple, associated with the identity and

functioning of the organism over its lifetime.

All these considera tions are partly speculative and demand further de-

tailed elaboration to tes t their per tinence. On the other hand, the status

conferred on our concepts of space and time in the constitution of objec-

tivity has to do with epistemological issues, and here it impacts at a really

profound level on the status of categories as basic as causality a nd on con-

ceptual pairs as important as that of local and global. These categories and

concepts are to a greater or lesser degree derived fr om the intuitive or the

theoretical representation of space and time and it is in connection with

both intuitive and theor e tical considerations that we must give an account

of the role of space and time in scientiﬁc explanation.

The mathematical organization of scientiﬁc theories confers our frame

of reference a status and a structure which is objective yet incr e asingly

counter-intuitive: that is to say, bas ic categories and derived conce pts are

both increasingly accurate a nd increasingly unfamiliar in character. We

have arrived at a stage where, alongside the development of mathema tical

conceptual constructions (whether in symbolic or diagrammatic form), new

structural int uitions are achieved. Thes e intuitions are inherently generated

by the mathematical system itself and further and further lacking in directly

empirical content.

The task confronting us today is the rational articulation of connections

between these new types of (theory-generated) intuition and the exp e ri-

mental results with which we are presented in the realm of physical and

biological phenomena. To recall an old distinction from hermeneutics, ex-

planation may make progress, but comprehension may be poorly equipped

to follow. We recall Thom remarking , in one of his barbed quips, that

quantum mechanics was unintelligible and that everything about it that

was rigorous was insigniﬁcant. Yet, we have made here a n extensive use

of ideas coming from quantum mechanics even to indirectly make more

intelligible some aspects of the living state of matter.

What is the connection between increasingly abstract spaces and times,

theoretically constructed and formally speciﬁed, and the intuitively given

ones which preside over the development and regulation of our own co gnitive

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Space and Time from Physics to Biology 147

capacities? The permanence of our vocabulary, while it constitutes an

index o f familiarity, certainly does not suﬃce to justify or explain these

relationships. It is to the existence of profound cognitive schemas and

invariants of our mental representation and to the way they are tr ansformed

that we must appeal. The two epistemic frames pr oposed (< space, group,

equivalence> and <time, semi-group, order>) suggest a way of approaching

that diﬃcult question, jointly to the other opposing pairs we examined

(lo c al/global, regular/singular . . . ).

We w ill return to these conceptual pairs and dualities, in particular

in reference to biology, in Chapters 5 and 6. The next chapter mostly

goes back to physics, in order to clarify the role of some key (conceptual)

construction principles and prepare the grounds for a further compara tive

exploration.

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Chapter 4

Invariances, Symmetries, and

Symmetry Breakings

4.1 A Major Structuring Principl e of Physics: The

Geodesic Principle

In physics, at least in the ﬁrst stage of rational reconstruction, all seems

to be organized according to a fundamental pr inciple which can be found

to underlie most formalizations: the geodesic principle. Informally, it is a

question of minimizing a length (or a surface), given a space and its met-

rics. We will now explain the variegated accomplishments of this principle

according to its ﬁeld of application. In particular, we w ill try to present it

in its full generality, as a unifying principle of conceptual construction, in

physics, in the sense proposed in Chapter 1.

The geodesic principle does indeed appear able to subsume most of the

theoretical s chemas currently known to physics, not in terms of their spe-

ciﬁc contents, but rather in terms of the c ommon principles which govern

them. It, moreover, has the advantage of being ex pressible in somewhat

ambivalent and clearly bijective terms, a c c ording to whether we are re-

lating it to the ma thema tical structures that it mobilizes or to the great

physical refer e nts that it modelizes, thus pr oviding a privileged example of

the relatio nship between ma thema tics and physics. These relationships are

so intimate that they have become core constitutive intera ctions between

and of mathematics and physics. Physically speaking, this pr inciple, which

took on various forms througho ut history, generalizes the optical principle

of the shortest path or the dynamic principle of least action in the form

under which it will ﬁnd its greatest extension. In fact, provided it under-

goes the adequate transformations, it corresponds to the idea according

to which any state of equilibrium maximizes or minimizes a cor rectly se-

lected state function (more speciﬁcally: free energy, or pa rtition function,

149

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150 MATHEMATICS AND THE NATURAL SCIENCES

which amounts to the same thing from the statistical mechanics viewpoint

. . . ). All processes take place while optimizing one or several quantities

(Lagrangian action, speed of entropy production, . . . ). As we will discuss

in length later on, a determina tion of this sort singularizes a sta te or a pro-

cess amo ng all possible states or trajectories according to spe c iﬁc criteria

of criticality (a minimum, a ma ximum, an optimal pa th, . . . ).

A preli minary example: Gali leo and inertia Galileo’s notio n of in-

ertia is an early instance of the geodesic principle, now fully understood

within the Hamiltonian frame we will refer to here. It provides by this an

example of the audacious spe c ulations that establishing a new construction

principle requires. First, it was preceded by confused but extremely deep

remarks by Bruno, and then it became a physical principle in the hands of

the Tuscan founding father of modern physics. The point is that any empir -

ical evidence suggests that physical bodies do stop if not pushed by a force.

Aristotle’s obse rvations were perfectly coherent with what we can witness

around us. Thus, the sun, in o rder to move around the Earth, needs to be

pulled, at all times, by divine horses or pushed by God’s hand or will (St.

Thomas). In no circumstances do we see a physical body moving forever,

along a vector of constant speed. Yet, by extrap olating from experiments,

but daring to push them towards a conceptual limit, Galileo proposed the

only pertinent principle, physical inertia. The physical principle constructs

knowledge by this sort of “phase transition,” as we called it, by which hu-

man imagination and symbolic culture go well beyond empirical evidence

and propose a uniﬁed conceptual fra me, at the “limit,” in a sense, which

makes us understands a large variety of phenomena, independently each of

them. In no way is the inertial principle “in the world,” yet it beautifully

constructs our knowledge as a form of novel objectivity.

The ﬁrst sectio n of this chapter presents the technical justiﬁcations for

our approach, which are at times more diﬃcult than the ensuing reﬂection.

These justiﬁcations are logically necessary to the reﬂection, but the a uton-

omy of the other sections (and chapters) should also enable a direc t access

to the consequences of the great principles of physics that we will evoke,

without the reader having to completely assimila te the concepts and no tions

introduced in this ﬁrst section. We will therefore ﬁrst and above all try to

present the geodesic pr inciple in its generality, in order to highlight its most

relevant characteristics. We will provide more substantial comments and

examples later on. The use of numerous technical terms will be unavoid-

able; we will try to deﬁne them as much as pos sible along the way. The

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Invariances, Symmetries, and Symmetry Breakings 151

object of this chapter is in fact to present, with a few detail s, the diﬀerent

aspects and a ce rtain number of applications of this orga nizing principle of

modern physics. For the reader, it would suﬃce, by means of the albeit

incomplete appreciation of this general framework, to see the stre ngth of

the geodesic principle: the speciﬁc technical details, in particular in sec tion

4.1.1 below, do not constitute a prerequisite for reading the rest of the book

(and, at ﬁrst reading, they may b e just gras ped in order to get even into

the subsequent sections of this chapter).

4.1.1 The physico-mathematical conceptual frame

Here a re some key hypotheses tha t underlie the approach we follow.

1) All fundamental laws of physics are the expression of a geodesic princi-

ple applied in the appropriate space. This is a very ge neral hypothesis,

corroborated by the formalisms of the various ﬁelds of physics, inas-

much a s one is less interested in the particular laws themselves, speciﬁc

to these ﬁelds, than in the principles which govern them and which

enable us to derive them ma thema tically. As we sa id, it is a question

of minimizing a length (or a surface), given a space and its metrics.

Mathematically speaking, this signiﬁes putting a calculus of variation

into eﬀect. In the ﬁeld of dynamics, the physics’ perspective on the

mathematical formulation consists in specifying that it is a Hamilto-

nian principle applied to an action, given a Lagrangian density.

But

it could also be, using a mor e static framework, a question of loo king

for the minimal surfaces corresp onding to an energy minimum for a

supe rﬁcial tension.

2) The essence of theoretical determination lies upon the identiﬁcation of

the adequate space and of its metric. Here again, we have the mathe-

matical formulation: the “objects” are supposed to live within a man-

Let’s recall here that in classical mechanics, Hamilton’s principle consists mainly in

deﬁning the trajectories as minimizing the Lagrangian action (that is, the time integral

of the Lagrangian). In the s implest case, the Hamiltonian, deﬁned in the space of posi-

tions and of moments, corresponds to the system’s total energy (kinetic plus potential),

whereas the Lagrangian, deﬁned in the space of posi tions and of velocities, corresponds

to the diﬀerence between these energies (kinetic energy minus potential energy). In ﬁeld

theory the Lagrangian may be obtained as the space integral of the Lagrangian density,

which expresses thus the space/local dependency of the Lagrangian. From the mathe-

matical point of view of symplectic geometry, the former resides in the manif old, which

is cotangent to the original manifold, while the latter resides in the manifold tangent

to this original manifold. In quantum physics, the magnitudes thus represented are not

only simple quantities, but operators.

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152 MATHEMATICS AND THE NATURAL SCIENCES

ifold endowed with a metric, and their behavior is determined by ex-

tremization of the latter. From a physical p oint of view, what we have

is essentially a precise determination of the system’s degr e e s of free-

dom and the explicit expression of the Lagrangian density associated

with it or, for example in statistical physics of equilibrium, it is the

determination of the partition function of a system.

Thus, the characterization of the eﬀective behavior of a sys tem is re-

duced to the construction of the relevant space and of the intended math-

ematical function, which may be roughly made in correspondence with the

observables, in physics. Be it a question of pheno menological La grangian

functions, partition functions, metrics as such, or path integrals (Feynman

integrals), the idea of extremization re mains the same and eﬀective deter-

mination is then nothing but a question of calculi, and is also capable of

responding to diﬀerent methods. What results from taking this principle

into account, that is, what are the most profound theoretical and c oncep-

tual consequences of this approach? In our view, and based upon its most

fecund dynamic interpretation, it is possible to identify the main co nse-

quences o f which the scope will ﬁnally prove to be quite extensive and

the gnosiological importance essential for the c urrent intelligibility of the

process of co nstruction of objectivity in contemporary physics:

i) The identiﬁcation of a metric (in the mathematical pe rspective) or of

the Lagrangian density (in physics) enables us to deﬁne the tra nsfor-

mations of symmetry which leave the equations of movement invariant.

The physical aspect is given by the trajectories deﬁned by the Euler-

Lagrange equations, that is (mathematical aspect), the corre sponding

variational equations.

It is therefore a question here o f characterizing

the system’s symmetry group (as symmetry transfor mations of a sys-

tem form a group), which preserves the Euler-Lagrang e equations or,

more generally, of identifying the gauge tra nsformations (as both local

and global transforma tions – of scale or reference system, say; see the

fo otnote in the premise of Chapter 3).

ii) To these transformations in symmetry correspond, by Noether’s the-

orems (Noether, 1918; Hill, 1951), invariants (mathematical aspect)

or conserved quantities (physical aspect) speciﬁc to the system under

The mathematical approach may be related to the physical, as the m etric element ds

may be written as a function of the Lagrangian L, that is as the integral of L over time

squared (the metric coeﬃcients g

are then the potentials). A geodesic thus becomes

the minimization of the Lagrangian, as a shortest path in that metric.