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 123

Unfortunately, anyone who observes tha t the range of biological phe-

nomena displays aspects which elude description in terms of current phys-

ical theory risks being bra nded an obs c urantist and accused of believing

in magic or to be a vitalist. The situation is not helped by the fact that

one does indeed encounter terminology of a magical-poetical ﬂavor in some

writings on this subject. Confronted with this position, some tough- minded

commentators cling to the notio n of a deterministic program (in the sense

of Laplace and Turing ) and see it encoded in the DNA or into the brain, as

the hardware on which the progr am, or rather a whole set of interlocking

programs, is run. As for the brain, others take up the issue of quantum

non-locality, locating its manifestatio n at the level of the microtubules of

neurons and claiming that this will turn out to fo rm the reductive basis of

consciousness. Other s a gain turn to the study of dynamical systems and

take this as the framework for modeling the evolution of neural networks

and the plasticity of their behaviors.

Clearly there a re very important diﬀerences between these approaches.

The ﬁrst of them nowadays comes within a hairsbreadth of b e ing a swindle.

It has long been clear that we s e e less and less evidence in current physics

of the kind of determinism embraced by Laplace and Turing, and even less

in biology. This is not to deny the importance of both Laplace and Tur ing

for rational mechanics and information sciences, r e sp e c tively.

The second approach, concerning quantum entanglement in cells o r in

the brain, sets out a challenge to be taken up, but is currently lacking in

exp e rimental evidence, or in linkages betwee n the scales of the structures

and systems involved in the hypothesis: between the a c tivity of neurons,

which are very large scale structures, and that of the quanta, intermediary

levels o f description are altogether lacking. The third approach is founded

on a strong body of evidence conce rning the workings of the brain – the

observed reinforcement of synaptic connections and, more generally, the

eﬀectiveness of the dynamical systems framework for the treatment of any

interactive system. Here pro gress has been re markable, yet the reduction

is performed towards a sp e c iﬁc physical theory: no novel con c e ptua l unity

is proposed. In particular, the formal neural brain ha s no body.

In these three approaches we also see the change of the notion of “deter-

minacy.” For Laplace (and for the sequential progra mming of computers)

any deterministic system is completely predictable.

Within dynamic s ys-

Turing in 1935 himself demonstrated that a Turing machine was subject to a kind of

unpredictability: one cannot decide the halting problem (whether the machine will halt

or not in executing a given program). In fact we can decide no “interesting” property

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124 MATHEMATICS AND THE NATURAL SCIENCES

tems theory, determinism does not necessarily imply predictability. Quan-

tum physics introduces a further and deep-going modiﬁcation of the con-

cept, v ia its dual, the notion of intrinsic (non-epistemic) indetermination.

In less than 120 years, our notions of what it is that determines w hat,

and our notion of w hat is a system evolving in time, have undergone a

profound shift. But we still have no equivalent general notions in biolo gy.

We cannot say in what manner DNA determines the ontogenesis of living

systems, nor in what way the state of a ner vous system determines its la ter

states. In an a ttempt to tackle these issues with the concepts of present

mathematical physics, researchers have entered the conceptual kitchen, so

to speak, and are busily drawing up a menu based on the recipes and cook-

ing utensils they have already mastered, a menu drafted, where possible, in

collaboration with the bio logists. This is a fair, pragmatic attitude, but to

make better pr ogress we stand a lso in need of a robust notion of biological

ﬁeld or theoretical determination at the level of the org anism, which is still

lacking.

3.2.4 Morphogenesis

Let us now turn again to the the notion of space appropriate to the study

of living s ystems. T his analysis is one key aspect of the connecting tissue

of this book as we try to comparatively develop it in the light of our un-

derstanding of the “construction principles” in mathematics and physics.

Symmetries and geodetic principles will be discussed here and mor e exten-

sively in Chapters 4 and 5, up to the extension of recent notions in physics

(the physics of “critca lity,” Chapter 6), which singularizes the living state

of matter.

of programs (Rice’s Theorem: see Rogers, 1967). However, this unpredictability only

becomes apparent in the limiting evolution of the system. The non-halting shows up

only when a system performs inﬁnitely many steps, and the undecidability of programs

is a property of functions inasmuch as they admit inﬁnite values and arguments. By con-

trast, the unpredictability of deterministic physical systems, investigated by Poincaré,

is manifested already in ﬁnite levels. Given the initial s tate (deﬁned with the due ap-

proximation), there exists a ﬁnite time (the Poincaré relaxation time) after which one

cannot predict the state in which the system will be found. Despite the well-known

undecidability results, classical computability theory thus conceives of a computation

as a deterministic process of calculation and as completely constrained to fol low a pre-

dictable evolution at any ﬁnite instant. Such a theory is a logical theory and Turing

machines are logical machines. In neither case are they to be thought of as constrained

by any physical li mitations. The issues of determinism and indeterminism i nvolved in

the halting problem are issues of logic, not of physics and not subject to the hazards of

physical approximation.

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Space and Time from Physics to Biology 125

One of the areas in which we see the richest use of geometrical concepts

in the study of living beings and their associated ecosystems is in the study

of morphogenesis. In this we include the study of the evolution of the

forms of living or ganisms and the inﬂuence of form on the structure of life

in general.

“The stability of living forms is geometric in character” (Thom, 1972,

p. 171). The topological complexity of a form is for Thom the locus of

its “meaning” and of its organization. Thom assigns an almost exclusive

explanatory role to topology: the topological evolution of the form of a

living individual provides the explanation for its biochemistry, rather than

the other way a round (Thom, 1972 , p. 175). T he form in question contains

information in two ways: it determines an equivalence class of top ological

forms under the action of a group of transformations; and it also supplies a

measure of the computational complexity of a system via the number and

evolution of its singularities. Here we can glimpse the idea of a “morpho-

genetic ﬁeld” which fashions living systems, in the cour se of their phylo-

genesis as well as their ontogenesis. A global structure and o perations of

a global character occupy center stage in Thom’s view. In the embryo, he

emphasizes, we already have the globa l pattern of the organism, from which

the spec ialization of organs and their function follows. As it ha s been said

(Jean, 1994, p. 270): plants form c e lls , not cells plants.

But just wha t is this “morpho genetic ﬁeld”? This expression could lead

us astray if we think in terms of the physical ﬁelds. The morphogenetic ﬁeld

must be thought of as in some sense containing all the known physical ﬁelds

at once, together with new ﬁelds characteristic of co-constituted organisms.

In particular, each ﬁeld – physical, biological, or cognitive, acts at a certain

level of or ganization, conceptu ally independent of others: the phenomenal

level and its conceptual structuring by our forms of scientiﬁc knowledge are

completely distinct. However, the individual organism achieves de facto

integration of this plurality of levels: its unity results from this integration

of physical, bio logical, and cognitive levels. These diﬀer e nt levels of struc-

ture a nd organization, analyz e d by quite diﬀerent scientiﬁc methods and

concepts, interact with each o ther via spatial and tempor al linkages. Each

level displays plasticity with respect to the others.

However, no current physical theory supplies the concepts needed to

describe these for ms of mutual action, control and constraint, operating

between the diﬀere nt levels: the ascending and descending linkages be-

tween them at all levels of the system and its biological, chemical, and

physical components and subsystems. Cybernetics, the ﬁrst theory of con-

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126 MATHEMATICS AND THE NATURAL SCIENCES

trol automation, has cer tainly furnished remarkable models of the linkages

involved in self-r e gulation. But these models have been located speciﬁcally

at the physical level, and are c onstrained by the range of the theoreti-

cal tools they employ, whereas living systems establish linkages between

conceptually wholly distinct levels of description (mo lecular, cellular , tis-

sues, organs, organism). Typically, the mathematics of orga n formation

(phillotaxis, formation of lungs, vascular systems, . . . ) optimizes exchanges

of energy. In contrast to this, the mathematics of ce llular networks (neural

in particular) deals with exchanges of “information” (mo stly gradients of

energy), a diﬀerent observable. These two areas do not talk to each other,

while it happens that c e llular tissues are part of organs which are integrated

in an organism.

Thom’s analysis, subsequently enriched by the work of ma ny other re-

searchers, is also directed at the physical aspec ts of the topological plas-

ticity of living forms, including those aspects induced by their “virtual”

interactions. His work deals with an extremely informative physis of living

systems, but still a physis. While, on the one side, it views the plasticity

seen in the evolution of living for ms as constrained by the dynamic ﬁelds

operating in their morphogenesis, on the other hand, topolog ical evolution

is regarded as developing within a physical schema which takes no account

of such phenomena as la tent potentials or the combinatory explosion of life

in all compatible directions. But, compatible with w hat? We do not mean

compatible with the forces acting on a system at a given instant, but also

with those it will expe rience. This poses a (mathematical) problem, which

is at the heart of the evolutionary and developmental plasticity of living

forms.

As it has been understood by thos e who have contributed to the most

fully worked-out aspects of the theory of morphogenesis, namely phyl-

lotaxis, it is possible to induce forms very similar to thos e se e n in phyl-

lotaxis by means of superconducting curr e nts imposed on a magnetic ﬁeld

(see Jean, 1994, p. 264). For instance, the Fibonacci sequence, which is

observed very frequently throughout the vegetable kingdom, can be repro-

duced by this method on a ny mesh of “soft o bjects” under repulsive forces

and stro ng deformations (ib., p. 265). In this sense, such an analys is does

indeed consider living forms with respect to their being as p urely physi-

cal systems – that is to say as bodies subject to the inﬂuence of physical

ﬁelds. But although an imp ortant and necessary inves tigation, this is not

exhaustive as an analysis of the forms of living systems.

Morphogenesis also has an important role to play in helping biology

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Space and Time from Physics to Biology 127

break out of the stranglehold of “genetic chauvinism” (that DNA provides

a complete description of embryogenesis or even ontogenesis). The latter, in

the writing of so me authors, takes the form of a near maniacal expr e ssion of

the Laplace-Turing vision of an absolutely deterministic causality, legislated

in advance by the initial conﬁguration of the system’s components: some

sort of homunculus residing in the genes, translated in the modern terms

of a coded homunculus. T his v ision of a closed future is strongly rooted in

currents relating genetics and sociobiology. This is unhappily congenial to

religious believers in predestination: one just repla c e s God for evolution a s

the typist on the molecular keyboard of the genetic prog ram and ontogeny is

completely understood (including marital ﬁdelity, see Young et al. (1999)).

In contrast to such a picture, Atlan replied: “the program of a living

organism is everywhere except in its genes”: certainly the p atterns and the

forms seen in phyllotaxis are not entirely in the genes. They are als o in

the structure of space and time and of physical matter and energy. The

genes do not contain all the information on the sy mmetries which are set

up in a system in interaction with its environment, such as are observed

in crystals and minerals (Jean, 1994 , p. 266 ). The so-called “program” for

the development of an organism is to be found in the interface between

its phylogenetic record (its genetic legacy) and its physical a nd biological

environment (its ecosystem).

An example of the greatest importance is pr ovided by the brain, which

in the course of ontogenesis manifests a developmental pattern, which is

both Darwinian a nd Lamarckian. The immense number of possible con-

nections among its neurons (each one of around 100 billion neurons has up

to 10,000 synaptic connections, maybe more) could not be (or at any rate

very little of it could be) encoded in the genes. Of the numerous connec-

tions established very rapidly during the growth of the fetus or the newbor n

child, most disappe ar through selection eﬀects (Edelman, 1987). On the

other hand, throughout the course of our entire life, stimuli lead the brain

to establish new connections a nd reinforce existing ones, jettisoning and

replacing existing connections as it does so, selecting certain neurons and

leaving others to die oﬀ. Cerebral plasticity, at all levels, is at the heart of

the continuity between phylogenesis and ontogenesis, and is what permits

continuing individual identity: “the structure of the nervous system carries

the material traces of its individual history” (Prochiantz, 1997).

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128 MATHEMATICS AND THE NATURAL SCIENCES

3.2.5 Information and geometric structure

In an epoch of free-ﬂoating bits, the picture of information (even of in-

telligence) as purely a sequence of bits enjoys great currency. T he digital

encoding of information is of great eﬀectiveness for certain purposes: once

encoded, such information can be sa feguarded and transmitted with unri-

valed accuracy and speed. No method is supe rior to that of bit-s torage in

the constr uction of digital computers and the networks they form, which

are now in the course of transforming our world.

Moreover, a number of notable mathematical results of the 1930s demon-

strated that all discret e encodings and their eﬀective treatment are equiva-

lent. Kleene, Turing, Church et al. demonstrated the equivalence of (very)

diﬀerent formalizatio ns of “computability”: the numerical functions calcu-

lated by using the systems of Herbrand, Curry, Gödel, Church, Kleene, and

Turing were the same. By means of an astonishing philosophical sleight of

hand, tra ding on the surprising and technically diﬃcult nature of these re-

sults, and inﬂuenced by the surrounding intellectual climate of formalist

and pos itivist ideas, the cla im was later made that any physical form in

which information is processed, and thus any biological form of information

processing or a ny form of intelligence, can be encoded in any such formal

system, thus it can be encoded in the form o f the strings of 1s a nd 0s used

in the memory stores of digital computers, see Longo (2007) for more on

some parodies of the Church-Turing thesis.

A quite diﬀerent way in w hich information can be thought of as struc-

tured, one involving geometric principles, is through equivalence cla sses of

continuo us deformations. These provide for the tra nsfer and processing

of much o f the information essential to the make-up of living beings and,

more generally, of physical systems. C ontinuous , diﬀerentiable or isometric

transformations and the regularities they preserve or fail to preserve may

help to structure and make intelligible living phenomena, as can be seen

not least in the geometric structure of DNA or of proteins and their evo-

lution. To these transformations the discrete and q uantitative structure of

bits of information serves as an addition; bits behave a s singularities and

thus as a possible measure of the topological complexity o f the geometric

structure. Information has both a qualitative and q uantitative nature. The

concentration o n only its quantitative, digital, nature has become a severe

limitation w hen information is assigned an explanatory role.

A frequent reaction is: yes, gr anted the role of these kinds of transfor-

mation and this kind of continuity, nonetheless, in the last instance, the

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Space and Time from Physics to Biology 129

geometric structur e s involved are reducible to very minimal discrete units.

Such a reaction hides many problems. Physical and mathematical pr in-

ciples prevent our modeling continuous and three-dimensional information

in discrete form. The notion of (Cartesian) dimension does no t apply in

the discrete frames of codings and programs. Inﬁnite discrete structures,

the natural numbers N in particula r, are isomorphic to any ﬁnite product:

N ≡ N

, by a computable isomorphism. That is, any ﬁnite string of in-

tegers can be encoded as an integer, and this is crucial for Gödel’s and

Turing’s approach to computability and their applications.

These isomorphisms make no sense in mathematical physics, as they

would simply destroy most of its theories: dimensional a nalysis, that is the

analysis of the number and type of va riables in a function f (x

, . . . , x

is cr ucial in physics and theories radically change when changing dimen-

sion. From the analysis of heat propagation (Poisson equations), whose

characteristics are very diﬀerent in one, two, or thre e dimensions, to the

“mean values theories,” which diﬀer radically from two, three, or four di-

mension, and a lot more. Not to quote relativity theory w here the uniﬁed

four -dimensional structure of space-time is crucial or string theory, where

intelligibility is given by moving to 10 or more dimensions. And math-

ematics proves it beautifully, in relation to the “natur al topology” on R,

the real numbers, that is in relation to the so-called Euclidean or interval

top ology. Recall ﬁrst that the interval topology is “natural” since it comes

from physical measure, which is, by principle, an interval. Then, and this

is fantastic, one can prove the following:

if A ⊂ R

and B ⊂ R

are open sets and A ≡ B, then n = m.

This theorem says that dimension is a topo logical invariant, when one

takes the natural topology in a s pace manifold, in the sense of Riemann.

This result is false when considering, for example, the discrete topology on

R, or, say, a weakly separated topology. This is a remarkable connection

between mathematics and physics, via measure: the interval of physical

measure, which yields the na turality of the Euclidean topology on the re-

als, ma kes dimension an invariant and gives a mathematical meaning to its

central role in physics. Thus, there is a mathematical gap between the con-

tinuum and the discrete, which is not, in general an approximation rela tion;

instead, they provide diﬀerent mathematical intelligibility of phenomena.

Now, the smallest living phenomena comprise dynamical systems (ther-

modynamic systems, systems with critical points). And here is a further

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130 MATHEMATICS AND THE NATURAL SCIENCES

reason for a discrete mesh laid down a priori to be suﬃcient for their

analysis. This is because sensitivity to initial conditions typically gener-

ates far-reaching consequences at or above the threshold of discernibility,

triggered by a variatio n below that of the smallest interval of the measure.

But what kind of discre tenes s are we really talking about here? On

the a ssumption we can push the encoding right down into the microphys-

ical rea lm (so as not to have to cover the whole earth with proce ssors) it

seems the discreteness in question will come from quantum physics and

will arise at the scale of the Planck length. One then e ncounters a fallacy,

well explained in Bitbol (2000a/b), of the same stamp as that involved

in the case of the formalist and mechanist re ductio n of mathematics to

formal manipulation (processing) of disc rete symbolic inputs. The refer-

ence to well-deﬁned and ultimate discrete level o f “material points” is the

sine qua non of La placian mechanics. No such appeal is possible in quan-

tum theory, because it is a theory of continuous (quantum) ﬁelds, where

Planck’s consta nt, the only pos sible referent for these notions of ultimate

discretization, has the dimensions of an action (energy times time), which

may be understood as a “ﬁbration” orthogonal to the continuum of space-

time. Moreover, quantum indetermination and the epistemological debate

which has raged around it involved assigning a role to the knowing subject

of a profoundly anti-mechanist kind. In approaching this topic armed with

a digitized version of microphysical structure one is lending suppor t to the

myth of Laplacian mechanism, as a matter of the bit-strings programmed

by formal laws of thought. This is the opposite of the the paradig ms pro-

posed by quantum mechanics.

We cannot renounce the mathematics of continua. The deformation

of geometric structure can reproduce information in analogue form. And

the analogy involved is intentional – one chooses what to represent or re-

produce in analogous form, one selects those aspects of the original form,

which are to undergo processing or simulation. The choice of analogy is

the outcome of a controlling vision or an aim, conceptually appropriate to

living systems, of a kind which is missing in physics, where the phenome-

nal arrow of time is oriented without backwards linkages; see, nonetheless,

Nove llo (2001). The fact that the reproduction a nd transforma tion of in-

formation via geometrical forms is analogue in character and may better

accommodate intentiona lity, appears to be just what is c rucial for biologi-

cal representation. T he eventual greater instability over time of geometrical

forms by comparison with binary bit-strings is actually an enriching factor,

because it correspo nds to the possibilities of evolutionary change. The anal-

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Space and Time from Physics to Biology 131

ysis of the geometric structuring of living systems (particularly of the br ain)

permits us to grasp a factor es sential in information: selective a nalogical

simulation carries with it evolutionary pos sibilities. By contrast with this,

the perfect stability of bit-by-bit information processing renders such an

elaboration impossible (whether this is a practical impossibility or one of

principle is unclear).

To co nclude: in studying living phenomena, from the most elementary

systems all the way up to cognitive agents, it is not so much a matter of

denying the important role played by formal and mechanical aspects (“bits”

are key singularities in relation to information) but rather of enriching

this analysis through the phenomenal richness of geometric structures and

their eﬀectiveness in information processing. O nce again, formalism and

mechanistic physicalism are seen to be not a variety of scientiﬁc reduction

of the kind we should expect to meet in scientiﬁc practice, but rather a

philosophical monomania, which ha s lost touch with the plurality of forms

accorded to our knowledge of and interaction with the world.

We mentioned the areas of morphogenesis and cellular networks as very

lively but far apart r e search areas. Work on bo th morphogenesis and the ar-

chitecture of dynamical neural networks (see e.g. Hertz et al., 199 1; Amar i

and Nagaoka, 2000, for the latter), despite its inco mpleteness , arising from

the limitations of a physicist’s, though neither formalist nor mechanist,

standpoint, at the very least sugge sts the richness of the geometrical (con-

tinuous) structures implicated in any account of the organization of living

systems.

The constitution of geometrical patterns of neural networks is a typical result of this

complex compositional activity and permanent dynamism, but it is not the only result.

See for instance the remarks in Edelman (1987) on the ﬁne structure of synaptic con-

nections and other aspects of neural plasticity, which go beyond those modeled by the

dynamics of neural networks. Consider also the interactive genesis of the forms of such

networks in the context of an ecosystem. Here the stimuli are of a physical or biological

nature, grounding the mental activity in the material of living systems. Animal intelli-

gence involves a dynamic of such forms distributed over many diﬀerent levels of structure

(from that of proteins to that of synapses to that of entire neural networks) all of w hich

are in mutual interaction. Its unity is that of a subtle and complex kind of interacting

and self-reacting ﬁeld which is still very diﬃcult to grasp.

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132 MATHEMATICS AND THE NATURAL SCIENCES

3.3 Spatiotemporal Determination and Biology

3.3.1 Biological aspects

The question of space has played a very important, even a foundational role

in biology: one which has not always received due appraisal. Take for ex-

ample the co nce pt of milieu interieur (internal environment) introduced by

Claude Bernard, which allowed an essential topological separation between

the interior and exterior of an organism. C onsider also the question of chi-

rality in biology, highlighted by Pasteur. In the wake of his experiments

on the tartra tes and the manner in which their biological activity diﬀered

depending on whether they coiled to the left or the right, he stated unhesi-

tatingly: “Life, as it is manifested to us, is a function of the asymmetry of

the universe and a consequence of it.”

Indeed Pasteur anticipated both developments in his own ﬁeld of scien-

tiﬁc inquiry and, mutatis mutandis, what later came about in physics with

the discovery of the a symmetry of matter and anti-matter, which co smol-

ogy now views as the precondition or the existence of the universe and of

the actual material structures we see all around us.

Biological structures a re subject to organizing processes leading to the

emergence of co mplex forms, such as those studied in developmental bi-

ology; furthermore, they display physiological functions which sustain the

part/whole mutual dependence, which mediate their integration as organ-

isms and regulate the linkages between the diﬀerent levels of organization

typical of orga nic existence. These facts clearly have a connection with

theories of the c ritical behavior of dynamical systems, such as that seen in

phase tr ansitions.

It was no t by chance that the ﬁrst mathematical models of biologi-

cal systems appealed to and borrowed from those of thermodynamics, in

particular models of cascade eﬀects in bifurcations of thermodynamical sys-

tems (see Nicolis, 1986; Nicolis and Prigogine, 1977), followed by models of

emergence of self-organized critical behavior (see Ha ken, 1978; Kauﬀman,

1993; Varela, 1989), and application of fra c tal geometry (see Mandelbrot,

1982; Bailly et al. 1989; Bouligand, 1989) and chaotic regimes (Babloy-

anz and Destexhe, 1993; Auger et al. 1989; Demongeot et al. 1989 ) to

an organic context. Alongside these developments, it had been clear that

the character and genesis of processes of formation could in many cas e s be

modeled using the elementary theory of catastrophes (Thom, 1980), and,

more generally, singularity theory. What clearly shows up in the analysis of