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

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Mathematical Concepts and Physical Objects 53

tempting to articulate internal space-times and external space-times. From

this point, we indeed witness the afore mentioned observation of “causal

laws” by these “mathematical structural organizations,” but which apply

this time to spaces and times presenting new characteristics.

As noted by Chevalley in his Present ation in the work by van Fraassen

(1989), this o bservatio n is massive, to an extent, in the analysis of contem-

porary physics and in its very functioning, o f “ substituting to the concept

of law that of symmetry,” thus e xtending the apprec iation of the author

who, while adopting what he calls the “semantic approach” in the analysis

of physics, does not hesitate to assert regarding symmetry that: “I consider

this concept as being the principal means of access to the world we create

in t heories.”

Such radicalism could surprise at ﬁrst sight but may, however, be nicely

explained if one realizes that it is an issue of considering essential elements

of the very process of the construction of physical objectivity and of the

determinations of the corresponding scientiﬁc objects. Indeed, as conserva-

tions of physical quantities are associated w ith the principles of rela tivity

and of symmetry and that all of physics base s itself on the measurement

of quantities related to properties which must remain stable in order to be

observed, one may go as far as ass e rting that these relativities and sym-

metries, while appea ring to reduce the possible information relative to the

systems under study, are constitutive of the very identity of these systems.

As if it were a question, to use the old vocabulary of medieval scholastics,

of identifying the primary qualities (that is, their essential identity struc-

tures) w hile leaving with the “laws” the c are of regulating their se c ondary

qualities (to which would correspond here their actual behavior). To con-

tinue a moment on this path, we could even go so far in the analysis of

the relationships b e tween symmetry and identity by considering that all

information is a (relative) breaking of symmetry and that, reciprocally, any

relative breaking of symmetry constitutes an objective element of informa-

tion. According to s uch a schema, it would then be relevant to consider

that to the metaphysical substa nce /form pair, to which was partially sub-

stituted during the scientiﬁc era the energy/information (or entropy) pair,

ﬁnally corresponds the symmetry/breaking of symmetry pa ir as constituent

of the identity of the scientiﬁc object, which we have just mentioned.

Let’s specify our argument concerning the relationships betwe e n c ausal

laws and mathematical structures of geometry. Relativistic theories – gen-

eral relativity in particular – cons titute the pr iv ileged do main where was

ﬁrst identiﬁed, in modern terms, the relevance of causal laws by their iden-

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54 MATHEMATICS AND THE NATURAL SCIENCES

tiﬁcation to structural organiza tions. This stems essentially from the in-

trinsic duality existing between the characterization of the geo metry of the

universe and that of energy-momentum within that universe. By this du-

ality a nd the putting into eﬀect of the principle o f invariance under the

diﬀerentiable transformations of space- time, the “forces” are relativized to

the nature of this geometry: they will even appear or disappear according

to the geometric na ture of the universe chosen a priori to describe phys-

ical behaviors. Now, it is similar for quantum physics, in gauge theories.

Here, gauge groups operate upon internal variables, such as in the case

of relativity, where the choice of local gauges and their changes enable us

to deﬁne, or conversely, to make disappear, the interactions characterizing

the reciprocal eﬀects of ﬁelds upon one another. For example, it is the

choice of the L orentz gauge which enables us to produce the potential for

electromagnetic interactions as corre lates to gauge invariances.

Consequently, if one considers that one of the modalities of expre ssion

and observa tion of the causal processes is to be found in the precise charac-

terization of the forces and ﬁelds “causing” the phenomena observed, then

it is apparent that this modality is profoundly thrown into question by the

eﬀects of these transformations. Not that the causal structure itself w ill

as a result be intrinsically subverted, but the description of its eﬀects is

profoundly relativized. This type of o bservatio n therefore leads to having

a more elaborate representation of causa lity than that resulting from the

ﬁrst intuition stemming from classical behaviors. Particularly, the causality

of contemporary physics seems much more ass ociated with the manifesta-

tion of a formal solidarity of the phenomena between themselves, as well

as between the phenomena and the referential frameworks chosen to de-

scribe them, than to an object’s “action” oriented towards another in inert

space-time, as classical mechanics could have accredited the idea. After

Kant, it already had cease d c laiming to r e stitute the functioning of reality

“as such,” whereas with contempor ary results it goes so far a s to present-

ing itself as technically dependent upon the models which account for the

phenomena under study. Causes, in this sense, become interactions and

these interactions themselves constitute the fabric of the universe of their

manifestations, its geometry: modifying this fabric appears to cause the

interactions to change; changing the interactions modiﬁes the fabric.

These considerations may extend to theories of the critical type in that,

among other things, the spontaneous breaking of symmetry partially sub-

vert Curie’s principle which stated that the sy mmetry of causes were to

be found in those of the eﬀects. And if we want to generalize the Curie

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Mathematical Concepts and Physical Objects 55

principle to the case of the eﬀects which manifest s ymmetry breaking rel-

ative to causes, then we are led to consider not only a singular experience

which manifests this breaking, but the entire class of equivalent expe riences

and their results. As breakings of symmetry are singularized at random by

ﬂuctuations w hich orient them (this is the fundamental hypothesis which

enables us to do without the existence of other “causes” which would not

have been considered in the problem), the acco unting for all possible ex-

periences contributes in eliminating this random character by averaging it

and contributes to restore the symmetry of potentialities in actuality and

in average. It nevertheless remains that the precise predictability of the

result of a given exper iment remains limited: if we know that it necessar-

ily belongs to the class of symmetries authorized by the Curie principle,

we do not know which possibility it actualizes , that precisely in which the

symmetry ﬁnds itself to be spontaneously broken and the usual causality

jeopardized.

To conclude, it therefore appears that, at the same time as causal laws

are replaced in their theoretical fecundity and their explanator y s c ope by

analyses of “geometric” structures and transformatio ns, it is the very con-

cept of causality which regains its s tatus of regulatory concept and w hich

moves away from the constitutive role we have more or less cons c iously con-

ferred to it. To return to the remarks we have already formulated a c c ording

to a diﬀerent per spe c tive, if the phenomenal explanation and the descrip-

tion of sensible manifestations continue in physics to have recourse to the

concept of cause, on the other hand, the constitution of identity and of

physical objectivity is increasingly related to the mathematical structures

which theorize them and now confer upon them predictive p ower. T his

provides renewed topicality to the considerations voiced by Weyl and his

friends (such as my interlocutor).

1.3.6 Towards the “cognitive subject”

Let’s return to another aspect of Longo ’s “Mathematical concepts and phys-

ical objects.” It co ntains remarks which may constitute the basis of a novel

inquiry into human cognition. L e t’s re port two of these indications which

deserve further r e ﬂec tions. He writes: “Foundational analysis, in mathe-

matics and in physics, must propose a scientiﬁc analysis of the cognitive

Note that the spontaneous symmetry breaking, in classical frames, has a random

origin, which, in contrast to what happens with i ntrinsic quantum ﬂuctuations, does not

throw causality as such into question, but concerns only observability (it is epistemic).

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56 MATHEMATICS AND THE NATURAL SCIENCES

subject and then highlight the objectivity of the construct ion of knowledge

within its reference frameworks or systems”. And further on, from a more

sp e c iﬁc angle: “The project of a cognitive analysis of the foundations of

mathematics requires an explanation of the cognitive subject, as living unity

of body and mind, living within intersubjectivity and within history. This

subject who traces on the phenomenal veil objects and structures, spaces and

concepts which are common to mathematics and to physics, lays outlines

which are not ‘already there,’ but which result from the interaction between

ourselves and the world.”

This appro ach, in a way, takes the dir e c tion opposite to Boole, Frege,

and their logicist and formalist upholders who wanted to ﬁnd in logic and

formalism “the laws of thought,” well, at least those governing mathemati-

cal thinking. Without denying the a dvances enabled by the developments

within these ﬁelds, Longo proposes on the other hand to see and ﬁnd es-

sential elements enabling the characterization and objective analysis of the

cognitive subject in the development of the practice and in mathemati-

cal s tructures themselves. He does not base this propositio n solely upon

mathematical rigor and demonstrative capacities (prevalence of proof prin-

ciples), but a lso on the conceptual stability they authorize as well as on

their aptitude to objectively thematize and categorize the quasi-kinesthetic

relationships to expe riences as primitive as those related to movement, to

space, to order (role of construction pr inciples). Beyond their spontaneous

language- related apprehensions (which we know to vary according to cul-

ture and civilization), inquiries on the basis of the analysis of “b e haviors”

and mathematical approaches would enable us in a way to stabilize these

exp e riences and intuitions by abstra c ting them, by objectivizing and uni-

versalizing them, and also by revealing the deepest of cognitive workings

which animate them.

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Chapter 2

Incompleteness and Indetermination in

Mathematics and Physics

Introduction

To continue the preceding discussion, this chapter addr e sses more speciﬁc

mathematical and physical topics, which, although being relatively techni-

cal, prese nt important philosophical issues. In particular, we will examine

the play between co nstruction principles and proof principles. In fact, the

epistemological issue related to the completeness of for mal theories and of

physical theories, as well as the “uncertainty principle” in quantum mechan-

ics, is extremely relevant for the scientiﬁc and foundational debate and it

relates to that play.

Thus, we will more closely try to understand to what extent the fact

of conceiving epistemology as the analysis o f a genesis presents the traits

of a foundational analysis, in mathematics and in physics. Since Frege,

as we have seen, many have attempted to disjoin the analysis of founda-

tions, essentially aiming to wrap mathematics in a fabric of absolute logical

certitudes, from the analysis of “origins” in the sense tha t can be found

in (Husserl, 1933 ), for instance. On the contrary, we aim to reaﬃrm the

“foundational” character of analysis within a human process, that is within

those actions in the world, which ground and constitute knowledge. As we

said, it is necessary to overco me the traditional distinction between founda-

tions and reconstruction of a genesis, in Husserl’s sense, becaus e scientiﬁc

certainty and objectivity, hence its founda tion, resides ﬁrst in the capacity

to expla in the choices and methods of knowledge to which one has recourse,

including their histo rical-constitutive dynamic. Or, the physicist would say,

to make explicit the reference system (physical and conceptual), the instru-

ments of measurement and of access to reality. To achieve the construction

of objectivities in science, it is necessary to know how to s et and to ex-

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58 MATHEMATICS AND THE NATURAL SCIENCES

plain these parameters on both the conceptual level as well as on the level

of praxis, both empirical and deductive. More than any other discipline,

quantum physics, to which we will return in part 2.2, shows us the impor-

tance of this explanation as well as the paradigmatic role it may play for

the fo undation of scientiﬁc knowledge.

2.1 The Cognitive Foundations of Mathematics: Human

Gestures in Proofs and Mathematical Incompleteness

of Formalisms

2.1.1 Introduction

The fo undational a nalysis of mathematics has been strictly linked to, and

often originated, philosophies of knowledge. Since Plato and Aristo tle, to

Saint Augustin and Descartes, Leibniz, Kant, Husserl, and Wittgens tein,

analyses of human knowledge have been largely indebted to insights into

mathematics, its proof methods and its conceptual constructions. In our

opinion this is due to the g rounding of mathematics in basic forms of knowl-

edge, in particular as constitutive elements of our active relation to space

and time; for others, the same logic underlies mathema tics as well as gen-

eral r e asoning, w hile emerging more clearly and soundly in mathematica l

practices. In this text we will focus on some “geo metric” judgments, which

ground proofs a nd concepts of mathematics in cognitive experiences. They

are “images,” in the broad sense of mental constructions of a ﬁgurative na-

ture: we will largely refer to the well-ordering of integer numbers (they

appear to our constructed imagination as spaced and ordered, one after

the other, along a p otentially inﬁnite line) and to the share d image of the

widthless continuous line, an a bstracted trajector y, as an invariant of the

practice of action in space and in time.

Their cognitive origin, possibly pre-human, will be hinted at, while fo-

cusing on their complexity and elementarity, as well as on their foundational

role in mathematics. The analysis will often refer to Châtelet (1993): his ap-

proach and his no tion of “mathematical gesture” (in short: a mental/bodily

image of/for action, see below) has partly inspired this chapter. However,

on one hand, we will try to address the cognitive origins of c onceptual ges-

tures, this being well beyond Châtelet’s project; on the other, we will apply

this conce pt to an analysis of re c e nt “concrete” mathematical incomplete-

ness results for lo gical formalisms. A critique of formalism and logicism

in the foundations of mathematics is an essential component of our episte-

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Incompleteness and Indetermination 59

mological approach; this approach is based on a genealogical analysis that

stresses the role of cognition and history in the foundations of mathematics,

which we consider to be a co-constituted tool in our eﬀort for making the

world intelligible.

2.1.2 Machines, body, and rationali t y

For one hundred years, hor des of ﬁnite sequences of signs with no signi-

ﬁcation have haunted the spaces of the foundations of mathematics and

cognition and indeed the spaces of rationality. Rules, which are ﬁnite se -

quences of ﬁnite sequences of signs as well, transform these se quences into

other sequences w ith no signiﬁcation. Perfect and certain, they are sup-

posed to transform the rational into the rational and stand as a paradigm

of rationality, since human rationality is in machines. “Sequence-matching”

reigns undisputed: when a sequence of meaningless s igns matches perfectly

with the sequence in the premise of one of the rules (the ﬁrst at hand, like

in Turing machines), it is transformed into its logic-formal consequences,

the sequence in the next line; this is the mechanical-elementary step of

computation and of reasoning. This step is certain s ince it is “out of us”;

its certainty does not depend on our action in the world, it is due to its

potential or eﬀective mechanizability.

All of that is quite great when one thinks of the transmission and elab-

oration of digital data, but, as for the foundations of mathematics (and

knowledge), a schizophrenic attitude is repeating itself. Man, who has in-

vented the wheel, excited by his genius invention, has probably once claimed

“my movement, the movement, is there, in the wheel . . . the wheel is com-

plete: I can go anywhere with it” (well, the wheel is great but as soon as

there is a stair . . . ); in this way, he thought he could place, or rediscover,

his own movement out of himself. In this way, the lever and the catapult

have become the paradigm of the arm and its action (Aristotle). Gears and

clocks’ strings coincide with body mechanisms, including br ain mechanisms

(Descartes); and the contraction of muscles is like the contraction of wet

strings (Cartes ian iatr o-mechanics in the XVIIth century, see Canguilhem,

2000).

But some claim the last machine, the computer, has b e e n invented by re-

ferring to human beings and their thinking, while it was not the case for the

lever and the ca tapult or the clock mechanisms and their springs. These are

not similar to our own body parts, they have not been designed as a model

of them. Here is the strong argument of formalists: this time the mathe-

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60 MATHEMATICS AND THE NATURAL SCIENCES

matical proof (Peano, Hilbert), in fact rationality, has been ﬁrst transferred

into something potentially mechanic. Then, on this basis, engineers have

produced machines. This is a strong argument, from a histo rical point of

view, but it leaves this functionality of human being (rationality, mathe-

matical proof), his intelligence, out of himself, out of his body, his brain,

his real-life experience. And schizophrenia remains. It has only preceded

(and allowed) the invention of the machine, of computer: it is upstream, in

the formalist paradigm of mathematica l deductio n, indeed cognition, s ince

man, in the minimal (elementary and simple) gesture of thinking, would be

supposed to transform ﬁnite sequences with no signiﬁcation into ﬁnite se-

quences with no signiﬁcation, by sequence matching and repl acement (from

Peano to Turing, see Longo (2007); some refer to Hobbes and Leibniz as

well).

But w here is this “human computer” whose elementary action of think-

ing would be so simple? As for the brain, the activity of the least neu-

ron is immensely complex: neurons have very diﬀerent sizes, they trigger

large biochemical cascades inside and outside their cellular structure, their

shapes and the form of their electrostatic ﬁeld change; moreover, their

activity is never isolated from a network, from a context of signiﬁcation,

from the world. In fac t, as for thinking, when we go from one sentence

to another one, with the simplest deduction (“if . . . then . . . ”), we are not

doing any sequence ma tching, but we move and deform huge networks of

signiﬁcation. Machine, with its very s imple logical gates, with its software

built on even more simple primitives, can only try to functionally imitate

our c ognitive activities. Machines do not “model” these activities, in a

physical-mathematical understanding of modeling – which means to pro-

pose a mathematical (and/or artiﬁcial) framework able to reproduce wha t

we consider the constitutive principles of the object modeled. But even the

functional imitation is easy to recognize, as we will more closely argue in

Chapter 5.

On the other hand, some algebraic calculi require purely mechanical

processes and are an integral part of mathematics. These calculi, we sha ll

argue, are the death of mathematics, of its meaning and its expressive

richness, if they are isolated from contexts of signiﬁcation and considered

as a general paradigm for it.

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Incompleteness and Indetermination 61

2.1.3 Ameba, motivity , and signiﬁcation

What else have in common clocks, fo rmalisms, and computers? All artifacts

are actually constituted by elementary and simple components: washers

and ropes, 0 and 1 sequences, logical g ates, all individually very simple,

are put together and associated into huge constructions that may reach a

very high complexity. In fact, complexity is the result of a construction

which superposes extremely simple constitutive elements: this ability to be

reproduced, to be accessible (that is, to be dismantled component by com-

ponent) is the strength of artiﬁcial constructions, which, by this, always

proceed “bo ttom-up.” On the contrary, the elementary biological compo-

nent, the cell, is e xtremely complex; it contains all the objective complexity

of life; it is elementary since once the ce ll is cut, it is not living anymore.

And embryogenesis is always “top-down.”

The contraposition between artiﬁcial and natural, just sketched ab ove

by referring to the e lementary and simple aspects of artifact as opposed

to the elementary and complex components of natural phenomena is the

core of our analysis: we will face it again in the complexity of linguistic

symbols, of living ce lls , of strings in quantum physics. At each phenomenal

level of this three level cla ssiﬁcation (human languages, b iological entities,

microphysics), rough yet historically rich, of the way the world appears to

us, the elementary seems extremely c omplex, perhaps the most complex in

the phenomenality and for sure the mo st diﬃcult to understand.

Moreover, a cell like the ameba or the paramecium changes internally

as well as in its relationships with the exter nal: it moves. This is essential

for life, from its action in space to cognitive phenomena since “motivity is

the original intentionality” (Merleau-Ponty, 1945). Now, in our opinion,

signiﬁcation is constituted by the interference of signal with an intentio nal

gesture, be this gesture “original” or not. In this way, gesture, which begins

in motor action, set the roots of signiﬁcation be tween the world and us, at

the interface of both. The chemical, thermal signal, which a ﬀects ameba

and cell, is “signiﬁcant” for the living, regarding its current internal change,

its action, and its movement. The neuron, reached by a synaptic discharge

which deforms its membrane and its electrostatic ﬁeld, reacts with a bio-

chemical cascade, with a subsequent deformation of its electrosta tic ﬁeld,

even by changing form and place of synaptic connections. In other words:

it reacts with an action, a gesture at its scale, with its internal and exter-

nal mobility; at its level, this reac tion is meaning. And the elementary,

minimal, living unit is preserved while the current action is modiﬁed by

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62 MATHEMATICS AND THE NATURAL SCIENCES

the signal (each neuron, as any living thing, is always a c ting ). This modi-

ﬁcation is at the roots of signiﬁcation. Of course, the neural network also

changes and the net of networks, and the brain as well, in a changing body.

This is the modiﬁed ac tivity of this entanglement and the awfully complex

coupling of organization levels which makes signiﬁcant the friction between

the living and the external world. The result is non-additive, indeed non-

compositional (it is not possible to rebuild it by assembling elementary

signiﬁcations “piece by piece”), since it requires the activity of the whole

network. Finally, perceptio n itself is equivalent to the diﬀerence between

an a c tive forecast and a signal (this analysis has gone from Merleau-Ponty,

1945 up to Bertho z , 1997). For this reason, perception leads to signiﬁca-

tion: it depends on prevision, which is an action, and ac c ompanies any

other ac tion. Perception is the res ult of interference between a signal and

an action or an anticipation. In short, there is no meaning without an

ongoing action.

As for man, signiﬁcation includes als o action within a communicating

community, interaction with others in a symbolic culture, rich of language,

gestures, and evocations. Thus, intentionality allows signiﬁcation, on the

basis of original intentionality: motion. This very intentionality which

envelops any object of thought, as an “aim.”

2.1.4 The abstract and the symbolic; the rigor

Through this constitutive route, entangled and complex in its origin, which

lies on the action of the living, humanity has a rrived to propo se our sym-

bolic culture. Its sy mbols are signiﬁcant, they refer to the world, and they

are in resonance with the world. Each symbol, each sentence has a huge

“correlation length” in the space of pres e nt and history. The correlation

space is almost a physical fact: that is why we use this term borrowed from

physics. This length describe s the possible distance of causal links: each

word is almost physically c orrelated, by an individual or a whole co mmu-

nity, to a huge set of words, acts, gestur e s, and real-life experiences. These

links co nstitute a “manifold” in space and time, in a mathematical sense

of manifold, since it is possible to give a geo metric structure to it (as has

been already under taken, in the modern analyses of signiﬁcation spaces,

see Victorri, 2002). There is no metaphysics of the ineﬀable, but rather a

concrete, material and symbolic reality of phylogenetic, ontogenetic, and

cultural complexity of the human be ing and his language (see also Ca-

diot and Visetti, 2001). This takes us far away from “thought as a formal