Bailly F., Longo G. Mathematics and the Natural Sciences: The Physical Singularity of Life
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x MATHEMATICS AND THE N ATURAL SCIENCES
categorization – of concepts as fundamental as those of space and time such
as they are addressed within the contempor ary framework of the natural
sciences.
To conclude and to summarize ourselves, in this book we attempt to
identify the organizing concepts of some physical and biological phenom-
ena by means of an analysis of the foundations of mathematics and of
physics, in the aim of unifying phenomena, of bringing different conceptual
universes into dialog. The analysis of the role of “order” and of symmetries
in the foundations o f mathematics w ill be linked to the majo r inva riants
and principles, among which is the g e odesic principle, which govern and
confer unity to the various physical theories. Particularly, we will attempt
to understand causal structures, a central element of physical intelligibility,
in terms of symmetries and their breaking. The importance of the mathe-
matical tool will als o be hig hlighted, enabling us to grasp the differences in
the models for physics and biology which are pr oposed by c ontinuous and
discrete ma thema tics, such as c omputational simulations.
In the case of biology, being particularly difficult and not as thoroughly
examined at a theoretical level, we will propose a “unification by concepts,”
an attempt which should always precede mathematization. This will con-
stitute an outline for unification also basing itself upon the highlighting
of conceptual differences, of complex points of passage, of technical irre-
ducibilities of one field to another. Indeed, a monist point of view such as
ours should not make us blind: there is no doubt, in our view, that “physical
matter” is unique and that there is nothing else in the universe. Neverthe-
less, the to ols for knowledge which humanity has constructed throughout
history in order to make intelligible natural phenomena, are not unified and
for good rea sons, reasons which are relevant to the very effectiveness of the
construction of the scientific objectivity with regard to different phenome-
nalities. And we cannot claim to unify them by means of a forced method-
ological and technical monism, according to w hich such a mathematical
method or physical theory, constructed around a very specific phenomenal
field, could make us understand everything. Even physics, as a theoretical
construction, is far from being unified: quantum mechanics and general rel-
ativity are not – yet – unified (the notions of quantum and relativistic field
differ). And, as we have mentioned earlier, physicists aim for unification,
not reductio n, meaning tha t they aim for a new notion of a field (either
of physical objects or of space-time) which will unify them, if necessary by
putting each theor y into p e rspective. Such is the case, for ins tance, with
the theory of superstrings which proposes new quantum “objects,” or with
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Preface xi
the radical changes in the very concepts of time and space, such as contem-
plated by non- c ommutative geometr y. And, since Copernicus and Galileo,
Leibniz and Newton – or since always in science – these theories, as for mi-
crophysics today, cons truct their own revolutionary conceptual frameworks,
which are in fact counter-intuitive, as well as constructing, in some cases
afterwards, their own mathematical tools.
In sho rt, we are monists in what concerns “matter” given that unity
can be found, we presume, in this ma tter which enters into friction with,
canalizes and opp oses our experiences and theorizing. However, unity is
not nece ssarily to be found within the existing theories it may be instead
constructed by novel perspectives. The dialog of the disciplines aims to
lead us towards new methodolo gical ideas a nd sy ntheses: coherence in the
intelligibility of the world is a difficult achievement. If it is possible, it is not
to be found, we insist, within the claimed unicity and completeness of the
theories currently available for addressing one or a nother of the historically
given phenomenal fields. It may be constructed instead by means of con-
ceptual bridges, sometimes of punctual reduction, capable of highlighting
points of contact and of friction, by common conceptual foundations. But
also by means of theoretical differentiations or dualities and, if possible, of
simultaneo us changes in perspective within vario us fields. Conceptualiza-
tions which are at the boundary of widespread physical theories are at the
center of our attempts in biology.
1
1
More philosophical reflections as well as technical work, for which this bo ok provides
the conceptual frame, may be downloaded from http://www.di.ens.fr/users/longo/
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Contents
Preface v
1. Mathematical Concepts and Physical Objects 1
1.1 On the Foundations of Mathematics. A First Inquiry . . 7
1.1.1 Terminological issues? . . . . . . . . . . . . . . . . 7
1.1.2 The genesis of mathematical str uctures and of their
relationships – a few conceptual analogies . . . . . 10
1.1.3 Formalization, calculation, meaning, subjectivity . 13
1.1.4 Between cognition and history: Towards new
structures of intelligibility . . . . . . . . . . . . . 17
1.2 Mathematical Concepts: A Constructive Approach . . . . 19
1.2.1 Genealogies of concepts . . . . . . . . . . . . . . . 19
1.2.2 The “transcendent” in physics and in mathematics 23
1.2.3 Laws, structures, and foundations . . . . . . . . . 31
1.2.4 Subject and objectivity . . . . . . . . . . . . . . . 37
1.2.5 From intuitionism to a renewed constructivism . . 40
1.3 Regarding Mathematical Co nce pts and Physical Objects . 44
1.3.1 “Friction” and the determination of physical
objects . . . . . . . . . . . . . . . . . . . . . . . . 45
1.3.2 The absolute and the relative in mathematics and
in physics . . . . . . . . . . . . . . . . . . . . . . . 47
1.3.3 On the two functions of language within the pro-
cess of objectification and the construction of
mathematical models in physics . . . . . . . . . . 48
1.3.4 From the rela tivity to reference universes to that of
these universes themselves as ge nerators of physi-
cal invariances . . . . . . . . . . . . . . . . . . . . 51
xiii
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xiv MATHEMATICS AND TH E NATURAL SCIENCES
1.3.5 Physical causality and mathematical s ymmetry . 52
1.3.6 Towards the “cognitive subject” . . . . . . . . . . 55
2. Incompleteness and Indetermination in Mathematics a nd Physics 57
2.1 The Cognitive Foundations of Mathematics: Human Ges-
tures in Proo fs and Mathematical Incompleteness of For-
malisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.1.1 Introductio n . . . . . . . . . . . . . . . . . . . . . 58
2.1.2 Machines, body, and rationality . . . . . . . . . . 59
2.1.3 Ameba, motivity, and signification . . . . . . . . . 61
2.1.4 The abstract and the symbolic; the rigor . . . . . 62
2.1.5 From the Platonist response to action and gesture 65
2.1.6 Intuition, gestures, and the numeric line . . . . . 69
2.1.7 Mathematical incompleteness of formalisms . . . . 73
2.1.8 Iterations and closures on the horizon . . . . . . . 75
2.1.9 Intuition . . . . . . . . . . . . . . . . . . . . . . . 78
2.1.10 Body gestures and the “cogito” . . . . . . . . . . . 82
2.1.11 Summary and conclusion of part 2.1 . . . . . . . . 83
2.2 Incompleteness, Uncertainty, and Infinity: Differences and
Similarities Between Physics and Mathematics . . . . . . 85
2.2.1 Completeness/incompleteness in physical theories 85
2.2.2 Finite/infinite in mathematics and physics . . . . 93
3. Space and Time from Physics to Biology 101
3.1 An Introduction to the Space and Time of Modern
Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.1.1 Taking leave of Laplace . . . . . . . . . . . . . . . 103
3.1.2 Three types of physical theory: Relativity, quan-
tum physics, and the theory of c ritical transitions
in dynamical sy stems . . . . . . . . . . . . . . . . 105
3.1.3 Some epistemological remarks . . . . . . . . . . . 11 1
3.2 Towards Biolog y: Space and Time in the “Field” of Living
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.2.1 The time of life . . . . . . . . . . . . . . . . . . . 113
3.2.2 More on Biological time . . . . . . . . . . . . . . . 115
3.2.3 Dynamics of the self-constitution of living sy stems 120
3.2.4 Morphogenesis . . . . . . . . . . . . . . . . . . . . 124
3.2.5 Information and geometric structure . . . . . . . . 128
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Contents xv
3.3 Spatiotemporal Determination and Biology . . . . . . . . 132
3.3.1 Biological aspects . . . . . . . . . . . . . . . . . . 132
3.3.2 Space: Laws of sc aling and of critical behavior.
The geometry o f biological functions . . . . . . . . 133
3.3.3 Three types of time . . . . . . . . . . . . . . . . . 136
3.3.4 Epistemological and mathematica l aspects . . . . 139
3.3.5 Some philosophy, to conclude . . . . . . . . . . . . 14 3
4. Invariances, Sy mmetries, and Symmetry Breakings 149
4.1 A Major Structuring Principle of Physics: The Geodesic
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.1.1 The physico-mathematical conceptual frame . . . 151
4.2 On the Role of Symmetries and of Their Breakings: From
Description to Determination . . . . . . . . . . . . . . . . 158
4.2.1 Symmetries, symmetry breaking, and logic . . . . 158
4.2.2 Symmetries, symmetry breaking, and determina-
tion of physical rea lity . . . . . . . . . . . . . . . 161
4.3 Invariance and Variability in Biology . . . . . . . . . . . . 165
4.3.1 A few abs tract invariances in biology: Homology,
analogy, allometry . . . . . . . . . . . . . . . . . . 1 65
4.3.2 Comments regarding the relationships between in-
variances and the conditions of possibility for life 169
4.4 Abo ut the Possible Recategorizations of the Notions of
Space and Time under the Current State of the Natural
Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5. Causes and Symmetries: The Continuum and the
Discrete in Mathematical Modeling 181
5.1 Causal Structures and Symmetries, in Physics . . . . . . 182
5.1.1 Symmetries as starting point for intelligibility . . 186
5.1.2 Time and causality in physics . . . . . . . . . . . 187
5.1.3 Symmetry breaking and fabrics of interaction . . 19 0
5.2 From the Continuum to the Discrete . . . . . . . . . . . . 195
5.2.1 Computer science and the philosophy of arithmetic 196
5.2.2 Laplace, digital rounding, and iteration . . . . . 198
5.2.3 Iteration and prediction . . . . . . . . . . . . . . . 201
5.2.4 Rules and the algorithm . . . . . . . . . . . . . . 203
5.3 Causalities in Biology . . . . . . . . . . . . . . . . . . . . 210
5.3.1 Basic representation . . . . . . . . . . . . . . . . . 211
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xvi MATHEMATICS AND TH E NATURAL SCIENCES
5.3.2 On contingent finality . . . . . . . . . . . . . . . . 2 15
5.3.3 “Causal” dynamics: Development, maturity, aging,
death . . . . . . . . . . . . . . . . . . . . . . . . . 216
5.3.4 Invariants of causal reduction in biology . . . . . 218
5.3.5 A few co mments and comparisons with physics . . 220
5.4 Synthesis and Conclusion . . . . . . . . . . . . . . . . . . 220
6. Extended Criticality: The Physical Singularity of Life
Phenomena 225
6.1 On Singularities and Criticality in Physics . . . . . . . . . 227
6.1.1 From gas to c rystal . . . . . . . . . . . . . . . . . 22 7
6.1.2 From the local to the global . . . . . . . . . . . . 229
6.1.3 Phase transitions in self-organized criticality and
“order for free” . . . . . . . . . . . . . . . . . . . . 231
6.2 Life as “Extended Critical Situation” . . . . . . . . . . . . 236
6.2.1 Extended critical situations: General approaches . 240
6.2.2 The extended critical situation: A few precisions
and complements . . . . . . . . . . . . . . . . . . 24 2
6.2.3 More on the rela tions to autopoiesis . . . . . . . . 244
6.2.4 Summary of the characteristics of the extended
critical s itua tion . . . . . . . . . . . . . . . . . . . 245
6.3 Integration, Regulation, and Causal Regimes . . . . . . . 246
6.4 Phase Spaces and Their Trajectories . . . . . . . . . . . . 250
6.5 Another View on Stability and Variability . . . . . . . . . 2 55
6.5.1 Biolons as attractors and individual trajectories . 255
7. Randomness and Determination in the Interplay between
the Continuum and the Discrete 259
7.1 Deterministic Chaos and Mathema tical Randomness: The
Case of Class ical Physics . . . . . . . . . . . . . . . . . . . 262
7.2 The Objectivity of Quantum Randomnes s . . . . . . . . . 265
7.2.1 Separability vs no n-separability . . . . . . . . . . 267
7.2.2 Possible objections . . . . . . . . . . . . . . . . . 269
7.2.3 Final remarks on quantum randomness . . . . . . 273
7.3 Determination and Co ntinuous Mathematics . . . . . . . 274
7.4 Conclusion: Towards Computability . . . . . . . . . . . . 278
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Contents xvii
8. Conclusion: Unification and Separation of Theories, or
the Importance of Negative Res ults 281
8.1 Foundational Analysis and Knowledge Construction . . . 281
8.2 The Importa nce of Negative Results . . . . . . . . . . . . 285
8.2.1 Changing frames . . . . . . . . . . . . . . . . . . . 289
8.3 Vitalism and Non-Realism . . . . . . . . . . . . . . . . . . 292
8.4 End and Opening . . . . . . . . . . . . . . . . . . . . . . . 297
Bibliography 299
Index 313
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Chapter 1
Mathematical Concepts and Physic al
Objects
Introduction
With this text, we will first of all propos e and dis c uss a distinction, internal
to mathematics, betwee n “construction principles” and “proof principles.”
In sho rt, it will be a question of grasping the difference betwee n the con-
struction of mathematical concepts a nd structur e s and the role of proof,
more or less formalized. The objective is also to analyze the methods of
physics from a similar viewpoint and, from the analogies and differences
that we shall bring to attention, to es tablish a parallel between the foun-
dations of mathematics and the foundations of physics.
When proposing a mathematical s tructure, for example the integer num-
bers or the real numbers, the Cartesian space or a Hilbert space, we use a
plurality of concepts often stemming from different conceptual exper ience s:
the construction o f the integers evokes the g e neralized s uccessor opera-
tion, but at the same time we make sure they ar e “well-ordered,” in space
or time. That is , that they form a strictly increasing sequence, with no
(backward) descending chains. This apparently obvious property (doesn’t
it appear so?) yields this well-ordered “line of integer numbers,” a non-
obvious logical prope rty, yet one we ea sily “see” within our mental space
(can’t you s e e it?). And we construct the rationals, as ratios of integers
modulo ratio equivalence, and then the real numbers, as convergent se-
quences (modulo equiconvergence), for example. The mathematician “sees”
this Cantor–Dedekind-styled construction of the continuum, the modern
real line and continuum, a remarkable and very difficult mathematical re-
construction of the pheno menal c ontinuum. It is nevertheless not unique:
different continua may be more effective for certain applications, albeit that
their structures are loca lly and globally very different, non-isomorphic to
1
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2 MATHEMATICS AND THE NATURAL SCIENCES
this very familiar s tandard continuum (see Bell, 1998). And this construc-
tion is so imp ortant that the “objectivity” of real numbers is “all there,” it
depends solely upon this very constructio n, based on the well-order of in-
tegers, the passage to the quotients (the rationals) and then the audacious
limit operation by Cantor (add all limits of converg ing sequences). One
could say as much a bout the most important set-theoretic constructions,
the cumulative hierarchies of se ts, the s e ts constructed from the empty set
(a key concept in mathematics) by the iterated exponent ope rations, and
so on. These conceptual constructions therefore obey well-explicated “prin-
ciples” (of construction, as a matter of fact): add one (the successor), order
and take limits in space (thus, iteration, the well-order of the integers, limits
of converging s e quences).
But how may one grasp the properties o f these mathematical structures?
How may one “prove them”? The great hypo thesis of logicism (Frege) as
well as of formalism (Hilbert’s program) has been that the logico-formal
proof principles could have completely described the pr operties of the most
impo rtant mathematical structures. Induction, particular ly, as a logical
principle (Frege ) or as a potentially mechanizable formal rule (Hilbert),
should have permitted us to demons trate all the properties of integers (for
Frege , the logic of induction coincided, simply, with the structure of the
integers – it should have been “categorical,” in modern terms). Now it
happens that logico- formal deduction is not even “complete,” as we will
recall (let’s put aside Frege’s implicit hypothesis of categoricity); particu-
larly, many of the integers’ “ c oncrete” properties elude it. We will evoke the
“concrete” results of incompleteness from the last decades: the existence of
quite interesting properties, demonstrably realized by the well-ordering of
integers, and which formal proof principles are unable to grasp. But that
also concerns the fundamental properties of sets, the continuum hypothe-
sis, and of the axio m of choice, for example, demonstrably true within the
framework of certain constructio ns, as shown by Gödel in 1938 , or demon-
strably false in others (constructed by Cohen in 1964), thus unattainable
by the sole means of formal axiomatics and deductions.
To summarize this, the distinction between “construction principles”
and “proof principles” shows that theorems of incompleteness pro hibit the
reduction (theoretical and epistemic) of the former to the latter (or also of
semantics – prolifer ating and generative – to strictly formalizing syntax).
Can we find, this time, and in what concerns the foundations of physics,
some re levance to such a distinction? In w hat would it consist and would