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Symétries: Pour une épistémologie aux intérfaces des disciplines Giuseppe Longo CREA, CNRS - Ecole Polytechnique et Cirphles, Ens, Paris http://www.di.ens.fr/users/longo F. Bailly, G. Longo. Mathematics and Natural Sciences. The physical singularity of Life. Imperial College, 2011 (Hermann, Paris 2006).

Symétries: Pour une épistémologie aux intérfaces des ... · Symétries: Pour une épistémologie aux intérfaces des disciplines Giuseppe Longo CREA, CNRS - Ecole Polytechnique

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Symétries: Pour une épistémologie aux

intérfaces des disciplines

Giuseppe Longo

CREA, CNRS - Ecole Polytechnique et Cirphles, Ens, Paris

http://www.di.ens.fr/users/longo

F. Bailly, G. Longo. Mathematics and Natural Sciences. The physical singularity of Life.

Imperial College, 2011 (Hermann, Paris 2006).

Mathematics a science of invariants and invariant-preserving transformations

2

Towards an explicitation of the “principles” underlying the scientific constructions of objectivity

Symmetries

Mathematics a science of invariants and invariant-preserving transformations

Symmetries, a double status: as invariants (e. g., regularities in space)

and as transformations (e. g., preserving regularities)

3

Symmetries (from H. Weyl, 1952)

•  The mathematical (naive) version: a transformation that preserves “some” properties of a figure (those you care of…)

•  … but also an invariant (e.g. a mirror symmetry preserves symmetries)

4

Symmetries (from H. Weyl, 1952)

•  The mathematical (naive) version: a transformation that preserves “some” properties of a figure (those you care of…)

•  … but also an invariant (e.g. a mirror symmetry preserves symmetries)

Symmetries (including translation symmetries):

5

Symmetries (from H. Weyl, 1952)

•  The mathematical (naive) version: a transformation that preserves “some” properties of a figure (those you care of…)

•  … but also an invariant (e.g. a mirror symmetry preserves symmetries)

Symmetries and symmetry breakings:

6

Equilibria of Gods

The Greek notion: balance, equilibrium …

Yet used as “symmetry” as well

Mathematics, in H. Weyl’s terms (1952): The symmetry of an image in space is

“a subgroup of the group of automorphisms”

Back and forwards: Maths, Physics and Epistemology:… An homage to Greece:

7

The constructive content of Euclid’s Axioms

Euclid’s Aithemata (Requests) are the least constructions required to do Geometry [Heath,1908]:

1. To draw a straight line from any point to any point.

2. To extend a finite straight line continuously in a straight line.

3. To draw a circle with any center and distance.

4. That all right angles are equal to one another.

5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles

8

Maximal Symmetry Principles These “Requests” are constructions performed by ruler and compass:

an abstract ruler and compass

1. To draw a straight line from any point to any point.

The most symmetric drawing: any other path would break symmetries (a geodetic)

Cf. Hilbert style’s axiom: “for any two points, there exists one and only one segment…”

In Euclid, existence is by construction, unicity by construction and symmetry (any other path would reduce the plane symmetries) 9

Maximal Symmetry Principles 2. To extend a finite straight line continuously in a straight line.

Preserving symmetries

3. To draw a circle with any center and distance.

The most symmetric way to enclose a point by a continuous line

4. That all right angles are equal to one another.

Equality (congruence) is obtained by rotations and translations (symmetries - automorphisms!)

Note: right angles are defined as producing the most symmetric situations of two crossing lines 10

Maximal Symmetry Principles 5. That, if a straight line falling on two straight lines make the

interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles

In Book 3 this is shown to be equivalent to:

5.Bis You can draw exactly one line by a point to a line on a plane The most symmetric situations:

the two possible negations (non-euclidean geometries) reduce the symmetries, on the euclidean plane (they have less automorphisms)

G. Longo. Theorems as Constructive Visions. Invited Lecture, Proceedings of ICMI 19 conference on

Proof and Proving, Taipei, Taiwan, May 10 - 15, 2009, (Hanna, de Villiers eds.) Springer, 2010. 11

The original evidence …

« The original evidence should not be confused with the evidence of the axioms; because the axioms are already the result of a formation of sense and have this formation already behind them »

[Husserl, The Origin of Geometry, 1933]

Symmetries: rotations, translations …

The original evidence …

« The original evidence should not be confused with the evidence of the axioms; because the axioms are already the result of a formation of sense and have this formation already behind them »

[Husserl, The Origin of Geometry, 1933]

Symmetries: rotations, translations …

There is no space in Eclidean geometry! Nor infinity, but … the « no tickness line »

L’infini dans le tableau et les nouvelles symétrie de l’espace

14

The invention of space

15 Giustiniano, S. Vitale, Ravenna

Lo spazio in Giotto, Assisi, 1297-1300 .

16

Lo spazio in Giotto, Assisi, 1297-1300 .

17

Le débat metaphysique

Saint Thomas et l’infini actuel de Dieu, au delà d’Aristote

Le problème de la grâce de Marie

L’évêque Templier (Paris 1275)

L’infini et les annonciations dans la peinture italienne (Sara Longo, 2012)

G. Longo. L'infini mathématique "in prospettiva" et les espaces des possibles. A paraître dans "Le formalisme en action : aspects mathématiques et philosophiques", (J. Benoist, T. Paul eds) Hermann, 2012 18

Infinity in Mathematics, via Paintings Early perspective in Italian Renaissance:

A. Lorenzetti “Annunciation”, God vs. Mary, 1344

Masaccio …. San Bernardino da Siena (1380-1444) : the Annunciation = l’incommesurabile nel misurabile 19

Infinity in Mathematics, via Paintings Projective Geometry Italian Painting, XV century: Brunelleschi,

L. B. Alberti, Della Pittura, 1435: Infinity “in the painting” (Daniel Arasse, “Histoires de peintures”, 2004; Sara Longo, Thèse Doct., ‘11):

Piero della Francesca: “Annunciations”, God vs. Mary, 1470

20

Infinity in Mathematics, via Paintings

Infinity “in the painting”: Beato Angelico (1400-1455)

21

L’infinito nel quadro: la geometria proiettiva

Piero della Francesca, 1466

Infinity in Mathematics, via Paintings

From infinity “in the painting” to the mathematics of infinity: Piero della Francesca De perspectiva pingendi (~1450)

23

San Sebastiano, Antonello da Messina: “prospettiva”,a “point of view”

24

Wermer (1632-75)

.

25

Infinity and the Geometry of Space

Copernic: the point of view of the Sun (Van Frassen, 1970)

Modern spaces: Descartes spaces + Galileo’s inertia

Desargues: the synthesis

The history of the concepts of infinity and of space cross each other.

The epistemological analyses are (also?) historical analyses … 26

"The powerful dogma of the fundamental break between epistemological clarification and historical explanation, as well as the psychological explicitation in the frame of sciences of mind, the dogma of the break between epitemological origin and genetic origin, this dogma, as long as the concept of "history", of "historical analysis" and of "genesis" are not limited in an inadmissible way, as they usually are, this dogma must be reversed from top to bottom"

[Husserl, The origin of Geometry, 1933].

27

From a Geometry of figures to non-euclidean spaces

1.  Constructions and Proofs in Euclid: Rotations and translations (automorphisms of the plane)

2. Fifth axiom, in space: closure under homotheties local = global structure of space (space of senses = physical space)

These “principles of constructions”, in Euclid, produce objects and give proofs

Non-euclidean geometries will negate 2, --> 28

Modern Geometry: preserving invariance

•  (Riemann-)Klein: a geometry is a set of invariance preserving transformations (the group of automorphisms)

•  Various non-euclidian geometries (commun property): no closure under homotheties (as automorphisms or symmetries)

•  Riemann’s key distinction: global vs. local: •  global (topological, dimension) •  local (metric), no homotheties

Local ≠ global, a fundamental symmetry breaking…

(General) Relativity will unify gravitation and inertia along geodetics in Riemann’s manifolds ….

Euclid, a singularity: Shocking … 29

After the « delirium » of non-euclidian geometry (Frege, 1884, and Hilbert, 1899 - 1922)

.

30

After the « delirium » of non-euclidian geometry (Frege, 1884, and Hilbert, 1899 - 1922)

A royal way out (ideas from Aristotle and others…):

1.  Isolate/propose explicit logical (Frege: contentual) Proof Principles

(e.g. rules for quantifiers, arithmetic induction …) 2. Focus on the concept of number (Arthmetic)

Hilbert’s perspective (formal) Cf. Euclid’s axiom vs. Hilbert’s version: 1.  Trace, construct … rotations, translations … 2.  For any …. There exists …. deduce ….

31

Symmetries and Dualities in Logic (or the Category Theoretic interpretation of intuitionistic misteries)

.

32

Symmetries and Dualities in Logic (or the Category Theoretic interpretation of intuitionistic misteries)

Diagrams in Categories: Equivalently defined by equations? Missing the point….

33

Symmetries and Dualities in Logic (or the Category Theoretic interpretation of intuitionistic misteries)

Diagrams in Categories: Equivalently defined by equations? Missing the point….

The job of the category-theorist: pull the symmetries out of the equations

Example: A and B (constructive logic conjunction “and”)

Corresponds to (is interpreted by): Cartesian product in categories 34

C

AxBA B

f gh

pA pB

Definition: In a category, the cartesian product of A and B is an objet A×B, with two morphisms pA: A×B ⇒ A , pB:A×B ⇒ B, such that for any f : C⇒ A and g : C⇒ Β, there exists exactly one h : C⇒ A×B such that the following diagram commutes:

A few equations or

35

or: Diagrams matter because of the symmetries they display

36

or: Diagrams matter because of the symmetries they display

An application: Duality as symmetry transformation: coproduct, the dual diagram of the product as intuitionistic disjunction (or)

product (and) co-product (or)

37

C

AxBA B

f gh

pA pB

C

AA+BA B

f g

q qB

h

The categorical insight into quantifiers by dualities (adjonctions)

Adjonction: Fonctors: F: C →D, G: D → C

And a family of bijections (a categorical duality or ‘symmetry’):

homD(FY,X) ≈ homC(Y,GX)

F ⎯| G

There exists…. For all… ∃…∀… the “logical meaning”?

The categorical insight into quantifiers by dualities ���(adjonctions)

There exists…. For all… ∃…∀…

A simple concept : the diagonal functor Δ

Left and right adjoints to the diagonal functor  (Lawvere, Grothendiek):

∃  |⎯ Δ ⎯| ∀ (…Girard’s LL, LS)

Lambek J., Scott P.J. Introduction to higher order categorical logic, Cambridge University Press 1986.

Asperti A., Longo G. Categories, Types and Structures, M.I.T. Press, 1991.

Towards Physics

40

Construction Principles vs. Proof Principles

Mathematics ⏐ Physics ---------------------------------------------------------------------------------

Construction principles: symmetries, (well-)ordering … (same)

--------------------------------------------------------------------------------- Logico/formal proofs ⏐ Empirical proofs

Knwoledge is in the “circulation” between the two

This distinction (and their interplay): a leading theme in Bailly-Longo’s Book 41

Archimedes’ Equilibrium Postulate

At equilibrium:

“Two weights on a balance are in equilibrium if and only if

for reasons of symmetry no rotation occurs”

More generally:

“Magnitudes are in equilibrium at distances reciprocally proportional to their weights”

42

From Archimedes to … XXth century Physics

Back and forth between Mathematics and Physics…

How to relate symmetries and physical “laws” ?

43

The geodetic principle From Hamilton to Noether and H. Weyl (in two pages)

Beyond finalism: “the light ray, a falling stone … aim at their targets”…

NO: they preserve the tangent of the movement (momentum): the lagrangian version (a consequence of Noether’s theorems)

(Hamilton’s) least action principle (action = energy×times): extremize the “action functional”

More formally: minimize a measure (length, surface..) in a suitable metric space (even an Hilbert Space - Schrödinger’s equation)

That is, it defines the trajectories as specific (unique) by minimizing the Lagrangian action (the time integral of the Lagrangian) 44

The geodetic principle and symmetries

Noether's theorems [Noether, 1918] pull out of the equations the continuous transformation of symmetries which preserve the equations of movement.

Formally: “If a Lagrangian is invariant under an n-parameter continuous transformation (in the sense that the Lagrangian function is invariant), then the theory possesses n conserved quantities”

To each of these transformations corresponds conserved physical quantities: the observables, as invariants.

(e.g. invariance w.r.t. translation in space ≈ conservation of the kinetic moment; invariance w.r.t. translation in time ≈ conservation of energy

[van Frassen, 1994; Bailly, Longo, 2010, ch. 4])

45

Some consequences of the geodetic principle

The conceptual priority of the least (or stationary) action in the intended space (also very abstract Hilbert…):

A radical change in the notion of “Law”: A physical law is the expression of a geodetics

in an adequate space and with its metric.

The geometrization of Physics and the physicalization of Geometry

(Riemann-Einstein; Poincaré for classical dynamics)

46

Summarizing by a “geometric rephrasing” •  Re-read Euclids axioms as symmetries (Greek core)

•  Dualities (as symmeries) in Logic and Category Theory

•  Conservation properties as symmetries Example: the optical principle of the shortest path; Note: Euclid’s straight line = a light ray

47

Summarizing by a “geometric rephrasing” •  Re-read Euclids axioms as symmetries (Greek core)

•  Dualities (as symmeries) in Logic and Category Theory

•  Conservation properties as symmetries Example: the optical principle of the shortest path; Note: Euclid’s straight line = a light ray

More: the notion of generic object: The mathematical and the physical “object” are invariant

(symmetric/interchangeable) w.r. to the intended law: they are theoretical and/or empirical invariants

(Galileo’s falling stone or Einstein’s planet or a photon…) 48

Foundational challenges in Biology

49

“Methodological” introduction

The physicalist dogma: « life can be explained on the basis of the existing laws of Physics » [Perutz, 1987].

(Galileo, Einstein, Bohr ?…) Do we want to replace it by the opposite dogma?

No, but aim at “unification” (not a priori “reduction”…) of possibly autonomous theories of different phenomenal levels.

Physics: “unifications/reductions”: •  Sub-lunar/supra-lunar (Newton) •  Thermodynamic/statistical physics (Boltzmann) •  Relativistic/Quantum fields … 50

Fundamental = Elementary ?

The “alphabetic” principles: The world is made out of “solid bricks”, one on top of the other… The organism is made out of molecules… a big bag…

Galileo and Einstein’s fundamental theories: no atoms, no quanta (later: unification) (in Logic: a predicativist fashion)

51

Fundamental = Elementary ?

The “alphabetic” principles: The world is made out of “solid bricks”, one on top of the other… The organism is made out of molecules… a big bag…

Galileo and Einstein’s fundamental theories: no atoms, no quanta (later: unification) (in Logic: a predicativist fashion)

Moreover:

Elementary = simple ? No:

Elementary and complex (Biology + Q. Phys) 52

More foundational challenges in Biology

•  Mathematics as a science of invariants and invariance preserving transformations

•  Computer Science: also a science of iteration (from primitive recursion to portability of software)

•  Biology: a science of structural stability and variability (the main invariant?)

53

Theories in Biology: changing observables and parameters?

From Physics to Biology by “Conceptual Dualities”

(or: changing principles by dualities or conceptual symmetries)

[Bailly, Longo, 2006-7-8-9-10]

54

Genericity in Mathematics

”For all … ” means ”for a generic … ”

The misleading role of (arithmetical) induction

From Euclid to Type Theory…

Longo et al. The Genericity Theorem in Type Theory, 1993 55

Genericity in Mathematics

”For all … ” means ”for a generic … ”

The misleading role of (arithmetical) induction

From Euclid to Type Theory…

The genericity of the physical object (an invariant of the Theory and of Experiments)

+ the mathematical specificity of the “dynamics”

Longo et al. The Genericity Theorem in Type Theory, 1993 56

Dualities in Physics vs Biology

1.  Physics: Specific trajectoires (geodetics) and generic objects

57

Dualities in Physics vs Biology

1. Physics: Specific trajectoires (geodetics) and generic objects Biology: generic trajectories (compatible) and specific objects

58

Dualities in Physics vs Biology

1. Physics: Specific trajectoires (geodetics) and generic objects Biology: generic trajectories (compatible) and specific objects

2. Physics: energy as operator Hf, time as parameter f(t,x) ; Biology: energy as parameter, time as operator “anti-entropy” (dual to entropy) [Bailly, Longo, 2009]

59

Dualities in Physics vs Biology

1. Physics: Specific trajectoires (geodetics) and generic objects Biology: generic trajectories (compatible) and specific objects

2. Physics: energy as operator Hf, time as parameter f(t,x) ; Biology: energy as parameter, time as operator “anti-entropy” (dual to entropy) [Bailly, Longo, 2009]

A radical change: No invariance of the “objects” (no obvious theoretical and empirical

symmetries, see Montévil Thesis for more …) no optimal paths in Evolution nor in Ontogenesis (ongoing work) 60

Methodological (and logical) premises: How to deal with this new observables?

Physical vs. Biological Theories in Bailly-Longo three (correlated) approaches:

Theoretical extensions (in the sense of Logic) of physical theories

61

CONSERVATIVE (?) EXTENSIONS

Examples from Logic: T ⊂ T’ = T+NewAxiom (T’ extends T)

Formal Arithmetic (PA) 1. PA + König’s Lemma (any infinite, finitely branching tree has an

infinite branch) is a strict, conservative extension: it proves more on infinite trees, but no more arithmetic statements.

2. PA + Axiom of infinity = Set Theory (Set) is a strict, non-conservative extension of PA, since Gödel ‘31:

an axiom of infinity allows to prove Consistency of PA (Coher).

Gödel’s Theorem: Set is not conservative over PA (or, PA ⏐-/- Coher ) 62

Physical vs. Biological Theories Ontological (bunches of molecules) vs. Theoretical issue.

What about considering extensions of Physical Theories by proper observables?

-  Critical transition not just on (mathematical) “points” -  Levels of biological organization (anti-entropy) -  Various forms of irreversibility of time (+ a two dimensional time)

Reduction to the physical (sub-)theories? Why not …

In Physics: unification (Newton vs Galileo; Thermodynamics (limit); Relativity/QM …)

Question: conservative extension? 63

Physical vs. Biological Theories in Bailly-Longo three (correlated) approaches:

1 - extended criticality (a physical oxymore), JBS, 16, 2, 2008.

2 - organization (a new observable) as anti-entropy, JBS, 17, 1, 2009.

3 - extra (irreversible) time and two dimensional time (not linear time), with M. Montévil, 2011.

Common point to the approaches in 1, 2 and 3: Strict “Consistent” extensions, in the sense of Logic, compatible with current physical theories (Thermodynamics), but not necessarely reducible:

1: contract the extension of criticality (from interval to point); 2: “=“ instead of “≤” in balance equations (anti-entropy goes to 0); 3: collapse the extra dimension (a time bifurcation).

Question: are they conservative? 64

Aléatoire et irreversibilité du temps, physique/biologique,

comme brisures de symétries

66

Physical Determination (Classical)

Laplace’s view: A) determination implies predictibility

and B) determination implies randomness

[Laplace, Philosophie des Probabilités, 1786]

67

Physical Determination (Classical)

Laplace’s view: A) determination implies predictibility (false: Poincaré, 1890)

and B) determination implies randomness (= determ. unpredictab., '' )

[Laplace, Philosophie des Probabilités, 1786]

Thus, Poincaré broadened determinism by including classical randomness: a fluctuation/perturbation below measure, may yield an observable effect, over time:

“et nous avons un phénomène aléatoire”, [Poincaré, 1902]

68

Physical Determination (Classical)

Laplace’s view: A) determination implies predictibility (false: Poincaré, 1890)

and B) determination implies randomness (= determ. unpredictab., '' )

[Laplace, Philosophie des Probabilités, 1786]

Turing, 1950: From the LCM to the DSM (Discrete State Machine):

my DSM is laplacian!

Turing, 1952: Morphogenesis as a “continuous dynamics”, non-linear

(“exponential drift”, dynamic unpreditability = classical random)

69

Consequences from Physical Determination (Classical)

Laplace’s view: A) determination implies predictibility (false: Poincaré, 1890)

and B) determination implies randomness (= determ. unpredictab., '' )

[Laplace, Philosophie des Probabilités, 1786] [J. Monod, Le hasard et la nécessité, 1970]

Consequences of the Laplacian view: the “DNA is a program” theory, since

any predictable determination is programmable

Randomness as deterministic unpredictability

Classical/Relativistic systems are State Determined Systems: randomness is an epistemic issue

Examples: dies, coin tossing, a double pendulum ... the Planetary System (Poincaré, 1890; Laskar, 1992)… finite

(short and long) time unpredictability (the dies, a SDS, ‘know’ where they go: along a geodetics, determined

by Hamilton’s principle).

Recall Laplace: •  infinitary daimon: OK (over space-time classical continua); •  determination implies predictability (except singularities): Wrong!

Quantum unpredictability as intrinsic indeterminism

Quantum Mechanics is not deterministic: intrinsic/objective role of probabilities in constituting the theory:

•  measure of conjugated variables; •  entanglement, no hidden variables.

Schrödinger’s idea: the equational determination of a “law of probability” (thus back to the indeterministic nature of QM)

Quantum Mechanics: you can’t even think of an infinitary daimon (key difference: measure of conjugate variables).

Recent survey/reflections: [Bailly, Longo, 2011], [Longo, Paul, 2008]

72

Summary: Physical Randomness

1 - Classical randomness = deterministic unpredictability Epistemic, e. g. since Poincaré:

non-linear dynamics and the interval of measure (dice)

2 - Quantum randomness : Objective or intrinsic (to the theory):

•  indetermination (position/momentum) •  entanglement (Bell inequalities: dice vs. entangled quanta)

Different probabilities, different theories of randomness...

In search for a unified theory!

73

Summary: Physical Randomness

1 - Classical randomness = deterministic unpredictability

2 - Quantum randomness : measure

Different probabilities, different theories of randomness...

Yet, common point:

Randomness is correlated to (co-present with) irreversiblity of time (classically: bifurcations … ; quantum: measure)

Cf also Thermodynamics: II principle; diffusion as random paths.

Some more philosophy

1 - Laplace (strong, fantastic) program: the (written) equational determination allows to deduce/predict completely the properties of the physical World

(Newton: “one has to write and solve equations” … )

Poincaré: No, it does not work (1892: deterministic unpredictability)

Some more philosophy

1 - Laplace (strong, fantastic) program: the (written) equational determination allows to deduce/predict completely the properties of the physical World

(Newton: “one has to write and solve equations” … )

Poincaré: No, it does not work (1892: deterministic unpredictability)

2 - Hilbert program: the finite axiomatic writing of Mathematics allows to formally deduce/predict completely the properties of Mathematics

Gödel: No, it does not work (1931: undecidability)

Provable correlations between consequences, as forms of randomness

Towards Biology 3 - Crick, Monod ….

“the finite string of DNA base letters A, C, T, G completely determine embryogenesis, ontogenesis … evolution”

More: « the DNA code ... is the program for the behavioral computer of this individual » (Mayr 1961)

And the two ways interactions DNA – proteome/cell/organisms/ecosystem ?

None (Crick’s central Dogma, 1958), or just « noise », « bad copies »

Randomness (= noise) is “laplacian” (extraneous to determination and theory)

The constitutive role of randomness in Biology

The Physical Singularity of the Living State of Matter

One of the crucial « change of perspective », in Biology:

Randomness implies variability implies diversity An essential component of structural stability

Compare: Randomness as intrinsic to Quantum Mechanics (change measure and the « structure of determination »)

The constitutive role of randomness in Biology

The Physical Singularity of the Living State of Matter

One of the crucial « change of perspective », in Biology:

Randomness implies variability implies diversity An essential component of structural stability

Compare: Randomness as intrinsic to Quantum Mechanics (change measure and the « structure of determination »)

Kupiec, 1983 …. Buiatti M., Longo G. Randomness and Multilevel Interactions in Biology, Ongoing

work.

79

Biological relevance of randomness

Randomness in molecular activities …. more :

Each mitosis (cell division): Asymmetric partitions of proteomes; differences in DNA copies; changes in membranes …

In multicellular organisms: varying reconstruction of tissues’ matrix (collegen structure, cell-to-cell connections)

Not « noise », « mistakes » in polymerase as a Turing’s program, but non-specificity and randomness is at the core not only of variability and diversity (the main biological invariants), but even of cell differentiation (in embrogenesis: sensitivity in a critical transition; e.g. variability in Zebrafisch, N. Peyreiras, ongoing).

80

Which form of randomness ?

81

Quantum Randomness in Biology

Quantum tunneling: non-zero probability of passing any physical barrier (cell respiration, Gray, 2003; destabilizing tautomeric enol forms – migration of a proton: Perez, 2010)

Quantum coherence: electron transport (in many biogical processes: Winkler, 2005)

Proton transfer (quantum probability): RNA mutations (G-C pairs: Ceron-Carrasco, 2009)

REFERENCES IN:

Buiatti M., Longo G. Randomness and Multilevel Interactions in Biology,, Downloadable

82

Classical Randomness in Biology

Non linear affects (molecular level):

•  Molecular enthalpic oscillations

•  Turbulence in the cytoplasm of Eukaryote cells

•  Empowered metabolic random activities by (water) “QED coherence” (Del Giudice, 2005; Plankar, 2011)

(see also J.-J. Kupiec, T. Heams, B. Laforge work)

83

Classical and Quantum Randomness in Biology

Molecular level: non linear dynamics (classical)

and quantum processes superpose

That is: They happen simultaneously and interfere (not analyzed in Physics)

Morover: a quantum effect may be amplified by a (classical non-linear)

dynamics

84

Proper (?) Biological Randomness 1

Randomness within other levels of organization in an organism:

•  [Besides: Molecular activities (classical+quantum randomness)]

•  Cellular dynamics and interactions in a tissue

•  Developmental dynamics (contact inhibition between cells: Soto et al., 1999)

•  Fractal bifurcations (mammary glands development, ongoing work)

85

Proper Biological Randomness 2:

Recall: since Poincaré, randomness as “planetary resonance” … Extended to general non-linear dynamics:

at one level of (mathematical) determination (far from equilibrium: Pollicott-Ruelle resonance, dynamical entropy

in open systems (Gaspard, 2007))

86

Proper Biological Randomness 2:

Recall: since Poincaré, randomness as “planetary resonance” … Extended to general non-linear dynamics:

at one level of (mathematical) determination (far from equilibrium: Pollicott-Ruelle resonance, dynamical entropy

in open systems (Gaspard, 2007))

Bio-resonance (Buiatti, Longo, 2011): Randomness between different levels of organization in an organism:

thus, resonance (as interference) between different levels of (mathematical) determination

The mathematical challenge: the Mathematics (of Physics) does not deal with heterogeneous structures (of determination)

87

Bio-resonance

Physical resonance (at equilibrium / far from equilibrium) is related to “destabilization” (growth of entropy or disorder)

Bio-resonance includes “integration and regulation”, thus it stabilizes and destabilizes

Examples:   The lungs …   In embryogenesis …   In “colonies” of Myxococcus Xanthus, a prokaryote, and

Dictyostelium discoideum, an eukaryote (Buiatti, Longo, 2011)

88

Randomness in critical transitions Life is (not only) a dynamics, a process, but an

extended (permanent, ongoing … in time, space ..) critical transition

(Bailly, Longo, Montévil: book and papers)

A critical interval, not just a (mathematical) point, as in Physics.

Key understanding: continual symmetry changes

In Physics, the determination of trajectories is given by symmetries (the conservation properties)

An biological (ontophylogenetic) trajectory is a cascade of symmetry changes.

The ‘double’ irreversibility of Biological Time

Increasing complexity (Gould) in evolution is the result of a random asymmetric diffusion

F. Bailly, G. Longo. Biological Organization and Anti-Entropy, in J. of Biological Systems, Vol. 17, n.1, 2009.

Evolution, morphogenesis and death are strictly irreversible, but their irreversibility is proper, it adds on top of the physical irreversibility of time (thermo-dynamical):

e. g., increasing order induces (also some) disorder.

Thesis (the role of randomness): a random event is (always) correlated to a symmetry breaking.

One more reason for an intrinsic, proper Biological Randomness. END

Some references (more on http://www.di.ens.fr/users/longo )

•  Bailly F., Longo G. Mathematics and the Natural Sciences. The Physical Singularity of Life. Imperial College Press, London, 2011 (français: Hermann, 2006).

•  Bailly F., Longo G., Randomness and Determination in the interplay between the Continuum and the Discrete, Mathematical Structures in Computer Science, 17(2), pp. 289-307, 2007.

•  Longo G., Palamidessi C., Paul T.. Some bridging results and challenges in classical, quantum and computational randomness. In "Randomness through Computation", H. Zenil (ed), World Sci., 2010.

•  Longo G., Paul T.. The Mathematics of Computing between Logic and Physics. Invited paper, "Computability in Context: Computation and Logic in the Real World ", (Cooper, Sorbi eds) Imperial College Press/World Scientific, 2011.

•  Buiatti M., Longo G. Randomness and Multilevel Interactions in Biology, In progress(downloadable http://www.di.ens.fr/users/longo).

An application to Biological Evolution and “Complexity”

or Randomness contributes to increasing

organization

Bailly F., Longo G. Biological Organization and Anti-Entropy, in J. of Biological Systems, Vol. 17, n. 1, 2009.

91

Evolution and “Complexity” The wrong image (progress?):

92

Growing complexity in Evolution?

Which “complexity”? Evolutionary complexity?

93

However: Gould’s growth of “morphological” complexity [Full House, 1989]

94

However: Gould’s growth of “morphological” complexity [Full House, 1989]

95

Random increase of complexity [Gould, 1989]

Asymmetric Diffusion Biased Increase

96

How to understand increasing complexity?

No way to explain this in terms of random mutations (only):

1.  DNA’s (genotype) random mutations statistically have probability 0 to cause globally increasing complexity of phenotype (examples: mayfly (ephemeral); equus…[Longo, Tendero, 2007])

2.  Darwin’s evolution is selection of the incompatible (“the best” makes no general sense)

3.  Greater probabilities of survival and reproduction do not imply greater complexity (bacteria, … lizard…) [Maynard-Smith, 1969]

Gould's idea: symmetry breaking in a diffusion… 97

Mathematical analysis as a distribution of Biomass (density) over Complexity (bio-organization)

F. Bailly, G. Longo. Biological organization and anti-entropy. J. Bio-systems, 17-1, 2009.

Derive Gould’s empirical curb from •  general (mathematical) principles, •  specify the phase space •  explicit (and correct) the time dependence

Write a suitable diffusion equation inspired by Schrödinger operatorial approach

Note: any diffusion is based on random paths! 98

Morphological Complexity along phylogenesis and embryogenesis

Specify (quantify) Gould’s informal “complexity” as morphological complexity K

K = αKc + βKm + γKf (α + β + γ = 1)

•  Kc (combinatorial complexity) = cellular combinatorics as differentiations between cellular lineages (tissues)

•  Km (phenotipic complexity) = topological forms and structures (e.g., connexity and fractal structures)

•  Kf (functional complexity) = metabolic relations, neuronal and cellular (interaction) networks

Main idea: formalize K as anti-entropy, -S ≠ negentropy (not 0-sum, coding dependent) …. in balance equations… 99

The theoretical frame: analogies .... by a conceptual analogy with Quantum Physics:

In Quantum Physics (a “wave diffusion” in Hilbert Spaces):

•  The determination is a dynamics of a law of probability: ih∂ψ/∂t = h2∂2 ψ/∂x2 + v ψ (Schrödinger Eq.)

In our approach to Complexity in Biological Evolution: •  The determination is a dynamics of a potential of variability:

∂f /∂t = Db∂2f/∂K2 + αbf

What is f ? a diffusion equation, in which spaces?

Random walks … 100

The theoretical frame: dualities .... by conceptual dualities with Quantum Physics:

In Quantum Physics (Schrödinger equation): •  Energy is an operator, H(f), the “main” physical observable.

•  Time is a parameter, f(x, t),

101

The theoretical frame: dualities .... by conceptual dualities with Quantum Physics:

In Quantum Physics (Schrödinger equation): •  Energy is an operator, H(f), the “main” physical observable.

•  Time is a parameter, f(x, t),

In our approach to Complexity in Biological Evolution:

•  Time is an operator, identified with entropy production σ •  Energy is a parameter, f(x, e) (e.g. energy as bio-mass in

scaling-allometric equations: Q = kM1/n)

Our f is the density of bio-mass over complexity K (and time t ): m(t, K) 102

A diffusion equation: ∂m/∂t = Db∂2m/∂K2 + αbm(t,K) (3)

A solution m(t,K) = (A/√t) exp(at)exp(-K2/4Dt)

models Gould’s asymmetric diagram for Complexity in Evolution (diffusion ⇒ random paths…), also along t : (biomass and the left wall for complexity, archeobacteria original formation)

F. Bailly, G. Longo. Biological Organization and Anti-Entropy… Next picture by Maël Montevil:

103 103

(Implementation by Maël Montevil; “ponctuated equilibria” smoothed out)

104

Conclusion

•  Physics and Mathematics share common Construction Principles (symmetries, order… geodetics), diverge as for Proof Principles

•  Some extensions by dualities, as conceptual symmetries (generic vs specific; energy vs time), w. r. to physical theoretizing govern our theoretical approaches in Biology.

•  Extended criticality is (theoretically) based on ongoing symmetry changes - a consequence:

The specificity of the individual organism is the history of the cascade of symmetry changes.

Conclusion: with some humour …

Answers to Leibniz metaphysical question: Why is there Something instead of Nothing ?

106

Conclusion: with some humour …

Answers to Leibniz metaphysical question: Why is there Something instead of Nothing ?

Quantum void, Highly Energetic: •  symmetry breaking: quantum fluctuation … Big-bang: matter

and energy •  Symmetry: anti-matter (negative energy) •  Symmetry breaking: less anti-matter than matter

107

Conclusion: with some humour …

Answers to Leibniz metaphysical question: Why is there Something instead of Nothing ?

Quantum void, Highly Energetic: •  symmetry breaking: quantum fluctuation … Big-bang: matter

and energy •  Symmetry: anti-matter (negative energy) •  Symmetry breaking: less anti-matter than matter

Why is there Life ?

Energy production: entropy 1.  Symmetry: anti-entropy (life as irreversible organization) 2.  Symmetry breaking: less anti-entropy than entropy 108

Some references (papers downloadable) http://www.di.ens.fr/users/longo or Google: Giuseppe Longo

•  Bailly F., Longo G. Mathematics and Natural Sciences. The physical singularity of Life. Imperial Coll. Press/World Sci., 2011 (en français : Hermann, Paris, 2006).

•  Longo G. The Inert vs. the Living State of Matter: Extended Criticality, Time Geometry, Anti-Entropy - an overview. Invited Lecture at the Seoul National University, Seoul and KAIST (South Korea), November, 2010. (to appear).

•  Bailly F., Longo G., M. Montevil. 2-dimensional geometry for biological time. To appear in Progress in Biophysics and Molecular Biology, 2011.

•  Bailly F., Longo G. Biological Organization and Anti-Entropy, in J. of Biological Systems, Vol. 17, n. 1, pp. 63-96, 2009.

•  Bailly F., Longo G. Extended Critical Situations, in J. of Biological Systems, Vol. 16, No. 2, 1-28, 2008.

•  Longo G., Montévil M. From Physics to Biology by Extending Criticality and Symmetry Breakings. Invited paper, special issue of Progress in Biophysics and Molecular Biology, 2011.