A. M. Turing : from the Logic Computing Machine (1936) to the … · 2014. 4. 4. · 5 5...

A. M. Turing : from the "Logic Computing Machine" (1936) to

the genesis of forms (1952), today

Giuseppe Longo

CNRS - ENS, Paris

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

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SOME HISTORY

The « foundational split »:

1 – Geometry and the relation to Physical space:

B. Riemann (habilitation, 1854; director: Gauss):

« ... the concept of rigid body and of a light ray, non longer need to be valid. Thus ... the foundation of metric relations must be found elsewhere, in cohesive forces that act on it». (cf. Euclid)

H. Weyl (1921): divination …

W. Clifford (1882): «In Physics nothing else takes place, but continuous variations of space curvature».

Local vs global, Euclid's V axiom ... Physics/Geometry: Einstein;Poincaré, Enriques, H. Weyl

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2 – The Logical/linguistic Turn

Frege [FA, 1884]: «The wildest visions of delirium ... remain so long as they refer to intuition, subject to the axioms of Geometry.»

Focus on the certainty of Arithmetic ... Forgetting space and time

Hilbert, 1899 – 1930: Potentially Mechanizable Formal Systems (the Decision- problem) Arithmetic: the core finitistic theory of the countable discrete

(consistency)

A different Philosophy of Knowledge (and of Nature): Arithmetic certainty (no reference to space nor to measurement)

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2 – The Logical/linguistic Turn

Frege [FA, 1884]: «The wildest visions of delirium ... remain so long as they refer to intuition, subject to the axioms of Geometry.»

Focus on the certainty of Arithmetic ... Forgetting space and time

Hilbert, 1899 – 1930: Potentially Mechanizable Formal Systems (the Decision- problem) Arithmetic: the core finitistic theory of the countable discrete

(consistency)

A different Philosophy of Knowledge (and of Nature): Arithmetic certainty (no reference to space nor to measurement)

Turing will contribute to both perspectives ...

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Turing's three major papers

1 - The Logical Computing Machine (LCM):"On Computable Numbers with an Application to the Entscheidung-sproblem", Proc. London Math. Soc. 42, 230-265, 1936.

2 - Imitating human intelligence:"Computing Machines and Intelligence", Mind, LIX, 1950.

3 - Modelling morphogenesis:"The Chemical Basis of Morphogenesis" Philo. Trans. Royal Soc., B237, 37-72, 1952.

From the Logical ('36), later the Discrete State Machine ('50), to the Continuous Dynamics ('52) for the generation of spatial forms.

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Turing '36: The “Human Computer”

Implementing Hilbert's program: - certainty in potentially mechanisable human reasoning;- its “completeness”.

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Turing '36: The “Human Computer”

LCM: “A man in the process of computing a real number” p. 3

“Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book”, p. 19

“The behaviour of the computer at any moment is determined by the symbols which he is observing and his “state of mind” at that moment. … Let us imagine the operations performed by the computer to be split up into “simple operations” which are so elementary that it is not easy to imagine them further divided” p. 20

“the state of mind of the computer” (p.20): Software/hardware

In order to disprove Hilbert's program of complete formal mechanization: How to be a “computer”? (Archimede, Einstein ...)

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The Human Computer, '36 - '50

“ ...the computer works by such a desultory manner that he never does more than one step at a sitting”. Turing '36, p. 22

“We may now construct a machine to do the work of this computer”p. 21

Turing '50:“The human computer is supposed to be following fixed rules; he has no authority to deviate from them in any detail.....”

“He has also an unlimited supply of paper on which he does his calculations. He may also do his multiplications and additions on a 'desk machine', but this is not important.”

“The idea behind digital computers may be explained by saying that these machines are intended to carry out any operations which could be done by a human computer” ’50, p. 436 8

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Turing '50: The Physics of the Imitation Game, From the Logical Computing Machine to the Discrete State Machine

“The digital computers considered in the last section may be classified amongst the ‘discrete state machines’ [DSM] … can in fact mimic the actions of a human computer very closely”

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Turing '50: The Physics of the Imitation Game, From the Logical Computing Machine to the Discrete State Machine

“The digital computers considered in the last section may be classified amongst the ‘discrete state machines’ [DSM] … can in fact mimic the actions of a human computer very closely”

The Brain ? Beyond Logic

“The nervous system is certainly not a discrete-state machine [DSM]. A small error in the information about the size of a nervous impulse impinging on a neuron, may make a large difference to the size of the outgoing impulse ….”

“In the nervous system chemical phenomena are at least as important as electrical.”

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Turing '50: Some (remarkable) mathematical physics

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Turing '50: Some (remarkable) mathematical physics

“In a DSM, given the initial state of the machine, it is always possible to predict all future states. This is reminiscent of Laplace's view.”

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Turing '50: Some (remarkable) mathematical physics

“In a DSM, given the initial state of the machine, it is always possible to predict all future states. This is reminiscent of Laplace's view.”

“The system of the 'universe as a whole' is such that quite small errors in the initial conditions can have an overwhelming effect at a later time.The displacement of a single electron by a billionth of a centimetre at one moment might make the difference between a man being killed by an avalanche a year later, or escaping. It is an essential property of the mechanical systems which we have called ‘discrete state machines’ that this phenomenon does not occur. Even when we consider the actual physical machines instead of the idealised machines … ”

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Turing ‘52: Morphogenesis

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Turing ‘52: Morphogenesis

A model of morphogenesis by “action/reaction/difusion”:

- a set of partial differential equations describing a continuous system (tissue – medium -, space, time …)- (the linear approximation of) a dynamical system highly sensitive to initial conditions (“the exponential drift”, p. 43).

“This model will be a simplification and an idealization, and consequently a falsification.” Not an “imitation”

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Exponential drift

“The investigation is chiefly concerned with the onset of instability”

“Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances” p. 37

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Exponential drift

“The investigation is chiefly concerned with the onset of instability”

“Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances” p. 37

“… the presence of irregularities, including statistical fluctuations in the numbers of molecules undergoing the various reactions, will, if the system has an appropriate kind of instability, result in this homogeneity disappearing”. p. 42.

“Thus there is an exponential drift away from the equilibrium condition. It will be appreciated that a drift away from the equilibrium occurs with almost any small displacement from the equilibrium condition”. p. 43 [Gordon et al.: unstable equilibrium]

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Catastrophic instability

“ …some qualitative conclusions about the effects of non-linear terms.

… it would result in the amplitude becoming infinite in a finite time. This phenomenon may be called 'catastrophic instability'.....”

(this may lead to halt the growth) p. 58-59

“The set of reactions chosen is such that the instability becomes 'catastrophic' when the second-order terms are taken into account, i.e. the growth of the waves tends to make the whole system more unstable than ever”. p. 64

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Catastrophic instability

“ …some qualitative conclusions about the effects of non-linear terms.

… it would result in the amplitude becoming infinite in a finite time. This phenomenon may be called 'catastrophic instability'.....”

(this may lead to halt the growth) p. 58-59

“The set of reactions chosen is such that the instability becomes 'catastrophic' when the second-order terms are taken into account, i.e. the growth of the waves tends to make the whole system more unstable than ever”. p. 64

In general: differential equations for spread of morfogen in a ring produce standing wave forming a whorl.

“Just” a material (hardware) dynamics of forms: ...

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Turing’s Morphogenesis: key aspects

1 – The role of Instable Equilibria:Instabilities in action-reaction-diffusion processes lead to differentiation of spatial patterns by symmetry breakings

2 – The role of randomness:

Initial random concentration of chemical morphogens are “amplified” by the dynamics:

E. g. two cells, with nearly the same amount of a morphogen inside, end up, by proliferation, with very different concentrations

“This breakdown of symmetry or homogeneity may be illustrated by the case of a pair of cells originally having the same, or very nearly the same, contents … [yield] an exponential drift away” (1952, p. 42-3).

(Today’s tentative extensions to cell differentiations: Gordon, 2011)

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Summary on Turing: from Logic to the DSM to Morphogenesis

1936: The Logical Computing Machine

Key mathematical distinction:

software / hardware (the instructions / the paper)

1950: Physically, a (laplacian) Discrete State Machine vs.

unpredictable (continuous) dynamics (the Universe, the Brain)

1952: A continuous dynamics of forms (crucial non-linear effects and the role of measurement – catastrophic instability, exponential drift):

Its “evolution” as continuous deformations of (just) hardware

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Morphogenesis in Embryogenesis

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Following Turing, beyond Turing

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Morphogenesis in Embryogenesis

Meinhardt (1976, 1997), variants of Turing’s equations:autocatalitic production of a substance u, an activator, v, of u in a field f.

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Morphogenesis in Embryogenesis

Meinhardt (1976, 1997), variants of Turing’s equations:autocatalitic production of a substance u, an activator, v, of u in a field f.

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Better models of dendritic growth: anysotropy and noise (Fleury, 1999)

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Morphogenesis in Embryogenesis

Formation of the vascular tree (Honda et al, 1997; see Fleury, 1999):

More refined analysis, several different stages:

1 - “Plenary plexus” (a mass) of very thin capillaries, by percolation of small blood islands, randomly distributed.

2a - Percolation without sprouting (arterial tree of chick embryo)

2b - Sprouting (emerging from existing vessels: adult wound healing)

“The flow is an essential feature for the formation of the large scale features of the vascular system … not taken into account by the RD (action/reaction/diffusion) models” (Fleury, 1999)

Formation of lungs: forced “respiration” at 1/3 of pregnancy (Champagnat et al, 2009)

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Morphogenesis in Embryogenesis

Vascular system, lungs, mammary glands ...

Following Turing, but well beyond Turing, “deterministic continuous dynamics” soundly model their genesis.

Physics dominates in the morphogenesis of organs where exchange or production of energy and/or matter:

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Morphogenesis in Embryogenesis

Vascular system, lungs, mammary glands ...

Following Turing, but well beyond Turing, “deterministic continuous dynamics” soundly model their genesis.

Physics dominates in the morphogenesis of organs where exchange or production of energy and/or matter:

However, organs are integrated in an organism that regulates them (hormonal cascades, neural system …) and this, since the zygote.

Organs are made out of tissues (matrix, networks), not generated by a flow of inert particles, but by a proliferation with variation of moving cells (ongoing work) STOP?

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Form organs to species (bauplans and more)

Extensions to Evolution of non-linear dynamics ‘a la Turing’:

Richard Gordon, 1975 – 2011Daniel Meinhardt, 1976 – 1997Vincent Fleury, 1990 – 2012::

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Form organs to species (bauplans and more)

Extensions to Evolution of non-linear dynamics ‘a la Turing’:

Richard Gordon, 1975 – 2011Daniel Meinhardt, 1976 – 1997Vincent Fleury, 1990 – 2012::

1. The dynamicists’ tree(dynamically determined

trajectories)

From S. J. Gould, Wonderful Life, 1989

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Morphogenesis in Evolution: well beyond Turing

“Les genes se servent sur l’etagere de la morphogenese” ̀ ́ ̀ ̀(Fleury, 2011)

The dominating “physical determination” of biological morphogenesis (very rich : non-linear ...):

D'Arcy Thompson, Waddington, Thom …

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Morphogenesis in Evolution: well beyond Turing

“Les genes se servent sur l’etagere de la morphogenese” ̀ ́ ̀ ̀(Fleury, 2011)

The dominating “physical determination” of biological morphogenesis (very rich : non-linear ...):

D'Arcy Thompson, Waddington, Thom …

Vertebrates: Tetrapodes, a necessity, as the dynamicists claim ?

Tetrapodes losing podia ?The New Zealand Kiwi losing wings?

Eyes: Amblyopsidae (cavefish) eyes formation stops during embryogenesis, before functioning … “vicariance” (motility of neurons/synapses, neural darwinism)

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The Burgess fauna, -500 mlys

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Random “exploration” of bauplans, never incompatible with physical dynamics, but not determined by them: (classical) physics only provides constraints.

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Random “exploration” of bauplans, never incompatible with physical dynamics, but not determined by them: (classical) physics only provides constraints. Add active proliferation and bio-contingency:1. Quantum+classical molecularRandomness2. Integration+regulation, within the organism and the ecosystem …3. “Bio-resonance” (Buiatti, Longo, ‘12)

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2. Gould S. J. et al.: Yes, todays' animals derive from a few bauplans (Darwin), but “specified” after massive selection of “dynamically canalized”, yet random structural explorations (including of bauplans).

1. The dynamicists’ tree(dynamically determined

trajectories)

From Gould, Wonderful Life,1989

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The End: Challenges for Morphogenesis in Evolution

The challenge:

(see Gould’s analysis of the Burgess fauna and of Precambrian Ediacara fauna)

Reduction of bauplans, yet increasing (number of species) diversity and “complexity”:

• number of “tissues”, • organ connected components, • networks, • countable complexity of interfaces (e.g. fractal dimensions)

(Gould, ‘89, ‘96 …; Bailly, Longo, Montévil, ‘08, ’11, ‘12)39

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Some references on Turing http://www.di.ens.fr/users/longo or Google: Giuseppe Longo

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

On Turing:Longo G., Critique of Computational Reason in the Natural Sciences, In

"Fundamental Concepts in Computer Science" (E. Gelenbe and J.-P. Kahane, eds.), Imperial College Press, pp. 43-70, 2009.

Lassègue J., Longo G., What is Turing’s Comparison between Mechanism and Writing Worth? Longo's invited lecture, "The Turing Centenary Conference (CiE 2012)", Cambridge, June 18 - 23, 2012.

On the genesis of forms in Biology:Longo G., Montévil M. From Physics to Biology by Extending Criticality and

Symmetry Breakings. Invited paper, Progress in Biophysics and Molecular Biology, 106(2):340 – 347, 2011.

Longo G., Montévil M. Randomness Increases Order in Biological Evolution. Invited paper, conference on ''Computations, Physics and Beyond'', Auckland, New Zealand, February 21-24, 2012; LNCS vol. 7318 (Dinneen et al. eds), pp. 289 - 308, Springer, 2012.