Alan M. Turing: dalla Macchina alla morfogenesi - Giuseppe Longo Centre Cavaillès, CNRS - Ens, Paris
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Alan M. Turing: dalla Macchina alla
morfogenesi
Giuseppe Longo
Centre Cavaillès, CNRS - Ens, Paris
http://www.di.ens.fr/users/longo
1Turing and the Foundations of Maths: SOME HISTORY
The « foundational split »:
1 – Geometry and the relation to Physical space:
B. Riemann (habilitation, 1854)
Physics/Geometry: Poincaré, Einstein, H. Weyl … Connes
The issue of measurement, departing from Laplace:
•GDS: the approximated “access” (to space, interval), Poincaré
•Relativity: the invariant speed of light, the space/time correlation,
Einstein
•QM: measurement of conjugated variables, Planck’s h.
2
2Turing and the Foundations of Maths: SOME HISTORY
The « foundational split »:
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.»
Hilbert, 1899 – 1930:
Potentially Mechanizable Formal Systems (the Decision- problem)
Arithmetic: the core finitistic theory of the countable discrete
(consistency)
Focus on Arithmetic certainty
(forgetting space, geometry, “access” ... Forgetting measurement)
3
3Turing and the Foundations of Maths: SOME HISTORY
The « foundational split »:
1 – Geometry of dynamical systems and the relation to Physical space
2 – The Logical/linguistic Turn
Turing will contribute to both perspectives
4
4Turing's three major papers
The Logical Computing Machine:
"On Computable Numbers with an Application to the Entscheidung-
sproblem", Proc. London Math. Soc. 42, 230-265, 1936.
Imitating human intelligence:
"Computing Machines and Intelligence", Mind, LIX, 1950.
Modeling 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.
1952: addressing life not by “discrete coding”, but as “continuous
deformations” … the opposite of the “DSM that iterates” 5
5"On Computable Numbers with an Application to the
Entscheidungsproblem”, 1936
The invention of (machine) computability by
- Searching for a Negative Result (a well defined non-computable
function)
- The human computer (Hilbert’s potential mechanazability of
hyman deduction).
6
6The Human Computer, '36 - '50
Turing '36: “ ...the computer works by such a desultory manner that
he never does more than one step at a sitting”. p. 22
“We may now construct a machine to do the work of this computer”p. 21
7
7The Human Computer, '36 - '50
Turing '36: “ ...the computer works by such a desultory manner that
he never does more than one step at a sitting”. p. 22
“We may now construct a machine to do the work of this computer”p. 21
Hardware vs. Software
8
8The Human Computer, '36 - '50
Turing '36: “ ...the computer works by such a desultory manner that
he never does more than one step at a sitting”. p. 22
“We may now construct a machine to do the work of this computer”p. 21
Hardware vs. Software
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 9 9Turing '50: The Imitation Game,
From the Logical Computing Machine to the Discrete State Machine
The Physical Machine
10Turing '50: The Imitation Game,
From the Logical Computing Machine to the Discrete State Machine
The Physical 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”
11Turing '50: The Imitation Game,
From the Logical Computing Machine to the Discrete State Machine
The Physical 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.”
12Turing '50: Some pearls in mathematical physics
13Turing '50: Some pearls in 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.”
14Turing '50: Some pearls in 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 … ”
[Prediction: in practice/ in principle, measurement]
15Turing '50: The Brain ?
“The nervous system is certainly not a discrete-state machine ..”
Yet, a game of imitation:
A (digital) computer, a woman, a man …
Are you a man, a woman?
Aim: cheat the examiner:
for example: Add 34957 to 70764;
answer: 105621
To be like (to imitate) a “machine”, a 1912 - 54
“woman”, a “man” …
16« I believe that in about fifty years' time it will be possible to programme computers … to make them play the imitation game so well that an average interrogator will not have more than 70 per cent. chance of making the right identification after five minutes of questioning. The original question, 'Can machines think ? ' I believe to be too meaningless to deserve discussion » (Turing 1952, p. 442)
Turing ‘52: Morphogenesis
Physics for Biology
18
18Turing ‘52: Morphogenesis
J. D. Murray (1990) on Turing’s ‘52 paper:
“One of the most important papers in theoretical biology
of this century.”
“… it took the mathematical world more than 20 years to
realise the wealth of fascinating problems posed by his
theory. What is even more astonishing is that it was closer
to 30 years before a significant number of experimental
biologists took serious notice of its implications and
potential applications in developmental biology, ecology
and epidemiology.”
19
19Turing ‘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”
20
20Exponential 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
21
21Exponential 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]22 22
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
23Catastrophic 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.
Non-linearity (Instability, Fluctuations, “critical transitions”…)
“determine” their forms.
“Just” a material (hardware) dynamics
24 of forms: ...25
26
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 (approximation, measurement)
“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” ('52, p. 42-3).
(Today’s tentative extensions to cell differentiations: Gordon, 2011)
27Turing and the chromosomes, 1952
“It is suggested that a system of chemical substances, called
morphogens, reacting together and diffusing through a tissue, is
adequate to account for the main phenomena of morphogenesis.
… The purpose of this paper is to discuss a possible mechanism
by which the genes of a zygote may determine the anatomical
structure of the resulting organism.” p. 37
28Turing and the chromosomes, 1952
Morphogens: “the system to be considered consists of a number of
chemical substances (morphogens) diffusing through a mass of tissue
of given geometrical form and reacting together within it.” (p. 40)
“… each morphogen moves from regions of greater to regions of less
concentration, at a rate proportional to the gradient of the
concentration”. (p. 40)
29Turing and the chromosomes, 1952
Morphogens: “the system to be considered consists of a number of
chemical substances (morphogens) diffusing through a mass of tissue
of given geometrical form and reacting together within it.” (p. 40)
“… each morphogen moves from regions of greater to regions of less
concentration, at a rate proportional to the gradient of the
concentration”. (p. 40)
Genes [as parts of chromosomes]: “the characteristic action of the
genes themselves is presumably chemical.” (p. 38)
“The genes might thus be said to influence the anatomical form of the
organism by determining the rates of those reactions which they
catalyze … they do not diffuse.” (p. 38)
No mention of “coding” nor “program” …
30Turing and the “coding” of the homunculus
Some history (preformation vs morphogenesis):
The (aristotelian) homunculus vs “dynamics of forms”, since
Cuvier (preformationism) vs Geoffroy Saint-Hilaire (morphogenesis)
31Turing and the “coding” of the homunculus
Some history (preformation vs morphogenesis):
The (aristotelian) homunculus vs “dynamics of forms”, since
Cuvier (preformationism) vs Geoffroy Saint-Hilaire (morphogenesis)
Preformation: the coding and instructions, in the chromosomes:
Delbruck 1940s, Schrödinger (“as Laplace’s daimon”, 1944)
The program: “The DNA is … the program for the behavioural
computer of each individual”. [E. Mayr, 1961]
Monod “Le hasard et la nécessité”, 1970:
32
randomness vs. determination (the program) = Laplace (1820)Turing and the “coding” of the homunculus
Morphogenesis as auto-constitutive dynamics: D’Arcy Thompson,
Waddington (Turing’s only references in biology)
• Turing’s correspondence with Waddington on morphogenesis.
• Turing against “predefined design” (Hodges, 1983 (Gandy));
Morphogenesis: Monod’s “mise en oeuvre d’un projet” (1970).
• Turing against Huxley’s new-synthesis (Darwin’s evolution focusing
only on chromosomes).
1952: The “constitutive dynamics” of organisms as continuous
deformations of (just) hardware (crucial non-linear effects).
33Turing’s descent
Organogenesis, Embryogenesis and Evolution
in terms of non-linear dynamics ‘a la Turing’:
Evely Fox-Keller, 1970
Richard Gordon, 1975 – 2011
Daniel Meinhardt, 1976 – 1997
Vincent Fleury, 1990 – 2012
:
:
[Réne Thom, 1978 – 1990]
34Turing’s Morphogenesis and the Computer
In Turing’s analysis, continuity of models crucially steps in:
- approximation (an open interval of measurement or of the initial/
border conditions)
- various forms of instability, criticality, symmetry breakings …
Key issue: Discret (space-time) dynamics are not an approximation
of non-linear continuous dynamics.
35
35Turing’s Morphogenesis and the Computer
In Turing’s analysis, continuity of models crucially steps in:
- approximation (an open interval of measurement or of the initial/
border conditions)
- various forms of instability, criticality, symmetry breakings …
Key issue: Discret (space-time) dynamics are not an approximation
of non-linear continuous dynamics.
“It might be possible, however, to treat a few particular cases in detail
with the aid of a digital computer.
The essential disadvantage of the method is that one only gets results
for particular cases” (Turing, 1952, p. 72)
36
36Turing’s Morphogenesis and the Computer
In Turing’s analysis, continuity of models crucially steps in:
- approximation (an open interval of measurement or of the initial/
border conditions)
- various forms of instability, criticality, symmetry breakings …
Key issue: Discret (space-time) dynamics are not an approximation
of non-linear continuous dynamics.
“It might be possible, however, to treat a few particular cases in detail
with the aid of a digital computer.
The essential disadvantage of the method is that one only gets results
for particular cases” (Turing, 1952, p. 72)
The discrete is not an approximation of continua:
sensitivity of the dynamics implies divergent trajectories, yet …
37
37Today’s Shadowing Theorem: the “reverse” approximation
Computational problem: the round-off
Shadowing Theorem for hyperbolic dynamical systems (D, f, m)
For any x0 and δ there is an ε such that, for any ε-approximated
(or rounded-off ≤ ε ) trajectory, there is one in the continuum
which goes δ -close to it, at each step.
Informally:
Given a “sufficiently regular” non-linear iterated function system,
any discrete (space-time) trajectory can be actually approximated
by a continuous one (but, in general not the converse!)
Or … there are so many continuous trajectories, that, given a
discrete trajectory, you can find a continuous one which goes
close to it, see:
Pilyugin, S.Yu. (1999). Shadowing in Dynamical Systems.
Lecture Notes in Math. 1706, Springer-Verlag,
38 Berlin.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):
Its “evolution” as continuous deformations of (just) hardware
39Morphogenesis in Embryogenesis
Following Turing, beyond Turing
40Morphogenesis 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.
41Morphogenesis 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.
Better models of dendritic
growth: anysotropy and
noise (Fleury, 1999)
42Morphogenesis 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
43 et al, 2009)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:
44Morphogenesis 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 45
cells.Form organs to species (bauplans and more)
Extensions to Evolution of non-linear dynamics ‘a la Turing’:
Richard Gordon, 1975 – 2011
Daniel Meinhardt, 1976 – 1997
Vincent Fleury, 1990 – 2012
:
:
46Form organs to species (bauplans and more)
Extensions to Evolution of non-linear dynamics ‘a la Turing’:
Richard Gordon, 1975 – 2011
Daniel Meinhardt, 1976 – 1997
Vincent Fleury, 1990 – 2012
:
:
1. The dynamicists’ tree
(dynamically determined
trajectories)
From S. J. Gould, Wonderful
Life, 1989
47Morphogenesis in Evolution: well beyond Turing
“Les gènes se servent sur l’étagère de la morphogenèse”
(Fleury, 2011)
The dominating “physical determination” of biological
morphogenesis (very rich : non-linear ...):
D'Arcy Thompson, Waddington, Thom …
48Morphogenesis in Evolution: well beyond Turing
“Les gènes se servent sur l’étagère de la morphogenèse”
(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)
49The Burgess fauna, -500 mlys
5051
Random “exploration” of bauplans, never
incompatible with physical dynamics, but
not determined by them: (classical)
physics only provides constraints.
52Random “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 molecular Randomness 2. Integration+regulation, within the organism and the ecosystem … 3. “Bio-resonance” (Buiatti, Longo, ‘12) 53
2. Gould S. J. et al.: Yes, todays'
animals derive from a few
1. The dynamicists’ tree
bauplans (Darwin), but
(dynamically determined
“specified” after massive
trajectories)
selection of “dynamically
From Gould, Wonderful Life,1989 canalized”, yet random structural
explorations (including of bauplans).
54The 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)
55Some references on Turing
http://www.di.ens.fr/users/longo or Google: Giuseppe Longo
• Hodges, A., 1983, Alan Turing: the Enigma, London: Burnett; New York:
Simon & Schuster; London: Vintage (1992); New York: Walker (2000).
• Hodges, A., 1997, Turing, a natural philosopher, London: Phoenix; New
York: Routledge (1999)
• Copeland, B. J.(ed.), 2004, The Essential Turing, Oxford: Clarendon Press
• Bailly F., Longo G. Mathematics and the Natural Sciences. The Physical
Singularity of Life. Imperial Coll. Press, London, 2011 (Hermann, 2006).
• Longo G., From exact sciences to life phenomena: following Schrödinger and
Turing on Programs, Life and Causality. In Information and Computation,
special issue, n. 207, pp. 545-558, 2009.
• 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.
56
56Some references
http://www.di.ens.fr/users/longo or Google: Giuseppe Longo
Bailly F., Longo G. Biological Organization and Anti-Entropy, in J. of Biological
Systems, Vol. 17, n. 1, 2009.
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. The Inert vs. the Living State of Matter: Extended Criticality, Time
Geometry, Anti-Entropy - an overview. Invited paper, for a special issue of
Frontiers in Fractal Physiology, to appear, 2012. (in print).
Longo G., Montévil M., Kauffman S. No entailing laws, but enablement in the
evolution of the biosphere. Invited Paper, Genetic and Evolutionary
Computation Conference, GECCO’12, July 7-11, 2012, Philadelphia (PA,
USA); proceedings, ACM 2012.
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. 57You can also read
