Bailly F., Longo G. Mathematics and the Natural Sciences: The Physical Singularity of Life
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January 12, 2011 9:34 World Scientific Book - 9in x 6in mathematics
Index
abstraction, mathematical, 63
allometry, 167
ameba, 61
analogy, 166
anti-foundationalism, 283
approximation
digital, 206, 207
geometric, 208
of measurement, 277
arrow of time, 189
Aspect, 87
asymmetrical reading of the formal
determination, 191, 192
asynchronous processing, 114
Atlan, 127
Automatic Theorem Proving, 78
autop oiesis, 6, 244
axiom of choice, 2
axiomatic system, 69
axiomatic, sense of, 283
Badiou, 28
Bak’s sand pile, 232
Bell inequalities, 87
Bernard, 132
biolon, 257
stability of, 255
Bitbol, 295
Boltzmann constant, 273
Brouwer, 41, 72
Brouwer’s solipsism, 41
calculation, the concept of, 14
Cantor, 2
capacity to forget, 42, 67
Cartesian “Ego”, 38
Cartesian cogito, 82
Cassirer, 24
Category Theory, 34, 40
causal law, 53
causality, 54, 55, 140, 161, 185
epistemic, 188
local/global, 246
material, 189
objective, 188
physical, 182
cause
efficient, 185, 190, 191
material, 185
chaos, deterministic, 262, 264, 286
Châtelet, 58, 76, 82
Chevalley, 194
chirality, 132
Church, 128
circularity, 35
classical ascription of probabilities,
275
classical statistics
Maxwell–Boltzmann, 272
cognition as sequence-matching, 59
cognitive foundations, 20
cognitive relativism, 24
coherent structure, 232, 233
concept, scientific, 46
313
January 12, 2011 9:34 World Scientific Book - 9in x 6in mathematics
314 MATHEMATICS AND THE NATURAL SCIENCES
conceptual renormalization, 145
Connes, 93, 109
construction
of objectivity, 162
the concept of, 12
contingent finality, 137, 212
continuity, topological, 96
continuous line, 68
continuum, 1, 80
continuum hypothesis, 15
correlation length, 62, 242
CPT theorem, 157
critical phenomena, 228
critical system, 142
critical theories, 193
critical transition, 229
criticality, 97
cybernetics, 125
Darwinian evolutionary theory, 121
decidability, 85
definition of mathematics, 64
degeneracy, 213
functional, 213
systemic, 213
determination
formal, 184
objective, 184
determination vs cause, 183
determinations involved in living
phenomena, 169
determinism, 123
digital simulation vs mathematical
modeling, 202
dimension as a topological invariant,
129
dimensional analysis, 129
discrete, definition of, 259
discrete logistic equation, 207
dualism, 173
dynamic system, 87
dynamical sy stem, 121
non-linear, 110
theories of, 112
emergence of complex structures, 231
energy, 133
entropy, 249
negative, 249
EPR argument, 87, 90, 108
equivalence of different formalizations
of computability, 128
equivalence relation, 159, 160
Euclid, 81, 93
Euclid’s axiomatics, 43
evolution of a calculus in physical
terms, 205
evolutionary explanation, 122
extended criticality, 120, 133, 235,
253, 293
Fermat’s last theorem, 77
Fibonacci sequence, 126
finalism, 248
in physics, 250
finite/infinite, 75, 95
flux
of energy, 217
of information, 217
force, 54
formal, 65
formalism, 2, 4, 7, 32, 98
formal incompleteness of, 73
foundation, 8, 36, 57
foundational analysis, aim of, 284
foundations, 21
of mathematics, 5, 17, 42, 56, 69,
81, 197
of physics, 5
Frege, 2, 21, 70
friction with the world, 45
functionalism, 105
functionalist account of cognition, 104
functor, 34
Galileo, 150
gauge theories, 54
Gauss, 69, 77
Gauss’s proof, 78
gene, 127
genesis, 21
geodesic, 137, 234
January 12, 2011 9:34 World Scientific Book - 9in x 6in mathematics
Index 315
geodesic p rin ciple, 149, 151, 154, 156
geometric judgment of well order, 75,
76
geometric vs algebraic app roach, 113
geometrization, 50, 52
geometry
as a human construction, 93
Euclidean, 156
for Husserl, 26
fractal g. of organs, 134
non-commutative, 93, 112
geometry of proof, 66, 84
gesture, 61, 66, 69, 80, 84
mathematical, 58
Girard, 41, 66, 84
Girard’s proof analysis, 41
Gödel, 26, 27, 32, 65
Gödel incompleteness theorem, 287
Gould, 137
Heisenberg algebra, 109
Heyting, 80
hidden variables in quantum
mechanics, 90
Hilbert, 2, 70, 85
homology, 166
homotheties, 22
Husserl, 22, 23, 34, 82
identification between algorithm and
law, 204
imitation, computational, 203
incompleteness, 85, 91
Gödel’s theorem of, 33
of quantum physics, 88
of the formal theory of numbers, 29
theorem, 73
indetermination, 91, 288
biological, 251, 289
quantum, 91
inertia, 150
infinite ML-random sequences, 278
infinity, actual, 94, 96
information, 53, 128, 218
as deformation of geometric
structure, 130
integers, 70, 71
integer, concept of, 76
sequence of, 70
well-order of, 71
integration, 247
intentionality, 119, 130
internal clocks, 136
internal/external, 145
articulation of space, 138
intuition, 79, 80
intuitionism, 41
invariance, 9, 18, 51, 194
gauge, 54
invariant, 4, 22, 29, 34, 42, 67, 68,
120, 138, 153
biological, 219, 253
gauge, 38
irreversibility
of thermody namic, 95
of time, 111, 113
iterative vs processual, 141
Kaluza–Klein theory, 107
Kant, 144
Kastler, 289
Kleene, 128
knowledge, exact, 20
lambda-calculus, 66
landscape paradigm, 156
language, n atural, 51
Laplace, 85, 103
Laplace’s hypothesis, 198
Laskar, 263
law
fundamental l. of physics, 151
logical, 282
notion of, 204
laws of conservation, 187
laws of thought, 21, 69
local vs global, 139
logicism, 2, 4
logicist dogma, 21
Lowenheim, 70
mathematical constitution of
January 12, 2011 9:34 World Scientific Book - 9in x 6in mathematics
316 MATHEMATICS AND THE NATURAL SCIENCES
phenomena, 107
mathematical models of biological
systems, 132
mathematical object, assumption of,
26
mathematical objectivity,
construction of, 18
mathematical stru ctures, 33
mathematization, 32, 49
mean field theories, 135
mechanica universalis, 104
monism, 173, 238
morphism, 34
morphogenesis, 125, 131
Tu rin g’s model of, 203
morphogenetic field, 125
negative result, 286
Nernst p rin ciple, 269
Newton, 49, 226
No ether’s theorem, 193
non-commutativity, 91
non-lo cal effects, 108
non-lo cality, 267, 268
as globality in classical system, 271
non-separability, 267, 268
non-standard analysis, 14
number line, 71, 76
number theory, 79
number, concept of, 72
objective, 47
optimality, 233
order parameter, 98
order, spontaneous formation of, 234
organizational enclosure, 237
orgon, 257
passing from local to global, 229, 230
Pasteur, 132
pathology, causes of, 214
Peano, 70
phase transition, 94, 98, 227
theories of, 110
physical singularity of life, 235
physical vs living phen omena, 173
Piero della Francesca, 76
Planck constant, reduction to 0, 271
Platonism, 65
Poincaré, 66
Poincaré–Berthoz isomorphism, 67
positivism, 5
power laws, 167
examples of, 167
predictability, 198
principle
Curie, 55
logical, 2
of construction, 2, 4, 6, 31, 32, 73,
89
of least action, 149
of optimization, 168
of proof, 2–5, 31, 73, 89
of the shortest path , 149
teleonomic, 6
proof, 84
property, 23, 25, 184
quantum mechanics, 107
quantum non-separability, 91
quantum statistics
Bose–Einstein, 272
Fermi–Dirac, 272
quantum theory of th e hydrogen
atom, 96
quantum vs classical statistics, 272
randomness
biological, 254
classical, 262, 269
epistemic, 264
epistemic notion of, 201
in algorithmic theory of
information, 201
Martin-Löf mathematical
definition, 278
ML-r., 279
objective, 265
objective quantum, 267
quantum r. as epistemic, 270
rationality, 59
realism, 40
January 12, 2011 9:34 World Scientific Book - 9in x 6in mathematics
Index 317
in mathematics, 28
reality, 45
reduction of biology to physics, 226
reference frame, 38, 40
referring function of language, 48, 50
regular vs singular, 142
regulation, 247
relativity, 54, 105
general, 53, 106
special, 105
relaxation processes, 136
renormalization, 95, 228
resistance to changes in viewpoint
due to accountability obligation,
290
philosophical, 289
Riemann, 34, 69, 93
rigor of proof, 63
rule, formal, 2
saccade, 67, 119
scale-invariant parameter, 133
scaling, 167
rules of, 219
self-organization, 231
critical, 232
semantics, 89
sensitivity to the initial conditions,
199
sequence matching, 60
Set Theory, 40, 81
signification, constitution of, 61
singularity, 142
Skolem, 70
SNARC effect, 71
space, 163
as group, 143
as th e form of external sense, 144
spatial dimensions for biology, 171
state, 184
subject, cognitive, 39
subjective, 47
super-symmetric theories, 109
symbol, 63
symmetry, 4, 52, 153, 160, 248
external, 192
internal, 193
symmetry breaking, 33, 52, 98, 155,
193
in critical phenomena, 155
spontaneous, 54, 154
syntax, 89
Teissier, 67
temporality, 35, 138, 142
anticipatory form of, 137
biological, 136
internal, 136
spatialized type of, 141
theorem, 43
proving of, 77
theoretical characterization of
physical objects, 184
thermodynamics, 110
Thom, 126
three-body problem, 85, 110, 113
three-dimensionality of living
systems, 134
time, 163
as semi-group, 143
as th e form of internal sense, 144
bi-dimensionality of, 243
genetic-structural, 117
of clock mechanisms, 115
of constitutive process, 116
phenomenal, 115, 118
relational, 117
top ological evolution, 126
transcendence, 23
transcendental constitution of
mathematical objectivities, 97
transportation process
diffusive, 242
propagative, 242
truth
ontological, 66
transcendental, 39
Tu rin g, 104, 128, 203
Tu rin g machine, 104
uncertainty principle, 91
undecidability, 86, 280
January 12, 2011 9:34 World Scientific Book - 9in x 6in mathematics
318 MATHEMATICS AND THE NATURAL SCIENCES
undecidability theorem, 73
unification, 226, 288
of biology with physics, 122
unpredictability, 280
of a deterministic system, 264
of chaotic dynamic systems, 86
of Poincaré, 86
of quantum theories, 87
van Fraassen, 53, 160
vision as palpation through looking,
81
wave–particle duality, 25
Weyl, 34, 37, 39, 51, 116
Wittgenstein, 22
writing, origins of, 83
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