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date: 24 August 2019

(p. 501) Index

(p. 501) Index

References to figures are in italic type.

Abel, Niels Henrik, 386–7, 400, 458, 459
abstraction, 62, 145
accidental
properties, 68, 493–4
statements, 151
Adanson, Michel, 268
adaptive zone, 278
Adolph Frederick, King of Sweden, 262
Aldrovandi, Ulisse, 261
Alembert, Jean d’, 444, 450
Alexandroff, Paul, 337
algebra, 66–7, 492–3
generality of, 397, 453–7
generic reasoning, 441
linear, 439–40, 462
structures, 224–5
algebraic curves, 300–1
Amaldi, Edoardo, 230–3, 246–9
analogy, 26–7, 129, 286, 301, 342, 449
anatomical, 372, 378
Aristotelian, 31, 129
Maxwell, 348, 352
analysis, 385–6
art of, 83, 90
derivation, 60, 82, 86, 330
anatomy, 21–2, 289–91, 360
Bichat, 360–70
homology, 287–91
Ranvier, 374–80
structure, 287, 288, 293–4, 362, 363, 370–1, 374, 377–80
surgical, 366–7
tissue concept, 364–70
Andronov, Aleksandr, 306–9, 321
anharmonic ratio, 73–6
anthyphairesis, 119, 129–32
Apollonius, 50–1, 57, 71
arbitrary functions, 217–19, 388
description of, 390–1
exemplification of, 391–3
Lagrange on, 395–8
purpose of, 392–4
reference to, 390
arcana, 92
archetypes, 22, 291, 292
Aristotle, 10–11, 18, 20, 121–2, 123
anthyphairesis, 120, 130–1
genera, 113, 260
on generality in mathematics, 127–9
katholou, 125–7
on scientific knowledge, 124–5, 136
zoology and biology, 258–60, 266–7, 287
arithmetic generality, 453–7
complex numbers, 483–4, 485–7
fundamental law of arithmetic, 487
attractors, 315
Banach spaces, 224, 225–7, 236–8, 305–6
Banach, Stefan, 16, 223
doctoral dissertation, 225–7
fixed point theorem, 251
norm, 233–5
beauty, 94–5
Becker, Oskar, 119, 122, 123, 130
Belon, Pierre, 20, [link] , 290
Bernard, Claude, 363, 374, 376–7, 379
Bernoulli, Daniel, 450
Bernoulli, Johann, 331
Bertrand, Joseph, 172, 206, 207–13
Poincaré and, 215–19
Bichat, Xavier, 28–30, 360–80
bilinear forms, 435–8, 454–5, 463–4
binomial nomenclature, 261–2
biology, 470, 471–2
birds of paradise, 273
Birkhoff, George David, 302, 311
Boerhaave, Herman, 263
Bonnet, Charles, 265
Borel, Emile, 216, 392, 404–8
Bowen, Robert, 319
Boyden, Alan, 294
Breger, Herbert, 339, 341–2
British Association for the Advancement of Science, 352
Buffon, Georges-Louis Leclerc de, 19–20, 265–7, 277, 281–2, 290
Needle problem, 210
Burian, Richard, 359, 382
calculus of limits, 184
Canguilhem, Georges, 362
canonical forms, 454, 456–7, 460–2
Cantor, Georg, 14, 175, 201–2, 312, 317–18
Cantor set, 317–18
Carnot, Nicolas, 53, 68n, 74
Cartwright, Nancy, 333, 339–40
case
general, 443–4
particular, 52, 61, 69, 77, 170
catenary, 329–31, 332–3
Cauchy, Augustin Louis, 30–1, 34, 67n, 400–3, 404–5, 435, 447, 449n, 451–3
Cavaillès, Jean, 17, 495
Cavalieri, Bonaventura, 103, 335
Cayley, Arthur, 441–2
celestial mechanics, 183–4, 186–7, 198–201, 213, 310, 314
Lagrange on, 450–1
probability in, 214–17
Villarceau on, 445
cell division, 471
cell theory, 375–7, 381
Cellucci, Carlo, 338–9
Cesalpino, Andrea, 260, 262–3
Chasles, Michel, 4–8, 30–2, 70–1
on ancient geometry, 50–6
Aperçu historique, 47–50
on descriptive geometry, 63–70
on early modern geometry, 56–63
geometric generality and, 70–7
theorem form, 77–85
chemistry, 493
Chevalley, Claude, 304
chimpanzees, 278
Christoffel, Elwin, 436, 438n
circle tangents, 415
cirripeds, 269
cladism, 274, 275, 277–8, 279, 286
class
mathematical, 227
taxonomic, 270–1, 279–80
classification
biological see zoology
of operations, 245
combinatorics, 9, 107
commensurability, 115, 119, 129
complex numbers, 483–4, 485–7
Dedekind, 495–7
properties, 493–5
Comte, Auguste, 369–70, 375, 380n
conciseness, 8–9, 95–8
concrete approaches, 62, 339–40, 351, 373
Condillac, Étienne Bonnot de, 367–8
conic sections, 56–61, 77–85
conjugation, 75–6, 78
connections, principle of, 290, 292, 294
constructivism, 137
contingent relationships, principle of, 37, 63–4, 68, 69, 86
continuity, principle of, 6, 30–1, 37, 54, 66–7, 334, 400–3
continuum, 140–1
converging methods principle, 378–9
cosine function, 96
Cournot, Antoine, 208–9
Cramer, Gabriel, 95
Crew, Francis Albert Eley, 293
cross-ratio see anharmonic ratio
curve-fitting, 155–60
Cuvier, Georges, 20, 268–9, 282, 291, 295, 371
Darwin, Charles, 19–20, 269–71, 292
Daubenton, Louis Jean-Marie, 21–2, 25
de Baggis, H. F., 306
De Beer, Gavin, 294, 295
Dedekind, Richard, 484, 490n, 495–7
Delbrück, Max, 477
demonstration, 10, 113, 115–17, 123–5
Desargues, Girard, 33, 52, 56–63, 65, 70, 71, 77–85
Desault, Pierre Joseph, 366, 370–1
Descartes, René, 5, 26, 32–3, 58, 69, 333, 335
Leibniz and, 333–5
tangents, 414–17
critique of Fermat’s method, 420–2, 427–30
descriptive geometry, 63–5, 86
Desult, Pierre-Joseph, 29, 360–1, 366n, 369, 370
determinants, 99–100, 107
differential equations, 144–5, 300, 331–2, 444–5
Dirichlet, Peter, 387–92, 443
distributive operations, 248
division, 106
DNA, 478
Dobzhansky, Theodosius, 272
d’Ocagne, Maurice, 206
Dorier, Jean-Luc, 228
doubly asymptotic points, 310
du Bois-Reymond, Emil, 391, 393–4, 395–6
Duerden, James Edwin, 293–4
Duhem, Pierre, 4, 27, 345
Duval, Mathias, 375, 378
dynamical systems, 23–5, 301–2
genericity, 317–20
stability, 305–9
(p. 503) Eddington, A. S., 469–70, 473
eidos, 260
electric potential, 345–6
electromagnetism
Lagrangian model, 350–1
Maxwell’s honeycomb, 349–50
elementary divisors theorem, 439, 455–6
embedded generality, 31, 406–8
entrenchment, 151, 157
epistemic values, 2, 7, 34, 39, 91, 434, 442, 489
epistemological value, 7, 10, 21, 33, 36, 39, 51
ergodic theory, 318–21
Euclid, 10, 113–14, 129–30
object domains, 114–16
proportion and, 116–18
reconstructive hypothesis, 118–21
Eudoxus, 122–3
evolution, 475–6
exhaustion method, 55
Faraday, Michael, 347–8, 354
fecundity, 101–3, 347
see also fruitfulness
Fermat, Pierre de, 32–3, 341, 423–4
tangents, 33, 417–20
Descartes’ criticism of, 420–3
universality, 425–7
Flourens, Jean-Pierre, 364, 368–9
force, 347–8, 352
form, 453
bilinear, 435–8, 454–5, 463–4
canonical, 454, 456–7, 460–2
formal logic, 146
fossils, 274
Foucault, Michel, 287, 380, 381
Fourier series, 388, 393
Fréchet, Maurice René, 240, 247–8, 338
Fredholm equations, 251–2
Frobenius, Ferdinand Georg, 460
fruitfulness, 9, 27, 40, 51, 60
see also fecundity
function
arbitrary, 217–19, 388
Cauchy on, 401–3
expressed as power series, 387, 396
fields, 247–8
sets, 226
spaces, 225–6, 234
theory, 389
Galen, 362
Gardies, Jean-Louis, 123n, 131
Garsault, François Alexandre Pierre de, 289
Gauss, Carl Friedrich, 36, 435, 483–4, 484–5, 494
Gaussian integers, 484–5
genera (zoological), 257–8, 261–2
biological concepts, 274–5
Buffon, 265–7
gradist concept, 272–3
Homo, 276–8
Lamarck and Cuvier, 267–9
Linnaean, 262–4
phylogenetic concept, 274
general case, 443–4
general entities, 18–21, 24, 39, 365
general solutions, 198–9
generality, 1–3, 449–53
algebraic, 397, 453–7
arithmetic, 453–7
biology, 475–6
continuity and, 400–3
embedded, 31, 406–8
Leibniz’ principle, 8
physical science, 143–5, 469–70
rigor and, 386–9
generalization, 335
biological, 359–60
by abstraction, 145–6
Leibnizian analysis, 335
generic reasoning, 34, 438–43, 441, 448, 457
genericity, 23–4, 303–4
dynamical systems, 305–9
homoclinic tangencies, 316–17
genetics, 293–5
genos, 260
genres, 10, 18–20, 65–6, 266
genus, Aristotle on, 124–5
Geoffroy Saint-Hilaire, Etienne, 20, 290–1
geometric curves, 26, 32–3
geometry, 6–7
ancient, 50–6
Euclid, 113–14
proof in, 54–5
descriptive, 63–5
early modern, 56–63
Kummer on, 492–3
Gergonne, Joseph Diaz, 86n, 192n
Gesellschaft Deutscher Naturforscher und Ärtzte, 2n
Gessner, Conrad, 260–1
Gnomon of the Zhou, 10
Goodman, Nelson, 148, 150–62
grades, 272
gradism, 272–3, 278
Grandi, Guido, 334
greatest common divisor (GCD) methods, 459, 497
group theory, 341, 453, 457
Gyldén, Hugo, 184, 202–3, 215–16, 219
Hacking, Ian, 158
Haller, Albrecht von, 28, 361, 364, 369, 381
Hamburger, Meyer, 447
harmony, 91–4
Hawkins, Thomas, 15, 434, 441–3
heat flux, 346
Heath, Thomas Little, 114, 115, 133–4
Hennig, Willi, 274, 275
Hermite, Charles, 435, 448, 453
Hilbert, David, 394, 497–8
Homo, 276–8
homoclinic points, 310–12, [link]
homoclinic tangencies, 316–18
homogeneity law, 141
homographic deformation, 70
homology, 20, 286–7
anatomy, 287–91
evolutionary theory and, 292–3
genetics, 293–5
honeycomb, 349
Hopf, Heinz, 337–8
Hullett, James, 152, 158, 161
Hunter, John, 364
Huxley, Julian, 272
ideal
elements, 69n, 497–8
models, 27
cell, 35
fluids, 347
objects, 258–9
induction
empirical, 146–7
Goodman’s New Riddle, 151–3, 158
mathematical, 136, 137–8, 142–3
physical, 136, 148–9, 152–3
infinite
sequences, 121
sets, 139
integral invariants, 185
integration, 144
interpolation, 136, 150
induction and, 152–4
intuition, geometric, 392
invariant factors, 464
invention, 98–9
involution, 52, 75–8
irrational numbers, 484
jasmine flower, 421–2, [link]
Jordan, Camille, 3, 33, 34, 398–400, 446–7, 457–8, 458–9
on bilinear forms, 435–8, 463–4
Kant, Immanuel, 334, 336
katholou, 125–7
see also universal
Kelvin, William Thomson, Lord, 37, 345–6
Kolmogorov, Andreï, 25, 318–20
Kolmogorov–Arnold–Moser theory, 320
Kries, Johannes von, 217–18
Kronecker, Leopold, 3–4, 436–8, 453–5, 458–9, 464, 497
invariant factors, 464
on Weierstrass’ theorem, 438–43
Kuhn, Thomas, 38
Kummer, Ernst, 36, 487–91
analogy, 491–2
complex numbers, 485–7
ideal numbers, 488–91
Kuratowski, Casimir, 338
Lagrange, Joseph-Louis, 7n, 30, 72, 197–8, 386, 395–8, 444
celestial mechanics, 448–9
electrodynamics, 350–1
on generality, 450
Lakatos, Imre, 192, 193n, 196
Lamarck, Jean-Baptiste, 268, 282
Lankester, Edwin Ray, 292
Laplace, Pierre-Simon, 183–4, 441, 442, 450, 453
Laugwitz, Detlef, 339
law
of formation, 9, 91, 94, 98, 105
of nature, 473–5, 476
lawlike hypotheses, 151–2
Lebesgue measure, 321
Lefschetz, Solomon, 306, 337–8
Leibniz, Gottfried, 8–9, 25–6, 62, 90–1, 341, 374–5
on analysis, 333–7
beauty, 94–5
canon of division, 106
on the catenary, 329–33
conciseness/simplicity, 95–8
harmony, 91–4
notation for determinants, 99–100
utility or fecundity, 101–5
linear
algebra, 439–40, 462
operations, 245–50
systems, 228
Linnaeus, Carl, 19, 21, 261, 262–5, 276–7, 279–80, 280–1, 371
logic, 146, 342–3
London Mathematical Society, 352
Lorenz attractors, 315
Louisa Ulrica, Queen of Sweden, 262
(p. 505) magnitude (Euclidean), 114, 130–1
Malacarne, Vicenzo, 372
many-bodies problem, 450–1
materialism, 342
matrix theory, 442, 462
Maxwell, James Clerk, 4, 26–8, 346–9
honeycomb model, 348–9
illustrative thinking, 353–4
ulterior reflections, 351–3
Mayr, Ernst, 272–3, 275
Mazur, Stanislaw, 238
measure, 119, 128, 132–3
mechanics, 12
mechanics see celestial mechanics
mechanics
generality in, 469–70
Poincaré on, 143–5
probability in, 216
Meigen, Johann Wilhelm, 276
membranes, 371, 381
Mersenne, Marin, 59, 60, 82, 421
method, 5
of exhaustion, 55
of tangents, 56, 413–30
of transmutation of figures, 63–5
microscopy, 372, 374, 375–6
mind, power of, 141–3
Mittag-Leffler, Gösta, 182, 190, 194, 197, 202
modus ponens, 139
Monge, Gaspard, 63–5, 86
Morgagni, Giovanni Battista, 372
Morse, Marston, 303, 308
Morse-Smale systems, 309, 310–13, 314
mosquitoes, 276
mystic hexagram, 60–1, 81
natural method, 371
natural numbers, 139
Natural System, 271
necessity, 476
neighborhoods, 244
neurons, 381
Newton, Isaac, 62, 81
Nine Chapters on Mathematical Procedures, 10
nomenclature (zoological), 261–2
normal forms, 454–5
notation, 9, 98–101
numbers
complex, 483–7, 493–7
irrational, 484
natural, 139
objects, Euclidean, 114–16, 116–18
openness, 237, 242
operations, 244, 246
Owen, Richard, 291, 292
Palis, Jacob, 315, 320
Pappus of Alexandria, 51, 52, 53, 74, 77–8
paradigms, 15, 40, 122, 329
particular cases, 52, 61, 69, 77, 170
particular solutions, 199
Pascal, Blaise, 59–60, 79, 81, 83–5, 367–8
pattern cladistics, 275
Peano, Giuseppe, 17, 228–30, 232–3
Peixoto, Mauricio, 305–8, 320, 321
periodic solutions, 185
permanent properties, 6, 493–5
permutability, 116
persistence, 312, 317–18, 320
philosophy, 342–3
Phragmén, Lars Edvard, 182, 189
PhyloCode, 278–9
phylogenies, 276
physics, see also celestial mechanics
physiology, 28, 361–4, 376–9
Pincherle, Salvatore, 230–2, 246, 249
Pinel, Philippe, 364
planetary motion see celestial mechanics
Plato, 18, 128, 130, 258–60
Platonism, 470
Pliny the Elder, 260
Poincaré, Henri, 11, 13, 135–6, 300, 310, 314–15, 408
arbitrary functions, 217–19
on generality, 136–45
interpolation, 147–50
on probability, 176–8, 205–7, 408
recurrence theorem, 170–2
vocabulary, 173–6
on rigor, 385–6, 403–4
Poincaré section, 310
Poncelet, Jean-Victor, 6, 30–1, 37, 48, 66–7
Ponder, Eric, 472
Pontryagin, Lev, 306–7, 309
power series, 100–1, 387, 396
pre-classification programs, 308
predictive generality, 135–6, 146–7
Priest, Graham, 159–61
prime numbers, 92–4, [link] , 489
principles
of connections, 50, 290, 292, 294
of contingent relationships, 37, 63–4, 68, 69, 86
of continuity, 6, 30–1, 37, 54, 66–7, 334, 400–3
of converging methods, 378–9
of homography, 70
of negligibility, 470
of sufficient reason, 154–5
of uniformity of nature, 154
of virtual velocities, 72
(p. 506) probability, 176–7, 197–201, 202–3
Bertrand on, 207–13
in celestial mechanics, 214–17
physics, 216
process cladistics, 275
Proclus, 122
proof
analysis, 389
Chasles on, 68–9
geometric, 54–5, 71
proof-generated concept, 194
properties
accidental, 493–4
intrinsic, 205
permanent, 6, 68–9, 493–4, 498
proportion, Euclid, 116–18
propositions, 50–1
Putnam, Hilary, 2n
Pythagoras’ theorem, 169–70
Quetelet, Adolphe, 71–2, 208–9
quinarianism, 269
Quiquet, Albert, 205
Ranvier, Joseph-Louis, 30, 360, 365, 374–80
Rashevsky, Nicolas, 35, 471, 476–7
Ray, John, 261, 266
reasoning, 137–42
general, 15
generic, 34, 438–43, 448, 457
geometrical, 64, 65, 67, 86
recurrence theorem, 170–2
changes between formulations, 196–7
exceptional trajectories, 172–6, 194–6
non-constructive proof, 203–5
Poincaré’s error in proof, 189–90
proofs, 178–81
reflexivity, 237
relevance, 307, 394
Renaut, Joseph Louis, 374, 378
Riemann, Bernhard, 389, 392
Riesz, Frigyes, 17, 233–5, 248
rigor, 386–9, 403–4
Abel on, 386–7
Roberval, Gilles de, 33, 421, 424
Robin, Charles, 375n, 376, 378
Ruelle, David, 315n, 319
Saint-Vincent, Grégoire, 20, 55
Schwartz, R., 152, 158, 162
scientific culture, 2–3
scientific knowledge, 124–5
secular equation, 439–40, 457–8
secular inequalities, 447–9, 450
set theory, 201–2
simplicity, 2n, 8, 34–6, 48, 51, 73, 454, 456
Leibniz on, 95–8, 334
Poincaré on, 152, 154, 158
Simpson, George Gaylord, 272, 278
Sinai, Yacov, 319
singularity theory, 303–4, 307–8
Smale horseshoe, 313
Smale, Steve, 308, 312–16, 321
sophism, 342
space (mathematical), 17
Banach, 16, 224, 225–7, 305–6
geometrical, 83, 142, 162
linear, 231–2
probability, 217–20
topology, 233–41
vector, 227–30
species, 18, 265
Speusippus, 128
spherical cows, 468–9
stability, 187, 450–1
confinement, recurrence and, 192–4
stability (structural), 305–9
structure
anatomical, 287, 288, 293–4, 362, 363, 370–1, 374, 377–80
mathematical, 224–5, 228, 230, 237–8, 399, 407–9, 460
of physical theory, 348–50, 352, 355
sufficient reason, 154–5
summum genus, 264–5
surgical anatomy, 366–7
Sylvester, James Joseph, 448, 451, 452
tacit knowledge, 339, 341, 438
tangents, 56, 414–17
Descartes’ method, 414–17
Fermat’s method, 417–20
taxonomy, 260–77
see also zoology
temperature gradient, 346
thickness, 318
Thom, René, 24, 303–4
Thompson, D’Arcy W., 470, 474
Thomson, William (Lord Kelvin), 27, 345–6
three-body problem, 185, 191–2, 198–201
tissue, 361, 364–70
classification, 370–4
topology, 233–40
Tournefort, Joseph Pitton de, 261
tractrix, 333
trajectory, 172–4
non-recurring, 194–5
trajectory surfaces, 184
transcendental curves, 329–30, 335–6
transformations, 58, 453
(p. 507) bilinear forms, 436, 447
dynamical systems, 301, 305, 314
geometrical, 65–6, 70–3, 83, 86
transmutation of figures, 63, 64, 65
transmutation theorem, 101–4
Tschirnhaus, Ehrenfried Walther von, 63, 98, 108
uniformity, 5, 6, 31, 33, 34, 55, 62
unity
by analogy, 128–9
of mathematics, 336, 342, 343
of plan, 287, 290–1
universal
demonstrations, 122–3, 128
knowledge, 125
propositions, 127, 128
theory, 131–2
see also katholou
universality, 62, 420–4
generality and, 427–30
utility, 95, 101–5
values, 38–40, 402
epistemic, 2, 7, 34, 39, 91, 434, 442, 489
epistemological, 7, 10, 21, 33, 36, 39, 51
Veblen, Oswald, 337, 338
vector space axioms, 227
vector spaces, 227–33
vibrating string problem, 443–4, 450
Vicq d’Azyr, Félix, 290, 367
Villarceau, Antoine Yvon, 208, 443–4
viruses, 276
Vitrac, Bernard, 120
Vuillemin, Jules, 337, 342
Weierstrass, Karl, 4, 34, 184, 434, 437–8, 441–3, 451–2, 453, 455
Weinstein, Alexander, 294
Whitney, Hassler, 303
Wiles, Andrew, 341
Wolff, Christian, 334
Zeuthen, Hieronymous Georg, 119
zoology, 19, 258–79