Borovik A. Shadows of the Truth: Metamathematics of Elementary Mathematics
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268 References
J’espère que nos neveux me sauront gré,
non seulement des choses que j’ai ici expliquées,
mais aussi de celles que j’ai omises volontairement,
afin de leur la isser le plaisir de les inventer.
Descartes, Géométrie.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
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ALEXANDRE V. BOROVIK
Index
abacist, 23, 24
dell’Abbaco, Paolo, 23
abstraction, 75
addition, 69
associativity of, 63
commutativity of, 70
componentwise, 8
of heterogeneous quantities, 8
Aharoni, Ron, ix, 7
Akilov, Gleb Pavlovich, 55
Alechina, Na tasha, ix, 168
algebra, 134, 167, 184, 185
linear, 8
Peano, 80, 83
tensor, 9
unary, 83
algorist, 24
algorithm
division, 5
Allegory of the Cave, vii
Altınel, Evren, 66
Altınel, Kuban, 66
Altınel, Tuna, ix, 15, 64, 65
analysis, 187
p-adic, 116
dimensional, 7, 11 , 24, 28, 32, 41
functional, 103
real, 99
Andrews, Paul, x
angle
trisection of, 172
approximation, 102
Archimedes, 83
Arcimboldo, Gius eppe, 2
Arcozzi, Nicola, ix, 168
arithmetic
Dedekind-Peano, 83
floating point, 76
mental, 181
Peano, 44
Arnold, Vladimir Igorevich, 32, 34,
36
Arnoux, Pierre, ix, 22, 67, 176
arrow, 82
article
definite, 13
indefinite, 13
Axiom
of Choice, 12
of Induction, 45, 79
axioms
Dedekind-Peano, 45, 61, 77
Baez, John, x, 1 31
Baldwin, John, x, 63, 64, 72, 136
Ball, Rowena, 32
Baron-Cohen, Simon, 16 4
Bass, Hyman, 95
Baumgartner, Bernhard, ix, 91
Belegradek, Oleg, x, 159
Bell, Alan, 3
Bell, Frances, ix, 161
Belolipetsky, Mikhail, ix, 124
Bogomolny, Alexander, ix, 182
Boroditsky, Lera, 218
Borovik, Anna, ix, 166, 214
bound
uniform Lips chitz, 102
Braden, Lawrence, ix, 173
Brahmagupta, 137
269
270 Index
Buchstaber, Victor Matveevich, 7 1,
72
bundle
jet, 16
principal, 16
vector, 16, 219
C
++
, 45, 77
calculus, 97, 99, 100, 103, 134
lambda, 131, 133–135
Caramuel y Lobkowitz, Juan, 118
Cariolaro, David, ix, 3, 191
carry, 113, 114, 119
category, 82, 221
Cauchy, Augustin-Louis, 98
Cézanne, Paul, 6
Chevallard, Yves, 97
Chorin, Alexander, 35, 36
Church, Alonzo, 132
CINDERELLA, 171
circle, 29, 186
2-coboundary, 115
2-cocycle, 11 4, 115, 119
code
Gray, 56
coefficient
binomial, 81
cofunctor, 221
cohomology, 17, 114
column, 10
combinatorics, 17
Commodore 64, 16 6
commutativity
of addition, 190
compass, 171
compasses, 172
composition
of morphisms, 82
Comte, August, 96, 97
cone, 29
congruence, 215
connective, 13
consequtive “and”, 13
exclusive “or”, 13
inclusive “or”, 13
parallel “and”, 13
constant
dimensionless, 35
gravitational, 12
construction
compasses and ruler, 17 2
elementary geometric, 186
continuity, 98, 100
Conway, John Horton, 12
Cook, Alex R., ix, 114
coset, 115
cosine, 100
counting, 189
sheep, 77
Currie, Iain, ix, 43
curve
level, 37
cylinder, 29
Da Vinci, Leonardo, 213
De Morgan, Augustus, 48, 49
decimal, 111, 113 , 195
decimal place, 115
decimal point, 114
Dedekind, Richard, 45, 98
definition, 17 , 186
formal, 126
recursive, 79
Deloro, Adrien, x, 124, 186
Denizhan, Ya
˘
gmur, ix, x, 166, 221
denominator, 89, 194
common, 87, 88
derivative, 101
covariant, 16
partial, 37
Descartes, 268
Di Scala , Antonio Jose, ix, 166
dice, 170
differentiation, 174, 2 08
digit
significant, 114
units, 172
Dikstra, Edsger, 135
dimension, 11
direction
cardinal, 218
dispensing, 1– 3
displacement, 8
Disquisitiones Arithmeticae, 81
distribution, 5, 189
dividend, 177
divisibility, 4
division, 3, 12, 181
long, 112
by zero, 194
long, 114, 119, 155, 173
with remainder, 4, 5, 155
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
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ALEXANDRE V. BOROVIK
Index 271
without remainder, 177
divisor, 177
on a variety, 220
Dolan, James, 1 31
domain, 120
Euclidean, 158, 160
integral, 160
unique factorization, 158, 160
Doyle, Peter, 12
drag, 30
Froude’s, 31
quadratic, 31, 4 0
Droste effect, 58
duality, 8
of vector spaces, 8
Dudeney, Henry Ernest, 184
Dumas, Alexandre, 19
duodecilion, 125
Dyer-Benn et, John, 54
econometrics, 8
economics
mathematical, 8
Eisenberg, Theodore, x, 68
Eisenkölbl, Theresia, x, 183, 192
electric charge, 11
Elements, 38, 46, 57, 99, 126
energy
flow, 32
kinetic, 33
spectrum, 33
Epstein, David, x, 184, 185
equation, 167
quadratic, 184
simultaneous linear equations,
185
Euclid, 28, 38, 39, 46, 57, 99, 126,
215
Euler, L eona rd, 81, 100
Euler, L eonhard, 117
exponentiation
modular, 54, 55
extremum
constrained, 37
F#, 12
factor
highest common, 89
factorial, 210
factorisation
prime, 4
factorization, 181
Farquharson, Joseph, 77, 81
Feynman, Richard, 78
figure
solid, 29
Fisher, Gwen, x, 71, 183
Flaubert, Gustave, 19, 23
Flick, Ritchie, x, 129
fluid
viscous, 31
flux, 174
folklore
mathematical, 103
Fon-Der-Flaass, Dmitrii Ger-
manovich, xi
force, 8, 11
form
quadratic, 104
formalism, 16
epsilon-delta, 99
Formula
Lagrange Interpolation, 142
formula
Moivre, 100
quadratic, 184
formulae
Weierstrass substitution, 153
fraction, 1, 14, 6 8, 72, 194
in lowest terms, 89
frame, 219
Fraser, Muriel, x
Freudenthal, Hans, 98
Fried, Michael N., x, 190, 20 5
Froude, Willia m, 30
Fuglsig, Nicolai, 50, 5 7
function
analytic, 76
continuous, 97
definable, 99
Euclidean, 158, 160
Euler’s, 54
exponential, 97, 191
hypergeometric, 17
piecewise monotone, 99
successor, 45
transition, 219
uniformly Lipschitz differen-
tiable, 102
functional, 132
functor, 2 21
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
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ALEXANDRE V. BOROVIK
272 Index
G., Swiatoslaw, x
Gaby, Alice, 218
Galatasaray Li sesi, 15
Galileo Galilei, 28
Gangl, Herbert, x, 1 00
Gardiner, Tony, x
Garfunkel, Solomon, x, 167, 173
Garry, Dan, x, 5
Gaudi, Antoni, 222
Gauss, Carl Friedrich, 81
Gauss, Karl, 222
Gelfand, Israel Moiseevich, 1, 17
GEOGEBRA, 171
geometry, 29
differential, 16
elementary, 186
Euclidean, 99, 215
integral, 17
non-euclidian, 186
origami, 172
geometry, synthetic differential,
101
Gerard, Olivier, x, 66, 176, 177,
179, 202
Gibbon, John D., x, 5
Gilbert, Anthony David, x
Gilbert, Tony D., 89
Givental, A lexander, x, 28, 38, 39
Glaser, Anton, 118
Gorcheva, Yordanka, x
Gould, Stephen, x
Gowers, Timothy, 57
Grad, Alex, x, 10, 14, 85
gradient, 37
graph, 219
Gray code
cyclic, 56
Grigorchuk, Rostislav Ivanovich, x
Gromov, Michael, vi, x
Grothendieck, Alexandre, 185
group
n-valued, 71, 72
cohomology, 116
commutative, 72
multivalued, 71
of coboundaries, 116
renormalization, 28
Grundlagen der Analysis, 59, 61,
72
Guinness, 50, 57
Günter Törner, x, 58
Guugu Yimithirr, 217, 218
Hadamard, Jacques Salomon, 215
Harrington, Leo, x, 163
Harriot, Thomas, 118
Harris, Michael, 150
Harris, Mitchell, x
Harte, Robin, x, 174
Heeffer, Albrecht, x, 23, 25
Henderson, David, 126
Henkin, Leon, 79, 8 0
Hersh, Reuben, 182
Hiebert, James, 44
highest common factor, 89
Hokusai, Katsushika, 32, 3 3
homomorphism, 142
augmentation, 10
Hooke, Robert, 222
Howard, Luke, 146
Howard, Toby, x, 114
Høyrup, Jens, x, 16 9
Hutchinson, Alan, x, 133, 183
Ibn Tufa il, Abu Bakr, 11
identity
Fibonacci, 1 49, 150
index, 16
induction, 49
mathematical, 48
successive, 49
infinitesimal, 100
infinity, 189, 194, 197
inheritance, 77
Initial Teaching Alphabet, 15, 16
integer
10-adic, 116, 119
2-adic, 1 20
p-adic, 116
Gaussian, 149
negative, 68
non-negative, 7 1
one-digit, 172
relative, 67
symbolic, 76
integral, 134, 153
surface, 31
integration, 97
interval
nested intervals, 100
invariance
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
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ALEXANDRE V. BOROVIK
Index 273
scale, 29
invariant, 29
inverse, 73 , 219
isometry, 175
JAVA, 5
Jefferies, David, x, 19 5
Johansson, Mika el , x
Jones, Alexander, 1 26
Kıral, E ren Mehmet, x, 3, 4
Kahane, Jean-Pierre, 95
Kant, Immanuel, 213
Kantor, Jean-Michel, x
Kaptanoglu, H. Turgay, x
Katz, Mikhai l, x, 101
Keisler, H. Jerome, 100
Kheyfits, Alexander, x
Khovanova, Tanya, x, 174
Khrushchev, Niki ta Sergeevich,
190
Khudaverdyan, Hovhannes, x, 30
Khudaverdyan, Hovik, x, 116, 151
Kimber, Elizabeth, v, x, 43, 80 , 123
Kirby, Jonathan, x, 92
Knuth, Donald, 101, 103
Kock, Anders, 101
Kolmogorov, A ndrei Nikolaevich,
32, 34–36
Komendantskaya, Ekaterina, x,
193
König, Dénis, 198
Kortenkamp, Ulrich, x, 168
Krutetsky, Vadim Andreevich, 111,
113
Kutateladze, Semen Samsonovich,
x, 126 , 134
Lagrange, Joseph Louis, 36
Lakoff, George, 99
Lama, Vishal, x
Landau, Edmund, 59, 61, 72, 79, 80
language
mathematical, 16, 17
programming, 5
Laurent, Pierre Alphonse, 7
Lauri, Joseph, x, 170
Law
Froude’s, 39–41
Froude’s of Steamship Compar-
isons, 30, 32, 39
Kolmogorov’s “5/3 Law”, 11
law
associative, 63
commutative, 65
Coulomb’s, 11
Kolmogorov’s, 35
Lawvere, William, 101
Leedham-Green, Charles, x, 180
LEGO, 3, 210
Leibniz, Gottfried, 118, 121
Lemma
Lemma
König’s, 1 98
length, 11
Levshin, Vla d imir, 21
lieue, 20
limit, 98–100, 116, 194
epsilon-delta definition of, 99
inductive, 87, 89
Lindemann, Ferdinand von
Lindemann, 173
Lindenbaum, Adolf, 12
Lindquist, Christer, 165
line
contour, 37
real, 100
straight, 126
width of, 171
Lins, Romulo, 3
Livshits, Michael, x, 102
Livshitz, Michael, x
logarithm, 97
logic
mathematical, 99
logic, intuitionist, 101
Lomas, Dennis, x
Löwe, Benedict, xi
lowest common multiple, 89
MacKinnon, Dan, x
Maltcev, Victor, x, 90, 164
Manin, Yuri Ivanovich, 29
Mann, Heinrich, 27, 189
map
bilinear, 9
Gauss, 222
linear, 10, 219
smooth, 222
mass, 11
Matematik Dünyası, xi
MATLAB, 75–77, 162
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
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ALEXANDRE V. BOROVIK
274 Index
matrix, 10, 219
orthogonal, 104
square, 10, 219
transition, 10, 219
Mazur, Barry, 156
McBride, Peter, 135, 136
McKerrell, Archie, x, 190
measure
Lebesgue, 134
mechanics, 29
Megyesi, Gábor, x, 1 16
minus, 63, 136
Molière, Jean-Baptiste, 113
monoid
symmetrization of, 67
monomial
Laurent, 8
Montessori Method, 127
Montessori, Maria, 125, 127
Moreno, Javier, x
morphism, 82
identity, 82
motion
projectile, 28
Müller, Thomas, xi
multiple
lowest common, 89
multiplica tion, 181
n-valued, 72
associativity of, 93
column, 112
long, 181
of fractions, 14
multiplier
Lagrange, 36, 37
multiset, 71, 72
Muranov, Alexey, x, 91, 93, 118
Neman, Azadeh, x, 99, 179
Nesin, Ali, x, xi, 180
Nesin, Aziz, 180
Nesterov, Mikhail Vasilyevich, 195
Neuberger, John William, x, 197
Neubüser, Joachim, x
Ni¸sanyan, Sevan, x, 3
Niven, Ivan Morton, 173
Noether, Emmy, 61
nomination, 125
nonillion, 125
Nosov, Nikolay, 16 8
notation, 16
index-free, 16
novendecillion, 12 5
Novikov, Sergei Petrovich, 71
number
“named”, 1, 7, 3 2
bisqu are, 149
complex, 6, 68, 72
decimal, 164
final, 193
finite, 198
integer, 6
largest, 196
last, 193
named, 77
natural, 45, 46, 193
negative, 65
odd, 3
polygonal, 83
prime, 4
rational, 6
real, 6, 68, 72
signed, 69
transcendental, 120
triangular, 83
wave, 33
number system
base-n, 119
numeral, 13, 69, 79
cardinal, 3, 78
distributive, 3, 4, 78
ordinal, 3, 78
sheep, 78
Núnez, Raphael, 98, 9 9, 102
O’Farrell, An thony, x, 184
object, 82
initia l, 83
source, 8 2
target, 82
Obsessive Compulsive Disorder,
165
Ockley, Simon, 11
octillion, 125
Olshansky, Alexander Yurievich, x,
199
Olshansky, Yuri Nikolaevich, 199
operation
binary, 5
operator, 132, 133
opposite, 64
optimization
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