(Note: numbers in italics refer to illustrations and photos)
Abel, Niels Henrik, 119, 161, 165, 173, 182, 183, 281
abnormal sets, 277–78
absolute convergence, 158, 174
abstraction, 9–17
of analysis, 146
arbitrariness and, 176
definitions of, 8, 10
existence of, 11, 19–20
in Greek mathematics, 10, 29–30, 46
induction and, 15–17
infinity as, 20–21
irrational numbers as direct consequence of, 78
knowledge and, 22–24
language problems and, 11
levels of, 9n, 11–12, 14, 61, 69, 78, 80, 102, 142
modern mathematics based on, 106–7
Vicious Infinite Regress and metaphysics of, 56
abstract set theory, 235n
axiomatic, 284–85
Cantor as father of, 5
naïve, 284–85
abstract thinking:
dangers of, 12–14
directed, 23n, 41
Absurdity, Law of, 27n
Academy of Plato, 59n
acceleration, 126
as function, 102
see also uniform acceleration
“Achilles v. the Tortoise” paradox, 48, 104n
Acta Mathematica, 181n
actuality, potentiality and, 65–66, 99, 121
addition, of transfinite numbers, 243
Ad Vitellionem paralipomena (Kepler), 99
Agrippa, Marcus Vipsenius, 49
Akademie der Wissenschaften, 169
Alberti, Leon Battista, 96
Alembert, Jean le Rond d’, 138n, 148n, 159–60, 165, 216
alephs, 242
algebra, 9n, 95, 105, 118
abstraction in, 61
derivation of term, 92n
algebraic functions, 103
algebraic numbers, 103n, 166
Al-jabra, 92n
al-Khowarizmi, 92n
“All Cretans Are Liars,” 275n
alogos, 77
analysis, 154, 165
abstractness of, 146
applications of, 146, 147
arithmetization of, 118–19, 180, 183, 197, 214–15, 216n
Bolzano and, 119–23
definition of, 117–18
explanatory power of, 146
Greek mathematics compared with, 214
integers and, 217
nonstandard, 136n
rigorous, 119, 152
transfinite math as consequence of, 176n
Weierstrass’s work in, 182, 187n, 188
analytic expression, 163n
analytic philosophy, 12, 63
Analytic Theory of Heat (Fourier), 161, 163–64, 172–73
Anaximander, 44, 45
“and,” 27n
antidifferentiation, 134n
Archimedes, 84, 86, 87, 216
areas:
of circles, 84–86, 84, 85
under curves, 127, 128n, 130–32, 131
in planetary orbits, 98
relative, 94
Aristotle, 12n, 24n, 40n, 49, 56n, 62–70, 85, 91, 99, 106, 114n, 139n, 140, 201, 224, 280n
arguments against Zeno’s Paradoxes made by, 62–68
birth and death dates of, 59n
Christian dogma influenced by, 87, 92
decline of influence of, 99, 101
on different senses of infinity, 63, 74n
on infinity, 44, 63–68, 74n, 87, 93, 99, 121, 196–97, 203, 204
influence of, 87
Plato as teacher of, 59n
Plato’s One Over Many argument disputed by, 58–59
Aristotle’s Wheel, 38n
arithmetic, 9, 61, 248n
Arithmetica infinitorum (Wallis), 19, 105
Arithmetization of Analysis, 118–19, 180, 183, 197, 214–15, 216n
arratos, 77
Arrow paradox, 139–43
Asia, mathematics in, 91–92
Astronomia nova (Kepler), 98
astronomy, 81n, 103
“axiom,” definition of term, 26
axiomatic set theory, 284–85
Babylonian mathematics, 29, 43, 54, 75
Bacon, Francis, 101
Barrow, Isaac, 127, 128
base counting systems, 29
Bell, E. T., 170n, 197
Benacerraf, P., 256–57
Berkeley, Bishop George, 138n
Berlin, University of, 172, 178, 181, 199, 216n, 228
Berlinski, David, 104, 107
Bernoulli, Daniel, 137n, 159, 162
Bernoulli, Jakob (Jacques), 95, 137n
Bernoulli, Johann (Jean), 95, 137n
Bernoulli’s Lemniscate, 19
Bessel Functions, 155n
Big Bang, number of ultranano-instants since, 17
binary mathematics, 216n
Binomial Theorem, 19n, 96, 105n, 106, 117, 128, 129, 133n
biuniqueness, 262n
bivalence, principle of, 26n
Black Holes, 229n
Boltzmann, Ludwig, suicide of, 5–6
Bolzano, Bernhard P., 62n, 87n, 137n, 164, 173, 176, 181, 183, 187, 188, 201, 206, 246, 260
analysis and, 119–20
metaphysics of work of, 124–25
transfinite math and, 120–25, 123, 124
Bolzano-Weierstrass Theorem, 120n, 188–89, 231, 234, 242
Boole, George, 237n
Borel, Félix Édouard Émile, 279n
boundary conditions, 155
bounds, limits and, 111–13
Boyer, Carl B., 8, 11, 98
brachistocrone, 158
Bradley, F. H., 49
Bremermann, H., 18
Bremermann’s Limit, 18
Brouwer, Luitzen E. J., 7, 217n, 225, 283n, 304n, 305n
Brunelleschi, Filippo, 96
Burali-Forti Paradox, 276n, 290, 296n
calculus, 91, 94n, 100, 118, 120, 125–45
applications of, 107, 136, 137
continuity and infinity/limit as basis of, 43
derivation of term, 29
development of, 125–36
differential, 19n, 68, 128n
functions in, 104
Fundamental Theorem of, 134–35
Greek mathematics and, 68
importance of, 126, 137
infinitesimal, 132n
infinitesimals as problematic in, 32, 130, 136, 137–45
Leibniz’s role in, 68, 104, 126–36, 137
Newton’s role in, 19, 68, 104n, 105, 126–36, 137
precursors to, 98
problems addressed by, 126–27
of variations, 182n
see also integral calculus; limits
Cantor, Georg F. L. P., 18, 20, 30, 32, 33, 39, 60n, 69, 79n, 81, 87, 90, 91, 100, 101, 103n, 109n, 114n, 118, 121, 122, 125, 146, 154, 167–72, 177–78, 180, 188, 199, 200, 201, 207n, 215
alleged Jewish background, 5, 167, 242n
on Aquinas, 93
background of, 5, 167, 170–72, 242n
birth of, 170
as closet Platonist, 62n
death of, 5, 167, 305
Dedekind and, 218–19, 223, 250, 254, 259, 263, 293
Dimension Proof of, 259–65
education of, 139, 171–72, 228
family home of, 168–69
historical context and, 42–43
on infinite sets, 218, 222, 228–37
irrational numbers theory of, 219–24
later preoccupations of, 291
mathematical legacy of, 5
mental problems and institutionalization of, 5–7, 167
ordinal theory of, 293–300
pencil sketch by, 169, 170–71
personality of, 169–70
Philosophy of the Infinite of, 12n
physical appearance of, 167, 168
quasi-religious pronouncements of, 41n, 124n
scholarly books on, 237n, 297n
Cantor, Georg W., Sr., 170–72, 178n
cantor, Latin meaning of, 5
Cantor-Dedekind Axiom, 201, 212n
Cantorians, 304
Cantor’s Paradox, 276, 290
cardinal numbers, cardinality, 93, 124
definition of, 192n, 247, 295n
transfinite, 247n, 248–65
Cartesian coordinate plane, 97, 101, 103
Cartesian Products, 287n, 288
catenary problems, 158
Cauchy, Augustin Louis, 119, 139, 152–53, 154n, 161, 164, 173, 174, 176n, 181, 183, 221n, 282
Cauchy Convergence Condition (3C), 153–54
Cause, God as First, 92
Cavalieri, Francesco Bonaventura, 127
Characteristic Triangle, 131n, 133
Chesterton, Gilbert Keith, 6, 8
Choice, Axiom of, 254n, 288–89, 300, 303, 305
Christian Church, 91, 138n
Aristotle’s influence on dogma of, 87, 92
Two New Sciences and, 99
Church, A., 302n
circles:
areas of, 84–86, 84, 85
concentric, 38–39, 38
squaring of, 116, 216n
circularity, 220, 221
Clairaut, Alexis Claude, 148n, 160
“classes,” as term for sets, 207n
“clear and distinct apprehension,” 24
Clinton, Bill, 30
closed forms, 129
coefficients, 115–16
Cohen, Paul J., 7, 289, 301, 302, 305n
Collected Works (Cantor), 93n
college math courses, 101, 119, 131, 135n, 136, 143, 187n
abstractions in, 61, 146
differential equations in, 150
insufficiencies of, 51–52
math’s foundational problems ignored in, 32
not required for understanding this book, 35n, 36n, 125n
Completeness, 284–85
complex numbers, 61, 103n, 166
complex plane, of Gauss, 177n
concentric circles, 38–39, 38
conditional convergence, 174
conditions, boundary and initial, 155
cones, volume of, 86
conics, 98–99
conic sections, 96
conjunction, relation of, definition and symbol of, 27n
Consistency, 284–85
Consistency of Math, 282
Consistency strength, 290–91
Constant of Integration, 151
Constructivism, 217n, 224–27, 230, 254, 259, 264, 282
continuity, 126, 127, 137, 183
conundrum of, 145
convergence and, 173
curves and, 84
definition of, 43, 120
in Galileo’s work, 102–3
mathematics of, 117
in modern analysis, 137n
Number Line and, 73, 79, 87, 206–13, 236
Oresme, Suiseth, and Grandi’s work on, 94–96
Principle of, 96, 99
Real Line and, 239, 246, 258
Weierstrass on, 186–90
see also motion; paradoxes, Zeno’s
“Continuity and Irrational Numbers” (Dedekind), 200, 204, 207, 217, 218
continuity paradox, 7, 48–56, 71–72
continuous, discrete and, 71, 104, 118
continuous functions, 104, 110–11, 118, 119–20, 160, 183–86, 188, 197–202, 216n
Continuum, 73n, 109, 177n, 212, 257, 263
Continuum Hypothesis, 291–93, 298, 300–305
“Contributions to the Founding of the Theory of Transfinite Numbers” (Cantor), 294n
Contributions to the Study of the Transfinite (Cantor), 93n, 244, 245n
“Contribution to the Theory of Manifolds/Aggregates/Sets, A” (Cantor), 265n
Conventionalism, 224n
convergence, 53n, 152–54, 165–66, 172–80, 183, 231
absolute, 158, 174
conditional, 174
definition of, 172–73
tests for, 173–74, 187n
uniform, 156–58, 187n
Weierstrass on, 187
convergent infinite series, 51, 74, 164
decimals as representation of, 79n, 191, 220–21
definition of, 113–14, 158
of Suiseth, 94–95
convergent sequences, 54, 220
coordinate geometry, Cartesian, 97, 101
Counter-Counter-Reformation, 101
counting, 248n
counting systems, 29
Cours d’analyse (Cauchy), 119, 152, 183
curves, 103, 127–29
areas under, 127, 128n, 130–32, 131
Greek problem with, 81n, 84
tangents to, 127
see also calculus
cut, 83n
Danzig, T., 101
data processing systems, Bremermann’s Limit and, 18
Dauben, Joseph W., 169, 234n, 245n
D.B.P.,see Divine Brotherhood of Pythagoras
De Analysi (Newton), 133
Decidability, 235n, 284n
decimal notation, invention of, 79n, 91
decimal numbers:
convergent series represented by, 79n, 191, 220–21
irrational numbers expressed as, 79–80
as numerals, 79n
periodic, 79–80
terminal, 79–80
decomposition, of sets, 240
Dedekind, Richard, 18, 30, 33, 62n, 69, 73, 81, 83n, 87n, 91, 93, 103n, 109n, 114n, 118, 121, 146, 178, 180, 182n, 199–200, 245n, 265n, 287n
Cantor and, 218–19, 223, 250, 254, 259, 263, 293
career of, 200
infinite sets and, 213–19
irrational numbers theory of, 120n, 199–202, 219n, 226, 227n, 244
Number Line and, 206–13
as phenomenologist, 203–4
Real Line and, 201–5
Dedekind Cut, 202
deduction, as basis of mathematical proof, 25, 147, 225
define, definition of, 222
definitions, impredicative, 27, 279
Della pictura (Alberti), 96
dense sets, 240
denumerability, 248, 249–59, 251, 252, 269–74
De Potentia Dei (Aquinas), 92n
derivatives, 116, 117, 120, 129
differentials and, 149–50
derived sets, 232, 233–35
Desargues, Gérard, 96–97, 101, 177n
Descartes, René, 24, 97, 101, 128n
Diagonal Proof, 251–55, 251, 252, 258, 302
Dichotomy, of Zeno, 48–56, 63–67, 71, 74n, 79, 85, 87, 94, 105, 112, 136, 137, 140n, 190–97
Diderot, Denis, 138n
differential calculus, 19n, 68, 116, 128n
differential equations, 162, 172
definition of, 150–55
physical laws expressed as, 137, 147
types of, 154–55
differentials, derivatives and, 149–50
differentiation, 127, 134
Diffusion Equation, 162n, 163
“digits,” numbers as, 29
dimensionality, 23–24
Dimension Proof, 259–65
Diogenes Laërtius, 139n
Dirichlet, Peter Gustav Lejeune, 119, 174–76, 178, 179, 181, 182, 200
discontinuities, 157–58, 229n
discontinuous functions, 110–11, 163n
Discourse on Method (Descartes), 101
discrete, continuous and, 71, 104, 118
divergence, 53, 165
divergent infinite series, 35–36
definition of, 114
of Euler, 95–96
modern social examples of, 56
of Oresme, 94–95
divergent sequence,53
definition of, 53n
Divine Brotherhood of Pythagoras (D.B.P.):
abstraction and, 45–46, 201
irrational numbers and, 75–78, 81, 82, 118, 198
on mathematics of music, 47
metaphysics of mathematics of, 46–47, 48
division, of transfinite numbers, 243
domain, 110
double connotations, problems arising from, 33
Du Bois-Reymond, Paul, 265
Dürer, Albrecht, 96
“dyslexic 6” symbol, 150
Egypt, ancient, mathematics in, 29, 43–44, 46–47, 78
Einstein, Albert, 23
Elements (Euclid), 82, 214
Exhaustion Property in, 85–86
Fifth Axiom of, 39–40
Parallel Axiom of, 37–38, 165
Proposition 20 of, 28–29
“ellipses,” definition of, 28n
elliptic functions, 182
elliptic integrals, 182
emergency glossaries, 109–18, 149–58, 239–40
empiricism, 101
empty set, 235, 239–40
Encyclopédie, 138n
enormous numbers, 17–18
entailment:
definition of, 26, 27n
symbol for, 26
Epimenides, 275n
epistemology, 12, 55, 100
epsilon numbers, 298
epsilontic proof, 187n
equality, between sets, 40n, 122, 246–47
Equivocation, Fallacy of, 57
Essays on the Theory of Numbers (Dedekind), 200n, 204n
eternity, 124
Eublides’ Paradox, 275n
Euclid, 126, 166, 198, 214
geometrical basis of reasoning of, 72
point as defined by, 73
see also Elements
Euclid’s Axioms, 26
Eudoxus, 54n, 88, 97, 104, 112, 128, 191n, 201, 261
irrational numbers and, 82–87, 213–14
Euler, Leonhard, 95, 149n, 152, 159, 160, 165
“Eureka,” 84, 216
even numbers, squares of, 77
everywhere-dense sets, 240
exceptional points, 158, 229n, 232
Excluded Middle, Law of (LEM), 27, 29, 36, 77, 141, 225n
definition of, 26
Exhaustion Property, 84–87, 88, 97, 112, 191n, 214n
expansion, 114–15, 154
exponentiation, of transfinite numbers, 243–44
Extended Complex Plane, 177n
extractability, 80n
Extreme Values Theorem, 189–90, 233, 234
facts, difference between theorems and, 25–26
Fermat, Pierre de, 127, 128
first number class, 265n
first species sets, 233, 240
fluxion, 138, 141
flying, group confidence and, 16n
forcing, 301n
Formalism, 282–84, 303
forms:
closed, 129
latitude of, 102
Plato’s Theory of, 46n, 58–59
Foundations of Arithmetic (Frege), 276
“Foundations of the Theory of Manifolds” (Cantor), 121n
Fourier, Jean Baptiste Joseph, 106, 161, 163–66, 172–80, 181
Fourier coefficients, 116
Fourier Integrals, 164
Fourier Series, 115–16, 155, 161–66, 172–80, 183, 229
General Convergence Problem of, 174, 178, 182, 198
fractions, 103n
irrational numbers not representable as, 79
ratios of integers as, 75
Fraenkel, Abraham A., 238n, 245n, 284n, 285
Frege, Gottlob, 7, 276, 282, 285n
function:
derivation of term, 104
Japanese ideogram for, 151n
functions, 105, 148, 201n
as abstractions, 61, 102
in calculus, 104, 129
classifications of, 103–4, 110–11, 115–16, 118
continuous, 104, 110–11, 118, 119–20, 160, 183–86, 188
definition of, 110, 156, 175
elliptic, 182
Galileo on, 101–3
limit of, 112
periodic, 115, 160
sequences of, 152, 173
symbolism for, 149
Vibrating String Problem and, 159
see also differential equations
fundamental sequences, 222–32
Galileo Galilei, 97, 98–103, 121, 126, 138n, 206, 246
on functions, 101–3
on infinitesimals, 99–100
on infinities, 100–101
Inquisition and, 98, 99
Paradox of, 39–40, 42, 100–101, 122, 136, 247–48
Galois, Évariste, 161n
Gauss, Karl Friedrich, 165, 200, 203
Gauss’s complex plane, 177n
Gedankenwelt, 204
General Convergence Problem of Fourier Series, 174, 178, 182, 198
general differential geometry, 177n
General Relativity, 97
general truths, 9n
Genesis, 44n
Geometriae Pars Universalis (Gregory), 128
geometric figures, weight-bearing properties of, 97–98
geometric series:
definition of, 113
summing of, 51, 104
Géometrie, La (Descartes), 97
geometry:
as basis of Greek mathematics, 72, 76, 76, 87, 106
Cartesian coordinate, 97, 101, 103
divergence of mathematics and, 78–79, 118, 164–65, 214
infinity in, 36–39
limit and to apeiron and, 45
non-Euclidian, 97, 165
projective, 97, 101
Riemannian, 97
Germany, Nazi, 167
glossaries, emergency, 109–18, 149–58, 239–40
God:
Aquinas’s arguments for existence of, 92
Grandi on Void and, 95
infinity and, 21, 99
integers as creation of, 216
mathematics as language of, 124
Gödel, Escher, Bach (Hofstadter), 49
Gödel, Kurt, 5, 7, 23, 62n, 124, 125, 225n, 257, 275–76, 283, 289, 300, 301, 302, 304, 305
Golden Age, of mathematics, 106
Golden Mean, 46, 78
Goris, E. Robert “Dr. G.,” 26, 34, 50, 78n, 111–12, 150, 151n, 160n, 184, 185, 212n, 228, 246, 258, 262, 287n, 295n
Braunschweiger sandwich spread and, 219n
Hankie of Death of, 89n, 273n
“private-sector thinking” term coined by, 23n
“surd” preferred by, 75n
Göttingen, 169
Grandi, Francesco Luigi Guido, 94–96
Grandi Series, 35–36, 95, 158, 165
Grattan-Guinness, Ivor, 182
Greece, ancient, mathematics of:
abstraction and, 10, 29–30, 44
aesthetics of, 45
end of, 87
geometric basis of, 72, 76, 76, 87, 106
infinity absent from, 24, 28, 44, 68, 108
irrational numbers not recognized in, 80–81
legacy of, 25
limitations of, 54, 70n, 81n, 84
mathematical entities viewed as preexisting in, 20
metaphysics and religion indistinct from, 44
modern analysis compared with, 214
paradoxes and, 7
precursors to, see Babylonian mathematics; Egypt, ancient, mathematics in
zero absent from, 24, 54, 70, 73, 142
see also specific mathematicians
Green’s Theorem, 229n
Gregory, James, 102n, 104–5, 128
Gregory of St. Vincent, 104–5
group confidence, flying and, 16n
Halle, University of, 12n, 219
Halle Nervenklinic, 167
Halley, Edmund, 138n
Hamilton, Sir William Rowan, 199
Handel, George Frederick, 5n
“hankie” proof, 89–90
Hardy, Godfrey Harold, 25, 41–42, 43, 45n, 51, 60
Harmonic Analysis, 115n
Harmonic Series, 94, 95
Heine, E. H., 178n, 199, 228–29, 230
Hermite, Charles, 62n, 166n
Hilbert, David, 7, 276n, 305n
on abstract nature of infinity, 20
Cantor’s transfinite math praised by, 6
as Formalist, 282–83
transfinite math championed by, 20
historical fudging, author’s admission of, 82
History of Mathematics, A (Boyer), 8n
Hobbes, Thomas, 19
Hochschulen, 181n
Hofstadter, Douglas, 49
Homogeneity, Preservation of, 287n
Hôpital, Guillaume François Antoine de l,’ 137n
hyperbolas, 99
hypercubes (tesseracts), 23–24
hyperreal numbers, 137n
Hypothèse du Continu (Sierpinski), 303n
“I Am Lying” paradox, 30, 275
Identity, Law of, 26
imaginary numbers, 103n, 166
impredicative definitions, 279
incommensurability paradox, 7
incommensurable magnitudes, 75–77, 82, 125
Incompleteness proofs, 125
Incompleteness Theorems, 276, 283–84, 300
Independence of the Continuum Hypothesis, 302, 304
Indian mathematics, 91–92
indirect proof, see reductio ad absurdum, proof by
Indivisibles, Method of, 127
induction, 165n
abstract justification and, 15–17
mathematical, 241, 249
Principle of (P.I.), 14–16, 25
scientific proof based on, 147
inference, definition and examples of, 26–27
infinite product, 114
infinite series, 104, 128, 129
actuality vs. potentiality and, 67–68
Oresme, Suiseth, and Grandi’s work on, 94–96
summing of, 97, 104–5, 113
see also convergent infinite series; divergent infinite series
infinite sets:
Cantor and, 218, 227, 228–37, 245–59, 292
comparisons of, 247–59
Dedekind and, 213–19
Power Set proofs and, 265–75
subsets of, 122, 125
see also transfinite math; transfinite numbers
infinitesimal calculus, 132n
infinitesimals, 19n, 108
description and importance of, 68–69
Galileo’s work on, 99–100
in Kepler’s calculations, 98
limits and, 119
multiplication by, 90
orders of, 99–100
as problematic in calculus, 32, 130, 136, 137–45
between zero and one, 73–74, 80, 87–90
infinity:
Aquinas on, 92–93
Aristotle on, 44, 63–68, 74n, 87, 93, 99, 121, 196–97, 203, 204
Cantor’s Philosophy of, 12n
degrees of, 18, 100, 103n, 218, 237
difference between enormous numbers and, 17–18
division by, 145
early mathematical applications of, 97
eternity and, 124
existence of, 32–33, 39, 92–93, 97, 205
in Galileo’s work, 100–101
in geometry, 36–39
God and, 21, 99
Greek mathematics and, 24, 28, 44, 68, 108
Greek word for, see to apeiron
infinity of, 74, 80
as largest source of problems in mathematics, 32
legalistic handling of, 39–40
as mathematical taboo, 7n
Number Line and, 74, 87–90
parabolas and, 99
paradox/advance dynamic of, 278–79
points and concentric circles and, 38–39, 38
points and lines and, 36–38, 37, 38
symbol for, 17–18, 19, 105
as ultimate abstraction, 20–21, 35
vanishing point at, 96–97
Zeno on, 71–72
see also transfinite math; transfinite numbers
initial conditions, 155
Inquisition, Galileo and, 98, 99
instants, ultranano, 17
integers, 11, 72, 103n
analysis and, 217
cardinality of, 250n
denumerability of, 269–74
Galileo’s Paradox and, 39–40, 42, 100–101
grade-school teaching about, 8–9
Peano’s Postulates and, 287
as potentially infinite, 67
rational numbers as ratios of,75
reality of, 21, 215–16, 226
see also negative integers
integral calculus, 68, 128n
Method of Exhaustion and, 86
quadrature and, 116
integrals, integration, 117, 127, 134, 175, 178
elliptic integrals, 182
Integration, Constant of, 151
Intermediate Values Theorem, 189n
International Congress of Mathematicians, Second, 6, 282
intersection, of sets, 239
intervals, on Real Line, 111
Introduction to Mathematical Logic (Mendelson), 289n
Intuitionism, 215, 217, 224, 225, 282, 283n
irrational numbers (surds), 61, 103n, 106, 125, 188n
arithmetic of, 81
Cantor’s theory of, 219–24
in decimal notation, 79–80
Dedekind’s theory of, 120n, 199–202, 219n, 226, 227n, 244
definition of, 81
discovery of, 75–78, 118
Eudoxus and, 82–87, 213
importance of, 78–79
not recognized in Greek mathematics, 80–81
on Number Line, 90, 206–13
on Real Line, 73n, 81, 198–202
undefined state of, 166
Islamic mathematics, 91–92
James, William, 49
Jesus, 291n
Joachim, Joseph, 170
justification, Principle of Induction and, 15–17
Kant, Immanuel, 101
Kazan University, 170
Kepler, Johannes, 96, 97–99, 103, 112, 126, 136
Kleene, Stephen C., 225n
Kline, Morris, 8n, 10, 108, 147, 152n, 165, 197
knowledge:
abstraction and, 22–24
nature of, 12
Vicious Infinite Regress and,55
Kossak, H., 199
Kronecker, Leopold, 170, 172, 178n, 200, 215–17, 224–27, 230, 231n, 245n, 259, 265, 279, 283n
Kronecker Delta Function, 216n
Kummer, Ernst Eduard, 172, 216n
LaGrange, Joseph Louis, 159, 160, 161n
Lamp Paradox, 36
LaPlace, Pierre Simon de, 159, 161n
Lavine, Shaughan, 198, 228
“Least Upper Bound Lemma,” 189
Lebesgue, Henri Léon, 279n
Legendre, Adrien Marie, 182n
Leibniz, Gottfried Wilhelm von, 49, 95, 98n, 101, 164, 206
calculus and, 68, 104, 126–36, 137
multiple careers of, 129
Leibnizians, 137, 150
LEM, see Excluded Middle, Law of
lemniscate, derivation of, 19
Leonardo da Vinci, 96
Limited Abstraction Principle, 277n, 279
limit ordinals, 297n
limit points, 120n, 188, 231, 234
limits, 43–45, 53n, 110n, 113, 136
bounds and, 111–13
convergence to, 85, 105
first description of, 104–5
of functions, 112
infinitesimals and, 119
to apeiron and, 45
variables and, 173
Weierstrass on, 144n, 183, 186–90, 197–202
Lindemann, Ferdinand, 199, 216n
linear perspective, 96
lines, 263
Egyptian concept of, 47n
limit and to apeiron and, 45
points and infinity and, 36–38, 37, 38
as simplest curves, 131, 135
Liouville, Joseph, 166n, 258
Lives and Opinions (Diogenes), 139n
Localization Theorem, 179, 227
Location Paradox, 56–57, 64–65, 70
Locke, John, 101
logic, 6, 8
two-valued, 26n
logos, two meanings of, 77
Maclaurin, Colin, 138n
Maimonedes, 92
malum in se, 45
Mancuso, Paolo, 225n
mapping, noncontinuous, 264n
Masaccio, 96
Math, Consistency of, 282
Mathematical Calendar, 200
mathematical induction, 241, 249
Mathematical Thought (Kline), 108
Mathematical Thought from Ancient to Modern Times (Kline), 8n
mathematicians:
early signs of greatness of, 171
marital status and, 200n
mental illness and, 5–7, 12, 42n
music as love of, 181
national origin and, 160n
nerd stereotype of, 7n
nobles as, 161n
Mathematician’s Apology, A (Hardy), 25
mathematics:
abstractions as modern basis of, 106–7
aesthetics in, 45n
ambiguity in, 71
applications of, 103, 107–8, 147, 158–59
creation vs. discovery of, 20
crises over foundations of, 7, 30, 71, 125, 130, 136, 137, 146, 282, 284n
deduction vs. induction in, 25, 147–48
divergence of geometry and, 78–79, 118, 164–65, 214
eighteenth-century problems in, 158–59
as formal system, 25–32
Golden Age of, 106
in grade-school, 9, 72, 248n
in high-school, 9n, 80n
as language of God, 124
metaphysics and, 10
natural language expression of, 33–35, 39, 70, 74n, 238n, 278n
pyramidical nature of, 43
science and, 107–8, 147–48, 158–59
“significant results” in, 41–42
universal vs. particular division in, 128
see also college math courses; Greece, ancient, mathematics of
Math Explosion, 107
math historians, 5, 8
Measurement of a Circle (Archimedes), 86
Measurement of the Volume of Barrels (Kepler), 98n
Mendelson, E., 289n
Méray, H. C. R., 199
Mercator, Nicolaus, 96
metafunctions, 151, 153n
metaphysics, 12, 22, 65
as implicit in all mathematical theories, 10
origin of term, 59n
Metaphysics (Aristotle), 59, 62, 92, 203
Mill, John Stuart, 49
mind, conception of the inconceivable by, 22
Mr. Chicken, Principle of Induction effect on, 15
Mittag-Leffler, Magnus Gösta, 181n
model-theoretic paradoxes, 290
modus tollens, 29
definition of, 27
monism, static, 48
monotonic series, 157
Morgan, Augustus De, 166
motel problem, 34–35, 70
motion:
as abstraction, 142–43
Galileo’s studies of, 102–3
Oresme’s graphing of, 94, 98
planetary, 96, 98, 159
projectile, 98–99
see also continuity; paradoxes, Zeno’s
multiplication, of transfinite numbers, 243
Multivariable Calculus, 150n
music:
D.B.P. mathematical analysis of, 47, 159
mathematicians’ love of, 181
N/A entities (neither zero nor nonzero), 142–43
naïve set theory, 284–85
natural language, expressing mathematics in, 33–35, 39, 70, 74n, 238n, 278n
“Nature and Meaning of Numbers, The” (Dedekind), 204, 214, 287n
navigation, 103
Nearchus I, 43
necessary condition, 27n
negative integers, 106
in Asian mathematics, 91
conceptual trouble with, 166
Number Line and, 73, 166
neighborhood, on Real Line, 111
Neoplatonism, 91
New Math, 248n
Newton, Sir Isaac, 96, 98n, 101, 106, 165
calculus and, 19, 68, 104n, 105, 126–36, 137
Newtonians, 138–39
Nicholas of Cusa, 99
nominative usage, 33, 60
noncontinuous mapping, 264n
non-Euclidian geometry, 165
Riemannian, 97, 177n, 263n
non-Platonists, 62
nonstandard analysis, 136n
Nonstandard Analysis (Robinson), 137n
normal sets, 277–78
“not-,” 27n
nouns, denotations and abstraction and, 11, 19–20
Novum Organum (Bacon), 101
number:
abstraction of, 61
Aristotlean, 67
predicative vs. nominative use of, 33, 60
religion of, 45
number class, first and second, 265 265–75
Number Line, 72–81, 73, 144, 227
continuity and, 73, 76, 87, 206–13
description of, 72–73
infinities and, 74, 87–90
irrational numbers on, 90
negative integers and, 73, 166
rational numbers on, 88–90, 236
Real Line compared with, 73n, 109, 201, 212
zero on, 73
numbers:
algebraic, 103n
called “digits,” 29
enormous, 17–18
as geometrical shapes, 72, 76, 76
prenominate, 75
reality of, 9, 60–62, 203–4, 215, 226, 282, 283
transcendental, 103n
Type Theory and, 280–81
see also specific types of numbers
number theory, 30, 118, 187n, 287n
numerals, definition of, 79n
odd numbers, squares of, 77
omicron, 91n
omnipotence, 21, 22
one:
infinitesimals between zero and, 73–74, 80, 87–90
.999... as equal to, 31–32, 79n, 253–54
as sum of Grandi Series, 35–36, 95
One Over Many (O.O.M.) argument, 58–59, 64, 277n
one-to-one correspondence, see sets, equality between
“On the Extension of a Proposition from the Theory of Trigonometric Series” (Cantor), 218n, 219, 229
“On the Representability...” (Riemann), 178n, 229
Operationalism, 224n
“or, if not true,” required use of, 26n
order-type, 293–95, 298–300
ordinal numbers, 125, 293–300
definition of, 192n, 295
ordinary differential equations, 154
Oresme, Nicole, 94–96, 98, 102
orthogonals, 96
oscillating series, 36, 156n
painting, linear perspective in, 96
parabolas, 98–99
paradoxes:
arising from expressing math in natural language, 33–35
Aristotle’s arguments on, 62–68
Cantor’s, 276, 290
dynamic between advances and, 278–79
of Galileo, 39–40, 42, 100–101, 122, 136, 247–48
immunity to, 290
legalistic handling of, 31–32, 39
regressus, 30–31
self-referential, 30–31, 275–91
Zeno’s, 47–62, 64–65, 70, 137, 139–45, 180
see also continuity paradox; incommensurability paradox
Paradoxes of the Infinite (Bolzano), 120n, 121–25, 260
Parallel Axiom, 37–38, 165, 302
Parmenides, 47, 48, 59
Parmenides (Plato), 47, 59
partial derivative, total differential and, 150
partial differential equations, 154–55, 159
partial sums, 36, 114n
Pascal’s Triangle, 106
Peano, Giuseppe, 282, 285n
Peano’s Postulates, 26, 60n, 287, 291
peras, 45, 75
periodic decimal numbers, 79–80
periodic functions, 115, 160
perspective, linear, 96
phenomenology, 203–4
philosophy:
analytic, 12, 63
mathematics as, 52
pre-Socratic, 47
Philosophy of the Infinite, 12n
physical laws, expressed as differential equations, 137, 147
Physics (Aristotle), 56n, 59n, 62, 63, 65, 71n, 139n, 140
Pi, 97, 114, 216n
P.I.,see induction, Principle of
planes, infinite number of points on, 88
planetary motion, 159
Kepler’s Laws of, 96, 98
Plato, 24n, 47, 49, 82
Academy of, 59n
birth and death dates of, 59n
D.B.P. and, 46n
metaphysical abstraction and, 10
One Over Many argument of, 58–59, 277n
Theory of Forms of, 46n, 58–59
Platonism, 62, 203, 226, 245n, 257, 304–5
Plotinus, 49
plurality, 48
Poincaré, Jules-Henri, 217n, 225, 257, 279, 281, 283n, 290, 291
point nine nine nine... (.999...), as equal to one, 31–32, 79n, 253–54
points, 70
collections of, 125
concentric circles and infinity and, 38–39, 38
Euclid’s definition of, 73
exceptional, 158, 229n, 232
infinite number of, 88
limit, 120n, 188, 231, 234
limit and to apeiron and, 45
lines and infinity and, 36–38, 37, 38
nature of, 56n
on Number Line, 87–90
set theory and, 263
point sets, 188, 232
point-set theory, 235n
point-set topology, 265n
Poisson, Siméon Denis, 163
polygons, in calculating areas of circles, 84–85, 84, 85
polynomian functions, 103
polytopes, 263
potentiality, actuality and, 65–66, 74n, 99, 121
power, collections of points and, 145
power series, 115, 183n
Power Set Axiom, 266n, 288
Power Sets, 265–75, 276, 292
Prague, University of, 119, 121
pre-Creation, 44n
predicative usage, 33, 60
prenominate numbers, 75
prime numbers, infinite number of, 28–29
“Principles of a Theory of Order-Types” (Cantor), 294n
private-sector thinking, 23n, 41
product, infinite, 114
projectile motion, 98–99
projective geometry, 97, 101
Prometheus, 7
proof:
deduction and, 25, 147, 225
requirements for, 25
proper subsets, 247
Proposition 20, 28–29
“Purely Analytical Proof...” (Bolzano), 119n
Putnam, H., 256–57
Pythagoras, 10, 45–46, 58, 124, 125, 159, 261
Pythagorean Theorem, 46–47, 75, 77
Pythagoreans, see Divine Brotherhood of Pythagoras
quadrature, 116, 128n
quantum theory, 18, 22
quia arguments, of Aquinas, 92
Quine, W. V. O., 285
“Racetrack, The” paradox, 48
Radical Doubt, 101
Ramsey, Frank P., 281n, 285
range, 110
ratio:
of different orders of infinitesimals, 100
Eudoxus’s definition of, 82, 213
rational numbers as, 75–76
rational numbers, 103n
denumerability and, 250–55, 251, 252, 258
derivation of term, 75
infinite number of, 88
on Number Line, 88–90, 236
as ratio of two line lengths, 76
set of real numbers compared with, 237, 239, 258–59
real function, 110
Realists, 304
Real Line, 81, 145, 188, 231, 258
Bolzano and, 122–24, 123, 124
components of, 111
continuity and, 239, 246, 258
Dedekind and, 201–5
definition of, 73n, 90, 109–10
Number Line compared with, 73n, 109, 201, 212
topology of, 109n
Weierstrass and, 197–99
realness, mathematical vs. real, 21
real numbers, 103n, 189n, 234
definition of, 213
Dimension Proof and, 259–65
nondenumerability of, 251n, 257–58
Real Line and, 73n, 198, 246
set of rational numbers compared with, 237, 239, 258–59
symbol for set of, 177n
undefined state of, 166–67
Reducibility, Axioms of, 281
reductio ad absurdum, proof by (indirect proof):
definition of, 27
examples of, 28–29, 77
regressus in infinitum, 49
regressus paradoxes, 30–31
Regularity, Axiom of, 289–90, 296n
Relativity, General, 97
religion, anthropology of, 21
repeating decimals, 31–32
Riemann, Georg Friedrich Bernhard, 176–80, 181, 182, 200, 227, 229, 260
Riemannian geometry, 97, 177n, 263n
Riemann’s Localization Theorem, 179
Riemann Sphere, 177n
rigorous analysis, 119
Roberval, Gilles Personne de, 127
Robinson, Abraham, 136n
Roman Empire, 87
fall of, 91
Rudolph II, Emperor, 98n
Russell, Bertrand, 7, 9n, 48, 52, 84, 87, 180, 183, 192n, 198, 199, 207n, 228, 249n, 258, 280–81, 290, 291, 305n
Russell’s Antinomy, 276–79, 290
Saussure, Ferdinand de la, 10–11, 107
Schaeffer, Theodor, 172n
schnitt device, 207–12, 215, 217, 218, 226, 244n, 265n
science:
empirical nature of, 147
gap between everyday perceptions and, 22
mathematics and, 107–8, 147–48, 158–59
Principle of Induction as basis of, 14–15
Scientific Revolution, 81, 101, 107
second number class, 265n
second species sets, 233, 240
Seeming vs. Truth, Ways of, 47
self-referential paradoxes, 30–31, 275–91
sequences:
definition of, 111
fundamental, 222–23
limits of, 112
series:
definition of, 113–15
power, 115, 183n
trigonometric, 116n, 147
see also Fourier Series; geometric series; infinite series
set-of-all-sets paradox, 276
sets:
all math entities as, 296
bracket symbols and, 231n
classes as term for, 207n
decomposition of, 240
derived, 232, 233–35
empty, 235, 239–40
equality between, 40n, 122, 246–47
of first species, 233, 240
intersection of, 239
point, 188, 232
of second species, 233, 240
self-referential paradoxes and, 275–91
union of, 239
set theory, 30, 60n, 93, 101, 122, 177n
abstract, 235n
basic axioms of, 285–91
breakdown of, 125
Cantor as father of, 227, 235n
cardinality vs. ordinality in, 124–25, 192n, 250n
Eudoxus and, 83
philosophical comments on, 256–57
point, 235n
revolutionary nature of, 263–64
see also abstract set theory; infinite sets
severability, 207–8
Sierpinski, Waclaw, 303n
“significant results,” 41–42, 51
sine waves, 160
singularity, 229n
Skolem, Thoralf, 238n, 285
Socrates, 47, 59n, 202
Special Ed Math, 9
square root:
of minus one, 166
of three, 80
of two, 78n, 81
squares:
Galileo’s Paradox and, 39–40, 42, 100–101
of odd and even numbers, 77
squaring:
of circles, 116, 216n
as geometrical process, 76, 76
static monism, 48
Statics (Stevin), 97–98
Stevin, Simon, 97–98, 99
Stifel, Michael, 81n
subsets, 266
definition of, 239–40, 247
derived sets and, 235
proper, 247
subtraction, of transfinite numbers, 243
sufficient condition, 27n
Suiseth, R., 94–96
Summa Theologiae (Aquinas), 92n, 93
sums, of series, 113–14
surd:
etymology of term, 75n
see irrational numbers
“Sur la convergence des séries trigonométriques,” 174–75
Syracuse, 887
tangents, 127, 128n
Tangents, Method of, 127, 128
Tarski, A., 281n, 285
Taylor, Brook, 138n
ten, importance to D.B.P. of, 46
tensor analysis, 97
terminal decimal numbers, 79–80
tesseracts (hypercubes), 23–24
tetration, 298
theorems:
definition of, 26n
difference between facts and, 25–26
Third Man, 59, 67, 224, 280n
Thomas Aquinas, Saint, 49, 92–94, 244
Thought, Laws of, 26
3C (Cauchy Convergence Condition), 153–54
3D shapes and space, infinite number of points in, 88
three, square root of, 80
time:
as Aristotlean infinity, 63
eternal, 124
relativity of, 22
smallest continuous interval of, 17
timekeeping, 103
to apeiron, 87, 279n
definition of, 44, 72
limit and, 45
Vicious Infinite Regress and, 55–56
“to be,” 57, 58n, 65
Tolstoy, Leo, 170
topological space, 109
topology, 97, 279
total differential, partial derivative and, 150
transcendental functions, 103
transcendental numbers, 103n, 166
transcomputational problems, 18
transfinite cardinals, 247n, 248–65
transfinite math, 62, 779
Bolzano and, 120–25, 123, 124
Cantor as father of, 5
importance of, 5
major figures in, 178n
mathematicians’ responses to, 5, 6, 20, 32, 118, 122, 182n, 259, 264, 265
roots of, 7, 125, 146–49, 176n, 236
“significant results” and, 42
see also infinite sets; infinity
transfinite numbers, 239
Cantor’s symbols for, 242
as numbers vs. sets, 242–43
performing operations on, 243–44
reality of, 242, 244–45
transfinite ordinals, 249n, 276n
Transfinitists, 304
triangles:
centers of gravity of, 97–98
Characteristic, 131n, 133
trigonometric series, 116n, 147, 148, 155–56, 160–61, 172–80
Uniqueness Theorem for, 199, 229–37
see also Fourier Series
trigonometry, 105, 106n
as analysis, 117
in Greek mathematics, 81n
infinite products in, 114
name for infinity’s symbol in, 19
truths, general, 9n
Truth vs. Seeming, Ways of, 47
two:
even result from multiplying by, 77
square root of, 78n, 81
as sum of particular infinite series, 94
Two New Sciences (Galileo), 98–103, 121, 124
Two-Valued Logic, 26n
Types, Theory of, 280–81
ultranano-instants, definition and number of, 17
unicorns, 19–20, 21, 41, 205
uniform acceleration, 94
uniform convergence, 156–58
Uniform-Convergence Test, 173–74
union, of sets, 239
Uniqueness Theorem, 199, 229–37
Unit Square, 76–77, 82, 202, 260
Unlimited Abstraction Principle, 277n, 279
Unmoved First Mover, 92
vanishing point, 96
variables, 138
as abstractions, 61
functions and, 102, 110
limits and, 173
separation of, 162
Varia Responsa (Viète), 97
variations, calculus of, 182n
velaria, 159
velocity:
as abstraction, 102
instantaneous, 126, 128n, 140
relative, 94
verbs, abstract nouns rooted in, 11
Vibrating String Problem, 159
Vicious Circles, 278, 280, 284
Vicious Infinite Regress (VIR), 49, 55–56, 62, 74n, 79
Vienna Conservatory, 170
Viète, François, 97
VNB axiomatic system, 285, 302
Void, 44, 45, 95
Wallis, John, 19, 96, 104–5, 128, 133n
Wave Equation, 155, 159–60, 162n
wave functions, 115–16
Weierstrass, Karl Theodor, 30, 69, 109n, 114n, 120, 137n, 161n, 178n, 180–99, 201, 203, 214, 220, 230, 233
background of, 181
Cantor as student of, 139, 172
on continuity and limits, 186–90
on continuous functions, 183–86
importance of work of, 182–83
on limits, 144n, 183, 186–90, 197–202
Zeno’s Dichotomy and,190–97
Weierstrass’s M-Test, 187n
well-ordering principle, 288–89
Weyl, Hermann, 225n
Whitehead, Alfred North, 25
Word Problems, 34n, 35, 50, 195
Zeno, 40n, 54n, 73, 87, 88, 201, 213, 250
Aristotle’s arguments against, 62–68
birth and death dates of, 59n
death of, 43
Dichotomy of, 48–56, 63–67, 71, 74n, 79, 85, 87, 94, 104, 112, 136, 137, 140n, 190–97
paradoxes of, 47–62, 64–65, 70, 137, 139–45, 180
on two types of infinity, 71–72
Zermelo, E., 7, 238n, 254n, 284, 285, 291
zero, 106, 243
absent from Greek mathematics, 24, 54, 70, 73, 142
ban on dividing by, 31
entities neither nonzero nor, 142–43
Harmonic Series and, 94
infinitesimals and, 68, 100
infinitesimals between one and, 73–74, 80, 87–90
invention of, 54, 91
on Number Line, 73
as sum of Grandi Series, 35–36, 95
symbol for, 91
ZFS axiomatic system, 285, 290, 302, 304