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Wolff P. Breakthroughs in mathematics
Wolff P.  Breakthroughs in mathematics







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: Breakthroughs in mathematics

: Wolff P.

: en

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ed2k: ed2k stats

: 1963

: 284

: 10.12.2004

: | | ID
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Absurd, reduction to the      6062 155
Addition      150
Addition in Peanos theory      166
Algebra, Booles symbolism and      271272 (see also Boole George)
Algebra, geometrical problems solved by      see Geometry (Descartes)
Analytic geometry      96 104
Analytic geometry, advantages of      109111
Angles in Euclids Elements      4445
Angles in Lobachevski      7477 8492
Angles, defined      16 4647
Angles, measured by Gauss      95
Angles, propositions      2631 37
Apollonius      96
Archimedes, life of      129
Archimedes, The Sand Reckoner      116126 129137
Aristarchus of Samos      117118 131
Arithmetic, geometry and      103104
Arithmetic, Peanos analysis of      164169 179 188
Arithmetic, postulates in      126127
Arithmetic, reduced to logic      188189 190 194 Numbers)
Bernoulli, Daniel      207
Bernoulli, Jean      207
Bolyai, John      80
Boole, George, accomplishment of      275
Boole, George, background of      268
Boole, George, The Laws of Thought      243266 268275
Boole, George, The Laws of Thought, classes of signs      246256 269
Boole, George, The Laws of Thought, derivation of laws of symbols      256266 275
Boole, George, The Laws of Thought, laws of symbols      247256 269271
Buffon, Comte Georges de      229
Cantor, Georg      140
Chiliagon, in Archimedes      118 133
Circles in Euclid      1517 (see also Great circles)
Classes in Boolean algebra, law of contradiction      265266 274
Classes in Boolean algebra, symbols for      247256 264265 269271
Classes in definition of number      169176 184188
Classes, hereditary      177 180 191
collections      see Classes
Coloring of maps      217
Common notions in Euclids Elements      18 5051
Conic sections, defined      96
Constructions in Descartes      97102 104110
Constructions in Euclids Elements      5259
Constructions, compared with Descartes      106107
Constructions, propositions      1721 2628 37
Continuity and Irrational Numbers (Dedekind)      138149 152160
Contradiction, principles of      265266 274
Converse domain      173
Counting, Archimedes on      116126 129137
Counting, decimal system and      137
Counting, Russell on      186 (see also Arithmetic; Numbers)
Curves, equations for      107109
Cutting and not-cutting lines      70 84
Decimal system, Archimedes substitute for      137
Dedekind, Continuity and Irrational Numbers      138149 152160
Dedekind, Richard, background of      152
Deductiveness as characteristic of mathematics      267268
Definitions in Euclids Elements      1617 4547
Definitions in Euclids Elements, of numbers      114115 126127 150
Descartes, Geometry      97102
Descartes, Geometry vs. Euclids method      104110
Descartes, Rene, background of      102103
Division by zero      151152
Division, number system and      150151
Domain, defined      173
Duality, law of, in Boole      266
Earth, diameter of, in Archimedes      117 132
Elements of Geometry      see Euclids Elements of Geometry
Euclid, his method vs. Descartes      109110
Euclid, significance of      43
Euclids Elements of Geometry Book I      1541 4362
Euclids Elements of Geometry Book I, common notions      18 5051
Euclids Elements of Geometry Book I, definitions      1617 4547
Euclids Elements of Geometry Book I, postulates      17 4750
Euclids Elements of Geometry Book I, Proposition 1      1719 5254
Euclids Elements of Geometry Book I, Proposition 10      26
Euclids Elements of Geometry Book I, Proposition 11      27
Euclids Elements of Geometry Book I, Proposition 12      28
Euclids Elements of Geometry Book I, Proposition 13      29
Euclids Elements of Geometry Book I, Proposition 14      2930
Euclids Elements of Geometry Book I, Proposition 15      3031
Euclids Elements of Geometry Book I, Proposition 16      3132 9192 94
Euclids Elements of Geometry Book I, Proposition 17      32
Euclids Elements of Geometry Book I, Proposition 18      33
Euclids Elements of Geometry Book I, Proposition 19      3334
Euclids Elements of Geometry Book I, Proposition 2      1920 5457
Euclids Elements of Geometry Book I, Proposition 20      34
Euclids Elements of Geometry Book I, Proposition 21      35
Euclids Elements of Geometry Book I, Proposition 22      36
Euclids Elements of Geometry Book I, Proposition 23      37
Euclids Elements of Geometry Book I, Proposition 24      3738
Euclids Elements of Geometry Book I, Proposition 25      3839
Euclids Elements of Geometry Book I, Proposition 26      3941
Euclids Elements of Geometry Book I, Proposition 27      6364 8182
Euclids Elements of Geometry Book I, Proposition 28      6465
Euclids Elements of Geometry Book I, Proposition 29      6566 8283
Euclids Elements of Geometry Book I, Proposition 3      2021 57
Euclids Elements of Geometry Book I, Proposition 30      6667
Euclids Elements of Geometry Book I, Proposition 31      67
Euclids Elements of Geometry Book I, Proposition 32      6768 85
Euclids Elements of Geometry Book I, Proposition 4      2122 5759
Euclids Elements of Geometry Book I, Proposition 47      108109
Euclids Elements of Geometry Book I, Proposition 5      2223 5960
Euclids Elements of Geometry Book I, Proposition 6      2324 6062
Euclids Elements of Geometry Book I, Proposition 7      24
Euclids Elements of Geometry Book I, Proposition 8      25
Euclids Elements of Geometry Book I, Proposition 9      26
Euclids Elements of Geometry Book II      15
Euclids Elements of Geometry Book III      15
Euclids Elements of Geometry Book IV      15
Euclids Elements of Geometry Book IX      15 103
Euclids Elements of Geometry Book IX Proposition 20      115116 127129
Euclids Elements of Geometry Book V      15 43
Euclids Elements of Geometry Book VI      15
Euclids Elements of Geometry Book VII      15 103
Euclids Elements of Geometry Book VII, definitions      114115 126127
Euclids Elements of Geometry Book VIII      15 103
Euclids Elements of Geometry Book X      15 43
Euclids Elements of Geometry Book XI      15
Euclids Elements of Geometry Book XII      15
Euclids Elements of Geometry Book XIII      15 43
Eudoxus      43 118
Euler on geometry of position (topology)      197206 207215
Euler, Leonhard, background of      205207
Extraordinary, defined      228
Fifth Postulate of Euclid      17
Fifth Postulate of Euclid in analytic geometry      109
Fifth Postulate of Euclid, Euclids first use of      8283
Fifth Postulate of Euclid, Saccheris critique of      7880
Fifth Postulate of Euclid, substitutes for      8384 94
Finitude and mathematical induction      175182 190194
Fractions      151
Frege, Gottlob      169 180n 181 188
Gamblers fallacy      238239
Gauss, Karl Friedrich      152
Gauss, measures angles      95
Gelon, King of Syracuse      116 129 130
Geodesics      9394
Geometry (Descartes)      97102
Geometry (Descartes) vs. Euclids method      104110
Geometry, analytic      96 104
Geometry, analytic, advantages of      109111
Geometry, Euclidean      see Euclids Elements of Geometry
Geometry, non-Euclidean      80
Geometry, non-Euclidean, Riemannian      93 94
Geometry, non-Euclidean, which is most suitable?      9495 (see also Lobachevski Nicholas)
Geometry, of position      see Topology
Geometry, postulates in      5253
Geometry, proof in      43
Geometry, proof in reduction to the absurd      6062
Geometry, proof in superimposition      58
Geometry, proof in synthetic vs. analytic method      109110
Geometry, thinking scientifically about      4043
Great circles, non-Euclidean geometry and      9194
Heine, E.      139
Heliocentric theory      117118 130132
Hereditary property of numbers      177 180 191
Induction, mathematical      165 177182 190194
Inductive numbers      182
Inductive property of numbers      177 191
Infinity of numbers in Archimedes      116126 129137
Infinity of numbers in Russell      179
Introduction to Mathematical Philosophy (Russell)      161194
Introduction to Mathematical Philosophy (Russell), definition of number      169176 184188
Introduction to Mathematical Philosophy (Russell), finitude and mathematical induction      175182 190194
Introduction to Mathematical Philosophy (Russell), series of natural numbers      161169
Irrational numbers, described by Dedekinds cut      145149 152160
Irrational numbers, discovered      149
Konigsberg, problem of      198206 207216
Lagrange, Joseph Louis      219
Language      see Logic
Laplace, Pierre Simon de, background of      230231
Laplace, Pierre Simon de, on probability      219230 233241
Laws of Thought, The (Boole)      243266 268275
Laws of Thought, The (Boole), classes of signs      246256 269
Laws of Thought, The (Boole), derivation of laws of symbols      256266 275
Laws of Thought, The (Boole), laws of symbols      247256 269271
Leibniz, Baron Gottfried von      197198 215 220
Lines, cutting and not-cutting      70 84
Lines, Euclids definition of      16 4546
Lines, parallel      see Parallel lines
Lines, skew      47
Lines, straight      see Straight lines
Lobachevski, Nicholas, background of      80
Lobachevski, The Theory of Parallels      6877
Lobachevski, The Theory of Parallels, analytic geometry and      109
Lobachevski, The Theory of Parallels, angles and triangles      7477 8492
Lobachevski, The Theory of Parallels, publication of      80
Lobachevski, The Theory of Parallels, substitutes postulate for Euclids Fifth      84
Logic, laws of symbols in Boole      247256 269271
Logic, laws of symbols in Boole, derivation of      256266 275
Logic, mathematics and, in Boole      242 267268
Logic, mathematics and, in Frege      188
Logic, mathematics and, in Peano      188
Logic, mathematics and, in Russell      188189 190 194
Logic, symbolic      268 275
Maps, coloring of      217
Marcellus      129
Mathematical induction      165 177182 190194
Mathematics, common characteristic of      267
Mecanique Celeste      230231
Mechanical brains      242
Mersenne, Marin      102
Moebius strip      215217
Multiplication      150
Non-Euclidean geometry      80
Non-Euclidean geometry of Lobachevski      6877
Non-Euclidean geometry of Lobachevski, analytic geometry and      109
Non-Euclidean geometry of Lobachevski, angles and triangles      7477 8492
Non-Euclidean geometry of Lobachevski, postulate substituted for Euclids Fifth      84
Non-Euclidean geometry of Lobachevski, published      80
Non-Euclidean geometry, Riemannian      93 94
Non-Euclidean geometry, which is most suitable?      9495
Numbers, denned by Euclid      114115 150
Numbers, denned by Russell      169176 184188
Numbers, inductive      182
Numbers, irrational, described by Dedekinds cut      145149 152160
Numbers, irrational, discovered      149
Numbers, natural      150
Numbers, natural, theory of      164169 176182 190194
Numbers, quantity of, in Archimedes      116126 129130
Numbers, quantity of, in Euclid      115116
Numbers, quantity of, in Russell      175182 190194
Numbers, rational, compared with points on straight line      141145 152158
Numbers, rational, defined      152
Numbers, rational, properties of      140142
Opposite, reduction to the      128129
Parallel lines in Euclids Elements      44
Parallel lines in Euclids Elements, defined      17 47
Parallel lines in Euclids Elements, propositions      6368 8083
Parallel lines in Lobachevski      84 9193
Parallel lines in Lobachevski, theorems      7074
Parallel lines, three possibilities for      94
Parallelograms in Euclid      44
Peano, Giuseppe      183
Peano, number theory of      164169 179 188
Pheidias      118
Planetary theory in Aristarchus and Archimedes      117118 131
Planetary theory, Laplaces work on      230231
Poincare, Henri      181
Point, defined in Euclids Elements      16 4546
Position, relative      see Topology
Posterity of numbers      177178 180 192193
Postulates in Descartes      109
Postulates in Euclids Elements      17 4750
Postulates in Euclids Elements, lacking in number theory      126127 (see also Fifth Postulate of Euclid)
Postulates, function of, in geometry      5253
Prime numbers in Euclid, defined      114
Prime numbers in Euclid, quantity of      114116 127129
probability      218241
Probability, a priori vs. a posteriori arguments      233
Probability, defined      224 234 235
Probability, gamblers fallacy in      238239
Probability, Laplace on      219230 233241
Probability, rules for      224230 234241
Probability, truth and      231
Progressions, Peanos theory and      167
Proofs, , reduction to the opposite      128129
Proofs, method of superimposition      58
Proofs, reduction to the absurd      6062 155
Proofs, synthetic vs. analytic method      109110
Propositions of Euclid      see Euclids Elements of Geometry
Pseudosphere      9394
Pythagoras      149 164
Pythagorean theorem of Euclid      108109
Q.E.F.      53
Rational numbers, compared with points on straight line      141145 152158
Rational numbers, defined      152
Rational numbers, properties of      140142
Reason, derivation of symbols of logic from laws of      256266 275
Reasonable degree of belief      220 232
Reduction in Russells definition of number      188
Reduction to the absurd      6062 155
Reduction to the opposite      128129
Relations, signs of, in Boole      252256
Relations, types of, in Russell      173
Relative position      see Topology
Riemann, Bernhard      80
Riemann, geometry of      93 94
Right angles, Euclids definition of      16 4647
Russell, Bertrand, background of      183184
Russell, finitude and mathematical induction      175182 190194
Russell, finitude and mathematical induction, series of rational numbers      161169
Russell, Introduction to Mathematical Philosophy      161194
Russell, Introduction to Mathematical Philosophy, definition of number      169176 184188
Saccheri, Girolamo, on Euclids Fifth Postulate      7880
Sand Reckoner, The (Archimedes)      116126 129137
Schnitt, Dedekinds      145149 152160
Science, object of      256
Sets      see Classes
Signs, classes of      246156 269
Signs, defined      244 (see also Laws of Thought The)
Similarity and class      174 185 188
Skew lines      47
Sphere, non-Euclidean geometry and      9194
Square roots      see Irrational numbers
Stadium (Greekmeasure), length of      132
Straight lines in Euclids Elements      44 5257
Straight lines in Euclids Elements, defined      16 4546
Straight lines in Euclids Elements, propositions      1721 2628
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