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Peter Wolff — Breakthroughs in mathematics
Peter Wolff — Breakthroughs in mathematics



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Íàçâàíèå: Breakthroughs in mathematics

Àâòîð: Peter Wolff

Àííîòàöèÿ:

The author wishes to thank the various publishers and
individuals who permitted the selections in this book
to be reprinted. Copyright notices and credits are
given on the first page of each selection. Thanks also
go to the author's assistant, Mary Florence Haugen,
for all her help and encouragement in conceiving and
completing this book.


ßçûê: en

Ðóáðèêà: Ìàòåìàòèêà/

Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö

ed2k: ed2k stats

Èçäàíèå: Library Rebound

Ãîä èçäàíèÿ: 1963

Êîëè÷åñòâî ñòðàíèö: 276

Äîáàâëåíà â êàòàëîã: 08.10.2012

Îïåðàöèè: Ïîëîæèòü íà ïîëêó | Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
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Ïðåäìåòíûé óêàçàòåëü
Absurd, reduction to the      60—62 155
Addition      150
Addition in Peano’s theory      166
Algebra, Boole’s symbolism and      271—72 (see also Boole George)
Algebra, geometrical problems solved by      (see Geometry (Descartes))
Analytic geometry      96 104
Angles in Euclid’s Elements      44—45
Angles in Euclid’s Elements, defined      16 46—47
Angles in Euclid’s Elements, propositions      26—31 37
Angles in Lobachevski      74—77 84—92
Angles, measured by Gauss      95
Apollonius      96
Archimedes, life of      129
Archimedes, The Sand Reckoner      116—26 129—37
Aristarchus of Samos      117—18 131
Arithmetic, geometry and      103—4
Arithmetic, Peano’s analysis of      164—69 179 188
Arithmetic, postulates in      126—27
Arithmetic, reduced to logic      188—89 190 194 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      243—66 268—75
Boole, George, The Laws of Thought, classes of signs      246—56 269
Boole, George, The Laws of Thought, derivation of laws of symbols      256—66 275
Boole, George, The Laws of Thought, laws of symbols      247—56 269—71
Buff on, Comte Georges de      229
Cantor, Georg      140
Chiliagon, in Archimedes      118 133
Circles in Euclid      15—17 (see also Great circles)
Classes in Boolean algebra in definition of number      169—76 184—88
Classes in Boolean algebra in definition of number Book II      15
Classes in Boolean algebra in definition of number Book III      15
Classes in Boolean algebra in definition of number Book IV      15
Classes in Boolean algebra in definition of number Book IX      15 103
Classes in Boolean algebra in definition of number Book IX, Proposition      20 115—16 127—29
Classes in Boolean algebra in definition of number Book V      15 43
Classes in Boolean algebra in definition of number Book VI      15
Classes in Boolean algebra in definition of number Book VII      15 103
Classes in Boolean algebra in definition of number Book VII, definitions      114—15 126—27
Classes in Boolean algebra in definition of number Book VIII      15 103
Classes in Boolean algebra in definition of number Book X      15 43
Classes in Boolean algebra in definition of number Book XI      15
Classes in Boolean algebra in definition of number Book XII      15
Classes in Boolean algebra in definition of number Book XIII      15 43
Classes in Boolean algebra in definition of number Proposition 23      37
Classes in Boolean algebra in definition of number Proposition 24      37—38
Classes in Boolean algebra in definition of number Proposition 25      38—39
Classes in Boolean algebra in definition of number Proposition 26      39—41
Classes in Boolean algebra in definition of number Proposition 27      63—64 81—82
Classes in Boolean algebra in definition of number Proposition 28      64—65
Classes in Boolean algebra in definition of number Proposition 29      65—66 82—83
Classes in Boolean algebra in definition of number Proposition 30      66—67
Classes in Boolean algebra in definition of number Proposition 31      67
Classes in Boolean algebra in definition of number Proposition 32      67—68 85
Classes in Boolean algebra in definition of number Proposition 47      108—9
Classes in Boolean algebra, law of contradiction      265—66 274
Classes in Boolean algebra, symbols for      247—56 264—65 269—71
collections      (see Classes)
Coloring of maps      217
Common notions in Euclid’s, Elements      18 50—51
Conic sections, defined      96
Constructions in Descartes      97—102 104—10
Constructions in Euclid’s Elements      52—59
Constructions in Euclid’s Elements, compared with Descartes      106—7
Constructions in Euclid’s Elements, propositions      17—21 26—28 37
Continuity and Irrational, Numbers (Dedekind)      138—49 152—60
Contradiction, principles of      265—66 274
Converse domain      173
Counting, Archimedes on      116—26 129—37
Counting, decimal system and      137
Counting, Russell on      186 (see also Arithmetic Numbers)
Curves, equations for      107—9
Cutting and not-cutting lines      70 84
Decimal system, Archimedes’ substitute for      137
Dedekind, Richard, background of      152
Dedekind, Richard, Continuity and Irrational Numbers      138—49 152—60
Deductiveness as characteristic of mathematics      267—68
Definitions in Euclid’s Elements      16—17 45—47
Definitions in Euclid’s Elements of numbers      114—15 126—27 150
Descartes, Rene, background of      102—3
Descartes, Rene, Geometry      97—102
Descartes, Rene, Geometry vs. Euclid’s method      104—10
Division by zero      151—52
Division, number system and      150—51
Domain, defined      173
Duality, law of, in Boole      266
Earth, diameter of, in Archimedes      117 132
Elements of Geometry      (see Euclid’s)
Euclid, his method vs. Descartes’      109—10
Euclid, significance of      43
Euclid’s Elements of Geometry - Book I      15—41 43—62
Euclid’s Elements of Geometry - Book I, common notions      18 50—51
Euclid’s Elements of Geometry - Book I, definitions      16—17 45—47
Euclid’s Elements of Geometry - Book I, postulates      17 47—50
Euclid’s Elements of Geometry - Book I, Proposition 1      17—19 52—54
Euclid’s Elements of Geometry - Book I, Proposition 10      26
Euclid’s Elements of Geometry - Book I, Proposition 11      27
Euclid’s Elements of Geometry - Book I, Proposition 12      28
Euclid’s Elements of Geometry - Book I, Proposition 13      29
Euclid’s Elements of Geometry - Book I, Proposition 14      29—30
Euclid’s Elements of Geometry - Book I, Proposition 15      30—31
Euclid’s Elements of Geometry - Book I, Proposition 16      31—32 91—92 94
Euclid’s Elements of Geometry - Book I, Proposition 17      32
Euclid’s Elements of Geometry - Book I, Proposition 18      33
Euclid’s Elements of Geometry - Book I, Proposition 19      33—34
Euclid’s Elements of Geometry - Book I, Proposition 2      19—20 54—57
Euclid’s Elements of Geometry - Book I, Proposition 20      34
Euclid’s Elements of Geometry - Book I, Proposition 21      35
Euclid’s Elements of Geometry - Book I, Proposition 22      36
Euclid’s Elements of Geometry - Book I, Proposition 3      20—21 57
Euclid’s Elements of Geometry - Book I, Proposition 4      21—22 57—59
Euclid’s Elements of Geometry - Book I, Proposition 5      22—23 59—60
Euclid’s Elements of Geometry - Book I, Proposition 6      23—24 60—62
Euclid’s Elements of Geometry - Book I, Proposition 7      24
Euclid’s Elements of Geometry - Book I, Proposition 8      25
Euclid’s Elements of Geometry - Book I, Proposition 9      26
Eudoxus      43 118
Euler, Leonhard on geometry of position (topology)      197—206 207—15
Euler, Leonhard, background of      205—7
Extraordinary, defined      228
Fifth Postulate of Euclid      17
Fifth Postulate of Euclid in analytic geometry      109
Fifth Postulate of Euclid, Euclid’s first use of      82—83
Fifth Postulate of Euclid, Saccheri’s critique of      78—80
Fifth Postulate of Euclid, substitutes for      83—84 94
Finitude and mathematical, induction      175—82 190—94
Fractions      151
Frege, Gottlob      169 180 1 181 188
Gambler’s fallacy      238—39
Gauss, Karl Friedrich      152
Gauss, Karl Friedrich, measures angles      95
Gelon, King of Syracuse      116 129 130
Geodesics      93—94
Geometry (Descartes)      97—102
Geometry (Descartes) vs. Euclid’s method      104—10
Geometry of position      (see Topology)
Geometry, analytic      96 104
Geometry, analytic advantages of      109—11
Geometry, Euclidean      (see Euclid’s Elements of Geometry)
Geometry, non — Euclidean      80
Geometry, non — Euclidean Riemannian      93 94
Geometry, non — Euclidean which is most suitable?      94—95 (see also Lobachevski Nicholas)
Geometry, postulates in      52—53
Geometry, proof in      43
Geometry, proof in reduction to the absurd      60—62
Geometry, proof in superimposition      58
Geometry, proof in synthetic vs. analytic method      109—10
Geometry, thinking scientifically about      40—43
Great circles, non-Euclidean, geometry and      91—94
Heine, E.      139
Heliocentric theory      117—18 130—32
Hereditary property of numbers      177 180 191
Induction, mathematical      165 177—82 190—94
Inductive numbers      182
Inductive property of numbers      177 191
Infinity of numbers in Archimedes      116—26 129—37
Infinity of numbers in Russell      179
Introduction to Mathematical, definition of number      169—76 184—88
Introduction to Mathematical, finitude and mathematical induction      175—82 190—94
Introduction to Mathematical, Philosophy (Russell)      161—94
Introduction to Mathematical, series of natural numbers      161—69
Irrational numbers, described by Dedekind’s cut      145—49 152—60
Irrational numbers, discovered      149
Konigsberg, problem of      198—206 207—16
Lagrange, Joseph Louis      219
Language      (see Logic)
Laplace, Pierre Simon de background of      230—31
Laplace, Pierre Simon on probability      219—30 233—41
Laws of Thought, classes of signs      246—56 269
Laws of Thought, derivation of laws of symbols      256—66 275
Laws of Thought, laws of symbols      247—56 269—71
Laws of Thought, The (Boole)      243—66 268—75
Leibniz, Baron Gottfried von      197—98 215 220
Lines, cutting and not-cutting      70 84
Lines, Euclid’s definition of      16 45—46
Lines, parallel      (see Parallel lines)
Lines, skew      47
Lines, straight      (see Straight lines)
Lobachevski, Nicholas, analytic geometry and      109
Lobachevski, Nicholas, angles and triangles      74—77 84—92
Lobachevski, Nicholas, background of      80
Lobachevski, Nicholas, publication of      80
Lobachevski, Nicholas, substitutes postulate for Euclid’s Fifth      84
Lobachevski, Nicholas, The Theory of Parallels      68—77
Logic, derivation of      256—66 275
Logic, laws of symbols in Boole      247—56 269—71
Logic, mathematics and symbolic      268 275
Logic, mathematics in Boole      242 267—68
Logic, mathematics in Frege      188
Logic, mathematics in Peano      188
Logic, mathematics in Russell      188—89 190 194
Maps, coloring of      217
Marcellus      129
Mathematical induction      165 177—82 190—94
Mathematics, common, characteristic of      267
Mecanique Celeste      230—31
Mechanical brains      242
Mersenne, Marin      102
Moebius strip      215—17
Multiplication      150 128—29
Multiplication synthetic vs. analytic method      109—10
Non — Euclidean geometry      80
Non — Euclidean geometry of Lobachevski      68—77
Non — Euclidean geometry which is most suitable      94—95
Non — Euclidean geometry, analytic geometry and      109
Non — Euclidean geometry, angles and triangles      74—77 84—92
Non — Euclidean geometry, postulate substituted for Euclid’s Fifth      84
Non — Euclidean geometry, published      80
Non — Euclidean geometry, Riemannian      93 94
Numbers, defined by Euclid      114—15 150
Numbers, defined by Russell      169—76 184—88
Numbers, inductive      182
Numbers, irrational described by Dedekind’s cut      145—49 152—60
Numbers, irrational discovered      149
Numbers, natural      150
Numbers, natural, theory of      164—69 176—82 190—94
Numbers, quantity of, in Archimedes      116—26 129—30
Numbers, quantity of, in Euclid      115—16
Numbers, quantity of, in Russell      175—82 190—94
Numbers, rational, compared with points on straight line      141—45 152—58
Numbers, rational, defined      152
Numbers, rational, properties of      140—42
Opposite, reduction to the      128—29
Parallel lines in Euclid’s Elements      44
Parallel lines in Lobachevski      84 91—93
Parallel lines in Lobachevski, theorems      70—74
Parallel lines, defined      17 47
Parallel lines, propositions      63—68 80—83
Parallel lines, three possibilities for      94
Parallelograms in Euclid      44
Peano, Giuseppe      183
Peano, Giuseppe, number theory of      164—69 179 188
Pheidias      118
Planetary theory in Aristarchus and Archimedes      117—18 131
Planetary theory, Laplace’s work on      230—31
Poincare, Henri      181
Point, defined in Euclid’s Elements      16 45—46
Position, relative      (see Topology)
Posterity of numbers      177—78 180 192—93
Postulates in Descartes      109
Postulates in Euclid’s Elements      17 47—50
Postulates, function of, in geometry      52—53
Postulates, lacking in number theory      126—27 (see also Fifth Postulate of Euclid)
Prime numbers in Euclid, defined      114
Prime numbers in Euclid, quantity of      114—16 127—29
probability      218—41
Probability a priori vs. a posteriori, arguments      233
Probability, defined      224 234 235
Probability, gambler’s fallacy in      238—39
Probability, Laplace on      219—30 233—41
Probability, rules for      224—30 234—41
Probability, truth and      231
Progressions, Peano’s theory and      167
Proofs, method of superimposition      58
Proofs, reduction to the absurd      60—62 155
Proofs, reduction to the opposite, hereditary      177 180 191
Propositions of Euclid      (see Euclid’s Elements of Geometry)
Pseudosphere      93—94
Pythagoras      149 164
Pythagorean theorem of Euclid      108—9
Q.E.F.      53
Rational numbers, compared with points on straight line      141—45 152—58
Rational numbers, defined      152
Rational numbers, properties of      140—42
Reason, derivation of symbols of logic from laws of      256—66 275
Reasonable degree of belief      220 232
Reduction in Russell’s definition of number      188
Reduction to the absurd      60—62 155
Reduction to the opposite      128—29
Relations, signs of, in Boole      252—56
Relations, types of, in Russell      173
Relative position      (see Topology)
Riemann, Bernhard      80
Riemann, geometry of      93 94
Right angles, Euclid’s definition of      16 46—47
Russell, Bertrand, background of      183—84
Russell, Bertrand, definition of number      169—76 184—88
Russell, Bertrand, induction      175—82 190—94
Russell, Bertrand, Introduction to Mathematical Philosophy      161—94
Russell, Bertrand, series of rational numbers      161—69
Saccheri, Girolamo, on Euclid’s Fifth Postulate      78—80
Sand Reckoner, The (Archimedes)      116—26 129—37
Schnitt, Dedekind’s      145—49 152—60
Science, object of      256
Sets      (see Classes)
Signs, classes of      246—56 269
Signs, defined      244 (see also Laws of Thought)
Similarity and class      174 185 188
Skew lines      47
Sphere, non-Euclidean geometry and      91—94
Square roots      (see Irrational numbers)
Stadium (Greekmeasure), length of      132
Straight lines in Euclid’s Elements      44 52—57
Straight lines in Lobachevski      70 76—77 84 92—94
Straight lines in Riemann      93 94
Straight lines, defined      16 45—46
Straight lines, propositions      17—21 26—28
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