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Peter Wolff Breakthroughs in mathematics
Peter Wolff  Breakthroughs in mathematics









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: Breakthroughs in mathematics

: Peter Wolff

:

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.


: en

: /

:

ed2k: ed2k stats

: Library Rebound

: 1963

: 276

: 08.10.2012

: | | ID
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Absurd, reduction to the      6062 155
Addition      150
Addition in Peanos theory      166
Algebra, Booles symbolism and      27172 (see also Boole George)
Algebra, geometrical problems solved by      (see Geometry (Descartes))
Analytic geometry      96 104
Angles in Euclids Elements      4445
Angles in Euclids Elements, defined      16 4647
Angles in Euclids Elements, propositions      2631 37
Angles in Lobachevski      7477 8492
Angles, measured by Gauss      95
Apollonius      96
Archimedes, life of      129
Archimedes, The Sand Reckoner      11626 12937
Aristarchus of Samos      11718 131
Arithmetic, geometry and      1034
Arithmetic, Peanos analysis of      16469 179 188
Arithmetic, postulates in      12627
Arithmetic, reduced to logic      18889 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      24366 26875
Boole, George, The Laws of Thought, classes of signs      24656 269
Boole, George, The Laws of Thought, derivation of laws of symbols      25666 275
Boole, George, The Laws of Thought, laws of symbols      24756 26971
Buff on, Comte Georges de      229
Cantor, Georg      140
Chiliagon, in Archimedes      118 133
Circles in Euclid      1517 (see also Great circles)
Classes in Boolean algebra in definition of number      16976 18488
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 11516 12729
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      11415 12627
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      3738
Classes in Boolean algebra in definition of number Proposition 25      3839
Classes in Boolean algebra in definition of number Proposition 26      3941
Classes in Boolean algebra in definition of number Proposition 27      6364 8182
Classes in Boolean algebra in definition of number Proposition 28      6465
Classes in Boolean algebra in definition of number Proposition 29      6566 8283
Classes in Boolean algebra in definition of number Proposition 30      6667
Classes in Boolean algebra in definition of number Proposition 31      67
Classes in Boolean algebra in definition of number Proposition 32      6768 85
Classes in Boolean algebra in definition of number Proposition 47      1089
Classes in Boolean algebra, law of contradiction      26566 274
Classes in Boolean algebra, symbols for      24756 26465 26971
collections      (see Classes)
Coloring of maps      217
Common notions in Euclids, Elements      18 5051
Conic sections, defined      96
Constructions in Descartes      97102 10410
Constructions in Euclids Elements      5259
Constructions in Euclids Elements, compared with Descartes      1067
Constructions in Euclids Elements, propositions      1721 2628 37
Continuity and Irrational, Numbers (Dedekind)      13849 15260
Contradiction, principles of      26566 274
Converse domain      173
Counting, Archimedes on      11626 12937
Counting, decimal system and      137
Counting, Russell on      186 (see also Arithmetic Numbers)
Curves, equations for      1079
Cutting and not-cutting lines      70 84
Decimal system, Archimedes substitute for      137
Dedekind, Richard, background of      152
Dedekind, Richard, Continuity and Irrational Numbers      13849 15260
Deductiveness as characteristic of mathematics      26768
Definitions in Euclids Elements      1617 4547
Definitions in Euclids Elements of numbers      11415 12627 150
Descartes, Rene, background of      1023
Descartes, Rene, Geometry      97102
Descartes, Rene, Geometry vs. Euclids method      10410
Division by zero      15152
Division, number system and      15051
Domain, defined      173
Duality, law of, in Boole      266
Earth, diameter of, in Archimedes      117 132
Elements of Geometry      (see Euclids)
Euclid, his method vs. Descartes      10910
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 3      2021 57
Euclids Elements of Geometry - Book I, Proposition 4      2122 5759
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
Eudoxus      43 118
Euler, Leonhard on geometry of position (topology)      197206 20715
Euler, Leonhard, background of      2057
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      17582 19094
Fractions      151
Frege, Gottlob      169 180 1 181 188
Gamblers fallacy      23839
Gauss, Karl Friedrich      152
Gauss, Karl Friedrich, measures angles      95
Gelon, King of Syracuse      116 129 130
Geodesics      9394
Geometry (Descartes)      97102
Geometry (Descartes) vs. Euclids method      10410
Geometry of position      (see Topology)
Geometry, analytic      96 104
Geometry, analytic advantages of      10911
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, 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      10910
Geometry, thinking scientifically about      4043
Great circles, non-Euclidean, geometry and      9194
Heine, E.      139
Heliocentric theory      11718 13032
Hereditary property of numbers      177 180 191
Induction, mathematical      165 17782 19094
Inductive numbers      182
Inductive property of numbers      177 191
Infinity of numbers in Archimedes      11626 12937
Infinity of numbers in Russell      179
Introduction to Mathematical, definition of number      16976 18488
Introduction to Mathematical, finitude and mathematical induction      17582 19094
Introduction to Mathematical, Philosophy (Russell)      16194
Introduction to Mathematical, series of natural numbers      16169
Irrational numbers, described by Dedekinds cut      14549 15260
Irrational numbers, discovered      149
Konigsberg, problem of      198206 20716
Lagrange, Joseph Louis      219
Language      (see Logic)
Laplace, Pierre Simon de background of      23031
Laplace, Pierre Simon on probability      21930 23341
Laws of Thought, classes of signs      24656 269
Laws of Thought, derivation of laws of symbols      25666 275
Laws of Thought, laws of symbols      24756 26971
Laws of Thought, The (Boole)      24366 26875
Leibniz, Baron Gottfried von      19798 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, analytic geometry and      109
Lobachevski, Nicholas, angles and triangles      7477 8492
Lobachevski, Nicholas, background of      80
Lobachevski, Nicholas, publication of      80
Lobachevski, Nicholas, substitutes postulate for Euclids Fifth      84
Lobachevski, Nicholas, The Theory of Parallels      6877
Logic, derivation of      25666 275
Logic, laws of symbols in Boole      24756 26971
Logic, mathematics and symbolic      268 275
Logic, mathematics in Boole      242 26768
Logic, mathematics in Frege      188
Logic, mathematics in Peano      188
Logic, mathematics in Russell      18889 190 194
Maps, coloring of      217
Marcellus      129
Mathematical induction      165 17782 19094
Mathematics, common, characteristic of      267
Mecanique Celeste      23031
Mechanical brains      242
Mersenne, Marin      102
Moebius strip      21517
Multiplication      150 12829
Multiplication synthetic vs. analytic method      10910
Non Euclidean geometry      80
Non Euclidean geometry of Lobachevski      6877
Non Euclidean geometry which is most suitable      9495
Non Euclidean geometry, analytic geometry and      109
Non Euclidean geometry, angles and triangles      7477 8492
Non Euclidean geometry, postulate substituted for Euclids Fifth      84
Non Euclidean geometry, published      80
Non Euclidean geometry, Riemannian      93 94
Numbers, defined by Euclid      11415 150
Numbers, defined by Russell      16976 18488
Numbers, inductive      182
Numbers, irrational described by Dedekinds cut      14549 15260
Numbers, irrational discovered      149
Numbers, natural      150
Numbers, natural, theory of      16469 17682 19094
Numbers, quantity of, in Archimedes      11626 12930
Numbers, quantity of, in Euclid      11516
Numbers, quantity of, in Russell      17582 19094
Numbers, rational, compared with points on straight line      14145 15258
Numbers, rational, defined      152
Numbers, rational, properties of      14042
Opposite, reduction to the      12829
Parallel lines in Euclids Elements      44
Parallel lines in Lobachevski      84 9193
Parallel lines in Lobachevski, theorems      7074
Parallel lines, defined      17 47
Parallel lines, propositions      6368 8083
Parallel lines, three possibilities for      94
Parallelograms in Euclid      44
Peano, Giuseppe      183
Peano, Giuseppe, number theory of      16469 179 188
Pheidias      118
Planetary theory in Aristarchus and Archimedes      11718 131
Planetary theory, Laplaces work on      23031
Poincare, Henri      181
Point, defined in Euclids Elements      16 4546
Position, relative      (see Topology)
Posterity of numbers      17778 180 19293
Postulates in Descartes      109
Postulates in Euclids Elements      17 4750
Postulates, function of, in geometry      5253
Postulates, lacking in number theory      12627 (see also Fifth Postulate of Euclid)
Prime numbers in Euclid, defined      114
Prime numbers in Euclid, quantity of      11416 12729
probability      21841
Probability a priori vs. a posteriori, arguments      233
Probability, defined      224 234 235
Probability, gamblers fallacy in      23839
Probability, Laplace on      21930 23341
Probability, rules for      22430 23441
Probability, truth and      231
Progressions, Peanos theory and      167
Proofs, method of superimposition      58
Proofs, reduction to the absurd      6062 155
Proofs, reduction to the opposite, hereditary      177 180 191
Propositions of Euclid      (see Euclids Elements of Geometry)
Pseudosphere      9394
Pythagoras      149 164
Pythagorean theorem of Euclid      1089
Q.E.F.      53
Rational numbers, compared with points on straight line      14145 15258
Rational numbers, defined      152
Rational numbers, properties of      14042
Reason, derivation of symbols of logic from laws of      25666 275
Reasonable degree of belief      220 232
Reduction in Russells definition of number      188
Reduction to the absurd      6062 155
Reduction to the opposite      12829
Relations, signs of, in Boole      25256
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      18384
Russell, Bertrand, definition of number      16976 18488
Russell, Bertrand, induction      17582 19094
Russell, Bertrand, Introduction to Mathematical Philosophy      16194
Russell, Bertrand, series of rational numbers      16169
Saccheri, Girolamo, on Euclids Fifth Postulate      7880
Sand Reckoner, The (Archimedes)      11626 12937
Schnitt, Dedekinds      14549 15260
Science, object of      256
Sets      (see Classes)
Signs, classes of      24656 269
Signs, defined      244 (see also Laws of Thought)
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 Lobachevski      70 7677 84 9294
Straight lines in Riemann      93 94
Straight lines, defined      16 4546
Straight lines, propositions      1721 2628
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