BIBLIOGRAPHY

In the following list of books we have not attempted to be comprehensive. New books are appearing rapidly, and the reader may very easily find some among them which he prefers. Also there are now several series of surveys on mathe­matics, sometimes translated from foreign languages and issued in English, particularly by American and British publishers. We have in mind the series by Random House and by the Mathematical Association of America, together with some excellent elementary texts from Russia. Also there are now several journals catering for the teaching of mathematics, such as Mathematics Teaching and the Mathematical Gazette, published in Britain, and the American Mathematical Monthly, published in the United States. These contain many interesting articles relating to small points in mathematics which have been found to lead to interesting teaching situations. At the same time they review the flood of relevant modern literature.

Perhaps we should draw attention also to books by mathematicians which might help to reveal the thought-processes of a mathematician; in particular, we mention the books by Hardy [51] and Littlewood [83], those of Bell [10, 12] and several articles in Newman [93].

[1] ALBERT, A. A. Structure of Algebras, American Mathematical Society Colloquium Publications no. 24 (1939).

[2] ALEXANDROFF, P. Elementary Concepts of Topology, Dover, New York (1961 ). [3] ALEXANDROFF, P. and HoPF, H. Topologie, Vol. I (Grundlehr. math. Wiss. 5)

Springer, Berlin (1935). [4] APOSTOL, T. M. Mathematical Analysis, Addison-Wesley, Reading, Mass.

(1960). [5] ARNOLD, B. H. Introduction to Concepts of Elementary Topology, Prentice­

Hall, Englewood Cliffs, N.J. (1964). [6] ARTIN, 1E. Galois Theory, 2nd edn, Notre Dame Mathematical Lectures

(1948). [7] AusLANDER, L. Differential Geometry, Harper and Row, New York (1967). [8] AusLANDER, L. and MACKENZIE, R. E. Introduction to Differentiable Manifolds,

McGraw-Hill, New York (1963). [9] BELL, A. W. and FLETCHER, T. J. Symmetry Groups, pamphlet no. 12 of

Mathematics Teaching, Association of Teachers of Mathematics (1964). [10] BELL, E. T. Men of Mathematics, Penguin Books, Harmondsworth, Middx

(1953). [11] BELL, E. T. Mathematics, Queen and Servant of Science, Bell, London (1952). [12] BELL, E. T. The Development of Mathematics, 2nd edn, McGraw-Hill, New

York (1945). [13] BERGE, C. The Theory of Graphs, Methuen, London (1962). [14] BING, R. H. Elementary Point-Set Theory, pamphlet no. 8 of American

Mathematical Monthly (1960). 617

618 BIBLIOGRAPHY

[15] BIRKHOFF, G. Lattice Theory, 3rd edn, American Mathematical Society Colloquium Publications no. 25 (1966).

[16] BIRKHOFF, G. and MAcLANE, S. A Survey of Modern Algebra, revised edn, Macmillan, London (1965).

[17] BoURBAKI, N. Elements de mathematique, Hermann, Paris (1939 on); the parts General Topology, Parts 1 and 2, and Theory of Sets are available in English: Addison-Wesley, Reading, Mass. (1967, 1968).

[18] BoURBAKI, N. Eliments d'histoire des mathematiques, Hermann, Paris. [19] BoYER, C. B. A History of Mathematics, Wiley, New York (1968). [20] BusACKER, R. G. and SAATY, T. L. Finite Graphs and Networks, McGraw­

Hill, New York (1965). [21] CAIRNS, S. S. Introductory Topology, Ronald, New York (1961). [22] COHEN, P. J. Set Theory and the Continuum Hypothesis, Benjamin, New York

(1966). [23] COHN, P. M. Linear Algebra, Routledge and Kegan Paul, London (1959). [24] CoURANT, R.DifferentialandintegralCalculus, 2 vols, Blackie, London(1952). [25] COURANT, R. and RoBBINS, H. What is Mathematics?, Oxford University

Press, London (1941). [26] CROWELL, R. H. and Fox, R. H. Introduction to Knot Theory, Ginn,

Waltham, Mass. (1963). [27] CUNDY, M. and RoLLETT, A. P. Mathematical Models, 2nd edn, Clarendon

Press, Oxford (1961). [28] DAVENPORT, H. The Higher Arithmetic, Hutchinson, London (1952). [29] DAVIS, M. Computability and Solvability, McGraw-Hill, New York (1958). [30] DIEUDONNE, J. Foundations of Modern Analysis, Academic Press, New York

(1960). [31] DODGSON, Rev. C. L. (Lewis Carroll) Symbolic Logic, reprint, Dover, New

York (1958). [32] DoRN, W. S. and GREENBERG, H. J. Mathematics and Computing, Wiley,

New York (1967). [33] DuFF, G. F. D. Partial Differential Equations, Clarendon Press, Oxford

(1956). [34] DuRELL, C. V. Projective Geometry, Macmillan, London (1931). [35] ElLENBERG, S. and STEENROD, N. Foundations of Algebraic Topology

Princeton University Press, Princeton, N.J. (1952). [36] FLANDERS, H. Differential Forms with Applications to the Physical Sciences,

Academic Press, New York (1963). [36a] FLETCHER, T. J. (editor) Some Lessons in Mathematics: A Handbook on the

Teaching of 'Modern Mathematics' by members of tht· Association of Teachers of Mathematics, Cambridge University Press, L01. don (1964).

[37] FoRT, M. K (editor) Topology of 3-Manifolds and Related TvtJics, Prentice­Hall, Englewood Cliffs, N.J. (1962).

[38] GARDNER, M. Scientific American Book of Puzzles and Diversions, Simon and Schuster, New York (1959).

[39] GARDNER, M. The Second Scientific American Book of Puzzles and Diversions, Simon and Schuster, New York (1961).

[40] GARDNER, M. New Mathematical Diversions from Scientific American, Simon and Schuster, New York (1966).

BIBLIOGRAPHY 619

[41] GASKELL, R. E. Engineering Mathematics, Staples, London (1960). [42] GELFAND, I. M., RAIKOV, D., and SHILOV, G. Commutative Normed Rings,

Chelsea, New York (1964). [43] GoDEL, K. On Formally Undecidable Propositions (English trans.), Oliver

and Boyd, Edinburgh (1962). [44] GoDEL, K. 'The consistency of the continuum hypothesis' Ann. Math.

Stud. 3 (1940). [45] GRIFFITHS, H. B. Topology, reprint from Mathematics Teaching, Association

of Teachers of Mathematics (1967). [46] HADWIGER, H., DEBRUNNER, H. and KLEE, V. Combinatorial Geometry in the

Plane, Holt and Rinehart, New York (1964). [47] HALL, D. W. and SPENCER, G. L. Elementary Topology, Chapman and Hall,

London (1955). [48] HALL, M. Theory of Groups, Macmillan, London (1959). [49] HALMOS, P. R. Finite-Dimensional Vector Spaces, Van Nostrand, Princeton,

N.J. (1958). [49a] HALMOS, P.R. Measure Theory, Van Nostrand, Princeton, N.J. (1950).

[50] HALMOS, P.R. Naive Set Theory, Van Nostrand, Princeton, N.J. (1960). [51] HARDY, G. H. A Mathematician's Apology, reprint of 1st edn with foreword

by C. P. Snow, Cambridge University Press, London (1967). [52] HARDY, G. H. Pure Mathematics, Cambridge University Press, London

(1965). [53] HARDY, G. H. Divergent Series, Clarendon Press, Oxford (1949). [54] HARDY, G. H. and WRIGHT, E. M. An Introduction to the Theory of Numbers,

Clarendon Press, Oxford (1959). [55] HAUSDORFF, F. Mengenlehre (1927); English trans., Set Theory, Dover,

New York (1957). [56] HERSTEIN, I. N. Topics in Algebra, Blaisdell, Waltham, Mass. (1964). [57] HILBERT, D. The Foundations of Geometry, English trans. by E. J. Townsend,

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43, Cambridge University Press, London (1953). [63] HILTON, P. J, Homotopy Theory and Duality, Nelson, London (1967). [64] HILTON, P. J. and WYLIE, S. Homology Theory, Cambridge University Press,

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University Press, London (1947-54). [67] HUREWICZ, W. and WALLMAN, H. Dimension Theory, Princeton University

Press, Princeton, N.J. (1961). 21*

620 BIBLIOGRAPHY

[68] JEFFREYS, B. S. and JEFFREYS, H. Methods of Mathematical Physics, 3rd edn, Cambridge University Press, London (1956).

[69] KELLEY, J. L. General Topology, Van Nostrand, Princeton, N.J. (1955). [70] KELLEY, J. L. Inttoduction to Modern Algebra, Van Nostrand, Princeton,

N.J. (1960). [71] KELLOGG, 0. D. Potential Theory, Chelsea, New York (1947). [72] KEMENY, J. G., SNELL, J. C., and THoMPSON, G. L. Introduction to Finite

Mathematics, Prentice-Hall, Englewood Cliffs, N.J. (1959). [73] KEMENY, J. G., SNELL, J. C., THOMPSON, G. L., and MIRKIL, H. Finite

Mathema_tical Structures, Prentice-Hall, Englewood Cliffs, N.J. (1959). [74] KEMENY, J. G. and SNELL, L. Mathematical Models in the Social Sciences,

Blaisdell, Waltham, Mass. (1962). [74a] KINGMAN, J, F. C. and TAYLOR, S. J. Introduction to Measure and Prob­

ability, Cambridge University Press, London (1966). [75] KLEIN, F. On Riemann's Theory of Algebraic Functions and Their Integrals,

Chelsea, New York (1963). [76] KLEIN, F. Elementary Mathematics from the Advanced Standpoint, Dover,

New York (1939). [77] KoLMOGOROFF, A. N. and FoMIN, S. V. Elements of the Theory of Functions

and Functional Analysis, Graylock, New York (1957). [78] KREYSZIG, E. Advanced Engineering Mathematics, Wiley, New York (1962). [79] LANG, S. Algebra, Addison-Wesley, Reading, Mass. (1967). [80] LEDERMANN, W. Introduction to the Theory of Finite Groups, 4th edn,

Oliver and Boyd, Edinburgh (1961). [81] LEFSCHETZ, S. Introduction to Topology, Princeton University Press,

Princeton, N.J. (1949). [82] LIGHTHILL, M. J, Introduction to Fourier Analysis and Generalised Functions,

Cambridge University Press, London (1958). [83] LITTLEWOOD, J. E. A Mathematician's Miscellany, Methuen, London (1953). [84] MAcLANE, S. Homology, Springer, New York (1963). [85] MAcLANE, S. and BIRKHOFF, G. Algebra, Macmillan, London (1967). [86] MICHIE, D. 'Trial and error' Penguin Science Surveys, Penguin Books,

Harmondsworth, Middx (1961). [87] MILNE-THOMSON, L. M. Theoretical Hydrodynamics, 4th edn, Macmillan,

London (1962). [88] MIRSKY, L. An Introduction to Linear Algebra, Clarendon Press, Oxford

(1955). [89] MITCHELL, B. Theory of Categories, Academic Press, New York (1965). [90] MOISE, E. E. Elementary Geometry from an Advanced Standpoint, Addison­

Wesley, Reading, Mass. (1963). [91] MoRSE, M. Calculus of Variations in the Large, American Mathematical

Society Colloquium Publications no. 18 (1934). [92] NAGEL, E. and NEWMAN, J. R. Godel's Proof, Routledge and Kegan Paul,

London (1959). [93] NEWMAN, J. R. (editor) The World of Mathematics, Simon and Schuster,

New York (1956). [94] NEWMAN, M. H. A. Elements of the Topology of Plane Sets of Points, Cam­

bridge University Press, London (1954).

BIBLIOGRAPHY 621

[95] NICKERSON, H. K, SPENCER, D. C., and STEENROD, N. Advanced Calculus, Van Nostrand, Princeton, N.J. (1959).

[96] O'HARA, C. W. and WARD, D. R. An Introduction to Projective Geometry, Clarendon Press, Oxford (1937).

[97] 0LDS, C. D. Continued Fractions, Random House, New York (1963). (98] OLMSTED, J. M. H. Intermediate Analysis, Appleton-Century-Crofts, New

York (1956). (99] OsGOOD, W. F. Functions of Real Variables, Chelsea, New York (1935).

[100] PAPY, G. Groups, Macmillan, London (1964). [101] PATTERSON, E. M. Topology, Oliver and Boyd, Edinburgh (1959). [102] PATTERSON, E. M. Vector Algebra, Oliver and Boyd, Edinburgh (1968). [103] PoSTNIKOV, M. M. Foundations of Galois Theory, Clarendon Press, Oxford

(1962). [104] QUADLING, D. A. Introduction to Analysis, Clarendon Press, Oxford (1959). [105] RADEMACHER, H. and TOEPLITZ, 0. The Enjoyment of Mathematics, Princeton

University Press, Princeton, N.J. ( 19 57). [106] REDISH, K. A. An Introduction to Computational Methods, E.U.P., London

(1961). [107] REuTER, G. E. H. An Introduction to Differential Equations, and Linear

Operators, Routledge and Kegan Paul, London (1958). [108] RossER,]. B. Logic for Mathematicians, McGraw-Hill, New York (1953). [109] RuDIN, W. Principles of Mathematical Analysis, 2nd edn, McGraw-Hill,

New York (1964). [110] RuDIN, W. Real and Complex Analysis, McGraw-Hill, New York (1966). [111] RussELL, B. History of Western Philosophy, Allen and Unwin, London (1957). [112] RussELL, B. and WHITEHEAD, A. N. Principia Mathematica, 3 vols, Cam-

bridge University Press, London (1910-13). [113] RYSER, H. J. Combinatorial Mathematics, Wiley, New York (1963). [114] SAATY, T. L. Lectures on Modern Mathematics, Vols I, II, III, Wiley, New

York (1963, 1964, 1966). [115] SAWYER, W. W. Prelude to Mathematics, Penguin Books, Harmondsworth,

Middx (1955). [116] ScHWARTZ, L. Introduction to the Theory of Distributions, Toronto University

Press, Toronto (1952). [117] SEIFERT, H. and THRELFALL, W. Lehrbuch der Topologie, Chelsea, New York

(1947). [118] SEIFERT, H. and THRELFALL, W. Variationsrechnung in Grossen, Chelsea,

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(1961). [121] SOMMERVILLE, D. M. Y. Analytical Conics, Bell, London (1937). [122] STEINER, H. G. 'Mathematisierung und Axiomatisierung einer politischen

Struktur' Der Mathematikunterricht 3 (1966). [123] STOLL, R. R. Sets, Logic and Axiomatic Theories, Freeman, San Francisco

(1962). [124] THOMPSON, D'ARcY W. On Growth and Form, abridged and edited by J. T.

Bonner, Cambridge University Press, London (1961).

622 BIBLIOGRAPHY

[125] THURSTON, H. A. The Number System, Interscience, New York (1959). [126] TuLLER, A. A Modern Introduction to Geometries, Van Nostrand, Princeton,

N.J. (1967). [127] TuRING, A. M. 'Computing machinery and intelligence' Mind 59, 433-460

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Waltham, Mass. (1960). [131] VANDER WAERDEN, B. L. Modern Algebra, Ungar, New York (1953). [132] WANG, HAo IBM Jl. Res. Dev. 4 (1960). [133] WASSERMAN, E. 'Chemical topology' Scient. Am. (Nov. 1962). [134] WEYL, H. Symmetry, Princeton University Press, Princeton, N.J. (1952). [135] WHITESITT, J. E. Boolean Algebra and its Applications, Addison-Wesley,

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of Atomic Spectra, Academic Press, New York (1959). [136] WILDER, R. L. Introduction to the Foundations of Mathematics, Wiley, New

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Abel, 128, 134, 366 abelian group, xiv, 128, 289 abelianizer, 307 abelianizing functor, 580 absolute value, 146, 469 abuse of language, 24 Adams, 404 addition formulae, 523 affine equivalence, 213 Alexander's horned sphere, 411 algebra, xiv

of limits, 460 of sets, 103

algebraic inversion theorem, 31 number, 34, 383, 402 topology, 439

algebraically closed, 399 algorithm, 157 altitude, 331 analysis, 451 analytic function, 537 angle, 245, 329 anticommutativity, 199 approximate integration, 532 arbitrarily small (family), 421 arc, 414 Archimedean order, 393 Archimedes, 382 area, 228 Argand diagram, 149, 400 argument (of function), 18 Aristotelian axiom, 71 arithmetic, xiii, 121 arithmetic mean, 342 associates, 151 associative law, 46, 288 associativity of composition, 27 atom, 351 augmented matrix, 325 automorphism, 133, 302 axial perspective, 265 axiom, 124, 242, 594

of choice, 96, 615 of Eudoxus, 344

B-algebra, 108 basis, 186, 312

INDEX

623

Beltrami, 600 bijection, 21 binary operation, 124 binomial

expansion with remainder, 528 theorem, 70

Bolyai, 600 Boole, 347 Boolean

algebra, xiv, 110, 347 ideal, 111

Borel sets, 353 bound variable, 610 boundary, 444

condition, 514 bounded

above, 67 sequence, 389

Brianchon's theorem, 270 Brouwer, 411

fixed-point theorem, 407 Burali-Forti paradox, 598

cr.functions, 559 Cairns, 447 calculus, xv, 451 canonical projection, 333 Cantor, 71, 90 cardinal, 91 Cartesian product, 38, 97, 414, 537

of categories, 581 casting out 9's, 136 category, xii, 571 Cauchy sequence, 386 Cauchy-Riemann equations, 557 Cayley numbers, 404 Cayley's theorem, 302 cell, 202 central perspective, 265 centralizer, 304 centre, 304, 593

of perspectivity, 277 of sphere, 189

chain rule, 478, 554 characteristic, 365 characteristic function, 107 Chinese remainder theorem, 144

624

choice function, 96 of representative, 57

chord, 331 Church, 135 circle, 189 circular points at infinity, 275 cis, 520 class, 598 closed, 127

interval, 453 manifold, 444 set, 418

closure, 421 codomain, 572, 573 Cohen (Paul), 102, 615 cohomology, 441 colllneation, 265 combination, 80, 82 combinatorially equivalent, 447 common

factor, 11 multiple, 11

commutative group, 128, 289 ring, 124

commutator, 304 subgroup, 304, 584

compact, 426, 433 polyhedron, 440

complement, 12 complementary function, 516 complementation, 348 complete, 398 completeness, 610

of IC, 401 completion, 398 components (of morphism), 588 complex numbers, 399 complex-valued function, 520 composition. (of functions), 26, 468 computer science, vii conformal, 557 congruence (mod m), 56, 138 congruent (mod m), 138 conic, 197, 268 conjugate subgroup, 302 connected (graph), 67 consistency, 600, 610 w-consistency, 614 constancy test, 486 constant

function, 19 homomorphism, 575

INDEX

continued fractions, 174 continuous, 417, 423, 466, 542 continuously differentiable, 559 continuum, 92

hypothesis, 101, 615 contour, 560 contractible, 43 5 contravariant, 77, 579 converge, 386, 464 convergent (to a continued fraction), 175 convex, 188 convolution, 503 co-ordinate, 38

charts, 554 plane, 4 system, 553

coproduct, 593 corollary, 1 correspondence, 19 coset, 305

of xmodR, 57 representatives, 58 space, 305

cosine, 247 countably infinite, 34, 92 counter-example, 16 counting, 33, 88 covariance, 440 covariant, 77, 440, 578 covering, 433 Cramer's rule, 208, 323 Cretan paradox, 598 critical point, 560 cross-cap, 445 cross-ratio, 279 crystallographic groups, 308 cube, 39 cubic, 367 cyclic

group, 293 permutation, 291

cylinder, 39

decimal, 393 decomposition

space, 430 theorem, 308

Dedekind, 343 deduction, 117 definite integral, 24, 495 definition, 1, 242 deformable, 435 degenerate line, 187 degree (of polynomial), 147, 356

De Moivre's theorem, 522 De Morgan laws, 12 denumerably infinite, 34 derivation, 363 derivative, 24, 363, 475, 545 derived group, 304 Desargues' theorem, 258, 265 Descartes, 600 determinant, 204, 319, 324 diagonal, 41

process (of Cantor), 92 diagram, 411 dictionary ordering, 53 differentiable, 474, 546

manifold, 554 differential, 479, 542, 549

coefficient, 4 7 4 equation, 514 geometry, 539

dimension, 186, 424 of vector space, 313

Diophantine equation, 144 Dirac ll-function, 501 direct

product, 142 sum, 130, 299, 315

direction, 254 ratios, 187

discriminant, 268, 295 disjoint sets, 9 distance, 328 distinguished element, 373 distribution, 500 distributive law, 40, 106 divisible, 10 division algebra, 455 divisors of zero, 132 domain, 18, 50, 573

of discourse, 571 dual

basis, 335 category, 593 of Boolean algebra, 111, 347 vector space, 333, 580

duality in projective geometry, 261 open - closed, 422

dummy suffix, 65 variables, 496

duplicating the cube, 134

edge {of graph), 51 eigenvalue, 319

INDEX

eigenvector, 319 element (of a set), 4 ellipse, 268 embed, 26 empty set, 3, 7 endomorphism, 302 entries (in a matrix), 203 Entscheidungsproblem, 610 epic, 585 epimorphism, 301 equality

of functions, 20 of sets, 6

equation, 14 equivalence, 21, 574

relation, 56 induced by a function, 58 induced by partitioning, 59

equivalent formulae, 114 objects, 574 sets, 32

Erlanger Program, 180, 278, 407 Euclid, 147, 157 Euclidean

axiom, 147 domain, 150 geometry, 249 norm, 147 topological space, 408

Euclid's algorithm, 157, 359 theorem (on primes), 162

Eudoxus, 372, 382 Euler, 68, 141, 407, 448

characteristic, 322, 44 7 exchange lemma, 312 excluded middle, 606 exhaustion, 233 existence

proof, 15 theorem (for hcf), 156

exponential function, 24, 509 growth, 513

exponentially bounded, 518 exponentiation {of cardinals), 93

factor, 10, 154 group, 306

factorial function, 80 factorization into primes, 162 Fermat, 142 Fermat's last theorem, 250

625

626

field, 131 of fractions, 402

finite cardinal, 93 sequence, 388 set, 33, 89 subcovering, 433

finite-dimensional, 312 five-colour theorem, 407 fixed point (of permutation), 82 flat, 315 flow chart, 210 forgetful functor, 579 formal system, 603 formula, 115, 604 four-colour problem, 407 Fourier series, 538 free

abelian group, 298 differential calculus, 410 group, 580 variable, 610

Frege, 90 frontier, 421 full subcategory, 576 function, 1, 18, 42

of several variables, 540 functional analysis, 455 functor, xii, 577

category, 582 fundamental

group, 439 sequence, 386 theorem of algebra, 363, 399 theorem of calculus, 495

Galois, 134, 366 theory, 134, 287

gamma function, 504 Gauss, 363, 371 Gaussian

integers, 148 sphere, 539

generalized function, 502 generator (of cyclic group), 293 generators

of group, 294, 577 of ideal, 153, 156 of vector space, 312

Gentzen, 610 genus, 445 geodesic, 601 geometric mean, 342 geometry, xiv, 179, 241

INDEX

G-equivalent, 283 Giidel, 101, 135, 569, 602

number, 612 Giidel's

proofs, 611 theorems, 608

gradient, 549 Gram-Schmidt process, 332 graph

of a function, 41 of a relation, 50

greatest lower bound, 350 Greek philosophy, 241 group, 129, 287

algebra, 404 representation, 319 table, 289

half-open interval, 453 Hamilton, 403 handle, 445 harmonic, 557

mean, 343 Hessian matrix, 560 higher derivatives, 482 hcf ( = highest common factor), 11, 1 54 Hilbert, 243, 282, 603 homeomorphic, 408 homeomorphism, 406,417 homogeneous

co-ordinates, 265 equations, 325 homology, 441 homomorphism, 48, 131, 137, 299, 329

of algebras, 455 of Boolean algebras, 352

homotopic, 435 rei x0 , 439

homotopy, 434 theory, 435

Hopf map, 443 hyperbola, 268 hyperbolic

functions, 512 geometry, 600

hypercomplex numbers, 402

ideal, xiv, 144, 152, 309 idempotent

laws, 104 operation, 348

identity, 14 function, 23

iff, xii

image, 20, 303 implication, 117 implicit function, 561 incidence relation, 259 inclusion, 4

map, 25 indefinite integral, 495 independence of representatives, 57 independent axiom, 602 index

of critical point, 560 of subgroup, 305

indexed family, 59 indexing set, 59 induction theorem, 55 inductive

definition, 64 method, 241 ordering, 98

inequalities, 338 inertia principle, 468 infinite

cardinal, 93 decimal, 393 integral, 504 limit, 462 product, 538 set, 33, 93

infinite-dimensional, 426 infinitesimal, 475 information theory, viii initial

conditions, 516 object, 575

injection, 21 inner

automorphism, 302 product, 190, 327

space, 328 integer, 4, 376 integral

domain, 146 norm, 146

integrand, 495 integration, 494

by parts, 500 intermediate value theorem, 469 intersection, 8 interval, 453 intuitionism, 15 invariants, 278 inverse, 47, 288

function, 30, 490 image, 26

INDEX

of relation, 52 inversion theorem, 28 invertible

element, 47 function, 30 morphism, 574

involution, 23 irreducible element, 163 isometry, 219, 329 isomorphic, 48, 301 isomorphism, 48, 111, 301, 574 iterative process, 530

Jacobian, 566 matrix, 549

Jordan arc, 225 content, 238 curve, 225, 410

theorem, 226

kernel, 111, 131, 144, 212, 303, 591 kernel-object, 592 Kirby, 448 Klein, 180, 282

bottle, 431 knot, 410

groups, 308 Koenigsberg bridges, 68, 407 Kronecker delta, 207

Lagrange remainder, 528 Lagrange's theorem, 165, 305 language of mathematics, 1 Laplace transform, 518 Laplace's equation, 547, 557 lattice, 202, 235, 350 law

of indices, 293, 511 of trichotomy, 5, 54

Lawvere, 599

627

Icm ( = least common multiple), 11, 160 least upper bound, 98, 344, 397 Leibniz, 451 Leibniz' theorem, 70, 482 lemma, 1 length, 188, 220, 330 level surfaces, 562 lexicographical ordering, 53 limit, 386, 458 line, 330

at infinity, 267 conic, 270

628

linear algebra, 310

of ~3 , 203 equations, 207, 322 function, 188 graph, 51, 67 operator, 514 ordering, 54, 338 programming, 330 transformation, 211, 315

linearly dependent, 185, 312 independen~ 185, 312

Liouville, 92 Lobachevsky, 600 local linearity, 543 locus (of path), 226 logarithm, 507 logarithmic function, 24 logic, 594

of geometry, 241 logical connective, 112 loop, 434 lower central series, 584

magic square, 138 magnification, 297 manifold, 442

with boundary, 444 map, mapping, 19, 417 mapped-interval theorem, 471 mathematical

induction, 62 logic, 594

matrix, 203, 217, 317 max, 469 maxima and minima, 529, 560 maximal

element, 97 ideal, 153

Mazur, 411 mean value theorem, 484, 499 measure theory, 220, 495, 539 meta-language, 603 meta-mathematics, 603 metric, 328

space, 414 metrizable, 419 m-fold zero, 364 Milnor, 447 min, 469 Mobius

band, 273, 431 transformation, 295

INDEX

model, 243, 441 of ~llJl2, 272

modulus, 146, 188 Modus Ponens, 118, 605 Moise, 447 monic, 585

polynomial, 359 monkey problem, 144 monomorphism, 111, 300 monotonic function, 471 morphism, 572 mostest theorem, 4 70 multiplication by scalars, 182 multiplicative

inverse, 47 system, 45

multiplicity (of a zero), 360

Napier, 507 natural

equivalence, 581 map, 60 numbers, 4 projection, 429 transformation, 581

negation, 15 negative integers, 376 neighbourhood, 420, 462 neutral element, 46, 288 Newton, 451 Newton's method, 366, 530 node (of graph), 51 non-commutativity of composition, 27 non-compact, 426 non-degenerate critical point, 560 non-Euclidean geometry, 331, 596, 600 non-metric space, 433 non-metrizable space, 433 non-negative integers, 4 non-orientable surface, 445 non-singular, 212, 300

matrix, 319 non-trivial solution, 210, 326 norm, 147, 188 normal

form, 445 subgroup, 304

normed domain, 150 null

sequence, 386 space, 316

nullhomotopic, 435 number system, xiv numerical analysis, viii

object, 572 object-language, 603 octonions, 404 one-one, 21 onto, 21 open

interval, 453 sentence, 14 set, 419

operator, 19 orbit, 82 order, 339

of group, 289 of point with respect to curve, 226 relation, 53

for cardinals, 94 ordered

field, 340 pair, 38,44 ring, 340 set, 53

ordering, 53, 33S subset, 339

order-preserving, 54, 340 ordinal

number, 101 series, 101

orientable surface, 445 orientation-reversing, 446 oriented area, 200 origin, 181 orthogonal, 219, 329

complement, 332 matrix, 336 transformation, 336

orthonormal basis, 332

pair, 38 Pappus' theorem, 267 parabola, 268 parallel, 331

directions, 254, 255 line and plane, 197 planes, 195 postulate, 331, 600 projection, 278

parallelogram law, 183 partial

derivative, 546 differential coefficients, 543 fractions, 171 ordering, 338 sums, 536

particular integral, 516

INDEX

partitioning, 58, 59 Pascal's

table, 83 theorem, 269

path, 220, 413 pathwise-connected, 413, 552 Peano, 55, 373, 424

axioms, 373, 608 mapping, 94

period, 489 permutation, 79 perpendicular, 331

directions, 254, 255 projection, 256 to two skew lines, 251

perspective, 258 perspectivity, 277 physical induction, 62 7T, 225 Picard's iteration process, 518 Pierce-De Morgan laws, 12, 348 pigeonhole principle, 66 place-holder, 14 plane, 194, 330

projective geometry, 259 Poincare, 424

conjecture, 444 point

in a category, 575 of inflexion, 530

polar, 271 pole, 271 polygon, 229 polygonal curve, 225 polyhedron, 239 polynomial, 34, 354, 455

equation, 133 form, 354 function, 354, 361

postulate, 242 power

series, 535 set, 7

predicate, 603 preservation of inequalities, 468 pretzel, 43 7 prime

element, 163 number, 10, 139

primitive, 495 polynomial, 168

principal ideal, 153 value, 525

629

630

principle of duality, 263 of mathematical induction, 63, 373

probability, 84 and statistics, viii

product in a category, 587 matrix, 217, 318 metric, 416

product-preserving functor, 589 projection, 41, 189

of IR3 onto IR, 181 projective geometry, 258 projectivity, 278 I proof, 1, 242, 610

game, 606 proper

factor, 154 ideal, 153 subset, 6

proposition, 1, 103 propositional calculus, 112 punctured disc, 413 Pythagoras, 382 Pythagoras' theorem, 330 Pythagorean metric, 415

quadratic equation, 367 form, 560 polynomial, 356 residue, 145

quantifiers, 14 quartic, 367 quaternions, 402 quintic, 366 quotient

field, 170, 171, 381 group, 306 map, 60 set, 57 space, 305, 315, 430

radius of convergence, 536 of sphere, 189

Rado (T.), 447 range, 18, 50, 573 rank

of abelian group, 308 of linear transformation, 320

rational function, 171, 402 numbers, 4, 379

INDEX

real algebra, 403, 454 numbers, 4, 388 projective plane, 260 projective space, 277

reciprocal duality, 266 function, 456

rectangular hyperbola, 197 rectifiable path, 222 recursive function, 609 reductio ad absurdum, 55, 119 reflection-inversion, 297 reflexive, 51 relation, 1, 50 relations (among generators), 577 remainder

in Taylor expansion, 526 theorem, 359

representation, 319 residue class, 136 restriction, 25 retract, 590 reverse (of path), 221 Riemann, 495, 539 right angle, 330 rigid motion, 219 rigidly equivalent, 219 ring, xiv, 124, 308

with integral norm, 146 Rolle's

conditions, 483 theorem, 483

root of equation, 133 of polynomial, 358

Rosser, 614 rotation, 245, 297 route (of linear graph), 67 Russell (Bertrand), 90 Russell paradox, 597

saddle point, 560 scalar product, 190, 193, 327 Schoenflies, 410 Schroeder-Bernstein theorem, 95 Schwartz (L.), 502 Schwartz paradox, 239 Schwarz' inequality, 328 second-order equation, 515 section (of rationals), 344 segmental path, 221 segmentally connected, 221 semantic paradox, 598

INDEX

semigroup, 574 separation (of IR3), 196 sequence, 386, 464 sequentially compact, 426 series, 535 set, 1, 3, 598

of functions, 73 theory, xiii

shuffles, 79 Siebenmann, 448 signed

area, 200 volume, 201, 215

simple dosed curve, 225 zero, 365

simply connected, 434 Simpson's rule, 534 simultaneous congruences, 143 sine, 247 singleton, 4 single-valued, 18 skew field, 403 Smale, 444, 449 small

category, 573 errors, 545

solid of genus p, 445 solid torus, 39 solution

domain, 358 set, 322 of congruences, 143

space-filling curve, 424 spanning set, 312 special norm, 172 sphere, 189, 331

with p handles, 445 spiral approach, ix, x splitting field, 369 square

matrix, 324 numbers, 69

squaring the circle, 229 stabilizer, 82 stationary points, 560 statistics, vii stereographic projection, 554 Stirling's formula, 84, 535 straight path, 221 strict indexing, 59 strictly decreasing, strictly increasing,

strictly monotonic, 455 strong axiom of choice, 97

structure, 243 constants, 403

subalgebra, 111, 454 subcategory, 576 subdivision (of polygon), 231 sub-field, 132 subgroup, 130, 293, 298 sub-ring, 127 subset, 5 subspace, 415 substitution, 354

function, 356 rule, 502

Sullivan, 448 sum (of series), 536 summability, 539 support, 500 surface, 442 switching circuit, 108 symmetric, 51

difference, 107 group, 82, 289

tangent, 269, 474 plane, 545

tautology, 118 Taylor expansion, 526, 559 tensors, 539 term

of formula, 609 of sequence, 386

term-by-term differentiation, 536

terminal object, 575 theorem, 1 theory of types, 599 thick torus, 39 topological

invariant, 434 space, 408, 419

topology, 282, 308 of JRn, XV, 406

torus, 39 totally ordered, 54 totient function, 141 transcendental

extension, 402 number, 92

transfinite arithmetic, 90 transformation, 19 transitive, 5, 51 translation, 191, 297, 315 transpose, 205, 335 trapezium rule, 533

631

632

triangle inequality, 193, 328, 415 of reference, 265

triangular numbers, 69 triangulation, 447 trigonometrical functions, 486, 523 trisecting an angle, 134 trivial

group, 575 homomorphism, 300, 575 solution, 210, 326

truth-table, 113 truth-value, 112 Turing, 135, 604, 611

underlie, 128 underlying functor, 579 unicursal graph, 67 unification, xii union, 8, 59 unique factorization domain, 170 uniqueness theorem (for hcf), 155 unit, 150, 574

circle, 128 matrix, 218 points (in II~P), 184

unity element, 124 universal mapping property, 585 universe of discourse, 3 unordered pair, 38 upper

bound,97, 396 semi-continuous, 430

INDEX

value, 18 vector, 191

addition, 182 equation ofline, 186 geometry, 181 product, 198 space, 310

IR3 , 181 subspace, 184, 314

vector-valued function, 548 Venn diagram, 5 vertex (of polygon), 229 visual perception, 450 volume, 200

Waring's problem, 165 Wedderburn, 404 Weierstrass, 455, 481 well-formed formula, 605 well-ordered, 54 Whitehead (J. H. C.), 447 wild arc, 413 Wilson's theorem, 142

Zeeman, 444,450 zero

characteristic, 365 morphism, 575 object, 575 of a function, 358 polynomial, 356

Zorn's lemma, 98

INDEX OF SPECIAL SYMBOLS

This index contains those symbols and notations which have a special meaning. We do not include notation (likef(x)) which is both standard and familiar to the reader. We have included very brief descriptions of the meanings of the symbols and a reference to a page number for those readers requiring further detail. Since some symbols defy alphabetical classification, we cannot avoid asking the reader occasionally to skim through several sections to find the symbol required. We draw particular attention to the last two items!

As;B A is included in B } 4 A:::?.B A contains B, B S:: A A;::;B equivalence of sets 32 A;::;B equivalence of objects of a category 574 A;::;B isomorphism of groups 301 A~B homeomorphism of spaces 413 AvB union of sets 8 AnB intersection of sets 8 A-B complement of B in A 12 A~B symmetric difference 107 AxB Cartesian product 38 #A cardinality of A 33 A/R set of cosets of A modulo equivalence relation R 57 [a] residue class of a 136 a= b (mod m) congruence modulo m 138 [a]m residue class, modulo m, of a 140 alb a divides b 154 afb a does not divide b 164 {an} sequence with an as nth term 416 (a, b) ordered pair 38 (a,b'} (a, b)

intervals on real line 21,453 <a, b) (a, b) a(\b vector product 198 ~ ring and abelian group axioms 124 d(G) area of G 228 AT transpose of the matrix A 335

BA set of functions from A to B 21, 73 IEB2 standard Boolean algebra 110 J1il axioms for Boolean algebra 347

c complement 12 card X cardinality of X 91 c cardinality of continuum (~) 92

633

634

e = (c1, c2, ca) c C[x] ~~ c. Cl(A) 'C(I) 'C"(I) 'C;(I)

'C(A) 'C'(A) ~at

detM :m deg dim..,X !2(I) Df de (f) !2.(I) !2p(A, jRm)

equ ~ U(!R2) e ea exp e iuc

f(A) ft> fA f-1 f-'(A) Fr(A) f':::!.g f(x) --->-l f(x)--->- oo ~

gof ~ glb

hs IHI H(U, V)

iff Im{f)

INDEX OF SPECIAL SYMBOLS

embedding of real into complex projective geometry 274 field of complex numbers 125 set of polynomials over C 125 axioms for commutative group 128 C with the point at infinity ( oo) 29 5 closure of A 421 set of functions continuous on the interval I 466 set of functions n times continuously differentiable on I 482 set of complex-valued functions n times continuously differentiable on I 524 set of functions continuous on A 542 set of functions r times continuously differentiable on A 559 axioms for a category 572

determinant of M 204 axioms for a distance function (metric) 328 degree of a polynomial 356 dimension of space X at point x 424 set of functions differentiable on the interval I 475 the derivative off 475 the differential off at c 480 set of complex-valued functions differentiable on I 524 set of functions A --->- jRm differentiable at p 549

equivalence of formulae 114 axioms of extension in plane projective geometry 262 IR2 embedded in projective geometry 267 embedding of IR2 in projective geometry 267 identity element of group G 299 exponential function 509 base of natural logarithms 511 axioms of plane Euclidean geometry 600

image of A under f 20 counterimage under f xi, 26 function from XA to YA induced by f: X--->- Y 75 function inverse to f 30 counterimage of A under f (conventional notation) 418 frontier of A 421 homotopy relation between maps 435 limit of function of a real variable 459 infinite limit of function of a real variable 462 formal model 604

composition off followed by g 27 group axioms 288, 289 greatest lower bound 350

function from BY to Bx induced by h: X--->- Y 76 set of quaternions 403 set of homomorphisms from U to V 317

if and only if xii image off 20

INDEX OF SPECIAL SYMBOLS 635

id identity function 23 I(A, B) set of injections A -->- B 77 0 standard unit interval 94 3J axioms for an ideal 152 IR" unit cube 213 3Jt) axioms for inner product 327 J identity function 454 Jrl• rational power of identity function 472

Ill locus of curve J 226

lub least upper bound 98,344 I at: axioms for a linear transformation 211 L(f) length of/ 223 lim Xn limit of a sequence 420 log x natural logarithm of real number x > 0 507 Log' (many-valued) logarithms of complex number ' 525

M(n,R) set of n x n-matrices over the ring R 126 ~jl properties of multiplication by scalars 183 ./(2 set of 2 x 2-matrices over IR 204 ~ properties of the determinant 206 1mn set of m x n-~atrices over 1R 317 Me(A,B) set of morphisms in e from A to B 572 M class of sets 598

N set of positive integers 4 Nn set of positive integers (1, 2, ... , n) 33 nP, number of injections N,--->- Nn 80 nC, number of subsets of r integers in Nn 80 ,1i axioms for an integral norm 146

121 empty set 7 ® order axioms 338 O(h) of the order of h 533 o(x) small compared with x 542

j)O// set of subsets of universe 7 pX power set of X 26 Perm A set (group) of permutations of A 82,289 p•q inner (scalar) product 190 P(a, b, c) parallelepiped determined by a, b, c 200 ~sa area axioms 228 gJ plane projective geometry 258 tllB axioms for plane projective geometry 259 &J(F) plane projective geometry over field F 260 &J(IR) plane projective geometry over IR 260 &J• dual projective geometry 262 &J(IC) plane projective geometry over IC 274 ll subgroup axioms 293 tll Peano axioms 373 p.c. path-connected 413 P :=:: q rel Xo homotopy of loops relative to base-point x0 439 [p] homotopy class of loop p 440 ~ parallel postulate 600

636 INDEX OF SPECIAL SYMBOLS

0 set of rationals 4 O[x] set of polynomials over 0 125 a subgroup axiom 298 <!Ill conditions on a family of compact subsets 429

IR" Euclidean n-space xiv IR real line, set of real numbers 4,388 jR2 Euclidean plane 4 IR+ set of positive real numbers 20 jR+ set of non-negative real numbers 25 IR[x] set of polynomials over IR 125 R set of rotations of unit circle 128 jR3 Euclidean 3-space 181 IRIJ:l>2 real projective plane 260 !RIJ:l>3 real projective 3-space 277 F;.(P1P2P3P4) cross-ratio 279 jRI set of functions I-->- IR 453 &t(I) set of functions on I satisfying Rolle's conditions 483

§1 unit circle 245 §2 unit sphere 272 S(A,B) set of surjections A -->- B 77 S.., set of formulae 115 s the world we live in 181 ~~ axioms for scalar product 193 s•(91') statement about 91' dual to s 263 s = (s., s2, s3) embedding of IR2 in complex projective geometry 275 §k-1 unit (k - 1) -sphere in IRk 415 z a part of mathematics 603 MT transpose of matrix M 205

0/1 universe of discourse 3 uiv u divides v 154 UFD unique factorization domain 170 UEB W direct sum (of vector subspaces) 315 -.<u, v>- inner (scalar) product of u and v 327,550 UL orthogonal complement of U 332 U,(x) r-neighbourhood of x 420 0/1 a covering of a space 433

w space of Euclidean vectors 191 ~~ axioms for vector product 199 Vol signed volume 201 vol unsigned volume 215 ~ vector space axioms 310

wff well-formed formula 604

{x} singleton set 4 xR a coset of a relation R 57 XfR quotient set of X by an equivalence relation R 57 ~ the polynomial x 355 X->-C x in a sufficiently small neighbourhood of c 459

x--+ f(x)

z z+ 1Ln 7L[x] 7L, ft'(J) 3 a F(=F1 )

rn A A 1r(X, x 0)

Ec EF EN <Pm w

N (aleph)

1

E

"' v 3 3! --,

& v

->--·-· =Cl)

~ :e • D => ...... § I

INDEX OF SPECIAL SYMBOLS 637

abbreviated way of describing a function which assigns to each x of an (understood) domain, an element f(x) of an (understood) range. Many writers replace the arrow by ~----;.-, or f"V->-. This avoids confusion where x is itself a set. 40

set of integers 4 set of non-negative integers 4 set of residue classes modulo n 58, 125 set of polynomials with integer coefficients 125 field of residue classes modulo prime p 365 set of complex-valued functions on I 520 elementary number theory 608

area vector 231 set of Gaussian integers 148 set of complex numbers a+ by -n, (a, bE 7L) 126 discriminant 203 lattice of points 215 fundamental group of X at x0 439 set of convergent sequences (in IQ) 388 set of Cauchy (=fundamental) sequences (in IQ) 388 set of null sequences (in IQ) 388 set of residues prime to m 140 ordinal of N with its usual order 101

a cardinal number 91

identity function 27,289

belongs to 4 does not belong to 4 for all 14 there exists 15 there exists a unique 251 contrary proposition, not 15, 112 ..• } or . . implies logtcal connectives 112

if and only if equivalence of propositions 103 is infinite 333 is homeomorphic to 413 is not homeomorphic to 425 ond of • p~f } indication of omission of proof

xii implies implies and is implied by a hard exercise 302 a difficult piece of text 373

Encounter with Mathematics

By L. Garding 1977. ix, 270p. 82 illus. cloth

The purpose of this text is to provide an historical, scientific, and cultural frame for the basic parts of mathematics encountered in college.

Nine chapters cover the topics of Number Theory, Geometry and Linear Algebra, Limiting Processes of Analysis and Topology, Differentiation and Integration, Series and Probability, and Applications. Each of these chapters moves from an historical introduction into a basic factual account, and finally into a presentation of the present state of the subject including, wherever possible, most recent research. Most end with passages from historical mathematical papers, as well as references to additional literature. Three remaining chapters deal with models and reality, the sociology, psychology, and teaching of mathematics, and the mathematics of the seventeenth century, providing a fuller historical background to infinitesimal calculus. Intended for beginning undergraduates, the text assumes background in high school or some college mathematics.

Undergraduate Texts in Mathematics

Apostol: Introduction to Analytic Number Theory. 1976. xii, 334 pages. 24 illus.

Childs: A Concrete Introduction to Higher Algebra. 1979. Approx. 336 pages. Approx. 7 ill us.

Chung: Elementary Probability Theory with Stochastic Processes. 1975. x, 325 pages. 36 illus.

Croom: Basic Concepts of Algebraic Topology. 1978. x, 177 pages. 46 illus.

Fleming: Functions of Several Variables. Second edition. 1977. xi, 411 pages. 96 illus.

Halmos: Finite-Dimensional Vector Spaces. Second edition. 1974. viii, 200 pages.

Halmos: Naive Set Theory. 1974. vii, 104 pages.

Hewitt: Numbers, Series, and Integrals. 1979. Approx. 450 pages.

Kemeny I Snell: Finite Markov Chains. 1976. ix, 210 pages.

Lax/ Burstein/ Lax: Calculus with Applications and Computing, Volume I. 1976. xi, 513 pages. 170 illus.

LeCuyer: College Mathematics with A Programming Language. 1978. xii, 420 pages. 126 illus. 64 diagrams.

Malitz: Introduction to Mathematical Logic. Set Theory - Computable Functions -Model Theory. 1979. Approx. 250 pages. Approx. 2 illus.

Prenowitz/ Jantosciak: The Theory of Join Spaces. A Contemporary Approach to Convex Sets and Linear Geometry. 1979. Approx. 350 pages. Approx. 400 ill us.

Priestley: Calculus: An Historical Approach. 1979. Approx. 409 pages. Approx. 269 ill us.

Protter/ Morrey: A First Course in Real Analysis. 1977. xii, 507 pages. 135 illus.

Sigler: Algebra. 1976. xii, 419 pages. 32 illus.

Singer/Thorpe: Lecture Notes on Elementary Topology and Geometry. 1976. viii, 232 pages. 109 illus.

Smith: Linear Algebra 1977. vii, 280 pages. 21 illus.

Thorpe: Elementary Topics in Differential Geometry. 1979. Approx. 250 pages. Approx. Ill ill us.

Wilson: Much Ado About Calculus. A Modern Treatment with Applications Prepared for Use with the Computer. 1979. Approx. 500 pages. Approx. 145 ill us.

Wyburn/Duda: Dynamic Topology. 1979. Approx. 175 pages. Approx. 20 ill us.