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Books
Casti, J.L., (1996) Five Golden Rules: Great Theories of 20th Century Mathematics - and Why They Matter, John Wiley & Sons, New York, Chichester & Brisbane, xiv + 235 pp. [This book and its sequel are nice, non-technical introductions to some important areas of mathematics.]
Casti, J.L., (2000) Five More Golden Rules: Knots, Codes, Chaos , and Other Great Theories of 20th Century Mathematics, John Wiley & Sons, New York, Chichester & Brisbane, iv + 268 pp.
Chartrand, G., (1985) Introductory Graph Theory, originally published in 1977 by Prindle, Weber & Schmidt under the title Graphs as Mathematical Models; Dover Publications, New York, xii + 294 pp.
Davis, M. (Ed.) (2004) The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Originally published in 1965 by Raven Press Books; Dover, Mineola, NY, 413 pp.
Devlin, K., (1999) Mathematics: The New Golden Age, Columbia University Press, New York, xi + 320 pp. [An easy to read tour through modern mathematics.]
Dowling, E.T., (2001) Introduction to Mathematical Economics, 3rd Edition; Schaum's Ouline Series; McGraw Hill, New York & London, viii + 523 pp. [If you're doing advanced undergrad economics or starting a postgrad program, and aren't confident with your maths, work your way through Dowling. You'll be glad you did.]
Garrity, T.A., (2002) All the Mathematics You Missed: But Need to Know for Graduate School, Cambridge University Press, Cambridge & New York, xvii + 347 pp. [Like it says ....]
Hoel, P.G., Port, S.C. and Stone, C.J., (1987) Introduction to Stochastic Processes, Waveland Press, Long Grove, IL, viii + 203 pp.
Keynes, J.M., (1921) A Treatise on Probability, London, Macmillan; Dover unabridged reproduction, 2004, Mineola, NY, ix + 466 pp.
Kolmogorov, A.N. and Fomin, S.V., (1975) Introductory Real Analysis, trans. & edited by Silverman, R.A.; Dover Publications, New York, xii + 401 pp.
[One of the best introductions to real analysis and a masterful translation of a Russian classic.]
Luenberger, D.G., (1969) Optimization by Vector Space Methods, John Wiley & Sons, New York, xiii + 326 pp. [Almost 40 years old, still in print and still expensive! One of the best books on optimization by far.]
Matiyasevich, Y.V., (1993) Hilbert's Tenth Problem, MIT Press, Cambridge, MA & London, xxii + 264 pp.
Mikosch, T., (1998) Elementary Stochastic Calculus with Finance in View, Advanced Series on Statistical Science & Applied Probability Vol. 6; World Scientific Publ., Singapore ; River Edge, N.J., ix + 212 pp.
Nagel, E. and Newman, J.R., (2001) Gödel's Proof, Revised Edition; New York University Press, New York & London, xxiii + 129 pp.
Pour-El, M.B. and Richards, J.I., (1989) Computability in Analysis and Physics, Springer-Verlag, Berlin, Heidelberg & New York, x + 206 pp.
Schey, H.M., (1997) Div, Grad, Curl and All That: An Informal Text on Vector Calculus, 3rd Edition; W. W. Norton, New York & London, viii + 164 pp.
Stewart, I., (1995) Concepts of Modern Mathematics, Dover Publications, New York, viii + 339 pp. [A wonderfully lucid introduction to modern mathematics.]
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Papers
Blum, L., Shub, M. and Smale, S., (1989) "On a Theory of Computation and Complexity over the Real Numbers: NP-Completeness, Recursive Functions and Universal Machines", Bulletin (New Series) of the American Mathematical Society, Vol. 21, No. 1, July, pp. 1-46.
Church, A., (1936) "An Unsolvable Problem of Elementary Number Theory", American Journal of Mathematics, Vol. 58, pp. 345-363; reprinted in The Undecidable, Davis, M. (Ed.), 1965, Raven Press, Hewlett, NY, pp. 89-107.
Corry, L., (1992) "Nicolas Bourbaki and the Concept of Mathematical Structure", Synthese, Vol. 92, No. 3, September, pp. 315-348.
da Costa, N.C.A. and Doria, F.A., (1991) "Undecidability and Incompleteness in Classical Mechanics", International Journal of Theoretical Physics, Vol. 30, No. 8, August, pp. 1041-1073.
Davis, M., (1973) "Hilbert's Tenth Problem is Unsolvable", American Mathematical Monthly, Vol. 80, No. 3, March, pp. 233-269.
da Costa, N.C.A. and Doria, F.A., (1994) "Gödel Incompleteness in Analysis, with an Application to the Forecasting Problem in the Social Sciences", Philosophia Naturalis, Vol. 31, pp. 1-24.
Gödel, K., (1931) "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I." Monatshefte für Mathematik und Physik, Vol. 38, No. 1, pp. 173-198; translated as "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I." In The Undecidable, Davis, M. (Ed.), 1965, Raven Press, Hewlett, NY, pp. 4-38.
Hoffman, K.L., (2000) "Combinatorial Optimization: Current Successes and Directions for the Future", Journal of Computational and Applied Mathematics, Vol. 124, No. 1-2, December, pp. 341-360.
Jones, J.P., (1974) "Recursive Undecidability: An Exposition", American Mathematical Monthly, Vol. 81, No. 7, August - September, pp. 724-738.
Jones, J.P. and Matiyasevich, Y.V., (1991) "Proof of Recursive Unsolvability of Hilbert's Tenth Problem", American Mathematical Monthly, Vol. 98, No. 8, October, pp. 689-709.
LeVeque, W.J., (1969) "A Brief Survey of Diophantine Equations", In Studies in Number Theory, MAA Studies in Mathematics, Vol. 6. ed. LeVeque, W.J.; Prentice-Hall for the Mathematical Association of America, Englewood Cliffs, NJ, pp. 4-24.
Mathias, A.R.D., (1992) "The Ignorance of Bourbaki", Mathematical Intelligencer, Vol. 14, No. 3, pp. 4-13.
Mathias, A.R.D., (1998) "Further Remarks on Bourbaki", Unpublished manuscript, 24 September, 7 pp.
Mathias, A.R.D., (2002) "A Term of Length 4,523,659,424,929", Synthese, Vol. 133, No. 1-2, October, pp. 75-86.
Matiyasevich, Y.V., (1970) "Solution of the Tenth Problem of Hilbert", Matematikai Lapok, Vol. 21, pp. 83-87.
Poonen, B., (2008) "Undecidability in Number Theory", Notices of the American Mathematical Society, Vol. 55, No. 3, March, pp. 344-350.
Pour-El, M.B. and Richards, J.I., (1983) "Computability and Noncomputability in Classical Analysis", Transactions of the American Mathematical Society, Vol. 275, No. 2, February, pp. 539-560.
Pour-El, M.B. and Richards, J.I., (1984) "Lp-Computability in Recursive Analysis", Proceedings of the American Mathematical Society, Vol. 92, No. 1, September, pp. 93-97.
Raatikainen, P., (2003) "Some Strongly Undecidable Natural Arithmetical Problems, with an Application to Intuitionistic Theories", Journal of Symbolic Logic, Vol. 68, No. 1, March, pp. 262-266.
Robinson, J., (1969) "Diophantine Decision Problems", In Studies in Number Theory, MAA Studies in Mathematics, Vol. 6. ed. LeVeque, W.J.; Prentice-Hall for the Mathematical Association of America, Englewood Cliffs, NJ, pp. 76-116.
Saari, D.G., (1995) "A Chaotic Exploration of Aggregation Paradoxes", SIAM Review, Vol. 37, No. 1, March, pp. 37-52.
Saari, D.G., (1995) "Mathematical Complexity of Simple Economics", Notices of the American Mathematical Society, Vol. 42, No. 2, February, pp. 222-230.
Saari, D.G., (2002) ""Mathematical Social Sciences;" An Oxymoron?" Lecture notes for a series of five lectures, 5-24 September (Unpublished Manuscript), Pacific Institute for the Mathematical Sciences, 52 pp.
Schwartz, J.T., (1962) "The Pernicious Influence of Mathematics on Science", In Discrete Thoughts: Essays on Mathematics, Science and Philosophy ed. Kac, M., Rota, G.-C. and Schwartz, J.T.; Revised Edition; Birkhäuser, Boston, Basel & Berlin, 1992, pp. 19-25.
Senechal, M., (1998) "The Continuing Silence of Bourbaki - An Interview with Pierre Cartier, June 18, 1997", Mathematical Intelligencer, Vol. 20, No. 1, Winter, pp. 22-28.
Smale, S., (1998) "Mathematical Problems for the Next Century", Mathematical Intelligencer, Vol. 20, No. 2, Spring, pp. 7-15.
Turing, A.M., (1936) "On Computable Numbers, with an Application to the Entscheidungsproblem", Proceedings, London Mathematical Society, Series 2, Vol. 42, pp. 230-265, correction (1937) Vol. 43, pp. 544-546.
Wulwick, N.J., (1995) "The Hamiltonian Formalism and Optimal Growth Theory", In Measurement, Quantification and Economic Analysis: Numeracy in Economics ed. Rima, I.H.; Routledge, London & New York, pp. 406-435.
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Quotes
On the implications of Kurt Gödel’s proof
"[W]ithin any formal description of arithmetic there must exist true statements that cannot be proved (incompleteness) and statements whose truth or falsity cannot be decided algorithmically (undecidability). A huge range of questions – mostly algebraic or combinatorial – has since been proved undecidable.
Ian Stewart describing the conclusions of Kurt Gödel’s (1931) proof in Stewart, I., (1991) "Deciding the Undecidable", Nature, Vol. 352, No. 6337, 22 August, pp. 664-665; p. 664.
On the dramatic developments in mathematics in the 20th century
"For the foundations of mathematics, and even the philosophy of its application to science, this century has been one of shattered illusions. Cosy assumption after cosy assumption has exploded in mathematicians faces."
Ian Stewart in Stewart, I., (1988) "The Ultimate in Undecidability", Nature, Vol. 332, No. 6160, 10 March, pp. 115-116.
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updated: 10 March 2008
Copyright © Brett Parris, 2008. All rights reserved.
This is a personal web page and does not necessarily reflect the views of Monash University.
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Copyright © Brett Parris, 2008. All rights reserved.
This is a personal web page and does not necessarily reflect the views of Monash University.
See the official disclaimer. Back to Top
