×

zbMATH — the first resource for mathematics

Responses to “Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics”, by A. Jaffe and F. Quinn. (English) Zbl 0803.01014
15 Mathematiker legen ihre Ansicht zu den Ausführungen von A. Jaffe und F. Quinn im Bull. Am. Math. Soc. 29, No. 1, 1-13 (1993; Zbl 0780.00001) dar. Sie äußern sich etwa zu den Fragen: Wozu ist Intuition in der Mathematik erfolgreich? Helfen versuchte Ansätze weiter? Wie zwingend sind Beweise? Hierzu werden Beispiele aus der Geschichte der Mathematik von Euler über Poincaré, Paul Lévy und Appel-Haken bis in die allerneueste Zeit aufgeführt. Die Ausführungen bezeugen das große Spektrum von Ansichten über die mathematische Forschung und bereichern durch etwa weniger bekannte Bemerkungen. Ferner wird der Zusammenhang von Mathematik und Physik erörtert.

MSC:
01A65 Development of contemporary mathematics
01A99 History of mathematics and mathematicians
00A30 Philosophy of mathematics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Arthur Jaffe and Frank Quinn, ”Theoretical mathematics”: toward a cultural synthesis of mathematics and theoretical physics, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 1 – 13. · Zbl 0780.00001
[2] Gregory J. Chaitin, Algorithmic information theory, Cambridge Tracts in Theoretical Computer Science, vol. 1, Cambridge University Press, Cambridge, 1987. With a foreword by J. T. Schwartz. · Zbl 0655.68003
[3] G. J. Chaitin, Information, randomness & incompleteness, 2nd ed., World Scientific Series in Computer Science, vol. 8, World Scientific Publishing Co., Inc., River Edge, NJ, 1990. Papers on algorithmic information theory. · Zbl 1015.00502
[4] G. J. Chaitin, Information-theoretic incompleteness, World Scientific Series in Computer Science, vol. 35, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. · Zbl 0776.68065
[5] G. J. Chaitin, Exhibiting randomness in arithmetic using Mathematica and C, IBM Research Report RC-18946, June 1993.
[6] G. J. Chaitin, Randomness in arithmetic and the decline and fall of reductionism in pure mathematics, Bull. European Assoc. for Theoret. Comput. Sci., no. 50 (July 1993). · Zbl 1023.68589
[7] G. J. Chaitin, The limits of mathematics–Course outline and software, IBM Research Report RC-19324, December 1993.
[8] G. J. Chaitin, Randomness and complexity in pure mathematics, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 4 (1994), no. 1, 3 – 15. · Zbl 0877.68068 · doi:10.1142/S0218127494000022 · doi.org
[9] K. Appel and W. Haken, Every planar map is four colorable. I. Discharging, Illinois J. Math. 21 (1977), no. 3, 429 – 490. K. Appel, W. Haken, and J. Koch, Every planar map is four colorable. II. Reducibility, Illinois J. Math. 21 (1977), no. 3, 491 – 567. K. Appel and W. Haken, The class check lists corresponding to the supplement: ”Every planar map is four colorable. I. Discharging” (Illinois J. Math. 21 (1977), no. 3, 429 – 490), Illinois J. Math. 21 (1977), no. 3, C1-C210. (microfiche supplement). K. Appel and W. Haken, Supplement to: ”Every planar map is four colorable. I. Discharging” (Illinois J. Math. 21 (1977), no. 3, 429 – 490) by Appel and Haken; ”II. Reducibility” (ibid. 21 (1977), no. 3, 491 – 567) by Appel, Haken and J. Koch, Illinois J. Math. 21 (1977), no. 3, 1 – 251. (microfiche supplement). Kenneth Appel and Wolfgang Haken, The solution of the four-color-map problem, Sci. Amer. 237 (1977), no. 4, 108 – 121, 152. , https://doi.org/10.1038/scientificamerican1077-108 Kenneth Appel and Wolfgang Haken, Every planar map is four colorable, J. Recreational Math. 9 (1976/77), no. 3, 161 – 169. · Zbl 0387.05009
[10] K. Appel and W. Haken, The four color proof suffices, Math. Intelligencer 8 (1986), no. 1, 10 – 20, 58. · Zbl 0578.05022 · doi:10.1007/BF03023914 · doi.org
[11] Steve Smale, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 1 – 36. · Zbl 0456.12012
[12] Heegard, Sur l’analysis situs, Bull. Soc. Math. France 44 (1916), 161-142. [Translation of: Forstudier til en topologisk teori för de algebraiske Sammenhang, dissertation, Copenhagen, Det Nordiske Forlag Ernst Bojesen, 1898.]
[13] Poincaré, Sur les nombres de Betti, Comptes Rendus Acad. Sci. (Paris) 128 (1899), 629-630. [Reprinted in Œuvres 6.] · JFM 30.0435.01
[14] Poincaré, Complément à l’analysis situs, Rendiconti Circolo Matematico Palermo 13 (1899), 285-343. [Reprinted in Œuvres 6.] · JFM 30.0435.02
[15] Poincaré, Deuxième complément à l’analysis situs, Proc. London. Math. Soc. 32 (1900), 277-308. [Reprinted in Œuvres 6.] · JFM 31.0477.10
[16] Poincaré, Cinquième complément à l’analysis situs, Rendiconti Circolo Matematico Palermo 18 (1904), 45-110. [Reprinted in Œuvres 6.] · JFM 35.0504.13
[17] Arthur Jaffe and Frank Quinn, ”Theoretical mathematics”: toward a cultural synthesis of mathematics and theoretical physics, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 1 – 13. · Zbl 0780.00001
[18] John Frank Adams, Infinite loop spaces, Annals of Mathematics Studies, vol. 90, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978.
[19] A. S. Schwarz, The partition function of degenerate quadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2 (1977/78), no. 3, 247 – 252. · Zbl 0383.70017 · doi:10.1007/BF00406412 · doi.org
[20] A. Schwarz, New topological invariants arising in the theory of quantized fields, Baku International Topological Conference, Abstracts (Part 2) Baku, 1987.
[21] Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351 – 399. · Zbl 0667.57005
[22] Arthur Jaffe and Frank Quinn, ”Theoretical mathematics”: toward a cultural synthesis of mathematics and theoretical physics, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 1 – 13. · Zbl 0780.00001
[23] B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. · Zbl 0177.17902
[24] Stephen Smale, Book Review: Catastrophe theory: Selected papers,, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1360 – 1368. · doi:10.1090/S0002-9904-1978-14580-7 · doi.org
[25] R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier, Grenoble 6 (1955 – 1956), 43 – 87 (French). · Zbl 0075.32104
[26] R.S. Zahler and H. Sussmann, Claims and accomplishments of applied catastrophe theory, Nature 269 (1977), 759-763.
[27] Correspondence on catastrophe theory, Nature 270 (1977), 381-384 and 658.
[28] E. C. Zeeman, Catastrophe theory, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977. Selected papers, 1972 – 1977.
[29] E. C. Zeeman, Controversy in science: on the ideas of Daniel Bernoulli and René Thom, Nieuw Arch. Wisk. (4) 11 (1993), no. 3, 257 – 282. · Zbl 0802.01008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.