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“The End of Proof”? The Integration of Different Mathematical Cultures as Experimental Mathematics Comes of Age

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Mathematical Cultures

Part of the book series: Trends in the History of Science ((TRENDSHISTORYSCIENCE))

Abstract

In this paper, the recent emergence of a professed “experimental” culture in mathematics during the past three decades is analysed based on an adaptation of Hans-Jörg Rheinberger’s notion of “experimental systems” that mesh into experimental cultures. In so doing, I approach the question of how distinct mathematical cultures can coexist and blend into a common understanding that allows for cultural convergence while preserving heterogeneity.

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Notes

  1. 1.

    Rheinberger (1997, p. 238).

  2. 2.

    On a similar note, Knorr Cetina defined epistemic cultures as “amalgam[s] of arrangements and mechanisms—bonded through affinity, necessity and historical coincidence—which in a given field, make up how we know what we know” (Knorr Cetina 1999, p. 1).

  3. 3.

    Recently, more work has been devoted to the cultural construction of scientific disciplines (see e.g. Lenoir 1997).

  4. 4.

    The movement to boycott the Elsevier publishing company was initiated by Gowers in 2012 (see Arnold and Cohn 2012; Hassink and Clark 2012). It has received considerable attention in the media, yet the discussions about online publishing dates back two decades, at least (see also Odlyzko 1995; Quinn 1995).

  5. 5.

    The challenge to the a priori status of mathematical knowledge is analysed in McEvoy (2008). The differences between individual and collective surveyability have led me to approach the issue of trust within mathematics from a perspective of social epistemology; see also Andersen et al. (2014).

  6. 6.

    Jaffe and Quinn (1993) as referred to above.

  7. 7.

    The statements in parentheses are quotations from Borwein and Bailey (2004, pp. 2–3).

  8. 8.

    See Borwein and Bailey (ibid., pp. 255–256) originally published as Borwein et al. (1996). For Medawar’s essay, see Medawar (1979, pp. 69–75).

  9. 9.

    Horgan would also publish the similarly provocative and controversial book “The End of Science” (Horgan 1996). In many disciplines in the natural sciences (such a climate science), the computer has made substantial shifts in epistemic practice, sometimes leading to entire new disciplines or new standards of data and explanation.

  10. 10.

    Using the BBP formula, extensive computations have determined individual digits of π using, for instance, approximately 140 CPU years during 1998–2000 to determine that the quadrillionth binary digit of π is 0, a record which was surpassed in 2010 when a 1000-node cluster used 500 CPU-years to determine that the two quadrillionth bit was also 0 (Sze 2010).

  11. 11.

    http://oeis.org/, see also Sloane (2003).

  12. 12.

    See https://groups.google.com/d/topic/comp.soft-sys.math.mathematica/guGsJSGwo-g/discussion.

  13. 13.

    One study which takes a commendable inclusive survey of actual practice is Colton (2007).

References

  • Andersen, L. E., Sørensen, H. K. and the PCSP group. (2014). Contemporary mathematics in practice: Social epistemology in the light of unsurveyability in mathematics. In preparation.

    Google Scholar 

  • Andrews, G. E. (1994). The death of proof? Semi-rigorous Mathematics? You’ve got to be kidding! The mathematical intelligencer, 16(4), 16–18.

    MathSciNet  MATH  Google Scholar 

  • Appel, K., Haken, W., & Koch, J. (1977). Every planar map is four colorable. Part II: reducibility. Illinois Journal of Mathematics, 21(3), 491–567.

    MathSciNet  MATH  Google Scholar 

  • Arnold, D. N., & Cohn, H. (2012). Mathematicians take a stand. Notices of the AMS, 59(6), 828–833.

    Google Scholar 

  • Atiyah, M., et al. (1994). Responses to “theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics”, by A. Jaffe and F. Quinn. Bulletin of the American Mathematical Society, 30(2), 178–207.

    Article  MathSciNet  MATH  Google Scholar 

  • Avigad, J. (2008). “Computers in mathematical inquiry”. In P. Mancosu (Ed.), The philosophy of mathematical practice (Chap. 11, pp. 302–316.). Oxford: Oxford University Press.

    Google Scholar 

  • Bailey, D. H., & Borwein, J. M. (2011). Exploratory experimentation and computation. Notices of the AMS, 58(10), 1410–1419.

    MathSciNet  MATH  Google Scholar 

  • Bailey, D. H., Borwein, J. M., et al. (August 28, 2014). Opportunities and challenges in 21st century experimental mathematical computation: ICERM workshop report. URL http://www.davidhbailey. com/dhbpapers/ICERM-2014.pdf (visited on 10/19/2014).

  • Baker, A. (2008). Experimental Mathematics. Erkenntnis, 68(3), 331–344.

    Article  MathSciNet  MATH  Google Scholar 

  • Billey, S. C., & Tenner, B. E. (2013). Fingerprint databases for theorems. Notices of the AMS, 60(8), 1034–1039.

    Article  MathSciNet  MATH  Google Scholar 

  • Borwein, J. M. (2012). Exploratory experimentation: Digitally-assisted discovery and proof. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education. The 19th ICMI study (Chap. 4, pp. 69–96). New ICMI Study Series 15. Berlin: Springer.

    Google Scholar 

  • Borwein, J., & Bailey, D. (2004). Mathematics by experiment: Plausible reasoning in the 21st century. Natick (MA): A K Peters.

    MATH  Google Scholar 

  • Borwein, J., Bailey, D., & Girgensohn, R. (2004). Experimentation in mathematics: Computational paths to discovery. Natick (MA): A K Peters.

    MATH  Google Scholar 

  • Borwein, J., Borwein, P., et al. (1996). Making sense of experimental mathematics. The mathematical intelligencer, 18(4), 12–18.

    MathSciNet  MATH  Google Scholar 

  • Borwein, J. M., & Jungić, V. (2012). Organic mathematics: Then and now. Notices of the AMS, 59(3), 416–419.

    MATH  Google Scholar 

  • Britz, T., & Rutherford, C. G. (2005). Covering radii are not matroid invariants. Discrete Mathematics, 296, 117–120.

    Article  MathSciNet  MATH  Google Scholar 

  • Campbell, D., et al. (1985). Experimental mathematics: The role of computation in nonlinear science. Communications of the ACM, 28(4), 374–384.

    Article  MathSciNet  Google Scholar 

  • Colton, S. (2007). Computational discovery in pure mathematics. In S. Dz̆eroski & L. Todorovski Computational discovery (pp. 175–201). Heidelberg: Springer.

    Google Scholar 

  • Cox, D. A. (1985). Gauss and the arithmetic-geometric mean. Notices of the AMS, 32(2), 147–151.

    MathSciNet  Google Scholar 

  • Cranshaw, J., & Kittur, A. (2011). The polymath project: Lessons from a successful online collaboration in mathematics. In Proceedings of the 2011 Annual Conference on Human Factors in Computing Systems (pp. 1865–1874). CHI ’11. Vancouver, BC, Canada: ACM.

    Google Scholar 

  • Detlefsen, M., & Luker, M. (1980). The four-color theorem and mathematical proof. The Journal of Philosophy, 77(12), 803–820.

    Article  Google Scholar 

  • Epple, M. (2001). Matematikkhistoriens former: Genier, ideer, institusjoner, matematiske verksteder - et metahistorisk essay. ARR Idéhistorisk Tidsskrift, 1–2, 94–108.

    Google Scholar 

  • Epple, M. (2004). Knot invariants in Vienna and Princeton during the 1920s: Epistemic configurations of mathematical research. Science in Context, 17(1/2), 131–164.

    Google Scholar 

  • Epple, M. (2011). Between timelessness and historiality: On the dynamics of the epistemic objects of mathematics. Isis, 102(3), 481–493.

    Google Scholar 

  • Epstein, D., & Levy, S. (2000). To the reader. Experimental Mathematics, 9(1), 1.

    Article  MathSciNet  Google Scholar 

  • Epstein, D., Levy, S., & de la Llave, R. (1992). About this journal. Experimental Mathematics, 1(1), 1–3.

    Google Scholar 

  • Gieryn, T. (1996). Review of Andrew Pickering, ‘The Mangle of Practice’, Chicago: University of Chicago Press, 1995. American Journal of Sociology, 102(2), 599–601.

    Article  Google Scholar 

  • Goldstein, L. J. (1973). A history of the prime number theorem. The American Mathematical Monthly, 80(6), 599–615.

    Article  MathSciNet  MATH  Google Scholar 

  • Gowers, W. T. (2000). The two cultures of mathematics. In V. Arnold et al. [s.l.] (Ed.), Mathematics: Frontiers and perspectives (pp. 65–78). American Mathematical Society.

    Google Scholar 

  • Gowers, T., Nielsen, M. (October 15, 2009). Massively collaborative mathematics. Nature, 461, 879–881.

    Google Scholar 

  • Hacking, I. (1992). ‘Style’ for historians and philosophers. Studies in the History and Philosophy of Science, 23(1), 1–20.

    Article  MathSciNet  Google Scholar 

  • Haspel, C., & Vasquez, A. (1982). Experimental mathematics using APL and graphics. APL ’82: Proceedings of the international conference on APL (pp. 121–127). Heidelberg, Germany: ACM.

    Chapter  Google Scholar 

  • Hassink, L., & Clark, D. (2012). Elsevier’s response to the mathematics community. Notices of the AMS, 59(6), 833–835.

    Google Scholar 

  • Heintz, B. (2000a). Die Innenwelt der Mathematik. Zur Kultur und Praxis einer beweisenden Disziplin. Wien and New York: Springer.

    Book  Google Scholar 

  • Heintz, B. (2000b). In der Mathematik ist ein Streit mit Sicherheit zu entscheiden. Perspektiven einer Soziologie der Mathematik”. Zeitschrift für Soziologie, 29(5), 339–360.

    Google Scholar 

  • Horgan, J. (1993). The death of proof. Scientific American, 269(4), 74–82.

    Article  MathSciNet  Google Scholar 

  • Horgan, J. (1996). The end of science: Facing the limits of knowledge in the twilight of the scientific age. Helix Books. Reading (Mass) etc.: Addison-Wesley Publishing Company, Inc.

    Google Scholar 

  • Jaffe, A., & Quinn, F. (April 1994). Response to comments on “Theoretical mathematics”. Bulletin of the American Mathematical Society, 30(2), 208–211.

    Google Scholar 

  • Jaffe, A., & Quinn, F. (1993). “Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical physics. Bulletin of the American Mathematical Society, 29(1), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  • Kitcher, P. (1985). The nature of mathematical knowledge. Oxford: Oxford University Press.

    Book  MATH  Google Scholar 

  • Knorr Cetina, K. (1999). Epistemic cultures. How the sciences make knowledge. Cambridge (Mass) and London: Harvard University Press.

    Google Scholar 

  • Kohlhase, M. (2014). Mathematical knowledge management: Transcending the management: Transcending the one-brain-barrier with theory graphs. Newsletter of the European Mathematical Society, 92, 22–27.

    MathSciNet  MATH  Google Scholar 

  • Lam, C. W., Thiel, L., & Swiercz, S. (1989). The non-existence of finite projective planes of order 10. Canadian Journal Mathematics, 41(6), 1117–1123.

    Article  MathSciNet  MATH  Google Scholar 

  • Lenoir, T. (1997). Instituting science. The cultural production of scientific disciplines. Writing Science. Stanford: Stanford University Press.

    Google Scholar 

  • MacKenzie, D. (2005). Computing and the cultures of proving. Philosophical Transactions of The Royal Society, A, 363, 2335–2350.

    Article  MathSciNet  MATH  Google Scholar 

  • Mancosu, P. (2010). Mathematical style. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy. Spring 2010. URL http://plato.stanford.edu/archives/spr2010/entries/mathematicalstyle/

  • Mandelbrot, B. B. (1982). The fractal geometry of nature. San Francisco: W. H. Freeman and Company.

    MATH  Google Scholar 

  • Marden, A. (1997). Fred Almgren and the geometry center. Experimental Mathematics, 6(1), 11–12.

    Article  MathSciNet  Google Scholar 

  • Mayhew, D., & Royle, G. F. (2008). Matroids with nine elements. Journal of Combinatorial Theory. B, 98, 415–431.

    Article  MathSciNet  MATH  Google Scholar 

  • McEvoy, M. (2008). The epistemological status of computer-assisted proofs. Philosophia Mathematica (3), 16, 374–387.

    Google Scholar 

  • McEvoy, M. (February 2013). Experimental mathematics, computers and the a priori. Synthese, 190(3), 397–412.

    Google Scholar 

  • Medawar, P. B. (1979). Advice to a Young scientist. San Francisco: Harper & Row.

    Google Scholar 

  • Nathanson, M. B. (2011). One, two, many: Individuality and collectivity in mathematics. The mathematical intelligencer, 33(1), 5–8.

    Article  MathSciNet  MATH  Google Scholar 

  • Odlyzko, A. M. (1995). Tragic loss or good riddance? The impending demise of traditional scholarly journals. Notices of the AMS, 42(1), 49–53.

    Google Scholar 

  • Pease, A., & Martin, U. (2012). Seventy four minutes of mathematics: An analysis of the third Mini-Polymath project. In Proceedings of AISB/IACAP 2012, Symposium on Mathematical Practice and Cognition II.

    Google Scholar 

  • Pickering, A. (Ed.). (1992). Science as practice and culture. Chicago & London: University of Chicago Press.

    Google Scholar 

  • Pickering, A. (Ed.). (1995). The mangle of practice. Chicago: University of Chicago Press.

    Google Scholar 

  • Pólya, G. (1957). How to solve it: A new aspect of mathematical method (2nd ed.). New York: Garden City Doubleday Anchor Books.

    MATH  Google Scholar 

  • Quinn, F. (1995). Roadkill on the electronic highway: The threat to the mathematical literature. Notices of the AMS, 42(1), 53–56.

    Google Scholar 

  • Rheinberger, H.-J. (1997). Toward a history of epistemic things. Synthesizing proteins in the test tube. Writing science. Stanford: Stanford University Press.

    Google Scholar 

  • Rheinberger, H.-J. (October 2012). From experimental systems to cultures of experimentation: Historical reflections. Unpublished manuscript of talk given in Aarhus, September 24, 2012.

    Google Scholar 

  • Sarvate, D., Wetzel, S., & Patterson, W. (2011). Analyzing massively collaborative mathematics projects. The Mathematical Intelligencer, 33(1), 9–18.

    Article  MathSciNet  MATH  Google Scholar 

  • Sloane, N. J. A. (2003). The on-line encyclopedia of integer sequences. Notices of the AMS, 50(8), 912–915.

    MathSciNet  MATH  Google Scholar 

  • Snow, C. P. (1959/1993). The two cultures. In S. Collini (Ed.), With an intro. Cambridge: Cambridge University Press.

    Google Scholar 

  • Sørensen, H. K. (2010a). Experimental mathematics in the 1990s: A second loss of certainty? Oberwolfach Reports, No., 12, 601–604.

    Google Scholar 

  • Sørensen, H. K. (2010b). Exploratory experimentation in experimental mathematics: A glimpse at the PSLQ algorithm. In B. Löwe & T. Müller (Eds.), PhiMSAMP. Philosophy of mathematics: Sociological aspects and mathematical practice. Texts in philosophy 11 (pp. 341–360). London: College Publications. URL http://www.lib.uni-bonn.de/PhiMSAMP/Book/.

  • Sørensen, H. K. (2013). Er det matematiske bevis ved at dø ud? In O. Høiris (Ed.), Fremtiden (pp. 99–132). Aarhus: Aarhus Universitetsforlag.

    Google Scholar 

  • Stöltzner, M. (2005). Theoretical mathematics: On the philosophical significance of the Jaffe-Quinn debate. In G. Boniolo, P. Budinich, M. Trobok (Eds.), The role of mathematics in the physical sciences. Interdisciplinary and philosophical aspects (pp. 197–222). Dordrecht: Springer.

    Google Scholar 

  • Swart, E. R. (1980). The philosophical implications of the four-colour problem. The American Mathematical Monthly, 87(9), 697–707.

    Article  MathSciNet  MATH  Google Scholar 

  • Sze, T.-W. (2010). The two quadrillionth bit of Pi is 0! Distributed computation of Pi with Apache Hadoop”. CoRR, vol. abs/1008.3171. URL http://arxiv.org/abs/1008.3171

  • The flyspeck project fact sheet. URL http://code.google.com/p/flyspeck/wiki/FlyspeckFactSheet (visited on October 23, 2014).

  • Tymoczko, T. (February 1979). The four-color problem and its philosophical significance. Journal of Philosophy, 76(2), 57–83. URL http://www.jstor.org/stable/2025976.

    Google Scholar 

  • Van Kerkhove, B., & Van Bendegem, J. P. (2008). Pi on earth, or mathematics in the real world. Erkenntnis, 68(3), 421–435.

    Article  MathSciNet  MATH  Google Scholar 

  • Zeilberger, D. (1994). Theorems for a price: Tomorrow’s semi-rigorous mathematical culture. The mathematical intelligencer, 16(4), 11–14, 76.

    Google Scholar 

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Acknowledgment

I am grateful for the excellent suggestions made by colleagues at the Centre for Science Studies and by participants in the “Mathematical Cultures” series of conferences. Some of this research has been presented in Sørensen (2010a) and Sørensen (2013).

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  1. Centre for Science Studies, Department of Mathematics, Aarhus University, Aarhus, Denmark

    Henrik Kragh Sørensen

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  1. Henrik Kragh Sørensen
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Correspondence to Henrik Kragh Sørensen .

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  1. School of Humanities, University of Hertfordshire, Hertfordshire, United Kingdom

    Brendan Larvor

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Sørensen, H.K. (2016). “The End of Proof”? The Integration of Different Mathematical Cultures as Experimental Mathematics Comes of Age. In: Larvor, B. (eds) Mathematical Cultures. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28582-5_9

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