Intellectual generosity and the reward structure of mathematics

Abstract

Prominent mathematician William Thurston was praised by other mathematicians for his intellectual generosity. But what does it mean to say Thurston was intellectually generous? And is being intellectually generous beneficial? To answer these questions I turn to virtue epistemology and, in particular, Roberts and Wood’s analysis of intellectual generosity (Intellectual virtues: an essay in regulative epistemology. Oxford University Press, Oxford, 2007). By appealing to Thurston’s own writings and interviewing mathematicians who knew and worked with him, I argue that Roberts and Wood’s analysis nicely captures the sense in which he was intellectually generous. I then argue that intellectual generosity is beneficial because it counteracts negative effects of the reward structure of mathematics that can stymie mathematical progress.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    For work on intellectual humility see e.g. Whitcomb et al. (2017), Church (2016), Roberts and Wood (2007, chp.9). For work on intellectual courage, see e.g. Baehr (2011), Roberts and Wood (2007, chp.8). For work on epistemic justice, see e.g. Fricker (2007) and Medina (2011).

  2. 2.

    We will see that intellectual generosity involves a concern with intrinsic over extrinsic intellectual goods. However I am agnostic on the question whether intrinsic or extrinsic goods are ultimately superior.

  3. 3.

    Although Franklin didn’t know this, her own data had been crucial to Watson and Crick’s work, but they had used it without her knowledge and without giving her credit (Roberts and Wood 2007, p. 298).

  4. 4.

    Perelman (2002, 2003a, b) proved the full geometrization conjecture, which included the Poincaré conjecture.

  5. 5.

    There were, however, some doubts about the reasons Thurston was slow at publishing his theorems and proofs. More on this below.

  6. 6.

    Nonetheless, some noted that Thurston was not perfect. For example, he would sometimes miss the deadlines for submitting letters or reports that were needed for job applications or promotions and did not take an active approach to advising students.

  7. 7.

    I am grateful to two anonymous reviewers for raising the issues considered below.

  8. 8.

    Thurston did, however, publish parts of it (Thurston 1994, p. 176).

  9. 9.

    Roberts and Wood did note that a concern with intrinsic over extrinsic intellectual goods characteristic of intellectual generosity can help researchers persevere where others might give up and so potentially generate new intrinsic intellectual goods in this manner (Roberts and Wood 2007, pp. 301–302). They also pointed to specific ways in which McClintock’s intellectual generosity to Creighton generated intrinsic intellectual goods (Roberts and Wood 2007, p. 302).

  10. 10.

    I am grateful to an anonymous reviewer for pointing out relevant connections between intellectual generosity and social epistemology.

  11. 11.

    The arguments that I make in this section are a modified and shortened form of arguments I make in Morris (n.d.).

  12. 12.

    See https://mathscinet.ams.org/mathscinet/msc/msc2010.html for the full list.

  13. 13.

    Such scenarios happen in practice: “Fields in the mathematical sciences are mature enough so that researchers know the capabilities and limitations of the tools provided by their field, and they are seeking tools from other areas” (National Research Council 2013, p. 97).

  14. 14.

    Thanks to Bonnie Gold for alerting me to Erdős’s generosity.

References

  1. Agol, I. (2019). Personal communication via email, July 2019.

  2. Alladi, K. (1998). Editorial. The Ramanujan Journal, 2(1), 5–6.

    Article  Google Scholar 

  3. Baehr, J. (2008). Four varieties of character-based virtue epistemology. The Southern Journal of Philosophy, 46(4), 469–502.

    Article  Google Scholar 

  4. Baehr, J. (2011). The inquiring mind: on intellectual virtues and virtue epistemology. Oxford: OUP.

    Google Scholar 

  5. Battaly, H. (2008). Virtue epistemology. Philosophy Compass, 3(4), 639–663.

    Article  Google Scholar 

  6. Benham, C. (2011). UC Davis mathematics newsletter. https://www.math.ucdavis.edu/files/6013/7176/9863/newsletter_2012-rev1-3.pdf Archived at https://perma.cc/6LKD-S4UX.

  7. Casadevall, A., & Fang, F. C. (2014). Specialized science. Infection and Immunity, 82(4), 1355–1360.

    Article  Google Scholar 

  8. Church, I. M. (2016). The doxastic account of intellectual humility. Logos & Episteme, 7(4), 413–433.

    Article  Google Scholar 

  9. Creighton, H. B., & McClintock, B. (1931). A correlation of cytological and genetical crossing-over in Zea mays. Proceedings of the National Academy of Sciences of the United States of America, 17(8), 492–497.

    Article  Google Scholar 

  10. Crowley, J. (2014). Putting a value on expository writing. https://sinews.siam.org/Details-Page/putting-a-value-on-expository-writing. Archived at https://perma.cc/H4GL-9F74. Accessed Sept 9, 2019.

  11. Dick, A. (1981). Emmy Noether 1882–1935. Boston: Birkhäuser.

    Google Scholar 

  12. Dunne, E., Hulek, K. (2019). MSC2020. https://msc2020.org/ Archived at https://perma.cc/BG4X-DPGW, n.d. Accessed 3 Dec, 2019.

  13. Eliashberg, Y. (2019). Personal communication via email, June 2019.

  14. Fricker, M. (2007). Epistemic Injustice: Power and the Ethics of Knowing. Oxford: Oxford University Press.

    Google Scholar 

  15. Gabai, D. (2013). Fine letters. https://www.math.princeton.edu/sites/default/files/2017-10/newsletter2013_2.pdf Archived at https://perma.cc/J9XB-WYKF.

  16. Gabai, D., & Kerckhoff, S. (2015). Thurston, William P. 1946–2012. Notices of the American Mathematical Society., 62(11), 1318–1332.

    Article  Google Scholar 

  17. Gabai, D., Kerckhoff, S., Minsky, Y., Mosher, L., Weeks, J., Farb, B., et al. (2016). 1946–2012, part II. Notices of the American Mathematical Society, 63(01), 31–41.

    Article  Google Scholar 

  18. Greco, J. (2002). Virtues in epistemology. In P. K. Moser (Ed.), The Oxford handbook of epistemology. New York: Oxford University Press.

    Google Scholar 

  19. Hass, J. (2019). Personal communication via email, June 2019.

  20. Heesen, R. (2018). Why the reward structure of science makes reproducibility problems inevitable. The Journal of Philosophy, 115(12), 661–674.

    Article  Google Scholar 

  21. Heesen, R. (2019). The credit incentive to be a maverick. Studies in History and Philosophy of Science, 76, 5–12.

    Article  Google Scholar 

  22. Hellman, H. (2006). Great feuds in mathematics: ten of the liveliest disputes ever. Hoboken: Wiley.

    Google Scholar 

  23. Jackson, A. (2006). Conjectures no more? Consensus forming on the proofs of the Poincaré and geometrization conjectures. Notices of the American Mathematical Society, 53(8), 897–901.

    Google Scholar 

  24. Jaffe, A., & Quinn, F. (1993). Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics. Bulletin of the American Mathematical Society, 29(1), 1–14.

    Article  Google Scholar 

  25. Krantz, S. G. (1997). A primer of mathematical writing: Being a disquisition on having your ideas recorded, typeset, published, read and appreciated. Providence: American Mathematical Society.

    Google Scholar 

  26. Lady, L (n.d.). How does one do mathematical research? (or maybe how not to). http://www.math.hawaii.edu/lee/how-to.html. Archived at https://perma.cc/LZE6-ETTD.

  27. Lin, T. (2016) Remembering Felix Browder, a nonlinear genius in a nonlinear world. The New Yorker, December 2016. Archived at https://perma.cc/UGA5-VQ38.

  28. McMullen, C. (1990). Iteration on Teichmüller space. Inventiones Mathematicae, 99(1), 425–454.

    Article  Google Scholar 

  29. Medina, J. (2011). The relevance of credibility excess in a proportional view of epistemic injustice: Differential epistemic authority and the social imaginary. Social Epistemology, 25(1), 15–35.

    Article  Google Scholar 

  30. Miller, C. B. (2018). Generosity: A preliminary account of a surprisingly neglected virtue. Metaphilosophy, 49(3), 216–245.

    Article  Google Scholar 

  31. Minsky, Y. (2019). Personal communication via email, July 2019.

  32. Morgan, J. W., & Shalen, P. B. (1984). Valuations, trees, and degenerations of hyperbolic structures, I. Annals of Mathematics, 120(3), 401–476.

    Article  Google Scholar 

  33. Morgan, J. W., & Shalen, P. B. (1988). Degenerations of hyperbolic structures, II: Measured laminations in 3-manifolds. Annals of Mathematics, 127(2), 403–456.

    Article  Google Scholar 

  34. Morgan, J. W., & Shalen, P. B. (1988). Degenerations of hyperbolic structures, III: Actions of 3-manifold groups on trees and thurston’s compactness theorem. Annals of Mathematics, 127(3), 457–519.

    Article  Google Scholar 

  35. Morris, R. L. Increasing specialization: Why we need to make mathematics more accessible. Unpublished manuscript, n.d.

  36. National Research Council. (2013). The Mathematical Sciences in 2025. The National Academies Press

  37. Odifreddi, P. (2006). The mathematical century: The 30 greatest problems of the last 100 years. Princeton: Princeton University Press.

    Google Scholar 

  38. Oreskovich, A.-M., Sagalovskiy, D. Math Club interview with Professor Curtis McMullen, n.d. http://www.math.harvard.edu/~ctm/expositions/html/interview.html. Archived at https://perma.cc/C9S7-N9SY.

  39. Perelman, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159 [math.DG], November 2002. URL https://arxiv.org/abs/math/0211159.

  40. Perelman, G. (2003a). Ricci flow with surgery on three-manifolds. arXiv:math/0303109 [math.DG], March 2003. URL https://arxiv.org/abs/math/0303109.

  41. Perelman, G. (2003b). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math/0307245 [math.DG], July 2003. URL https://arxiv.org/abs/math/0307245.

  42. Roberts, R. C., & Wood., W. J. (2007). Intellectual virtues: an essay in regulative epistemology. Oxford: Oxford University Press.

    Google Scholar 

  43. Rota, G.-C. (1991). The concept of mathematical truth. The Review of Metaphysics, 44(3), 483–494.

    Google Scholar 

  44. Rota, G.-C. (1997a). The phenomenology of mathematical beauty. Synthese, 111(2), 171–182.

    Article  Google Scholar 

  45. Rota, G.-C. (1997b). Ten lessons I wish I had been taught. In G.-C. Rota & F. Palombi (Eds.), Indiscrete Thoughts (pp. 195–203). Boston, MA: Birkhäuser.

    Google Scholar 

  46. Sosa, E. (1991). Reliabilism and intellectual virtue. In W. P. Alston (Ed.), Knowledge in perspective: Selected essays in epistemology (pp. 131–146). Cambridge: Cambridge University Press.

    Google Scholar 

  47. Sosa, E. (1991). Intellectual virtue in perspective. In W. P. Alston (Ed.), Knowledge in perspective: Selected essays in epistemology (pp. 270–294). Cambridge: Cambridge University Press.

    Google Scholar 

  48. Stein, H. (1988). Logos, logic, and logistiké: Some philosophical remarks on the nineteenth century transformation of mathematics. History and Philosophy of Modern Mathematics, 11, 238–259.

    Google Scholar 

  49. Stephan, P. (2012). How economics shapes science. Harvard: Harvard University Press.

    Google Scholar 

  50. Stewart, I. (2013). Visions of infinity: The great mathematical problems. New York: Basic Books.

    Google Scholar 

  51. Strevens, M. (2003). The role of the priority rule in science. The Journal of Philosophy, 100(2), 55–79.

    Article  Google Scholar 

  52. Thoma, J. (2015). The epistemic division of labor revisited. Philosophy of Science, 82(3), 454–472.

    Article  Google Scholar 

  53. Thurston, W. P. (2010). What’s a mathematician to do? https://mathoverflow.net/questions/43690/whats-a-mathematician-to-do/44213. Archived at https://perma.cc/XVA8-8L6D.

  54. Thurston, W.. (2002). Geometry and topology of three-manifolds. Available online at http://library.msri.org/books/gt3m/.

  55. Thurston, W. P. (1982). Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bulletin of the American Mathematical Society, 6(3), 357.

    Article  Google Scholar 

  56. Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30, 161–177.

    Article  Google Scholar 

  57. Truesdell, C. (2007). Leonard Euler, supreme geometer. In D. William (Ed.), The genius of Euler: Reflections on his life and work (pp. 13–42). Washington, D.C: MAA.

    Google Scholar 

  58. Turri, J., Alfano, M., Greco, J. (2018). Virtue epistemology. In E.N Zalta (ed), The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, summer 2018 edition. https://plato.stanford.edu/archives/sum2018/entries/epistemology-virtue/.

  59. Waldo Dunnington, G., Gray, J., & Dohse, F.-E. (2004). Carl Friedrich Gauss: Titan of Science. Washington, D.C: MAA.

    Google Scholar 

  60. Whitcomb, D., Battaly, H., Baehr, J., & Howard-Snyder, D. (2017). Intellectual humility: Owning our limitations. Philosophy and Phenomenological Research, 94(3), 509–539.

    Article  Google Scholar 

  61. Zagzebski, L. T. (1996). Virtues of the mind. Cambridge: Cambridge University Press.

    Google Scholar 

  62. Zollman, K. J. S. (2019). The scientific Ponzi scheme. Unpublished manuscript available on the PhilSci-Archive http://philsci-archive.pitt.edu/16264/.

Download references

Acknowledgements

I am very grateful to Ian Agol, Yakov Eliashberg, Yacin Hamami, Bonnie Gold, Joel Hass, Erich Kummerfeld, Curtis McMullen, Yair Minsky and Alan Weinstein. I am also very grateful to participants at both the Mathematics in Practice conference held at Stanford University in May 2019 and the MidWest Philosophy of Mathematics Workshop held at Notre Dame in November 2019. Finally I am very grateful to three anonymous reviewers for their many helpful comments and suggestions. This work was partly undertaken while I held a Postdoctoral Scholarship at the Suppes Center for History and Philosophy of Science at Stanford University.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Rebecca Lea Morris.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Morris, R.L. Intellectual generosity and the reward structure of mathematics. Synthese (2020). https://doi.org/10.1007/s11229-020-02660-w

Download citation

Keywords

  • Virtue epistemology
  • Intellectual virtues
  • Intellectual generosity
  • Mathematical practice