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Recalcitrant Disagreement in Mathematics: An “Endless and Depressing Controversy” in the History of Italian Algebraic Geometry

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Fig. 1
Fig. 2

Notes

  1. It is true that, as an anonymous referee pointed out, very few mathematicians would be able to state the “relevant axioms.” This points to a divergence between this characterization of proof with actual mathematical proofs, whose starting points are subject-specific acceptable starting points, rather than (foundational) axioms. Still, mathematicians do generally acknowledge that such “relevant axioms” could be in principle made explicit – that is why this characterization will do for this context. The more so because, as it will be clear shortly, we won’t focus on disagreement concerning axioms.

  2. It is not entirely clear what “perfect rigor” is, and even less clear whether that has been attained (Burgess and De Toffoli 2022; Paseau 2016). Remember that, of course, many of the proofs that were deemed to be rigorous at the time of Poincaré’s writing, would not be acceptable nowadays. Nevertheless, it is important that still at the time, an ideal of rigorous proof was shared by many mathematical communities. We will return on this issue later.

  3. See, for example, (Maddy 2011).

  4. Arguably, however, some of the axioms of set theory “force themselves upon us as true” (Gödel 1964, 271).

  5. Indeed, foundational disagreement can also concern the choice of a logic.

  6. See (Paseau 2015). Note that, notwithstanding the (missleading) name, probabilistic proofs are not genuine (deductive) proofs at all.

  7. This is the “surveyability” requirement for proofs (Tymoczko 1979).

  8. For instance, De Toffoli (2023) argues that in some cases diagrams are essential to the proof in which they figure. That is, that there are plausible criteria of identity for proofs such that when eliminating the diagrams from certain diagrammatic proofs we would inevitably transform such proofs into different ones.

  9. This type of disagreement can also be seen to be linked to CONCEPTION OF PROOF-DISAGREEMENT. According to Wagner (2022, 5) the “problem of consensus [over the correctness of putative proofs] and individuation of proof are intertwined.” However, for the scope of this argument, we prefer to keep them separate.

  10. Thanks to Fenner Tanswell for encouraging us to think of this case as a hybrid type of disagreement.

  11. See (De Toffoli 2022, 256) for an analysis of this case from an epistemological perspective.

  12. Note that the correctness of this reaction is accepted even from those epistemologists who advocate that in general one should stick to one’s own original position in face of disagreement (Kelly 2010, 199).

  13. This is, however, not always the case. Vladimir Voevodsky, long before finding a mistake in his own results that granted him the Fields medal, was made aware of a counterexample. He wrongly believed that his results were in good standing and that the putative counterexample came from a fallacious argument (Voevodsky 2014).

  14. In order for this process of self-correction to be possible, mathematical arguments must be shareable among practitioners, see (De Toffoli 2021a).

  15. See (Goldstein 2013) discussing the ubiquity of PP-DISAGREEMENT in 17th century mathematics.

  16. See (Brigaglia and Ciliberto 1998, 300).

  17. Our translation from Italian.

  18. This is not, according at least to Severi and Enriques, incompatible with mathematical rigor. Indeed, both mathematicians identified two types of rigor, one connected with the possibility of always adding more details and another, consisting in describing the mathematical facts faithfully. For a discussion, see (De Toffoli and Fontanari 2022).

  19. They discuss a related result, the Theorem of Completeness of the Characteristic Series. The Fundamental Theorem under discussion is one of the main consequences of such Completeness Theorem.

  20. Our translation from Italian.

  21. This letter is stored in the Beniamino Segre Archives at the California Institute of Technology and reproduced (and partially translated) by Babbitt and Goodstein (2011).

  22. Note that the situation might change radically due to technological innovations related to new computer proof assistants. These tools are making the formalization of mathematics more and more manageable (Avigad 2018).

  23. These and related questions are addressed in (Burgess and De Toffoli 2022).

  24. This definition is taken from the governmental website: https://www.usa.gov/federal-agencies/u-s-courts-of-appeal.

  25. We report them here briefly, but the interested reader should consult (Wagner 2022).

  26. See (Avigad 2021).

  27. For a pluralist conception of rigor, see (Tanswell forthcoming).

  28. There are important critiques to this view. Here are two objections that we find most relevant: (i) not all formal systems will do (a silly example of what won’t do is an inconsistent system), (ii) the overgeneration problem discussed in (Tanswell 2015): to a single p-proof, many formal proofs can be associated.

  29. Note that this view is inspired by Burgess’s, but departs from his view in this respect.

  30. For a discussion of these two types of rigor and their connection with different notions of objectivity in mathematics, see (De Toffoli and Fontanari 2022).

  31. The distinction between rigor and acceptability for p-proofs parallels Miranda Fricker’s (1998) distinction between indicator properties and working indicator properties for good informants, which she develops in her investigation of the phenomenon of epistemic injustice. According to Fricker, a good informant is both competent and trustworthy – this is what constitutes rational authority. The indicator properties are properties that reliably indicate rational authority. Instead, working indicator properties “are those properties actually used in a given practice to indicate rational authority, and which may or may not be so reliable” (ibid. 168).

  32. For a contemporary case, one might point to symplectic geometry (Hartnett 2017).

  33. One such exception being Fesenko (2019).

  34. See (Klarreich 2018) for a description of the case and (Aberdein 2023) for an insightful philosophical analysis.

  35. This is related to the problem of how we should identify proofs, which is related to CONCEPTION OF PROOF-DISAGREEMENT. See Footnote 8. Thanks to one of the anonymous referees for this clarification.

  36. We do not want to imply there is not but simply that our view does not entail there is one.

  37. See (Fallis 2003).

  38. Our translation from Italian.

  39. See (Ciliberto and Sallent Del Colombo 2018, 15).

  40. Our translation from Italian.

  41. Our translation from Italian.

  42. Our translation from Italian.

  43. Our translation from Italian.

  44. Our translation from Italian.

  45. These can be numbered among “mathematical virtues” (Aberdein et al. 2021)

  46. Our translation from Italian.

  47. It was then fixed by Andrew Wiles in about one year with the help of his former student Richard Taylor.

References

  • Aberdein A (2023) “Deep Disagreement in Mathematics.” Global Philosophy 33(17):1–27

    Google Scholar 

  • Aberdein A, Rittberg C, Tanswell F (2021) “Virtue theory of mathematical practices: an introduction.” Synthese 199(3–4):10167–10180

    Google Scholar 

  • Atiyah M, Borel A, Chaitin GJ, Friedan D, Glimm J, Gray JJ, Hirsch MW, Mac Lane S, Mandelbrot B, and RuelleD (1994) “Responses to: A. Jaffe and F. Quinn, ‘Theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics’.” Bulletin of the American Mathematical Society 30 (2): 178–207

  • Avigad J (2018) “The Mechanization of Mathematics.” Notices of the American Mathematical Society 65(6):681–690

    Google Scholar 

  • Avigad J (2021) “Reliability of mathematical inference.” Synthese 198:7377–7399

    Google Scholar 

  • Azzouni J (2006) “How and Why Mathematics Is Unique as a Social Practice.” In 18 Unconventional Essays on the Nature of Mathematics, edited by Reuben Hersh, 201–219. Springer

  • Babbitt D, Goodstein J (2011) “Federigo Enriques’s Quest to prove the ‘Completeness theorem.’” Notices of the American Mathematical Society 58:240–249

    Google Scholar 

  • Barany M (2018) “The Fields Medal should return to its roots.” Nature 553:271–273

    Article  Google Scholar 

  • Bell J, Hellman G (2006) “Pluralism and the Foundations of Mathematics.” In Scientific Pluralism, edited by Kenneth Waters, Herbert Feigl, Stephen H. Kellert and Helen Longino, 64–79. University of Minnesota Press

  • Brigaglia A, Ciliberto C (1998) “Geometria Algebrica.” In La matematica italiana dopo l’unità. Gli anni tra le due guerre mondiali, edited by Simonetta di Sieno, Angelo Guerraggio and Pietro Nastasi, 185–320. Marcos y Marcos

  • Burgess J (2015) Rigor and structure. Oxford University Press

  • Burgess J, De Toffoli S (2022) “What Is Mathematical Rigor?” Aphex 25:1–17

    Google Scholar 

  • Castelnuovo G (1889) “Numero delle involuzioni razionali giacenti sopra una curva di dato genere.” Rendiconti R. Accademia dei Lincei 5(4)

  • Castelnuovo G (1928) “La geometria algebrica e la scuola italiana.” Congresso Internazionale dei Matematici: 3–10 settembre 1928, 1928

  • Castelnuovo G (1930) “Luigi Cremona nel centenario della nascita.” Rendiconti R. Accademia dei Lincei 6(12):613–618

    Google Scholar 

  • Christensen D (2009) “Disagreement as evidence: the epistemology of controversy.” Philosophical Compass 4(5):756–767

    Article  Google Scholar 

  • Ciliberto C, Sallent Del Colombo E (2018) “Francesco Severi: il suo pensiero matematico e politico prima e dopo la Grande Guerra.” Preprint, ArXiv: 1807.05769

  • Clarke-Doane J (2020) Morality and Mathematics. Oxford University Press

  • De Toffoli S (2021a) “Groundwork for a fallibilist account of mathematics.” Philosophical Quarterly 71(4):823–844

    Article  Google Scholar 

  • De Toffoli S (2021b) “Reconciling Rigor and Intuition.” Erkenntnis 86:1783–1802

    Article  Google Scholar 

  • De Toffoli S, Fontanari C (2022) “Objectivity and Rigor in Classical Italian Algebraic Geometry” Noesis 38:195–212

    Google Scholar 

  • De Toffoli S (2022) “Intersubjective Propositional Justification.” In Propositional and Doxastic Justification: New Essays on their Nature and Significance, edited by Paul Silva Jr and Luis R. G., Oliveira, 241–262. Routledge

  • De Toffoli, S (2023) “Who’s Afraid of Mathematical Diagrams?” Philosopher’s Imprint 23(1): 1–20

  • Easwaran K (2015) “Rebutting and undercutting in mathematics.” Philosophical Perspectives 29(1):146–162

    Article  Google Scholar 

  • Elga A (2007) “Reflection and Disagreement.” Nous 41(3):478–502

    Article  Google Scholar 

  • Enriques F (1942) “Sui sistemi continui di curve appartenenti ad una superficie algebrica.” Commentarii Mathematici Helvetici 15:227–237

    Article  Google Scholar 

  • Enriques, F (1949) Le superfici algebriche. Zanichelli

  • Fallis D (2003) “Intentional Gaps in Mathematical Proofs.” Synthese 134:45–69

    Google Scholar 

  • Fesenko I (2019) “About certain aspects of the study and dissemination of Shinichi Mochizuki’s IUT Theory.” Manuscript

  • Fontanari C (2023) “Guido Castelnuovo and his heritage: geometry, combinatorics, teaching.” In Algebraic Geometry between Tradition and Future, edited by G. Bini. Springer

  • Fricker M (1998) “Rational authority and social power: Towards a truly social epistemology.” Proceedings of the Aristotelian Society 98 (2): 159–177

  • Gödel K (1964) “What is Cantor’s Continuum Problem?” In Philosophy of Mathematics: Selected Readings, edited by Paul Benacerraf and Hilary Putnam, 258–273. Englewood Cliffs: Prentice-Hall

  • Goldstein C (2013) “Routine controversies: Mathematical Challenges in Mersenne’s correspondence.” Revue d’histoire des sciences 66(2):249–273

    Article  Google Scholar 

  • Hartnett K (2017) “A fight to fix geometry’s foundations.” Quanta Magazine

  • 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

  • Kelly T (2010) “Peer disagreement and higher order evidence.” In Social Epistemology: Essential Readings, edited by Alvin I. Goldman and Dennis Whitcomb, 183–217. Oxford University Press

  • Klarreich E (2018) “Titans of Mathematics Clash over Epic Proof of ABC Conjecture.” Quanta Magazine

  • Lakatos I (1976) Proofs and refutations. Cambridge University Press

  • Maddy P (2011) Defending the Axioms: on the philosophical foundations of set theory. Oxford University Press

  • Mumford D (1966) Lectures on curves on an algebraic surface. Princeton University Press

  • Mumford D (2011) “Intuition and rigor and Enriques’s quest.” Notices of the American Mathematical Society 58:250–260

    Google Scholar 

  • Nasar S, Gruber D (2006) “The Clash Over the Poincaré Conjecture” The New Yorker

  • Paseau A (2015) “Knowledge of Mathematics without Proof.” The British Journal for the Philosophy of Science 66(4):775–799

    Article  Google Scholar 

  • Paseau A (2016) “What’s the Point of Complete Rigour?” Mind 125(497):177–207

    Google Scholar 

  • Poincaré H (1889) “La logique et l’intuition.” L’enseignement mathématique 1(5):157–162

    Google Scholar 

  • Rittberg C, Tanswell F, Van Bendegem JP (2020) “Epistemic injustice in mathematics.” Synthese 197(9):3875–3904

  • Segre B (1938) “Un teorema fondamentale della geometria sulle superficie algebriche ed il principio di spezzamento.” Annali di Matematica Pura ed Applicata 17(1):107–126

    Article  Google Scholar 

  • Severi F (1942) “Intorno ai sistemi continui di curve sopra una superficie algebrica.” Commentarii Mathematici Helvetici 15:238–248

    Article  Google Scholar 

  • Severi F (1944) “Sul teorema fondamentale dei sistemi continui di curve sopra una superficie algebrica.” Annali di Matematica 23:149–181

    Article  Google Scholar 

  • Severi F (1948) Fondamenti di geometria algebrica. CADAM

    Google Scholar 

  • Severi F (1958) Geometria dei sistemi algebrici sopra una superficie e sopra una varietà algebrica. Edizioni Cremonese

    Google Scholar 

  • Tanswell F (2015) “A problem with the dependence of Informal Proofs on formal proofs.” Philosophia Mathematica 23(3):295–310

    Article  Google Scholar 

  • Tanswell F (forthcoming) Mathematical Rigour and Informal Proof. Cambridge Elements: Cambridge University Press

  • Tymoczko T (1979) “The four-color problem and its philosophical significance.” The Journal of Philosophy 76(2):57–83

    Article  Google Scholar 

  • Voevodsky V (2014) “The Origins and Motivations of Univalent Foundations.” The Institute Letter (The Institute for Advanced Studies)

  • Wagner R (2017) Making and breaking Mathematical sense: histories and philosophies of Mathematical Practice. Princeton University Press

  • Wagner R (2022) “Mathematical consensus: a research program.” Axiomathes 32(3):1185–1204

    Article  Google Scholar 

  • Zariski O (1935) Algebraic surfaces. Second Supplemented Edition ed. Springer

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Acknowledgements

We wish to thank the two anonymous referees for their detailed comments that helped us clarify our positions. Thanks are also due to Fenner Tanswell for insightful suggestions. A recent draft was discussed during a reading group at IUSS Pavia – we are particularly grateful to Andrea Sereni and Guido Tana for their feedback. This research project was partially supported by GNSAGA of INdAM and by PRIN 2017 “Moduli Theory and Birational Classification.”

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  1. Scuola Universitaria Superiore IUSS Pavia, Linguistics and Philosophy IUSS Center, Palazzo del Broletto, Piazza della Vittoria, 15, 27100, Pavia, PV, Italy

    Silvia De Toffoli

  2. Department of Mathematics, University of Trento, Via Sommarive 14, 38123, Povo, TN, Italy

    Claudio Fontanari

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  1. Silvia De Toffoli
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Correspondence to Silvia De Toffoli.

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De Toffoli, S., Fontanari, C. Recalcitrant Disagreement in Mathematics: An “Endless and Depressing Controversy” in the History of Italian Algebraic Geometry. glob. Philosophy 33, 38 (2023). https://doi.org/10.1007/s10516-023-09691-1

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