Ancient Greek Mathematical Proofs and Metareasoning

  1. Valente, Mario Bacelar 1
  1. 1 Pablo de Olavide University, Seville, Spain
Book:
Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques

Publisher: Birkhäuser

ISSN: 2662-8503,  2662-8511

ISBN: 9783031461927,  9783031461934

Year of publication: 2024

Pages: 15-33

Type: Book chapter

DOI: 10.1007/978-3-031-46193-4_2 GOOGLE SCHOLAR lock_openOpen access editor

Sustainable development goals

Abstract

We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, we go through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feeling-knowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness.

Bibliographic References

  • Ackerman R (2019) Heuristic cues for meta-reasoning judgments: review and methodology. Psychological Topics 28, 1–20.
  • Ackerman R, Thompson VA (2017) Meta-reasoning: monitoring and control of thinking and reasoning. Trends in Cognitive Sciences 21, 607–617.
  • Ashton Z (2021). Audience role in mathematical proof development. Synthese 198, 6251–6275.
  • Bang D, Frith CD (2017) Making better decisions in groups. Royal Society Open Science 4, 170193.
  • Bartley JE, Boeving ER, Riedel MC, Bottenhorn KL, Salo T, Eickhoff SB, Brewe E, Sutherland MT, Laird AR (2018) Meta-analytic evidence for a core problem solving network across multiple representational domains. Neuroscience and Biobehavioral Review 92, 318–337.
  • Beck M, Geoghegan R (2010) The art of proof: Basic training for deeper mathematics. New York: Springer.
  • Cunningham DW (2012). A logical introduction to proof. New York: Springer.
  • Dal Magro T, García Pérez MJ (2019). On Euclidean diagrams and geometrical knowledge. Theoria 34, 255–276.
  • Dal Magro T, Valente M (2021). On the representational role of Euclidean diagrams: representing qua samples. Synthese 199, 3739–3760.
  • Dutilh Novaes C (2016). Reductio ad absurdum from a dialogical perspective. Philosophical Studies 173, 2605–2628.
  • Dutilh Novaes C (2018). A dialogical conception of explanation in mathematical proofs. In: Ernest P (ed) The philosophy of mathematics education today. Cham: Springer, p. 81–98.
  • Dutilh Novaes C (2020) The dialogical roots of deduction. Cambridge: Cambridge University Press.
  • Efklides A (2006) Metacognition and affect: what can metacognitive experiences tell us about the learning process? Educational Research Review 1, 3–14.
  • Ferreirós J, García Pérez MJ (2020). Beyond natural geometry: on the nature of proto-geometry. Philosophical Psychology 33, 181–205.
  • Fiedler K, Ackerman R, Scarampi C (2019) Metacognition: monitoring and controlling one’s own knowledge, reasoning and decisions. In: Sternberg RJ, Funke J (eds) The psychology of human thought: an introduction. Heidelberg: Heidelberg University Publishing, p. 89–110.
  • Fitzpatrick R (2008). Euclid’s Elements of geometry. Morrisville: Lulu. (Online version available at http://farside.ph.utexas.edu/books/Euclid/Euclid.html)
  • Frankish K (2018) Inner speech and outer thought. In: Langland-Hassan P, Vicente A (eds) Inner speech: new voices. Oxford: Oxford University Press. (Author’s preprint, available via: https://nbviewer.jupyter.org/github/k0711/kf_articles/blob/master/Frankish_Inner%20speech%20and%20outer%20thought_eprint.pdf)
  • Freksa C, Barkowsky T, Falomir Z, van de Ven J (2019). Geometric problem solving with strings and pins. Spatial Cognition & Computation 19, 46–68.
  • Giardino V (2013). A practice-based approach to diagrams. In: Aminrouche M, Shin S (eds) Visual reasoning with diagrams. Basel: Birkhäuser, p. 135–151.
  • Giaquinto M (2007). Visual thinking in mathematics. New York: Oxford University Press.
  • Heath T L (1956) The thirteen Books of the Elements. New York: Dover Publications.
  • Heath TL (1981). A history of Greek mathematics. New York: Dover Publications.
  • Hohol M, Miłkowski M (2019). Cognitive artifacts for geometric reasoning. Foundations of Science 24, 657–680.
  • Hohol M (2020). Foundations of geometric cognition. New York: Routledge.
  • Høyrup J (2019a) Hippocrates of Chios – his Elements and his lunes: A critique of circular reasoning. AIMS mathematics 5, 158–184.
  • Høyrup J (2019b) From the practice of explanation to the ideology of demonstration: an informal essay. In: Schubring G (ed) Interfaces between mathematical practices and mathematical education. Cham: Springer, p. 27–46.
  • Joyce DE (1998). Euclid’s Elements. http://aleph0.clarku.edu/~djoyce/java/elements/elements.html.
  • Krantz SG (2011) The proof is in the pudding: The changing nature of mathematical proof. New York: Springer.
  • Levelt WJM (1989) Speaking: From intention to articulation. MIT Press.
  • Magnani L (2013). Thinking through drawing. The Knowledge Engineering Review 28, 303–326.
  • Manders K (2008). Diagram-based geometric practice. In: Mancosu P (ed) The philosophy of mathematical practice. New York: Oxford University Press, p. 65–79.
  • Mueller I (2006) Greek mathematics to the time of Euclid. In: Gill ML, Pellegrin P (eds) A companion to ancient philosophy. Malden: Blackwell Publishing, p. 686–718.
  • Netz R (1999) The shaping of deduction in Greek mathematics: a study in cognitive history. Cambridge: Cambridge University Press.
  • Netz R (2004) Eudemus of Rhodes, Hippocrates of Chios and the earliest form of a Greek mathematical text. Centaurus 2004, 243–286.
  • Rav Y (2007) A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica 15, 291–320.
  • Rouault M, McWilliams A, Allen MG, Fleming SM (2018) Human metacognition across domains: insights from individual differences and neuroimaging. Personality Neuroscience 1, 1–13.
  • Silver I, Mellers BA, Tetlock PE (2021) Wise teamwork: collective confidence calibration predicts the effectiveness of group discussion. Journal of Experimental Social Psychology 96, 104157.
  • Tall D, Yevdokimov O, Koichu B, Whiteley W, Kondratieva M, Cheng, YH (2021) Cognitive development of proof. In: Hanna G, de Villiers M (eds) Proof and proving in mathematical education. Cham: Springer, p. 13–49.
  • Vaccaro AG, Fleming SM (2018) Thinking about thinking: a coordinate-based meta-analysis of neuroimaging studies of metacognitive judgements. Brain and Neuroscience Advances 2, 1–14.
  • Vitrac B (2012). The Euclidean ideal of proof in the Elements and philological uncertainties of Heiberg’s edition of the text. In: Chemla K (ed) The history of mathematical proof in ancient traditions. Cambridge: Cambridge University Press, p. 69–134.