Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-5b777bbd6c-7sgmh Total loading time: 0 Render date: 2025-06-18T19:17:31.769Z Has data issue: false hasContentIssue false
Series:   Elements in the Philosophy of Mathematics

Philosophy of Mathematics from the Pythagoreans to Euclid

Published online by Cambridge University Press:  21 April 2025

Barbara M. Sattler
Barbara M. Sattler
Affiliation:
Ruhr University Bochum

Summary

This Element looks at the very beginning of the philosophy of mathematics in Western thought. It covers the first reflections on attempts to untie mathematics from its practical usage in administration, commerce, and land-surveying and discusses the first ideas to see mathematical structures as constituents underlying the physical world in the Pythagoreans. The first two sections focus on the epistemic status of mathematical knowledge in relation to philosophical knowledge and on the various ontological positions ancient Greek philosophers in early and classical times ascribe to mathematical objects – from independent and separate entities to mere abstractions and idealisations. Section 3 discusses the paradigmatic role mathematical deductions have played for philosophy, the role of mathematical diagrams, and mathematical methods of interest for philosophers. Section 4, finally, investigates a couple of individual concepts that are fundamental for both philosophy and mathematics, such as infinity.
Get access

Keywords

deductionsprinciplesnumbers and geometrical objectsPlatoAristotleEuclid
Type
Element
Information
Series: Elements in the Philosophy of Mathematics
Online ISBN: 9781009122788
Publisher: Cambridge University Press
Print publication: 22 May 2025

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Element purchase

Temporarily unavailable

Bibliography

Primary Sources

Annas, Julia (1976), Aristotle’s Metaphysics Books M and N, Clarendon Press.Google Scholar
Aristotle (1995), The Complete Works of Aristotle: The Revised Oxford Translation, edited by Barnes, Jonathan, Princeton University Press.Google Scholar
Barnes, Jonathan (1993), Aristotle’s Posterior Analytics (second edition), Clarendon Press.Google Scholar
Diels, Hermann, and Kranz, Walther (eds.) (1951), Fragmente der Vorsokratiker, Weidmannsche Verlagsbuchhandlung.Google Scholar
Heath, Thomas (1956), Euclid, The Thirteen Books of the Elements, translated with introduction and commentary, volumes I–III, Dover.Google Scholar
Huffman, Carl (1993), Philolaus of Croton: Pythagorean and Presocratic, Cambridge University Press.Google Scholar
Huffman, Carl (2005), Archytas of Tarentum, Cambridge University Press.CrossRefGoogle Scholar
Hussey, Edward (1993), Aristotle’s Physics Books III and IV, Clarendon Press.Google Scholar
Kirk, Geoffrey S., Raven, John E., and Schofield, Malcolm (eds.) (1983), The Presocratic Philosophers (second edition), Cambridge University Press. [KRS]CrossRefGoogle Scholar
Lee, Henry D. P. (ed.) (1967), Zeno of Elea, translation with notes, Hakkert.Google Scholar
Plato, (1997), Complete Works, ed. Cooper, John, Hackett.Google Scholar
Vitrac, Bernard (1990–2001), Euclide d’Alexandrie, Les Éléments, volumes I–IV, Presses Universitaires de France.Google Scholar

Secondary Sources

Acerbi, Fabio (2013), ‘Aristotle and Euclid’s Postulates’, Classical Quarterly 63(2), 680–685. DOI: https://doi.org/10.1017/S0009838813000177.CrossRefGoogle Scholar
Acerbi, Fabio (2020), ‘Mathematical Generality, Letter-Labels, and All That’, Phronesis 65(1), 27–75. DOI: https://doi.org/10.1163/15685284-12342029.CrossRefGoogle Scholar
Annas, Julia (1987), ‘Die Gegenstände der Mathematik bei Aristoteles’, in Graeser, Andreas (ed.), Mathematics and Metaphysics in Aristotle, P. Haupt Publishers, pp. 131–147.Google Scholar
Balme, David M. (1987), ‘The Place of Biology in Aristotle’s Philosophy’, in Gotthelf, Allan & Lennox, James G. (eds.), Philosophical Issues in Aristotle’s Biology, Cambridge University Press, pp. 9–20.CrossRefGoogle Scholar
Baron, Samuel (2021), ‘Mathematical Explanation: A Pythagorean Proposal’, The British Journal for the Philosophy of Science, 75(3), 1–29.Google Scholar
Broadie, Sarah (2021), Plato’s Sun-Like Good, Cambridge University Press.CrossRefGoogle Scholar
Burkert, Walter (1972), Lore and Science in Ancient Pythagoreanism, trans. Edwin L. Minar, Cambridge University Press.Google Scholar
Burnyeat, Myles (2000), ‘Plato on Why Mathematics is Good for the Soul’, in Smiley, Timothy (ed.), Mathematics and Necessity: Essays in the History of Philosophy, The British Academy, pp. 1–81.Google Scholar
Chemla, Karine (2024), ‘The Dark Side of the History of Proof’, in Sriraman, Bharath (ed.), Handbook of the History and Philosophy of Mathematical Practice, Springer, pp. 2137–2161. DOI: https://doi.org/10.1007/978-3-030-19071-2_81-1.CrossRefGoogle Scholar
Cleary, John (1985), ‘On the Terminology of “Abstraction” in Aristotle’, Phronesis, 30(1), 13–45.CrossRefGoogle Scholar
Cuomo, Serafina (2001), Ancient Mathematics, London: Routledge.Google Scholar
De Risi, Vincenzo (2021), ‘Euclid’s Common Notions and the Theory of Equivalence’, Foundations of Science 26, 301–324.CrossRefGoogle Scholar
Harari, Orna (2003), ‘The Concept of Existence and the Role of Constructions in Euclid’s Elements’, Archive for History of Exact Sciences 57, pp. 1–23. DOI: https://doi.org/10.1007/s004070200053.CrossRefGoogle Scholar
Harari, Orna (2008), ‘Proclus’ Account of Explanatory Demonstrations in Mathematics and Its Context’, Archiv für Geschichte der Philosophie 90, 137–164.CrossRefGoogle Scholar
Heath, Thomas L. (1921), A History of Greek Mathematics, Oxford University Press.Google Scholar
Hellman, Geoffrey, and Shapiro, Stewart (2018), Varieties of Continua, Oxford University Press.CrossRefGoogle Scholar
Horsten, Leon (2022), ‘Philosophy of Mathematics’, The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), https://plato.stanford.edu/archives/spr2022/entries/philosophy-mathematics/.Google Scholar
Jope, James (1972), ‘Subordinate Demonstrative Science in the Sixth Book of Aristotle’s Physics’, Classical Quarterly 22, 279–292.CrossRefGoogle Scholar
Klein, Jacob (1968), Greek Mathematical Thought and the Origin of Algebra, trans. Eva Brann, Massachusetts Institute of Technology Press.Google Scholar
Knorr, Wilbur Richard (1975), The Evolution of the Euclidean Elements, Reidl.CrossRefGoogle Scholar
Lang, Marc (2021), ‘What Could Mathematics Be for It to Function in Distinctively Mathematical Scientific Explanations?’, Studies in History and Philosophy of Science 87, 44–53.CrossRefGoogle Scholar
Lear, Jonathan (1982), ‘Aristotle’s Philosophy of Mathematics’, Philosophical Review 91, 161–192.CrossRefGoogle Scholar
Lennox, James (2021), ‘Aristotle’s Biology’, The Stanford Encyclopedia of Philosophy (Fall 2021 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/fall2021/entries/aristotle-biology/.Google Scholar
Mancosu, Paolo, Poggiolesi, Francesca, and Pincock, Christopher (2023), ‘Mathematical Explanation’, The Stanford Encyclopedia of Philosophy, Edward N. Zalta & Uri Nodelman (eds.), https://plato.stanford.edu/archives/fall2023/entries/mathematics-explanation/.Google Scholar
Manders, Kenneth (2008), ‘The Euclidean Diagram’, in Mancosu, Paolo (ed.), Philosophy of Mathematical Practice, Clarendon Press, pp. 112–183.Google Scholar
McKirahan, Richard D. (1992), Principles and Proofs: Aristotle’s Theory of Demonstrative Science, Princeton University Press.Google Scholar
Mendell, Henry (2004), ‘Aristotle and Mathematics’, Stanford Encyclopaedia of Philosophy, Edward N. Zalta (ed.), https://plato.stanford.edu/archives/spr2017/entries/aristotle-mathematics/.Google Scholar
Mendell, Henry (2022), ‘Plato on Mathematics’, in David Ebrey and Richard Kraut (eds.), The Cambridge Companion to Plato, pp. 358–398, Cambridge University Press.Google Scholar
Menn, Stephen (2002), ‘Plato and the Method of Analysis’, Phronesis 47(3), 193–223. DOI: https://doi.org/10.1163/15685280260458127.CrossRefGoogle Scholar
Mueller, Ian (1969), ‘Euclid’s Elements and the Axiomatic Method’, The British Journal for the Philosophy of Science 20, 289–309.CrossRefGoogle Scholar
Mueller, Ian (1974), ‘Greek Mathematics and Greek Logic’, in Corcoran, John (ed.), Ancient Logic and Its Modern Interpretations, Synthese Historical Library, vol. 9, pp. 35–70, Springer.CrossRefGoogle Scholar
Mueller, Ian (1980), ‘Ascending to Problems: Astronomy and Harmonics in Republic VII’, in Anton, John Peter (ed.), Science and the Sciences in Plato, Delmar, pp. 103–122.Google Scholar
Mueller, Ian (1981), Philosophy of Mathematics and Deductive Structure in Euclid’s Elements, MIT Press.Google Scholar
Netz, Reviel (1999), The Shaping of Deduction in Greek Mathematics, Cambridge University Press.CrossRefGoogle Scholar
Neugebauer, Otto (1936), Studien zur Geschichte der antiken Algebra III (Quellen und Studien zur Geschichte der Mathematik, Abteilung B: Studien 3).Google Scholar
Neugebauer, Otto (1969), Vorlesungen über Geschichte der antiken mathematischen Wissenschaften 1, Vorgriechische Mathematik (second edition). Grundlehren der mathematischen Wissenschaften, vol. 43, Springer.CrossRefGoogle Scholar
Proclus, (1992), A Commentary on the First Book of Euclid’s Elements, trans. Glenn R. Morrow, Princeton University Press.Google Scholar
Restrepo, Guillermo, and Villaveces, José (2012), ‘Mathematical Thinking in Chemistry’, Hyle 18(1), 3–22.Google Scholar
Salmon, Wesley (1980), Space, Time and Motion: A Philosophical Introduction, Dickenson Publishing Company.Google Scholar
Sattler, Barbara M. (2012), ‘A Likely Account of Necessity, Plato’s Receptacle as a Physical and Metaphysical Basis of Space’, Journal of the History of Philosophy 50(2), 159–195.CrossRefGoogle Scholar
Sattler, Barbara M. (2019), ‘The Notion of Continuity in Parmenides’, Philosophical Inquiry 43(1/2), 40–53.CrossRefGoogle Scholar
Sattler, Barbara M. (2020a), The Concept of Motion in Ancient Greek Thought, Cambridge University Press.CrossRefGoogle Scholar
Sattler, Barbara M. (2020b), ‘Divisibility or Indivisibility: The Notion of Continuity from the Presocratics to Aristotle’, in Shapiro, Stewart and Hellman, Geoffrey (eds.), The History of Continuity: Philosophical and Mathematical Perspectives, pp. 6–26, Oxford University Press.CrossRefGoogle Scholar
Sattler, Barbara M. (2021), ‘Paradoxes as Philosophical Method and their Zenonian Origins’, Proceedings of the Aristotelian Society 121(2), 153–181.CrossRefGoogle Scholar
Scott, Dominic (2006), Plato’s Meno, Cambridge University Press.Google Scholar
Simplicius (2002), On Aristotle Physics 3, trans. James O. Urmson, Bloomsbury.Google Scholar
Simplicius (2011), On Aristotle Physics 1.3–4, trans. C. C. W. Taylor and Pamela M. Huby, Bloomsbury.Google Scholar
Szabó, A. (2004), ‘Wie ist die Mathematik zu einer deduktiven Wissenschaft geworden?’, in Christianidis, Jean (ed.), Classics in the History of Greek Mathematics, pp. 45–80. Springer (first published in 1956).CrossRefGoogle Scholar
Paul, Tannery (1884), ‘Sur l’authenticité des axiomes d’Euclide’, Bulletin des Sciences Mathématiques et Astronomiques, Series 2, 8(1), 162–175.Google Scholar
Tannery, Paul(1887), La geometrie grecque, Gauthier-Villars.Google Scholar
Unguru, Sabetai (1975), ‘On the Need to Rewrite the History of Greek Mathematics’, Archive for History of Exact Sciences 15, 67–114.CrossRefGoogle Scholar
von Fritz, Kurt (1955), ‘Die Archai in der griechischen Mathematik’, Archiv für Begriffsgeschichte 1, 12–103.Google Scholar
von Fritz, Kurt (1970), ‘The Discovery of Incommensurability by Hippasus of Metapontum’, in Furley, David and Allen, Reginald E. (eds.), Studies in Presocratic Philosophy 1, 211–231, Routledge.Google Scholar
Zeuthen, Hieronymus G. (1896), Geschichte der Mathematik in Altertum und Mittelalter, A. D. Höst.Google Scholar

Save element to Kindle

To save this element to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Philosophy of Mathematics from the Pythagoreans to Euclid
Available formats
×

Save element to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Philosophy of Mathematics from the Pythagoreans to Euclid
Available formats
×

Save element to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Philosophy of Mathematics from the Pythagoreans to Euclid
Available formats
×