Element contents
Medieval Finitism
Published online by Cambridge University Press: 09 January 2025
Summary
Discussing various versions of two medieval arguments for the impossibility of infinity, this Element sheds light on early stages of the evolution of the notion of INFINITIES OF DIFFERENT SIZES. The first argument is called 'the Equality Argument' and relies on the premise that all infinities are equal. The second argument is called 'the Mapping Argument' and relies on the assumption that if one thing is mapped/ superposed upon another thing and neither exceeds the other, the two things are equal to each other. Although these arguments were initially proposed in the context of discussions against the possibility of infinities, they have played pivotal roles in the historical evolution of the notion of INFINITIES OF DIFFERENT SIZES.
Information
- Type
- Element
- Information
- Online ISBN: 9781009047623Publisher: Cambridge University PressPrint publication: 30 January 2025
References
Primary Sources
Aquinas, Thomas (2006) Summa Theologiae Ia. 2–11, Volume 2, Existence and Nature of God. Edited and translated by McDermott, T.. Cambridge: Cambridge University Press.Google Scholar
Aristotle, (1984) The Complete Works of Aristotle: The Revised Oxford Translation. Edited by Barnes, J.. Princeton, NJ: Princeton University Press.Google Scholar
Bacon, Roger (1928) The Opus Majus of Roger Bacon. Translated by R. B. Burke. Two Vols. Philadelphia: University of Pennsylvania Press.Google Scholar
Bacon, Roger (2012) Opus Tertium, in Fr. Rogeri Bacon Opera Quædam Hactenus Inedita. Edited by Brewer, J. S.. Cambridge: Cambridge University Press, pp. 3–310.CrossRefGoogle Scholar
al-Baghdādī, Abu-l-Barakāt (1938) al-Muʿtabar fi-l-ḥikma [The Considered]. Edited by al-ʿAlawī al-Ḥaḍramī, A., al-Yamānī, Aḥmad Ibn Muḥammad, and al-Musāwī, Zayn al-ʿĀbidīn. Three Vols. Hyderabad: Dāʾirat al-maʿārif al-ʿuthmāniyya.Google Scholar
al-Baghdādī, ʿAbd al-Qāhir (1981) Uṣūl al-dīn [The Principles of Religion]. Beirut: Dār al-kutub al-ʿilmiyya.Google Scholar
Ibn al-Marzubān, Bahmanyār (1996) Al-Taḥṣīl [The Attainment]. Edited by Muṭahharī, M.. Tehran: Tehran University Press.Google Scholar
Bonaventure, (1882) Commentaria in Quatuor Libros Sententiarum [Commentary on the Four Books of the Sentences], in Opera Omnia. Edited by the Fathers of Collegium St. Bonaventure. Vol. 1. Quaracchi: ex typographia Collegii St. Bonaventurae.Google Scholar
Bonaventure, (1964) On the Eternity of the World (De Aeternitate Mundi). Translated by P. M. Byrne. Milwaukee, WI: Marquette University Press.Google Scholar
Crescas, Ḥasdai (2018) Light of the Lord (Or Hashem). Translated by R. Weiss. Oxford: Oxford University Press.Google Scholar
al-Dawānī, Jalāl al-Dīn (2013) Risāla fī ithbāt al-wājib [A Treatise on the Proof of the Necessary]. Edited by Ghūsh, M. A. Abū. Amman: Dār al-nūr al-mubīn lil-dirāsāt wa-l-nashr.Google Scholar
Euclid, (1908) The Thirteen Books of Euclid‘s Elements. Translated by T. L. Heath. Cambridge: Cambridge University Press.Google Scholar
al-Fārābī, (1985). Fuṣūṣ al-ḥikam [Bezels of Wisdom]. Edited by ĀL Yāsīn, M. Ḥ.. Second Ed. Qum: Bīdār Press.Google Scholar
al-Ghazālī, (2000a) Maqāṣid al-falāsifa [The Aims of the Philosophers]. Edited by Bījū, M.. Damascus: Maṭbaʿa al-ṣabāḥ.Google Scholar
al-Ghazālī, (2000b) The Incoherence of the Philosophers. Edited and translated by Marmura, M. E.. Provo, UT: Brigham Young University Press.Google Scholar
al-Ghazālī, (2013) Moderation in Belief. Translated with an interpretive essay and notes by Yaqub, A. M.. Chicago, IL: The University of Chicago Press.Google Scholar
Ibn ʿ Adī, Yaḥyā (1988) ‘Maqāla fī ghayr al-mutanāhī’, in his Maqālat Yaḥyā Ibn ʿAdī al-falsafiyya. Edited by Khalīfāt, S.. Amman: University of Jordan Press, pp. 135–40.Google Scholar
Bājja, Ibn (1991) Sharḥ al-samāʿ al-ṭabīʿī li-Arasṭāṭālīs [Commentary on Aristotle‘s Physics]. Edited by Fakhrī, M.. Second Ed. Beirut: Dār al-nahār li-l-nashr.Google Scholar
Paquda, Ibn, Ben Joseph, Bachya (1996) Duties of the Heart. Translated into Hebrew by Ibn Tibbon and into English by Daniel Haberman. New York: Feldheim.Google Scholar
[Averroes, Ibn Rushd] (1987) Averroes’ Tahafut al-Tahafut (The Incoherence of the Incoherence). Volumes I and II. Translated by S. Van Den Bergh. Cambridge: Cambridge University Press.Google Scholar
Rushd, Ibn (1998) Tahāfut al-Tahāfut [The Incoherence of The Incoherence]. Edited by al-Jābirī, M. A.. Beirut: Markaz al-dirasāt al-waḥda al-ʿarabiyya.Google Scholar
Sīnā, Ibn (1959) Al-Shifāʾ, Al-Manṭiq, Al-Maqūlāt [The Categories of The Healing]. Edited by Qanawātī, J., al-Khuḍayrī, M. M., al-Ihwānī, A. F., and Zāyid, S.. Cairo: Al-Hayʾa al-ʿāmma li-shuʾūn al-maṭābiʿ al-amīriyya.Google Scholar
Sīnā, Ibn (1985) Al-Najāt [The Salvation]. Edited by Dānishpazhūh, M. T.. Tehran: Tehran University Press.Google Scholar
[Ibn Sīnā] Avicenna (2009) The Physics of the Healing. Edited and translated by McGinnis, J.. Provo, UT: Brigham Young University Press.Google Scholar
al-Khayyāt, (1957) Kitāb al-intiṣār wa-l-radd ʿalā Ibn al-Rawandī al-mulḥid [The Book of Victory]. Edited and translated into French by A. N. Nader. Beirut: Catholic Press.Google Scholar
al-Kindī, (2012) The Philosophical Works of al-Kindī. Translated by P. Adamson and P. E. Pormann. Karachi: Oxford University Press.Google Scholar
Lucretius, (2001) On the Nature of Things. Translated by M. F. Smith. Indianapolis, IN: Hackett.Google Scholar
Maimonides, Moses (1967) The Guide of the Perplexed. Translated by S. Pines. Chicago, IL: The University of Chicago Press.Google Scholar
Neckam, Alexandri (1863) De Naturis Rerum [On the Nature of Things]. Edited by Wright, T.. London: Longman, Roberts, and Green.Google Scholar
Philoponus, John (2004) Against Proclus on the Eternity of the World 1–5. Translated by M. Share. London: Bloomsbury.Google Scholar
Philoponus, John (2014) Against Aristotle on the Eternity of the World. Translated by C. Wildberg. London: Bloomsbury.Google Scholar
Plutarch, (1976) Against the Stoics on Common Conceptions, in Moralia, Volume XIII: Part 2: Stoic Essays. Edited and translated by H. Cherniss. Loeb Classical Library 470. Cambridge, MA: Harvard University Press.Google Scholar
al-Rāzī, Fakhr al-Dīn (1990) Al-Mabāḥith al-mashriqiyya fī ʿilm al-ilāhiyyāt wa-l-ṭabīʿiyyāt [Eastern Investigations]. Edited by al-Baghdādī, M.. Two Vols. Beirut: Dār al-kitāb al-ʿarabī.Google Scholar
al-Rāzī, Fakhr al-Dīn (1994). Sharḥ al-ʿuyūn al-ḥikma [Commentary on Fountains of Wisdom]. Edited by Hejazi, M. and Saqa, A. A.. Tehran: Muʾassisa li-l-ṭabāʾa al-nashr.Google Scholar
al-Shahrastānī, Muḥammad (1846) Kitab al-ilal wa al-niḥal: Books of Religions and Philosophical Sects. Edited by Cureton, W.. Two Vols. London: Society for the Publication of Oriental Texts.Google Scholar
al-Shahrastānī, Muḥammad (1934) The Summa Philosophiae of al-Shahrastānī: Nihāyat al-iqdām fī ʿilm al-kalām [The End of Steps in Theology]. Edited and translated by Guillaume, Alfred. London: Oxford University Press.Google Scholar
al-Shahrastānī, Muḥammad (2001) Struggling with the Philosopher: A Refutation of Avicenna’s Metaphysics. Edited and translated by Mayer, T., and Madelung, W.. London: I.B.Tauris.Google Scholar
al-Suhrawardī, Shihāb al-Dīn (1999) The Philosophy of Illumination. Edited and translated by Walbridge, J., and Ziai, H.. Provo, UT: Brigham Young University Press.Google Scholar
al-Tabrīzī, Muḥammad (1981) Sharḥ-i bīst-u-panj muqqadama dar ithbāt-i bārī taʿālā [Commentary on the Twenty-Five Premises for the Proof of the Existence of God]. Edited by Mohaghegh, M.. Translated into Persian by S. J. Sajjadi. Tehran: The Institute of Islamic Studies (University of Tehran-McGill University).Google Scholar
al-Ṭūsī, Naṣīr al-Dīn (2004) al-Maṣāriʿ al-Mūṣāriʿ [The Wrestlings Down of the Wrestler]. Edited by Madelung, W.. Tehran: The Institute of Islamic Studies (University of Tehran-McGill University).Google Scholar
William of Auvergne (1998) The Universe of Creatures. Selections Translated from the Latin with an Introduction and Notes by Teske, R. J.. Milwaukee, WI: Marquette University Press.Google Scholar
Secondary Sources
Abdel Meguid, A. (2018) ‘Al-Kindī’s Argument for the Finitude of Time in His Critique of Aristotle’s Theory of the Eternity of the World in the Treatise on First Philosophy: The Role of the Perceiving Soul and the Relation between Sensation and Intellection’, Journal of Islamic Studies, 29(3), pp. 323–56.CrossRefGoogle Scholar
Beşikci, M. N. (2022) ‘Mehmed Emīn Üsküdārīʾs Sharḥ al-Barahīn al-khamsa on Infinity’, Nazariyat, 8(1), pp. 101–43.CrossRefGoogle Scholar
Beşikci, M. N. (forthcoming) ‘A New Perspective on an Old Debate: Ismāʿīl Gelenbevī on Divine Knowledge and the Mapping Demonstration’, in Qureshi, Y. (ed.) An 18th-Century Ottoman Polymath: Ismāʿīl Gelenbevī. Berlin: De Gruyter.Google Scholar
Bowin, J. (2007) ‘Aristotelian Infinity’, Oxford Studies in Ancient Philosophy, 32, pp. 233–50.Google Scholar
Brown, P. (1965) ‘A Medieval Analysis of Infinity’, Journal of the History of Philosophy, 3(2), pp. 242–43.CrossRefGoogle Scholar
Busard, H. L. L. (1983) The First Latin Translation of Euclid’s Elements Commonly Ascribed to Adelard of Bath. Toronto: Pontifical Institute of Medieval Studies.Google Scholar
Cohoe, C. (2013) ‘There Must Be a First: Why Thomas Aquinas Rejects Infinite, Essentially Ordered, Causal Series’, British Journal for the History of Philosophy, 21(5), pp. 838–56.CrossRefGoogle Scholar
Coope, U. (2012) ‘Aristotle on the Infinite’, in Shield, C. (ed.), The Oxford Handbook of Aristotle. Oxford: Oxford University Press, pp. 267–86.Google Scholar
Cooper, J. (2016) ‘Aristotelian Infinites’, Oxford Studies in Ancient Philosophy, 51, pp. 161–206.Google Scholar
Davenport, A. (1999) Measure of a Different Greatness: The Intensive Infinite, 1250–1650. Leiden: Brill.CrossRefGoogle Scholar
Davidson, H. A. (1969) ‘John Philoponus as a Source of Medieval Islamic and Jewish Proofs of Creation’, Journal of the American Oriental Society, 89(2), pp. 357–91.CrossRefGoogle Scholar
Davidson, H. A. (1987) Proofs for Eternity, Creation and the Existence of God in Medieval Islamic and Jewish Philosophy. New York: Oxford University Press.Google Scholar
Dhanani, A. (2015) ‘The Impact of Ibn Sīnā’s Critique of Atomism on Subsequent Kalām Discussions of Atomism’, Arabic Sciences and Philosophy, 25(1), pp. 79–104.CrossRefGoogle Scholar
Duhem, P. (1985) Medieval Cosmology Theories of Infinity: Place, Time, Void, and the Plurality of Worlds. Edited and translated by Ariew, R.. Chicago: The University of Chicago Press.Google Scholar
Geach, P. T. (1967) ‘Infinity in Scholastic Philosophy’, in Lakatos, I. (ed.) Problems in the Philosophy of Mathematics. Amsterdam: North-Holland, pp. 41–42.Google Scholar
Grant, E. (ed.) (1974) A Source Book in Medieval Science. Cambridge, MA: Harvard University Press.Google Scholar
Gutas, D. (2014) Avicenna and the Aristotelian Tradition. Second Ed. Leiden: Brill.CrossRefGoogle Scholar
Hintikka, J. (1966) ‘Aristotelian Infinity’, The Philosophical Review, 75(2), pp. 197–218.CrossRefGoogle Scholar
Hodges, W. (2012) ‘Affirmative and Negative in Ibn Sīnā’, in Novaes, C. D., and Hjortland, O. T. (eds.) Insolubles and Consequences: Essays in Honour of Stephen Read. London: College Publications, pp. 119–34.Google Scholar
İskenderoğlu, M. (2002) Fakhr-al-Dīn al-Rāzī and Thomas Aquinas on the Question of the Eternity of the World. Leiden: Brill.CrossRefGoogle Scholar
Kohler, G. (2006) ‘Medieval Infinities in Mathematics and the Contribution of Gersonides’, History of Philosophy Quarterly, 23(2), pp. 95–116.Google Scholar
Kouremenos, T. (1995) Aristotle on Mathematical Infinity. Stuttgart: Franz Steiner Verlag.Google Scholar
Kretzmann, N. (1982) ‘Syncategoremata, Sophismata, Exponibilia’, in Kretzmann, N., Kenny, A., and Pinborg, J. (eds.) Cambridge History of Later Medieval Philosophy. Cambridge: Cambridge University Press, pp. 211–45.CrossRefGoogle Scholar
Lammer, A. (2018) ‘Two Sixth/Twelfth-Century Hardliners on Creation and Divine Eternity: al-Šahrastānī and Abū l-Barakāt al-Baġdādī on God’s Priority over the World’, in Al Ghouz, A. (ed.). Islamic Philosophy from the 12th to the 14th Century. Bonn: Bonn University Press.Google Scholar
Lear, J. (1980) ‘Aristotelian Infinity’, Proceedings of the Aristotelian Society, 80, pp. 187–210.CrossRefGoogle Scholar
Mancosu, P. (2009) ‘Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor’s Theory of Infinite Number Inevitable?’, The Review of Symbolic Logic, 2(4), pp. 612–46.CrossRefGoogle Scholar
Mancosu, P. (forthcoming) The Wilderness of Infinity: Robert Grosseteste, William of Auvergne and Mathematical Infinity in the Thirteenth Century.Google Scholar
Marmura, M. E. (1960) ‘Avicenna and the Problem of the Infinite Number of Souls’, Mediaeval Studies, 22, 232–39.CrossRefGoogle Scholar
Mayer, T. (2012) ‘The Absurdities of Infinite Time: Shahrastānī’s Critique of Ibn Sīnā and Ṭūṣī’s Defence’, in Hansberger, R., Afifi al-Akiti, M., and Burnett, C. (eds.) Medieval Arabic Thought: Essays in Honour of Fritz Zimmermann. London: The Warburg Institute, pp. 103–34.Google Scholar
McGinnis, J. (2010). ‘Avicennan Infinity: A Select History of the Infinite through Avicenna’, Documenti e studi sulla tradizione filosofica medievale, 21, pp. 199–221.Google Scholar
McGinnis, J. (2014). ‘The Eternity of the World: Proofs and Problems in Aristotle, Avicenna, and Aquinas’, American Catholic Philosophical Quarterly, 88(2), pp. 271–88.CrossRefGoogle Scholar
Monnot, G. (1996) ‘Al-Shahrastānī’, in Fleet, K., Krämer, G., Matringe, D, Nawas, J., and Rowson, E. (eds.) Encyclopedia of Islam. Brill: Leiden, pp. 214–16.Google Scholar
Murdoch, J. E. (1964) ‘Superposition, Congruence and Continuity in the Middle Ages’, in Cohen, I. B., and Taton, R. (eds.) Mélanges Alexandre Koyré I: L’Aventure de la Science. Paris: Hermann, pp. 416–41.Google Scholar
Murdoch, J. E. (1968) ‘The Equality of Infinites in the Middle Ages’, Actes du XI congrès international d’histoire des sciences, vol. III. Paris, pp. 171–74.Google Scholar
Murdoch, J. E. (1969) ‘Mathesis in Philosophiam Scholasticam Introducta: The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology’, in Arts libéraux et philosophie au moyen âge, Actes du lVe congrés international de philosophie médiévale. Montréal: Librairie philosophique J. Vrin, pp. 215–56.Google Scholar
Murdoch, J. E. (1981) ‘Mathematics and Infinity in the Later Middle Ages’, Proceedings of the American Catholic Philosophical Association, 55, pp. 40–58.CrossRefGoogle Scholar
Murdoch, J. E. (1982) ‘Infinity and Continuity’, in Kretzmann, N., Kenny, A., and Pinborg, J. (eds.) The Cambridge History of Later Medieval Philosophy. Cambridge: Cambridge University Press, pp. 564–91.Google Scholar
Pines, S. (1972) ‘An Arabic Summary of a Lost Work of John Philoponus’, Israel Oriental Studies, 2 pp. 320–52 (repr. in Pines, S. (1986) Studies in Arabic Versions of Greek Texts and in Medieval Science. Leiden: Brill, pp. 294–326).Google Scholar
Pines, S. (1997) Studies in Islamic Atomism. Translated by S. Schwarz and edited by Langermann, T.. Jerusalem: The Magnes Press.Google Scholar
Reck, E. (2023) ‘Dedekind’s Contributions to the Foundations of Mathematics’, in Zalta, E. N., and Nodelman, U. (eds.) The Stanford Encyclopedia of Philosophy (Winter 2023 Edition). https://plato.stanford.edu/archives/win2023/entries/dedekind-foundations/.Google Scholar
Rescher, N., and Khatchadourian, H. (1965) ‘Al-Kindī’s Epistle on the Finitude of the Universe’. Isis 56(4), pp. 426–33.Google Scholar
Rioux, J. W. (2023) Thomas Aquinas’ Mathematical Realism. Cham: Palgrave Macmillan.CrossRefGoogle Scholar
Shamsi, F. A. (1975) ‘Epistle on What Cannot Be Infinite and of What Infinity May Be Attributed’, Islamic Studies, 14(2), pp. 123–44.Google Scholar
Shihadeh, A. (2011) ‘Khojazāda on al-Ghazālī’s Criticism of the Philosophers’ Proof of the Existence of God’ in Yücedoğru, T., Koloğlu, O. Ş., Kılavuz, U. M., and Gömbeyaz, K. (eds.) International Symposium on Khojazāda. Bursa: Bursa Metropolitan Municipality, pp. 141–60.Google Scholar
Studtmann, P. (2004) ‘Aristotle’s Category of Quantity: A Unified Interpretation’, Apeiron, 37(1), pp. 69–91.CrossRefGoogle Scholar
Świętorzecka, K., Pietruszczak, A., and Zarepour, M. S. (forthcoming) ‘A Formal Reconstruction of Avicenna’s Proof of the Sincere’.Google Scholar
Sylla, E. D. (2021) ‘Infinity and Continuity Thomas Bradwardine and His Contemporaries’, in Shapiro, S., and Hellman, G. (eds.) The History of Continua: Philosophical and Mathematical Perspectives. Oxford: Oxford University Press, pp. 49–81.Google Scholar
Teske, R. J. (1990) ‘William of Auvergne on the Eternity of the World’, The Modern Schoolman, 67(3), pp. 187–205.CrossRefGoogle Scholar
Teske, R. J. (1995) ‘William of Auvergne’s Arguments for the Newness of the World’, Mediaevalia: Textos e Estudos, 7–8, pp. 287–302.Google Scholar
Teske, R. J. (2000) ‘William of Auvergne on Time and Eternity’, Traditio, 55, pp. 125–41.CrossRefGoogle Scholar
Thom, P. (2008) ‘Al-Fārābī on Indefinite and Privative Names’, Arabic Sciences and Philosophy, 18(2), pp. 193–209.CrossRefGoogle Scholar
Uckelman, S. L. (2015) ‘The Logic of Categorematic and Syncategorematic Infinity’, Synthese, 192(8), pp. 2361–77.CrossRefGoogle Scholar
van Ess, J. (2017). Theology and Society in the Second and Third Centuries of the Hijra. Translated by G. Goldbloom. Vol. 2. Leiden: Brill.Google Scholar
Wolfson, H. A. (1929) Crescas’ Critique of Aristotle. Cambridge, MA: Harvard University Press.Google Scholar
Zarepour, M. S. (2016) ‘Avicenna on the Nature of Mathematical Objects’, Dialogue, 55(3), pp. 511–36.CrossRefGoogle Scholar
Zarepour, M. S. (2020) ‘Avicenna on Mathematical Infinity’, Archiv für Geschichte der Philosophie, 102(3), pp. 379–425.CrossRefGoogle Scholar
Zarepour, M. S. (2022a) ‘Arabic and Islamic Philosophy of Mathematics’, in Zalta, E. N., and Nodelman, U. (eds.) The Stanford Encyclopedia of Philosophy (Summer 2022 Edition). https://plato.stanford.edu/archives/sum2022/entries/arabic-islamic-phil-math/.Google Scholar
Zarepour, M. S. (2022b) Necessary Existence and Monotheism. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Zarepour, M. S. (forthcoming) ‘The Evolution of the Mapping Argument in Avicenna’s Oeuvre’.Google Scholar
Accessibility standard: Unknown
Why this information is here
This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.Accessibility Information
Accessibility compliance for the HTML of this element is currently unknown
and may be updated in the future.
- 2
- Cited by