Mathematics, History, and Philosophy: An Introduction
In this short introduction, the nine chapters that comprise the section “Mathematics, History and Philosophy” of the Handbook of the Mathematics of the Arts and Sciences are synthesized with the purpose of unraveling interdisciplinary, cultural, and some
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Abstract In this short introduction, the nine chapters that comprise the section “Mathematics, History and Philosophy” of the Handbook of the Mathematics of the Arts and Sciences are synthesized with the purpose of unraveling interdisciplinary, cultural, and some elusive connections between these disciplines.
Keywords History of mathematics · History of philosophy · Nyaya · Pythagoreanism · Platonism · Interdisciplinary studies
Mathematics is intertwined with philosophy since time immemorial. One can trace it back to the Pythagoreans if one wants to follow Greek roots as a starting point for unfolding the history of mathematics. Other scholarships give Babylonian and Egyptian roots, but it is difficult to determine whether their development and study of mathematics was connected to their worldview or any known philosophy. Arguably, there are entire tomes devoted to this topic that are already part of the literature. Following Greek roots, we quickly find ourselves in the realm of Aristotelian logic that is still a cornerstone of modern mathematics. This development leads to works in mathematical argumentation as evidenced in the Italian prelate St. Anselm of Canterbury’s (1033–1109) “ontological argument” in the Proslogion, which was later revised by Leibniz and Gödel (Umland and Sriraman 2018). Other sources usually pursue the historical connection of logic that binds mathematics and philosophy. What then is the purpose of this section in the
B. Sriraman () Department of Mathematical Sciences, University of Montana, Missoula, MT, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences, https://doi.org/10.1007/978-3-319-70658-0_142-1
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Handbook of the Mathematics of the Arts and Sciences? The nine chapters that comprise this section of the Handbook try to pursue the historical relationship of mathematics and philosophy in strands that show interdisciplinary, cultural, and elusive relationships. Avigad (2018) summarizes this elusive relationship in his commentary to Descartes’ treatise Regulae ad Directionem Ingenii. Rule no.14 of Descartes’ treatise states “philosophy is a matter of discovering general truths by finding properties that are shared by disparate objects, in order to understand the features that they have in common. This requires comparing the degrees to which the properties occur” (Avigad 2018). One can question what these properties allude to as well as the implicit notion of truth. This would lead us into a deep discussion of similarities and differences of “methods” used by mathematicians and philosophers to determine truth, predicated on a suitable definition of “objects.” For instance, Aristotle who can be considered both as a mathematician and a philosopher used five different concepts in his treatment of mathematical objects including those that are perceptible from which one could abstract properties. However, are there other relationships not concerned with ontological and epistemolo
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