Abstract
This chapter addresses ontological questions as formulated in ancient Greece. Mathematics is about something, but it is unclear what kind of objects mathematics is dealing with. The chapter examines different suggestions. According to Plato, mathematics is about immutable entities that constitute a world of ideas. This world is real—although not palpable to our senses. We cannot sense mathematical objects, but we can grasp them by means of our rationality. This rationality, then, is a unique system of perception through which we reach beyond the capabilities of our senses and “see” the objects in the world of ideas, including the real mathematical objects.
The location of mathematical objects in a world of ideas is not merely an expression of Plato’s personal way of thinking. Platonism in the philosophy of mathematics reoccurs time and time again. For example, great mathematicians and logicians such as Frege and Gödel voiced Platonist notions. Thus the chapter addresses Platonism after Plato, Platonism before Plato, as well as Plato’s Platonism. Furthermore, the chapter examines the idea of axiomatisation, and how this structures Euclid’s Elements. This work got a paradigmatic significance, not only with respect to the formulation of mathematical knowledge, but with respect to knowledge in general.
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- 1.
Thus, it has been proven that the number of primes smaller than x can be approximated with the expression \( \frac{x}{\log x-1} \). See The Prime Pages for elaboration.
- 2.
See Iamblichus (1818).
- 3.
We know from Babylonian math tablets handed down to us that the Babylonians knew about Pythagoras’ theorem as early as 1000 years before Pythagoras, and that, furthermore, on the “Yale tablet,” one has found an approximation to \( \sqrt{2} \) that deviates from our value only on the 9th decimal place! For elaboration see the website Babylonian Pythagoras.
- 4.
In the Pythagorean world picture, Earth was conceived of as a globe in space, so at that time not everybody worried that sailors might fall over the edge.
- 5.
See the MacTutor History of Mathematics Archive (1999).
- 6.
The dialogue Timaeus tells how the four elements in the Greek world picture—earth, water, fire and air—can be understood as mathematical atoms or units. The four elements are today called the Platonic solids and are defined as a convex polyhedron whose side faces are all congruent regular polygons, in such a way that the same number of faces meet in each corner. It can be shown that there are five such solids. Earth was associated with the hexahedron (the cube), fire with the tetrahedron, air with the octahedron, and water with the icosahedron. The fifth solid, the dodecahedron, Plato associated with the form of the universe itself. See the Internet Classics Archive (Timaeus).
- 7.
See Plato, The Republic, Book VI, (The Internet Classics Archive, The Republic).
- 8.
See Plato, The Republic , Book VI, (The Internet Classics Archive, The Republic).
- 9.
See Plato, The Republic, Book VII, (The Internet Classics Archive, The Republic).
- 10.
In doing so, we are deviating from Aristotle’s terminology. While he uses the term “definition” as we do here, he refers to “general preconditions” as “axioms,” and “axioms” as “postulates.” However, we prefer to employ the terminology that has since become standard in mathematics and in philosophy of mathematics.
- 11.
Euclid compiled his Elements from the works of a number of his predecessors, among them Hippocrates of Chios, who flourished c. 460 BC (not to be confused with the physician Hippocrates of Cos (c. 460–377 BC)). The latest compiler before Euclid was Theudius. His textbook was used in the Academy and very likely the one used by Aristotle. The older Elements were, however, immediately superseded by Euclid’s and since forgotten. See Encyclopaedia Britannica (Euclid).
- 12.
See the online edition of Euclid’s Elements : Joyce (1998).
- 13.
See Joyce (1998).
References
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Encyclopaedia Britannica. Euclid. Retrieved from http://www.britannica.com/oscar/print?articleId=33178&fullArticle=true&tocId=2173.
Hersh, R. (1998). What is mathematics, really? London, UK: Vintage.
Iamblichus. (1818). The life of Pythagoras. (Translated into English by T. Taylor). Krotona, Hollywood, CA: Theosophical Publishing House.
Joyce, D. E. (1998). Euclid’s elements. Retrieved from http://aleph0.clarku.edu/~djoyce/java/elements/elements.html
MacTutor History of Mathematics Archive. (1999). Retrieved from http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Plato.html
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Ravn, O., Skovsmose, O. (2019). Mathematics in Eternity. In: Connecting Humans to Equations . History of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-01337-0_1
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