Henry Mendell

There is a fundamental tension in Greek philosophies of mathematics arising from the descriptive origins of mathematics and the deductive power of formal systems . Sophists rejected the epistemology; Platonists the ontology. Aristotle tries to preserve both by adopting a mathematical realism and grounding epistemology in experience. Hence there are three central questions in Aristotle's philosophy of mathematics: what are the objects of mathematics, how can we have knowledge of them, and how can we get knowledge of the physical world from knowledge of mathematical entities? My dissertation is in two parts: ontology and epistemology. ;Since Aristotle conceives of a substance as a subject for other entities which are in turn subjects for other entities, etc., mathematical understanding involves treating substances qua triangle, i.e. treating the triangle which belongs to substance as if it were substance. I call this view epistemological nominalism. By making the objects of mathematical material potential divisions, Aristotle can claim that geometry is true. However, in lower sciences such as meteorology idealizations become necessary. I examine Aristotle's treatments of different types of quantities. Since Aristotle's views on mathematics depend on his views on substance, I emphasize developmental features. Although I am sceptical about Aristotle endorsing universal mathematics, the importance of analogy in his middle period may arise from the Eudoxan theory of proportions. ;In Part II, Aristotle's treatment of mathematical elements is seen in a broad context. I compare treatments of first principles, distinctions between theorem and problem, and views on the basic objects of geometry in Aristotle, Archimedes, Euclid, and others. The Aristotelian elements are inadequate for doing geometry, but do not constitute all his geometrical hypotheses. Aristotle also thinks that mathematical demonstrations are syllogisms. The solution to both anomalies is that whereas demonstration constitutes the explanatory core of mathematical argument, induction provides the premises for deduction. Since induction requires imagination and hence perception, the diagram as image becomes the source of actual knowledge, knowledge of the universal in the particular. The epistemology is fully grounded in perception. However problems in Aristotle's account show that the tension is not resolved successfully

In his Metrica, Hero provides four procedures for finding the area of a circular segment (with b the base of the segment and h its height): an Ancient method for when the segment is smaller than a semicircle, $$(b + h)/2 \, \cdot \, h$$ ( b + h ) / 2 · h ; a Revision, $$(b + h)/2 \, \cdot \, h + (b/2)^{2} /14$$ ( b + h ) / 2 · h + ( b / 2 ) 2 / 14 ; a quasi-Archimedean method (said to be inspired by the quadrature of the parabola) for cases where b is more than triple h, $${\raise0.5ex\hbox{$\scriptstyle 4$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 3$}}(h \, \cdot \, b/2)$$ 4 / 3 ( h · b / 2 ) ; and a method of Subtraction using the Revised method, for when it is larger than a semicircle. He gives superficial arguments that the Ancient method presumes $$\pi = 3$$ π = 3 and the Revision, $$\pi = {\raise0.5ex\hbox{$\scriptstyle {22}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 7$}}$$ π = 22 / 7. We are left with many questions. How ancient is the Ancient? Why did anyone think it worked? Why would anyone revise it in just this way? In addition, why did Hero think the Revised method did not work when $$b > 3\;h$$ b > 3 h? I show that a fifth century BCE Uruk tablet employs the Ancient method, but possibly with very strange consequences, and that a Ptolemaic Egyptian papyrus that checks this method by comparing the area of a circle calculated from the sum of a regular inscribed polygon and the areas of the segments on its sides as determined by the Ancient method with the area of the circle as calculated from its diameter correctly sees that the calculations do not quite gel in the case of a triangle but do in the case of a square. Both traditions probably could also calculate the area of a segment on an inscribed regular polygon by subtracting the area of the polygon from the area of the circle and dividing by the number of sides of the polygon. I then derive two theorems about pairs of segments, that the reviser of the Ancient method should have known, that explain each method, why they work when they do and do not when they do not, and which lead to a curious generalization of the Revised method. Hero’s comment is right, but not for the reasons he gives. I conclude with an exploration of Hero’s restrictions of the Revised method and Hero’s two alternative methods.

The discussion of mathematical knowledge and its relation to the construction of an appropriate diagram in Aristotle's Metaphysics Θ 9. 1051 a21—33 is an important, if compressed, account of Aristotle's most mature thoughts on mathematical knowledge. The discussion of what sort of previous knowledge one must have for understanding a theorem recalls the discussion at An. Post. A 1. 71 a 17–21, where the epistemological point is similar and the examples the same. The first example, that the interior angles of a triangle equal two right angles, appears no less than thirty times in the corpus . The example of the angle inscribed in a semicircle being a right angle also occurs at An. Post. B 11. 94a 27–34, but in a very different context from its two companions. Illustrations of both theorems provided clear stock examples for Aristotle