The Art of the Intelligible (survey of mathematics in its conceptual development)

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Greek mathematicians tend to conceive of number arithmos as a plurality of units. Their conception involves:. For Aristotle and his contemporaries there are several fundamental problems in understanding number and arithmetic:. Aristotle presents three Academic solutions to these problems.


Units are comparable if they can be counted together such as the ten cows in the field. Units are not comparable, if it is conceptually impossible to count them together a less intuitive notion. Greeks used an Egyptian system of fractions. Additionally, Greeks used systems of measure, as we do, with units of measure being divided up into more refined units of measure. This feature of measure may be reflected in Plato's observation Rep.

Since Aristotle esp. Metaphysics x.

Hence, in the case discrete quantities, such as cows, the unit is very precise, one cow. In the case of continuous quantities, the most precise unit of time is the time it takes for the fixed stars the fastest things in the universe to move the smallest perceptible distance.

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But a point or indivisible with position removed or a cow qua unit, i. Aristotle's treatment of time Physics iv. Aristotle defines time as the number or count of change and then proceeds to distinguish two senses of number, what is counted e.

Time is number in the first sense, not as so-many changes, but as so-much change as measured by a unit of change. Aristotle clarifies the distinction between what is counted or is countable and that by which we count. These five black cats number as what is counted are different from these five brown cats, but their number that by which we count is the same. But what are the numbers by which we count? Aristotle says nothing, but we may speculate that the five by which we count is the single formal explanation of what makes the five black cats five and what makes the five brown cats five.

Hence, Aristotle probably subscribes to an Aristotelian version of the distinction between intermediate and Form-numbers. Aristotle's discussion of time also gives us some insight into the unity problem.

Does philosophy still need mathematics and vice versa? | Aeon Essays

What gives the five black cats unity is just that they can be treated as a unity. From this it follows for Aristotle that there can be no number without mind. Nothing is countable unless there exists a counter. It is often supposed, for Aristotle, mathematical explanation plays no role in the study of nature, especially in biology. This conception is, most of all, a product of anti-scholastics of the late Renaissance, who sought to draw the greatest chasm between their own mechanism and scholasticism.

Mathematics plays a vital role in both. The principal way in which mathematics enters into biological explanation is through hypothetical necessity:. Z may be a constraint determined by a mathematical fact. For example, animals by nature do not have an odd number of feet. For if one had an odd number of feet, it would walk awkardly or the feet would have to be of different lengths De incessuanimalium 9. To see this, imagine an isosceles triangle with a altitude drawn.

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Aristotle famously rejects the infinite in mathematics and in physics, with some notable exceptions. He defines it thus:. Implicit in this notion is an unending series of magnitudes, which will be achieved either by dividing a magnitude the infinite by division or by adding a magnitude to it the infinite by addition.

This is why he conceives of the infinite as pertaining to material explanation, as it is indeterminate and involves potential cutting or joining cf. Section 7. Aristotle argues that in the case of magnitudes, an infinitely large magnitude and an infinitely small magnitude cannot exist.

The Art of the Intelligible

In fact, he thinks that universe is finite in size. He also agrees with Anaxagoras, that given any magnitude, it is possible to take a smaller. Hence, he allows that there are infinite magnitudes in a different sense. Since it is always possible to divide a magnitude, the series of division is unending and so is infinite. This is a potential, but never actual infinite. For each division potentially exists. Similarly, since it is always possible to add to a finite magnitude that is smaller than the whole universe continually smaller magnitudes, there is a potential infinite in addition.

That series too need never end. For example, if I add to some magnitude a foot board, and then less than a half a foot, and then less than a fourth, and so forth, the total amount added will never exceed two feet. Aristotle claims that the mathematician never needs any other notion of the infinite. However, since Aristotle believes that the universe has no beginning and is eternal, it follows that in the past there have been an infinite number of days. Hence, his rejection of the actual infinite in the case of magnitude does not seem to extend to the concept of time.

As philosophers usually do, Aristotle cites simple or familiar examples from contemporary mathematics, although we should keep in mind that even basic geometry such as we find in Euclid's Elements would have been advanced studies. If we attend carefully to his examples, we can even see an emerging picture of elementary geometry as taught in the Academy. In the supplement are provided twenty-five of his favorite propositions the list is not exhaustive. Aristotle also makes some mathematical claims that are genuinely problematic.

Was he ignorant of contemporary work? Why does he ignore some of the great problems of his time?

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  5. Is there any reason why Aristotle should be expected, for example to refer to conic sections? Nonetheless, Aristotle does engage in some original and difficult mathematics. Certainly, in this Aristotle was more an active mathematician than his mentor, Plato. The standard English translation is given first, and, where appropirate, other more idiomatic translations. Greek is in parentheses. Aristotle Aristotle, General Topics: logic. Aristotle and Mathematics First published Fri Mar 26, Introduction 2.

    Demonstration and Mathematics 5. What Mathematical Sciences Study: 4 Puzzles 7. Aristotle's Treatment of Mathematical Objects 7. Universal Mathematics 9. Place and Continuity of Magnitudes Unit monas and Number arithmos Mathematics and Hypothetical Necessity The Infinite apeiron Introduction The late fifth and fourth centuries B. The Structure of a Mathematical Science: First Principles Aristotle's discussions on the best format for a deductive science in the Posterior Analytics reflect the practice of contemporary mathematics as taught and practiced in Plato's Academy, discussions there about the nature of mathematical sciences, and Aristotle's own discoveries in logic.

    Aristotle's list here includes the most general principles such as non-contradiction and excluded middle, and principles more specific to mathematicals, e. Aristotle divides posits thesis into two types, definitions and hypotheses: A hypothesis hupothesis asserts one part of a contradiction, e. For more information, see the following supplementary document: Aristotle and First Principles in Greek Mathematics 3.