Aristotle was a Greek philosopher and polymath. He was born in 384 BC in Stagira. After the early death of his parents, he was brought up by his guardians. When he was seventeen, he moved to Athens and continued his studies under Plato. He was married twice. His first wife was Pythias, whom he married in Lesbos and had a daughter with her. But after Pythias’s death, he, later on, married Herpyllis who bore him a son. After Plato died in 338 BC, Aristotle went to Macedonia, where he taught Alexander the Great, who was just 13 years old at that time. In 335 BC, Aristotle came back to Athens, where he opened up his library, Lyceum. He spent the rest of his life in this library doing teaching and research work. There is a large collection of writings of Aristotle and his students, preserved in this library. His major works include topics of physics, mathematics, zoology, biology, logic, poetry, music, astronomy and many more. All his work is divided into two categories. One that was published by him and are now lost, and the other category include those works that are now with us, which were not published during his lifetime. After Alexander’s death, anti-Macedonian feelings aroused in Athens that forced Aristotle to shift to Chalcis (his mother’s family estate), on Euboea. He died here because of the failure of digestive organs. He also left a will in which he asked to be buried next to his wife. After his death, all his work remained unnoticed for a long time. In the 1st century, Theophrastus saved his library Lyceum and passed his legacy onto his student Neleus. Neleus saved all his work from moisture and insects. And it was Apellicon who took it back to Athens in 100 BC.
1. Aristotle’s Deductive Approach in Mathematics
Aristotle discussed two major concerns for the nature of mathematics. In one, he mentions that there must be some unprovable principles to avoid infinite regresses. And in the other, he mentions that the proofs should be explanatory. He mentioned two ways to start a proof, i.e., by axioms or by posits. Axioms are the statements that are accepted as true without any proof, whereas Posits are divided into two types, i.e., hypothesis and definitions.
The hypothesis is also known as a supposition or “the proposed explanation based on limited evidence,” whereas the definition is the exact meaning of the word. According to him, axioms form the base for the proofs.
2. Aristotle’s Criticism of Plato’s Mathematical Cosmology
Aristotle criticizes Plato’s mathematical cosmology by claiming that the study of the motion of the sublunary body is Physics rather than mathematics. Therefore, he accepted Eudoxean astronomy as a helping hand in the study of the motion of heavenly bodies. As a part of the theory of natural places, Aristotle mentions that all heavenly bodies fall towards the centre of the universe, and they seem to follow mathematical symmetry. So, the similarities and differences between the thoughts of Aristotle and Plato can be judged by the difference in the conceptual application of physics and mathematics.
3. Problems About Mathematical Objects
However, Aristotle rejected Plato’s mathematical cosmology, but he has gained the deepest knowledge of science from him only. On one hand, Aristotle treats mathematical sciences as a model of scientific knowledge, and on the other hand, he agrees with Plato’s assumption that authentic knowledge must have a real object. These two statements raise questions about the existence of mathematical objects.
4. Ontological Status of Mathematical Objects
For the treatment of mathematical objects, Aristotle quoted five concepts in his discussions:-
Removal
By/From/Or/Through removal, Aristotle refers to mathematical objects as things. Usage of different expressions relates it to Topic of Definitions, where one analyses the result after adding and deleting a term from an expression. He believed that the facts that are not a part of science should be removed.
Precision
Aristotle claims that the sciences that have more properties eliminated are more precise. Like the science of Kinematics, which include uniform motion only is more precise than those involving non-uniform motion in addition.
As Separated
The language of mathematics is legal because we can conceive the changing magnitudes in different ways. Substances remained the only real thing to Aristotle. According to him, the most important characteristic of a substance is that they are separate.
X Qua (hêi) Y
Possible meanings of this Greek word are ‘where’, ‘in the manner that’, ‘by the means of the fact that. Some suggested that the meaning of this word is ‘because’. Like X Qua Y should be understood as ‘X in the respect that X is Y’ or ‘ X precisely because X is Y’.
Intelligible matter
Since perceptible matter is not a part of the object, one needs to have a notion of the matter of the object. Since an object should be made of several elements to distinguish it from other objects of the same form, it will have matter. He calls these matters intelligible matter or mathematical matter. The parts of a mathematical object, which do not occur in the definition of the object, e.g., the acute angle does not occur in the definition of right angle but is a part of it. So it is a non-perceptible material part of the angle.
5. Aristotle’s Dialectical Method
Aristotle claims that dialects provide a path to the first principles of philosophical sciences. There is a list of aporia given in ‘Metaphysics’ out of which three aporias are related to the problem of mathematical objects. The way he had treated every problem in his first philosophy is aporetic as he didn’t try to break the resulting deadlock. We cannot say that ‘Analytics’ serves as a true guide for Aristotle’s way of thinking and doing physics because in ‘Metaphysics’ he had made some methodological statements that diverged completely from what he said in ‘Analytics’.
6. Universal Mathematics
There is an analogy of mathematics suggested by Aristotle, which mentions that there is a super science of mathematics that covers all continuous magnitudes and discrete quantities such as numbers. He claims that although mathematicians had proved theorems such as a:b=c:d⇒a:c=b:d, separately for numbers, lines, planes, and solids, now we have one universal proof for all. On the contrary, ancient Greek mathematicians show no proof of Aristotelian universal mathematics.
7. Aristotle’s View of Infinite Divisibility and Continuity
Some philosophers in Plato’s academy reported that lines constitute indivisible magnitude, whether it is a finite number or infinite number. Aristotle denied this hypothesis by building a theory of continuity and infinite divisibility. He claimed that a line comprising of an infinite number of potential points is equivalent to saying that a line can be divided anywhere on it, bringing potential points to actuality. He further explained that the continuity of a line lies in the fact that any actual point within the line will hold together the line segments on either side. Aristotle mentions continuous proportions that in a:b=b:c, b is one magnitude, but it is used twice.
8. Unit (monas) and Number (arithmos)
Greek mathematicians believed that numbers are the plural of units. According to them, a number is made up of countable figures (unit). One is not a number, it is a unit that will construct a number. Later on, some questions arose on this concept such as overlapping problems. This problem says that what is the guarantee that when 3 is added to 5 then the correct result is not 5,6 or 7, namely some units in this 3 are not also in 5; however, Aristotle explained this problem by mentioning that we can compare units if they can be counted together like 7 pens on the table. Units are incomparable if it is not possible to count them together abstractly like the number 5 is not a combination of 5 units, but it is the 5th unit in the series of units.
9. Measure
Greek mathematicians used a system in which all fractions are proper parts, that we now call unit fractions. 2/3 is an exception, otherwise, all other fractions can be represented in the form 1/n. For example, 2/5 can be expressed as a sum of 1/3 and 1/15. Greeks used the same system of measurement as we do like 1 foot is 12 inches. However, Aristotle shared a different view in which he had mentioned the problem of sorting out precise units of measurement. In the case of distinct quantities, like 7 pens, the unit is very precise,i.e., one pen but in the case of continuous quantities, like the most precise unit for time is, the time it takes for the fixed stars to move the smallest distance.
10. Time
Aristotle defined time as the number or count of change. He further distinguished two types of numbers, what is counted and by which we count. For example, 10 cows in a ground, here, the number counted is 10 and by which it is counted is 1 cow. In his view, time is a number where the change is measured by a unit of change.
11. Aristotle on the Infinite
Aristotle denied the existing theory of infinity in mathematics and physics and defined it in his terms. He mentioned,
infinite is something for which it is always possible to take something outside.’
According to him, in the case of magnitudes, infinitely small or large magnitudes do not exist. He also supported Anaxagoras thought, that from any given magnitude, it is always possible to take smaller. Hence, his remark about infinite magnitudes was in a different sense. Aristotle believed that the universe has no end and beginning. This implies that in ancient times, there had been an infinite number of days, which doesn’t go well with his rejection of the theory of infinite magnitudes.
12. Aristotle and Greek Mathematics
Aristotle quoted examples from contemporary mathematics. He occasionally used a sort of mathematics that includes arithmetical operations and metrological geometry. Most of his examples were cited from the sort of mathematics that was associated with Greek, like construction of figures from given figures, and proving that these figures have properties. He had also provided his 25 favourite propositions. It can be said that he was engaged in difficult mathematics and was a more active mathematician than his mentor, Plato.