# Xenocrates’s Contribution in Mathematics Index of Article (Click to Jump)

## 1. Formal and Mathematical Numbers

Aristotle draws a fundamental difference between formal numbers and mathematical numbers. Aristotle agrees with Plato in this context. He says that both types of numbers are composed units but formal numbers constitute strange units such that units in one formal number cannot be combined with units in another number. Also, there is only one formal number for numbers 2, 3, 4,…etc. In his view, mathematical numbers are those that mathematicians work upon, for example, in performing arithmetic operations. Units of these numbers can be combined and can be added and subtracted easily. Speusippus (Plato’s nephew rejected the concept of formal and mathematical numbers). Aristotle mentions that Xenocrates was of the view that, “the forms and the numbers have the same nature” that is, the formal numbers and the mathematical numbers have the same nature. He believed that the distinction between both types of numbers in unnecessary and formal numbers can be used as mathematical numbers in calculations. He even called formal numbers, mathematical numbers.

The Greek mathematicians had their own way of understanding addition. For Plato and Speusippus, adding 2 and 3 means, adding units of 2 to a disjoint group of units of 3. Aristotle also understood addition in the same way but with different ontologies. According to Xenocrates, if you want to add 2 and 3, you actually cannot do this but taking 3 steps on the series will land us on formal number 5. However, there is no evidence to support this concept of addition. That’s why Aristotle criticized Xenocrates saying, he ends up making mathematical problems impossible and if his way of handling addition is correct then didn’t explain the concept much.

## 3. Xenocrates on Indivisibility

Aristotle assigns Xenocrates’s idea of indivisibility to Plato. On contrary, Alexander credited Xenocrates for a better understanding of Indivisibility than Plato. According to Xenocrates, what is not divisible, add infinity to it, but the division will stop at a point where it cannot be divided further. He thought that he can apply the same concept on a line that Aristotle is applying to a man. According to Aristotle, a man is indivisible, i.e., if you divide a man into two parts, you will not get two men. Xenocrates applied this notion on the line and said, after a certain point, dividing a line further will not yield lines. Maybe because of these remarks by Xenocrates, Aristotle mentioned his position as unmathematical.

## 4. Xenocrates Universe

Xenocrates described the universe as arranged in sequence: (1) forms(numbers);(2) lines; (3) planes; (4) solids; (5) solids in motion, i.e. astronomical bodies; …; (n) ordinary perceptible things. There is a contrast between Xenocrates and Speusippus in the explanation of the universe. Xenocrates universe is more ordered. Theophrastus mentioned that Pythagoreans and Platonists did not give a full story about the construction of the universe, they just go so far and stop and no one other than Xenocrates had placed all the things in order unlike, perceptible and intelligible, i.e., mathematical. Aristotle claimed, that Xenocrates universe showed continuity.

## 5. Xenocrates Definition of Triangles

It was believed that Xenocrates took ‘unit’ as male gods and ‘dyad’ as female gods. He also thought heavenly bodies to be gods. Plutarch discussed gods, daimones, and men. Xenocrates associated them with different types of triangles. He mentioned gods as equilateral triangles; daimones with an isosceles triangle; and men with scalene triangles. Because isosceles triangles are intermediate between equilateral ones and scalene ones, so daimones are intermediate between gods and men. In the view of Plutarch, Xenocrates daimones exist in two phases, i.e., good and bad. It is possible that they might have something to do with the explanation of the existence of evil.