Xenocrates was a Greek philosopher and mathematician, who was born in 396 BC in Chalcedon. In 376 BC, he went to Athens to enter Plato’s academy as a student. He also accompanied Plato to Syracuse, after the death of Dionysius. Also, After Plato’s death, Xenocrates and Aristotle both were invited to Assos and Xenocrates resided there for five years. After Plato’s death, his nephew Speusippus became head of the academy but in 340BC, he brought Xenocrates back to Athens and asked him to become his successor. An election took place between Xenocrates, Menedemus of Pyrrha, and Heraclides Ponticus but Xenocrates defeated them by a few votes. He remained head of the academy for the rest of his life. Although he resided in Athens for many years, he resented accepting the dominance of Macedonia. Many scholars believed that Xenocrates was not an original thinker, as a head of Plato’s academy he used to promote Plato’s teachings only. Diogenes Laertius mentioned the names of two of his books, ‘On numbers’ and ‘The theory of numbers’. He believed that matter is an indivisible quantity which shows that he had the idea of atomic theory. Also, like Pythagoras, he accepted the importance of numbers in philosophy. In his view, we humans die twice, once on the Earth and the second time on the moon when the mind separates from the soul and travels to the sun. Plutarch wrote about Xenocrates that he tried to calculate the total number of syllables that can be made out of 26 alphabets. According to Plutarch, Xenocrates got the total number of ways to 1,002,000,000,000. If this figure is true, then this would be the first attempt at solving a combinatorial problem in the history of mathematics. None of his writing survived. They were actually never published, the single copies in his hands were deposited in the academy. The academy was destroyed when Athens was attacked by Sulla’ troops in 86 BC. Xenocrates died in 314 BC and the leadership of the Academy passed to Polemo.

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## 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.

## 2. Views about Addition

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.