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Xenocrates (; ; c. 396/5314/3 BC) of Chalcedon was a Greek philosopher,
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, and leader ( scholarch) of the
Platonic Academy The Academy (), variously known as Plato's Academy, or the Platonic Academy, was founded in Classical Athens, Athens by Plato ''wikt:circa, circa'' 387 BC. The academy is regarded as the first institution of higher education in the west, where ...
from 339/8 to 314/3 BC. His teachings followed those of
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
, which he attempted to define more closely, often with mathematical elements. He distinguished three forms of being: the sensible, the intelligible, and a third compounded of the two, to which correspond respectively,
sense A sense is a biological system used by an organism for sensation, the process of gathering information about the surroundings through the detection of Stimulus (physiology), stimuli. Although, in some cultures, five human senses were traditio ...
,
intellect Intellect is a faculty of the human mind that enables reasoning, abstraction, conceptualization, and judgment. It enables the discernment of truth and falsehood, as well as higher-order thinking beyond immediate perception. Intellect is dis ...
and
opinion An opinion is a judgement, viewpoint, or statement that is not conclusive, as opposed to facts, which are true statements. Definition A given opinion may deal with subjective matters in which there is no conclusive finding, or it may deal ...
. He considered unity and duality to be
gods A deity or god is a supernatural being considered to be sacred and worthy of worship due to having authority over some aspect of the universe and/or life. The ''Oxford Dictionary of English'' defines ''deity'' as a God (male deity), god or god ...
which rule the
universe The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
, and the soul a self-moving
number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
.
God In monotheistic belief systems, God is usually viewed as the supreme being, creator, and principal object of faith. In polytheistic belief systems, a god is "a spirit or being believed to have created, or for controlling some part of the un ...
pervades all things, and there are daemonical powers, intermediate between the
divine Divinity (from Latin ) refers to the quality, presence, or nature of that which is divine—a term that, before the rise of monotheism, evoked a broad and dynamic field of sacred power. In the ancient world, divinity was not limited to a singl ...
and the mortal, which consist in conditions of the soul. He held that
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
objects and the Platonic Ideas are identical, unlike Plato who distinguished them. In
ethics Ethics is the philosophy, philosophical study of Morality, moral phenomena. Also called moral philosophy, it investigates Normativity, normative questions about what people ought to do or which behavior is morally right. Its main branches inclu ...
, he taught that
virtue A virtue () is a trait of excellence, including traits that may be morality, moral, social, or intellectual. The cultivation and refinement of virtue is held to be the "good of humanity" and thus is Value (ethics), valued as an Telos, end purpos ...
produces
happiness Happiness is a complex and multifaceted emotion that encompasses a range of positive feelings, from contentment to intense joy. It is often associated with positive life experiences, such as achieving goals, spending time with loved ones, ...
, but external goods can minister to it and enable it to effect its purpose.


Life

Xenocrates was a native of Chalcedon. By the most probable calculation he was born 396/5 BC, and died 314/3 BC at the age of 82. His father was named Agathon () or Agathanor ().Suda Encyclopedia, xi,42
/ref> Moving to
Athens Athens ( ) is the Capital city, capital and List of cities and towns in Greece, largest city of Greece. A significant coastal urban area in the Mediterranean, Athens is also the capital of the Attica (region), Attica region and is the southe ...
in early youth, he became the pupil of Aeschines Socraticus, but subsequently joined himself to
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
, whom he accompanied to
Sicily Sicily (Italian language, Italian and ), officially the Sicilian Region (), is an island in the central Mediterranean Sea, south of the Italian Peninsula in continental Europe and is one of the 20 regions of Italy, regions of Italy. With 4. ...
in 361. Upon his master's death, he paid a visit with
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
to Hermias of Atarneus. In 339/8 BC, Xenocrates succeeded
Speusippus Speusippus (; ; c. 408 – 339/8 BC) was an ancient Greece, ancient Greek philosopher. Speusippus was Plato's nephew by his sister Potone. After Plato's death, c. 348 BC, Speusippus inherited the Platonic Academy, Academy, near age 60, and remai ...
in the presidency of the school, defeating his competitors Menedemus of Pyrrha and Heraclides Ponticus by a few votes. On three occasions he was member of an Athenian legation, once to
Philip Philip, also Phillip, is a male name derived from the Macedonian Old Koine language, Greek (''Philippos'', lit. "horse-loving" or "fond of horses"), from a compound of (''philos'', "dear", "loved", "loving") and (''hippos'', "horse"). Prominen ...
, twice to Antipater. Xenocrates resented the Macedonian influence then dominant at Athens. Soon after the death of
Demosthenes Demosthenes (; ; ; 384 – 12 October 322 BC) was a Greek statesman and orator in ancient Athens. His orations constitute a significant expression of contemporary Athenian intellectual prowess and provide insight into the politics and cu ...
(c. 322 BC), he declined the citizenship offered to him at the insistence of Phocion as a reward for his services in negotiating peace with Antipater after Athens' unsuccessful rebellion. The settlement was reached "at the price of a constitutional change: thousands of poor Athenians were disenfranchised," and Xenocrates said "that he did not want to become a citizen within a constitution he had struggled to prevent". Being unable to pay the tax levied upon resident aliens, he is said to have been saved only by the courage of the orator Lycurgus, or even to have been bought by Demetrius Phalereus, and then emancipated. In 314/3, he died from hitting his head, after tripping over a bronze pot in his house. Xenocrates was succeeded as scholarch by Polemon, whom he had reclaimed from a life of profligacy. Besides Polemon, the statesman Phocion, Chaeron (tyrant of Pellene), the academic Crantor, the Stoic Zeno and
Epicurus Epicurus (, ; ; 341–270 BC) was an Greek philosophy, ancient Greek philosopher who founded Epicureanism, a highly influential school of philosophy that asserted that philosophy's purpose is to attain as well as to help others attain tranqui ...
are said to have frequented his lectures. Wanting in quickness of apprehension and natural grace he compensated by persevering and thorough-going industry, pure benevolence, purity of morals, unselfishness, and a moral earnestness, which compelled esteem and trust even from the Athenians of his own age. Xenocrates adhered closely to the Platonist doctrine, and he is accounted the typical representative of the Old Academy. In his writings, which were numerous, he seems to have covered nearly the whole of the Academic program; but
metaphysics Metaphysics is the branch of philosophy that examines the basic structure of reality. It is traditionally seen as the study of mind-independent features of the world, but some theorists view it as an inquiry into the conceptual framework of ...
and
ethics Ethics is the philosophy, philosophical study of Morality, moral phenomena. Also called moral philosophy, it investigates Normativity, normative questions about what people ought to do or which behavior is morally right. Its main branches inclu ...
were the subjects which principally engaged his thoughts. He is said to have made more explicit the division of philosophy into the three parts of
Physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
,
Dialectic Dialectic (; ), also known as the dialectical method, refers originally to dialogue between people holding different points of view about a subject but wishing to arrive at the truth through reasoned argument. Dialectic resembles debate, but the ...
and
Ethics Ethics is the philosophy, philosophical study of Morality, moral phenomena. Also called moral philosophy, it investigates Normativity, normative questions about what people ought to do or which behavior is morally right. Its main branches inclu ...
.


Writings

With a comprehensive work on
Dialectic Dialectic (; ), also known as the dialectical method, refers originally to dialogue between people holding different points of view about a subject but wishing to arrive at the truth through reasoned argument. Dialectic resembles debate, but the ...
() there were also separate treatises ''On
Knowledge Knowledge is an Declarative knowledge, awareness of facts, a Knowledge by acquaintance, familiarity with individuals and situations, or a Procedural knowledge, practical skill. Knowledge of facts, also called propositional knowledge, is oft ...
'', ''On Knowledgibility'' (), ''On Divisions'' (), ''On Genera and Species'' (), ''On Ideas'' (), ''On the Opposite'' (), and others, to which probably the work ''On Mediate Thought'' () also belonged. Two works by Xenocrates on Physics are mentioned (), as are also books ''On the
Gods A deity or god is a supernatural being considered to be sacred and worthy of worship due to having authority over some aspect of the universe and/or life. The ''Oxford Dictionary of English'' defines ''deity'' as a God (male deity), god or god ...
'' (), ''On the Existent'' (), ''On the One'' (), ''On the Indefinite'' (), ''On the Soul'' (), ''On the Emotions'' () ''On
Memory Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembe ...
'' (), etc. In like manner, with the more general
Ethical Ethics is the philosophical study of moral phenomena. Also called moral philosophy, it investigates normative questions about what people ought to do or which behavior is morally right. Its main branches include normative ethics, applied e ...
treatises ''On
Happiness Happiness is a complex and multifaceted emotion that encompasses a range of positive feelings, from contentment to intense joy. It is often associated with positive life experiences, such as achieving goals, spending time with loved ones, ...
'' (), and ''On
Virtue A virtue () is a trait of excellence, including traits that may be morality, moral, social, or intellectual. The cultivation and refinement of virtue is held to be the "good of humanity" and thus is Value (ethics), valued as an Telos, end purpos ...
'' () there were connected separate books on individual Virtues, on the Voluntary, etc. His four books on Royalty he had addressed to
Alexander Alexander () is a male name of Greek origin. The most prominent bearer of the name is Alexander the Great, the king of the Ancient Greek kingdom of Macedonia who created one of the largest empires in ancient history. Variants listed here ar ...
(). Besides these he had written treatises ''On the State'' (), ''On the Power of
Law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior, with its precise definition a matter of longstanding debate. It has been variously described as a science and as the ar ...
'' (), etc., as well as upon
Geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
,
Arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
, and
Astrology Astrology is a range of Divination, divinatory practices, recognized as pseudoscientific since the 18th century, that propose that information about human affairs and terrestrial events may be discerned by studying the apparent positions ...
. Besides philosophical treatises, he wrote poetry () and paraenesis.


Philosophy


Epistemology

Xenocrates made a more definite division between the three departments of philosophy, than
Speusippus Speusippus (; ; c. 408 – 339/8 BC) was an ancient Greece, ancient Greek philosopher. Speusippus was Plato's nephew by his sister Potone. After Plato's death, c. 348 BC, Speusippus inherited the Platonic Academy, Academy, near age 60, and remai ...
, but at the same time abandoned Plato's heuristic method of conducting through doubts (), and adopted instead a mode of bringing forward his doctrines in which they were developed dogmatically. Xenocrates recognized three grades of cognition, each appropriated to a region of its own: knowledge, sensation, and opinion. He referred knowledge () to that essence which is the object of pure thought, and is not included in the phenomenal world; sensation () to that which passes into the world of phenomena; opinion () to that essence which is at once the object of sensuous perception, and, mathematically, of pure reason - the essence of
heaven Heaven, or the Heavens, is a common Religious cosmology, religious cosmological or supernatural place where beings such as deity, deities, angels, souls, saints, or Veneration of the dead, venerated ancestors are said to originate, be throne, ...
or the
star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s; so that he conceived of in a higher sense, and endeavored, more definitely than Plato, to exhibit mathematics as mediating between knowledge and sensuous perception. All three modes of apprehension partake of truth; but in what manner scientific perception () did so, we unfortunately do not learn. Even here Xenocrates's preference for symbolic modes of sensualising or denoting appears: he connected the above three stages of knowledge with the three Fates: Atropos, Clotho, and Lachesis. We know nothing further about the mode in which Xenocrates carried out his
dialectic Dialectic (; ), also known as the dialectical method, refers originally to dialogue between people holding different points of view about a subject but wishing to arrive at the truth through reasoned argument. Dialectic resembles debate, but the ...
, as it is probable that what was peculiar to Aristotelian logic did not remain unnoticed in it, for it can hardly be doubted that the division of the existent into the absolutely existent, and the relatively existent, attributed to Xenocrates, was opposed to the Aristotelian table of categories.


Metaphysics

We know from
Plutarch Plutarch (; , ''Ploútarchos'', ; – 120s) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo (Delphi), Temple of Apollo in Delphi. He is known primarily for his ''Parallel Lives'', ...
that Xenocrates, if he did not explain the Platonic construction of the world-soul as Crantor after him did, nevertheless drew heavily on the '' Timaeus''; and further that he was at the head of those who, regarding the universe as unoriginated and imperishable, looked upon the chronological succession in the Platonic theory as a form in which to denote the relations of conceptual succession. Plutarch unfortunately, does not give us any further details, and contented himself with describing the well-known assumption of Xenocrates, that the soul is a self-moving number. Probably we should connect with this the statement that Xenocrates called unity and duality (''monas'' and ''duas'') deities, and characterised the former as the first male existence, ruling in heaven, as father and
Zeus Zeus (, ) is the chief deity of the List of Greek deities, Greek pantheon. He is a sky father, sky and thunder god in ancient Greek religion and Greek mythology, mythology, who rules as king of the gods on Mount Olympus. Zeus is the child ...
, as uneven number and spirit; the latter as female, as the mother of the gods, and as the soul of the universe which reigns over the mutable world under heaven, or, as others have it, that he named the Zeus who ever remains like himself, governing in the sphere of the immutable, the highest; the one who rules over the mutable, sublunary world, the last, or outermost. If, like other Platonists, he designated the material principle as undefined duality, the world-soul was probably described by him as the first defined duality, the conditioning or defining principle of every separate definitude in the sphere of the material and changeable, but not extending beyond it. He appears to have called it in the highest sense the individual soul, in a derivative sense a self-moving number, that is, the first number endowed with motion. To this world-soul Zeus, or the world-spirit, has entrusted - in what degree and in what extent, we do not learn - dominion over that which is liable to motion and change. The divine power of the world-soul is then again represented, in the different spheres of the universe, as infusing soul into the planets, Sun and Moon, - in a purer form, in the shape of Olympic gods. As a sublunary daemonical power (as
Hera In ancient Greek religion, Hera (; ; in Ionic Greek, Ionic and Homeric Greek) is the goddess of marriage, women, and family, and the protector of women during childbirth. In Greek mythology, she is queen of the twelve Olympians and Mount Oly ...
,
Poseidon Poseidon (; ) is one of the twelve Olympians in ancient Greek religion and mythology, presiding over the sea, storms, earthquakes and horses.Burkert 1985pp. 136–139 He was the protector of seafarers and the guardian of many Hellenic cit ...
,
Demeter In ancient Greek religion and Greek mythology, mythology, Demeter (; Attic Greek, Attic: ''Dēmḗtēr'' ; Doric Greek, Doric: ''Dāmā́tēr'') is the Twelve Olympians, Olympian goddess of the harvest and agriculture, presiding over cro ...
), it dwells in the elements, and these daemonical natures, midway between gods and men, are related to them as the
isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides ...
triangle is to the equilateral and the scalene. The divine world-soul which reigns over the whole domain of sublunary changes he appears to have designated as the last Zeus, the last divine activity. It is not until we get to the sphere of the separate daemonical powers of
nature Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...
that the opposition between
good and evil In philosophy, religion, and psychology, "good and evil" is a common dichotomy. In religions with Manichaeism, Manichaean and Abrahamic influence, evil is perceived as the dualistic cosmology, dualistic antagonistic opposite of good, in which ...
begins, and the daemonical power is appeased by means of a stubbornness which it finds there congenial to it; the good daemonical power makes happy those in whom it takes up its abode, the bad ruins them; for eudaimonia is the indwelling of a good daemon, the opposite the indwelling of a bad one. How Xenocrates tried to establish and connect scientifically these assumptions, which appear to be taken chiefly from his books on the nature of the gods, we do not learn, and can only discover the one fundamental idea at the basis of them, that all grades of existence are penetrated by divine power, and that this grows less and less energetic in proportion as it descends to the perishable and individual. Hence he also appears to have maintained that as far as consciousness extends, so far also extends an intuition of that all-ruling divine power, of which he represented even irrational animals as partaking. But neither the thick nor the thin, to the different combinations of which he appears to have tried to refer the various grades of material existence, were regarded by him as in themselves partaking of soul; doubtless because he referred them immediately to the divine activity, and was far from attempting to reconcile the duality of the ''principia'', or to resolve them into an original unity. Hence too he was for proving the incorporeality of the soul by the fact that it is not nourished as the body is. It is probable, that, after the example of
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
, he designated the divine ''principium'' as alone indivisible, and remaining like itself; the material, as the divisible, partaking of multiformity, and different, and that from the union of the two, or from the limitation of the unlimited by the absolute unity, he deduced number, and for that reason called the soul of the universe, like that of individual beings, a self-moving number, which, by virtue of its twofold root in the same and the different, shares equally in permanence and motion, and attains to consciousness by means of the reconciliation of this opposition.
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
, in his ''
Metaphysics Metaphysics is the branch of philosophy that examines the basic structure of reality. It is traditionally seen as the study of mind-independent features of the world, but some theorists view it as an inquiry into the conceptual framework of ...
'', recognized amongst contemporary Platonists three principal views concerning the ''ideal'' numbers, and their relation to the ideas and to ''mathematical''
number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
s: #those who, like Plato, distinguished ''ideal'' and ''mathematical'' numbers; #those who, like Xenocrates, identified ''ideal'' numbers with ''mathematical'' numbers #those who, like Speusippus, postulated ''mathematical'' numbers only Aristotle has much to say against the Xenocratean interpretation of the theory, and in particular points out that, if the ideal numbers are made up of arithmetical units, they not only cease to be principles, but also become subject to arithmetical operations. In the derivation of things according to the series of the numbers he seems to have gone further than any of his predecessors. He approximated to the
Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek co ...
in this, that (as is clear from his explanation of the soul) he regarded number as the conditioning principle of consciousness, and consequently of knowledge also; he thought it necessary, however, to supply what was wanting in the Pythagorean assumption by the more accurate definition, borrowed from Plato, that it is only insofar as number reconciles the opposition between the same and the different, and has raised itself to self-motion, that it is soul. We find a similar attempt at the supplementation of the Platonic doctrine in Xenocrates's assumption of indivisible lines. In them he thought he had discovered what, according to Plato, God alone knows, and he among men who is loved by him, namely, the elements or principia of the Platonic triangles. He seems to have described them as first, original lines, and in a similar sense to have spoken of original plain figures and bodies, convinced that the ''principia'' of the existent should be sought not in the material, not in the divisible which attains to the condition of a phenomenon, but merely in the ideal definitude of form. He may very well, in accordance with this, have regarded the point as a merely subjectively admissible presupposition, and a passage of Aristotle respecting this assumption should perhaps be referred to him.


Ethics

The information on his Ethics is scanty. He tried to supplement the Platonic doctrine at various points, and at the same time to give it a more direct applicability to life. He distinguished from the good and the bad something which is neither good nor bad. Following the ideas of his Academic predecessors, he viewed the good as that which should be striven after for itself, that is, which has value in itself, while the bad is the opposite of this. Consequently, that which is neither good nor bad is what in itself is neither to be striven after nor to be avoided, but derives value or the opposite according as it serves as means for what is good or bad, or rather, is used by us for that purpose. While, however, Xenocrates (and with him Speusippus and the other philosophers of the older Academy) would not accept that these intermediate things, such as
health Health has a variety of definitions, which have been used for different purposes over time. In general, it refers to physical and emotional well-being, especially that associated with normal functioning of the human body, absent of disease, p ...
,
beauty Beauty is commonly described as a feature of objects that makes them pleasure, pleasurable to perceive. Such objects include landscapes, sunsets, humans and works of art. Beauty, art and taste are the main subjects of aesthetics, one of the fie ...
, fame, good fortune, etc. were valuable in themselves, he did not accept that they were absolutely worthless or indifferent. According, therefore, as what belongs to the intermediate region is adapted to bring about or to hinder the good, Xenocrates appears to have designated it as good or evil, probably with the proviso, that by misuse what is good might become evil, and vice versa, that by virtue, what is evil might become good. Still he maintained that virtue alone is valuable in itself, and that the value of every thing else is conditional. According to this, happiness should coincide with the consciousness of virtue, though its reference to the relations of human life requires the additional condition, that it is only in the enjoyment of the good things and circumstances originally designed for it by nature that it attains to completion; to these good things, however, sensuous gratification does not belong. In this sense he on the one hand denoted (perfect) happiness as the possession of personal virtue, and the capabilities adapted to it, and therefore reckoned among its constituent elements, besides moral actions conditions and facilities, those movements and relations also without which external good things cannot be attained, and on the other hand did not allow that wisdom, understood as the science of first causes or intelligible essence, or as theoretical understanding, is by itself the true wisdom which should be striven after by people, and therefore seems to have regarded this human wisdom as at the same time exerted in investigating, defining, and applying. How decidedly he insisted not only on the recognition of the unconditional nature of moral excellence, but on morality of thought, is shown by his declaration, that it comes to the same thing whether one casts longing eyes, or sets one's feet upon the property of others. His moral earnestness is also expressed in the warning that the ears of children should be guarded against the poison of immoral speeches.


Mathematics

Xenocrates is known to have written a book ''On Numbers'', and a ''Theory of Numbers'', besides books on
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
.
Plutarch Plutarch (; , ''Ploútarchos'', ; – 120s) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo (Delphi), Temple of Apollo in Delphi. He is known primarily for his ''Parallel Lives'', ...
writes that Xenocrates once attempted to find the total number of syllables that could be made from the letters of the alphabet. According to Plutarch, Xenocrates result was 1,002,000,000,000 (a " myriad-and-twenty times a myriad-myriad"). This possibly represents the first instance that a combinatorial problem involving
permutations In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example ...
was attempted. Xenocrates also supported the idea of "indivisible lines" (and magnitudes) in order to counter Zeno's paradoxes.Simplicius, ''in Arist. Phys.''


See also

*'' On Indivisible Lines''


References


Bibliography

* Bartninkas, V. (2023).
Traditional and Cosmic Gods in Later Plato and the Early Academy
' Cambridge: Cambridge University Press, 211–238. * * * * * * Attribution *


External links

*

{{Authority control 4th-century BC Greek philosophers Academic philosophers Ancient Athenian philosophers Ancient Chalcedonians Ancient Greek metaphysicians Greek educators Metic philosophers in Classical Athens 390s BC births 310s BC deaths Philosophers and tutors of Alexander the Great 4th-century BC Greek mathematicians