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Philolaus

Philolaus is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Philolaus rather than just read about it. In short: Philolaus (; Ancient Greek: Φιλόλαος, Philólaos; c. 470 – c. 385 BC) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece.

Philolaus — main illustration
Philolaus — illustration

Key takeaways

  • Philolaus belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Philolaus to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Philolaus from memory before moving on to harder problems.

Reference excerpt

Philolaus (; Ancient Greek: Φιλόλαος, Philólaos; c. 470 – c. 385 BC) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece. Philolaus has been called one of three most prominent figures in the Pythagorean tradition and the most outstanding figure in the Pythagorean school. Pythagoras developed a school of philosophy that was dominated by both mathematics and mysticism. Most of what is known today about the Pythagorean astronomical system is derived from Philolaus's views. He may have been the first to write about Pythagorean doctrine. According to Böckh (1819), who cites Nicomachus, Philolaus was the successor of Pythagoras. He argued that at the foundation of everything is the part played by the limiting and limitless, which combine in a harmony. With his assertions that the Earth was not the center of the universe (geocentrism), he is credited with the earliest known discussion of concepts in the development of heliocentrism, the theory that the Earth is not the center of the Universe, but rather that the Sun is. Philolaus discussed a Central Fire as the center of the universe and that spheres (including the Sun) revolved around it.

Biography Various reports give the birthplace of Philolaus as either Croton, Tarentum, or Metapontum—all part of Magna Graecia (the name of the coastal areas of Southern Italy on the Tarentine Gulf that were colonized extensively by Greek settlers). It is most likely that he came from Croton. He migrated to Greece, perhaps while fleeing the second burning of the Pythagorean meeting-place around 454 BC. According to Plato's Phaedo, he was the instructor of Simmias and Cebes at Thebes, around the time the Phaedo takes place, in 399 BC. That would make him a contemporary of Socrates, and would agree with the statement that Philolaus and Democritus were contemporaries. The writings of much later authors are the sources of further reports about his life. They are scattered and of dubious value in reconstructing his life. Apparently, he lived for some time at Heraclea, where he was the pupil of Aresas, or (as Plutarch calls him) Arcesus. Diogenes Laërtius is the only authority for the claim that shortly after the death of Socrates, Plato traveled to Italy where he met with Philolaus and Eurytus. The pupils of Philolaus were said to have included Xenophilus, Phanto, Echecrates, Diocles, and Polymnastus. As to his death, Diogenes Laërtius reports a dubious story that Philolaus was put to death at Croton on account of being suspected of wanting to be the tyrant; a story which Diogenes Laërtius even chose to put into verse.

Writings

Pythagoras and his earliest successors do not appear to have committed any of their doctrines to writing. According to Porphyrius (Vit. Pyth. p. 40) Lysis and Archippus collected in a written form some of the principal Pythagorean doctrines, which were handed down as heirlooms in their families, under strict injunctions that they should not be made public. But amid the different and inconsistent accounts of the matter, the first publication of the Pythagorean doctrines is pretty uniformly attributed to Philolaus. In one source, Diogenes Laërtius speaks of Philolaus composing one book, but elsewhere he speaks of three books, as do Aulus Gellius and Iamblichus. It might have been one treatise divided into three books. Plato is said to have procured a copy of his book. Later, it was claimed that Plato composed much of his Timaeus based upon the book by Philolaus. One of the works of Philolaus was called On Nature. It seems to be the same work that Stobaeus calls On the World and from which he has preserved a series of passages. Other writers refer to a work entitled Bacchae, which may have been another name for the same work, and which may originate from Arignote. However, it has been mentioned that Proclus describes the Bacchae as a book for teaching theology by means of mathematics. It appears, in fact, from this, as well as from the extant fragments, that the first book of the work contained a general account of the origin and arrangement of the universe. The second book appears to have been an exposition of the nature of numbers, which in the Pythagorean theory are the essence and source of all things. He composed a work on the Pythagorean philosophy in three books, which Plato is said to have procured at the cost of 100 minae through Dion of Syracuse, who purchased it from Philolaus, who was at the time in deep poverty. Other versions of the story represent Plato as purchasing it himself from Philolaus or his relatives when in Sicily. Out of the materials which he derived from these books Plato is said to have composed his Timaeus. But in the age of Plato the leading features of the Pythagorean doctrines had long ceased to be a secret; and if Philolaus taught the Pythagorean doctrines at Thebes, he was hardly likely to feel much reluctance in publishing them; and amid the conflicting and improbable accounts preserved in the authorities above referred to, little more can be regarded as trustworthy, except that Philolaus was the first who published a book on the Pythagorean doctrines, and that Plato read and made use of it. Historians from the Stanford Encyclopedia of Philosophy, Chapter Philolaus' Book: Genuine Fragments and Testimonia, noted the following:

It is implied that these books were not by Philolaus himself, and it seems likely that the statement refers to three spurious works assigned to Pythagoras at D.L. VIII 6 (Burkert 1972a, 224–5). The story of Plato's purchase of these books from Philolaus was probably invented to authenticate the three forged treatises of Pythagoras. Burkert's arguments (1972a, 238–277), supported by further study (Huffman 1993), have led to a consensus that some 11 fragments are genuine (Frs. 1–6, 6a, 7, 13, 16 and 17 in the numbering of Huffman 1993) and derive from Philolaus' book On Nature (Barnes 1982; Kahn 1993 and 2001; Kirk, Raven and Schofield 1983; Nussbaum 1979; Zhmud 1997). Fragments 1, 6a and 13 are identified as coming from the book On Nature by ancient sources. Stobaeus cites fragments 2 and 4–7 as coming from a work On the Cosmos, but this appears to be an alternate title for On Nature, which probably arose because the chapter heading in Stobaeus under which the fragments are cited is 'On the Cosmos.'

Philosophy

… excerpt ends here. Continue reading the full article.

Illustrations

Philolaus illustration
Philolaus: Discussion of Philolaus in a book by  Charles Peter Mason (1870)
Discussion of Philolaus in a book by Charles Peter Mason (1870)
Philolaus: A page from Theorica Musicae (1492) [24]
A page from Theorica Musicae (1492) [24]

Worked examples

Example 1 — a first encounter with Philolaus

Start with the simplest possible case. Write down what Philolaus claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Philolaus before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Philolaus ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Philolaus

In research
Philolaus appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Philolaus in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Philolaus is common in secondary-school and first-year university syllabi. It links to neighbouring topics 380s BC deaths, 470s BC births, 4th-century BC Greek mathematicians, so understanding it makes those chapters shorter.
In everyday life
Look for Philolaus outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Philolaus in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Philolaus means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Philolaus out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Philolaus in simple terms?

Philolaus (; Ancient Greek: Φιλόλαος, Philólaos; c. 470 – c. 385 BC) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece.

Why does Philolaus matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Philolaus?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Philolaus.

Tags

  • 380s BC deaths
  • 470s BC births
  • 4th-century BC Greek mathematicians
  • 5th-century BC Greek mathematicians
  • 5th-century BC Greek philosophers
  • Ancient Crotonians
  • Ancient Greek physicists
  • Ancient Greek political refugees
  • Ancient Thebes, Greece
  • Presocratic philosophers
  • Pythagoreans of Magna Graecia

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