Philosophers

Thales:

  • Year

    • 624BC - 547BC

  • Sometimes credited with inventing science(Particularly Physics)

  • Born and died in Miletus(modern Turkey)

  • Wealthy

    • Had a chance as a young man to study in Egypt, where he learned geometry

  • First to challenge and ask “why is this true”

  • He calculated the height of the pyramids by comparing the shadow of the pyramid to the shadow of a stick with known height (Similar triangles)

  • Also used similar triangles to determine how far offshore Persian ships were during on of their attacks

  • Predicted solar eclipse

  • All is water

  • Wrote 3 books

    • Solstice

    • Astronomy

    • Equinox

  • Theorems

    • A circle is divided by any diameter into 2 equal parts

    • The angles at the base of an isosceles are equal

    • When 2 lines intersect, the opposite angles at the intersection point are equal

    • Two triangles are congruent if they have one side and two angles equal

    • In a triangle ABC, if the segment DE is parallel to BC, then |AB| / |AD|

    • The sum of the interior angles of a triangle is equal to 2 right angles

    • An angle formed by 3 points on a circle is a right angle if and only if a line through 2 of the points is a diameter

Pythagoras:

  • Year

    • 570BC-500BC

  • From Samos

  • First Pure mathematician

  • Founder of a cult

  • Wealthy

    • Joined priesthood, went to Egypt to learn geometry

  • 530BC , was taken as POW back to Persia where he learned Mesopotamian mathematics and reached “perfection” in music, astronomy, mysticism, and mathematics

  • 520BC , went back to Samos, island was devastated by corruption and oppression.

  • 515BC, went to Croton to found the Pythagorean Brotherhood

  • Elites of the brotherhood were part of an inner circle called “Mathematicians”

  • Mathematicians were vegetarians, only had communal property and worshiped Pythagoras

  • There were women in the group who were considered equals to men

Pythagoreans:

  • At its deepest level, reality is mathematical

  • Philosophy should be used for spiritual purification

  • The soul rises to union with the Divine (Life after death)

  • Certain symbols have mystical significance (eg the pentagram)

  • All members must observe strict loyalty secrecy

  • All is number

Heraclitus:

  • Year

    • 500 BC

  • Believed that the world is full of opposites

  • Everything always moving

  • “You can never step in the same river twice”

Parmenides:

  • Year

    • 500 BC

  • Debated with Socrates

  • Respected by Plato

  • Teacher of Zeno

Zeno of Elea:

  • Year

    • 490 BC- 430 BC

  • Student of Parmenides

  • Wrote 40 Paradoxes

  • 4 Paradoxes made into Aristotle

    • Moving from 0 to 1

    • Achilles and the Tortoise

    • The Arrow and motion

    • The Stadium, Rows A,B,C

Mathematics in Athens

Socrates:

  • Year

    • 469 BC - 399 BC

  • Interested in morality

  • Made logical arguments that were hard to beat

Hippias:

  • Year

    • 420 BC

  • He was a Sophist

  • Discovered the Quadratrix

    • Used to trisect an angle and square a circle, however, these solutions were not considered acceptable

Plato:

  • Year

    • 430 BC - 350 BC

  • Founded Plato’s Academy

    • Year

      • 387BC - 529 BC

  • Student of Socrates

  • Soldier who fought the Spartans

  • Left Athens after death of Socrates

  • Returned for good in 387 BC to found the academy

    • “Let no one ignorant of Geometry enter here”

  • Geometry can only be done with straight edge and compass

Platonism:

  • Ideal forms of math do not exist in nature

  • Most things in the real world are approximations

  • Is equality truly infinitely transitive

Hippocrates of Chios:

  • Year

    • 470 BC - 410 BC

  • Moved to Athens around 430 BC, learned the math of the Pythagoreans

  • Wrote the first version of the elements

  • First to square a region with curved sides

Platos Students

Theatetus:

  • Year

    • 417 BC - 369 BC

  • Called a natural philosopher and influenced Plato

  • Book X of element is believed to be his work

  • Generalized the theorem that square roots of non-square numbers up to 17 are irrational to those of all non-square numbers

  • Proved there are 5 regular convex polyhedra and he constructed the octahedron and the icosahedron

Eudoxus:

  • Year

    • 408 BC - 355 BC

  • Born in Cnidos and studied under Archytas ( Leader of the small remaining group of Pythagoreans)

  • Studied at the academy for 2 months

  • Created his own school in 368 BC

Aristotle:

  • Year

    • 384 BC - 322 BC

  • From Macedonia

  • Went to academy 367 BC

  • When Plato Died Eudoxus became head of the Academy and Aristotle stayed as professor

  • Personal tutor of Alexander

  • 335 Found his own school that worked in conjunction with the Academy

  • Major contribution to mathematical logic

Euclid:

  • Year

    • 325 BC

  • Known for the Elements

  • Collected works of his predecessors and corrected proofs

  • Taught at the Library

  • Constructions are all straight edge and compass

  • Could only use constructible numbers

Archimedes:

  • Years

    • 287 BC - 212 BC

  • Born and lived in Syracuse

  • Studied in Alexandria when young

  • Considered the greatest physicist and mathematician of the ancient world