MATH 3010 ID1

Aristotle

(384 to 322 BCE) A Greek philosopher and student of Plato during the Classical period in Ancient Greece. He is credited with the earliest study of formal logic and called the "Father of Western Philosophy. Spent his life providing complex analysis of the world around him; He established a library in the Lyceum which helped him to produce many of his hundreds of books on papyrus scrolls; An axiom (axiôma) is a statement worthy of acceptance and is needed prior to learning anything. Aristotle's list here includes the most general principles such as non-contradiction and excluded middle, and principles more specific to mathematicals, e.g., when equals taken from equals the remainders are equal.

Moscow Papyrus

used in Ancient Egyptian mathematics, written around 1850 BCE; divided into 25 problems in 1930; smaller but older than the Rhind

Rhind Papyrus

used in Ancient Egyptian mathematics; after Henry Rhind; a copy of it shows rational numbers being converted to exact and concise Egyptian fractions

Plato

(born around 247 to 222 BC and died in around 347) An Athenian philosopher during the Classical period in Ancient Greece. He was the innovator of written dialogue and dialectic forms in philosophy. Plato is known as the Founder of Western political philosophy. His most famous contribution is the theory of Forms known by pure reason; idealistic utopia in his symposium; In the Meno, Socrates uses a geometrical example to expound Plato's view that knowledge in this latter sense is acquired by recollection. Socrates elicits a fact concerning a geometrical construction from a slave boy, who could not have otherwise known the fact (due to the slave boy's lack of education). The knowledge must be present, Socrates concludes, in an eternal, non-experiential form.

Zeno

(of Elea) 5th century BCE; known mostly for propounding a number of ingenious paradoxes; most famous - show that motion is impossible by bringing to light apparent or latent contradictions in ordinary assumptions regarding its occurrence

Pythagoreans

Pythagoras' followers, separated into two schools (akousmatikoi and mathēmatikoi); some worshiped Pythagoras; everything could be explained in terms of numbers (integers and ratios); star symbol

Hippasus of Metapontum

Pythagorean philosopher; sometimes credited with the discovery of irrational numbers but overall little is known; helped identify fire as the first element in the universe. Some say that he was drowned after revealing a mathematical secret of the Pythagorean brotherhood.

Incommensurable Numbers

Two numbers are incommensurable with each other if and only if their ratio cannot be written as a rational number (i.e., quotient of two integers). In other words, their ratio is irrational; "not co-measurable"

Socrates

classic Greek (Athenian) philosopher; one of the founders of Western philosophy; first moral philosopher of Western ethical tradition; no writings - Plato's dialogues

Triangular Numbers

counts objects arranged in an equilateral triangle. The nth triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n

Thales of Miletus

one of the seven sages of Greece; many regard as the first Western philosopher; materialism & naturalism; search for universality - "first scientist"

Gnomon

part of a sundial that casts a shadow (the triangular blade on it) In geometry, a gnomon is a plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram; or, more generally, a figure that, added to a given figure, makes a larger figure of the same shape.

Pentagonal Number

similar to triangular numbers but for a pentagon (but not rotationally similar); pn = (3n2 - n) / 2 for n≥1 :1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532

Sophist

specific type of teacher in ancient Greece; in general, specialized in using the tools of philosophy and rhetoric; "one who does wisdom" or "wise man"

Pythagoras of Samos

Ancient Ionian Greek philosopher; founded Pythagoreanism; influenced Plato & Aristotle & thus Western philosophy; transmigration of souls (enter into a new body); the Pythagorean theorem, founder of Pythagorean brotherhood Pythagorean tuning, the five regular solids, the Theory of Proportions, the sphericity of the Earth (570 - around 490 BC)

Plimpton 322

Babylonian clay tablet, written around 1800 BCE; has two of the three numbers in a Pythagorean triple (a and b for a2 + b2 = c2)

Perfect Number

a positive integer that is equal to the sum of its positive divisors, excluding the number itself (for example, 1+2+3 = 6 so 6 is a perfect number); appeared in Euclid's writings; 6, 28, 496, ...Or 1+2+4+17 = 28 If (2^P - 1) is prime, then 2^n-1 * (1^P - 1)