The Saga of Mathematics: Greek Contributions from Plato to Ptolemy

Plato's Doctrine of Ideas and Universals

• The World of Reality: This doctrine asserts a view of reality consisting of two distinct worlds:

  • The Everyday World: Perceived by the senses, this is the world of change, appearance, and imperfect knowledge.

  • The World of Ideas: Perceived by reason, this is the world of permanence, reality, and true knowledge.

• Examples of Ideas:

  • Justice: This is an Idea that is only imperfectly reflected in human efforts to be just.

  • Two: This is an Idea that is participated in by every pair of material objects.

• Definition of Universals: A universal is an abstract object or term which ranges over particular things. This was a common concern for philosophers such as Plato and Aristotle.

• The Classic Problem of Universals: This involves whether abstract objects exist in a realm that is independent of human thought.

  • Realists: Argue that abstract objects do exist independently.

Plato's Extreme Realism and Epistemology

• Plato ($ca. 428-348 BC$): He was the first and most extreme realist. He argued that universals are forms existing in their own spiritual realm lying outside of space and time.

• The Concept of Participation: Individual objects (e.g., a specific dog) participate in the universal form (e.g., "dogness").

• Intellect vs. Senses: A universal can only be known by the intellect, never the senses. They are timeless, perfect patterns of "Being."

• Deceptive Phenomena: The world around us consists of blurred, shadowy copies of these forms, which constitute deceptive phenomena.

• Metaphysics and Epistemology:

  • Knowledge of a particular object requires access to unchanging universals.

  • Particulars are merely manifestations of the forms.

• The Problem of Representation: Plato's theory faces the challenge of explaining how universals are represented in particulars and how a universal can reside in a particular.

Aristotle's Critique and Theory of Universals

• Aristotle ($ca. 384-322$): He criticized Plato for introducing an unneeded aspect of "separateness" to the universal.

• Critique of Substance: Aristotle attacked Plato for holding that a universal was both a property and a substance.

• Dependency on Particulars: Aristotle believed universals did not exist independently of particulars. They are only present in specific things encountered through experience.

• Shared Attributes: Like Plato, Aristotle believed universals (e.g., "color") exist independently of human thought, but not in a spiritual realm. Instead, they are found in shared attributes.

  • Example of Greenness: "Greenness" is found within the class of all green objects, like grass and trees.

  • Essence: The essence of "dog" resides in each individual dog.

Plato and the Philosophy of Geometry

• Necessary Connections: Geometry establishes necessary connections between the forms of polygons.

• Eternal Truths: Geometric study is not about a specific rectangular table or a circular clock, but rather the discovery of truths regarding the perfect circle and perfect rectangle.

• The Academy: Plato founded this school of philosophy. Legend says the portal bore the inscription: "Let no one destitute of geometry enter my doors."

• Solid Geometry: Plato and his school emphasized solid geometry. He regarded it as the "first essential in the training of philosophers" due to its abstract nature.

• Application to Astronomy: In The Republic, Plato suggests a science of solid objects is necessary to consider objects in motion, such as astronomy.

The Platonic Solids

• Polyhedra: A polyhedron is a solid bounded by plane polygons.

  • Faces: The bounding polygons.

  • Edges: The intersection of faces.

  • Vertices: Points where three or more edges intersect.

• Regular Polyhedron: A solid where the faces are identical regular polygons.

• The Five Regular Solids: Only five are possible, each associated by Plato with an element of the universe:

  • Cube: Associated with Earth.

  • Tetrahedron: Associated with Fire.

  • Octahedron: Associated with Air.

  • Icosahedron: Associated with Water.

  • Dodecahedron: Associated with the Universe.

• Symmetry and Theory: Late in life, in the treatise Timaeus, Plato expounded a "theory of everything" based on these five symmetrical arrangements of points in space.

Euclid of Alexandria and "The Elements"

• Euclid ($ca. 325-265 BC$): Little is known of his personal life; information comes primarily from Proclus.

• Training and Alexandria: He likely trained in Athens, leaving by $322 B.C.$ due to the turmoil after Alexander the Great's death. He found refuge in Alexandria.

• The Museum: Founded by King Ptolemy Soter within the palace park, it was the first national university. Euclid was its first teacher of mathematics.

• The Library: Contained $600,000$ papyrus rolls.

• Anecdotes of Euclid:

  • The Royal Road: When King Ptolemy asked for a shorter way to master geometry, Euclid replied, "There is no royal road to geometry."

  • The Three Pence: A student asked what he would gain by learning geometry; Euclid instructed a slave to give him threepence, "since he must make gain out of what he learns."

• The Elements: A compilation of knowledge that served as the center of mathematical teaching for $2000$ years. The first printed edition appeared in $1482$.

Structure and Content of Euclid's Elements

• Overview: Contains $13$ books and more than $450$ propositions.

• Books I to VI (Plane Geometry):

  • I and II: Basic properties of triangles, parallels, parallelograms, rectangles, and squares.

  • III and IV: Properties and problems of the circle.

  • V: Work of Eudoxus on proportion applied to commensurable and incommensurable magnitudes.

  • VI: Applications of results from Book V to plane geometry.

• Books VII to IX (Number Theory):

  • VII: Self-contained introduction including the Euclidean algorithm for the GCD.

  • VIII: Numbers in geometrical progression.

  • IX: Continued number theory results.

• Book X: Theory of irrational numbers.

• Books XI to XIII (3D Geometry):

  • XII: Results showing circles are to one another as the squares of their diameters, and spheres are as the cubes of their diameters.

  • XIII: Properties of the five regular polyhedra and the proof that only precisely five exist.

Axioms, Postulates, and the Fifth Postulate

• Definitions: The Elements begins with $23$ definitions.

  • Definition 1: "A point is that which has no part."

  • Definition 23: "Parallel straight lines are straight lines which are in the same plane, and if extended indefinitely in both directions, do not meet in either direction."

• Axiom Groups: Divided into Postulates and Common Notions.

  • Common Notions: Self-evident truths applying to all sciences.

  • Postulates: Assumptions specific to the subject (geometry) that are not self-evident.

• The Five Common Notions:

  1. Things which equal the same thing also equal one another.

  2. If equals are added to equals, then the wholes are equal.

  3. If equals are subtracted from equals, then the remainders are equal.

  4. Things which coincide with one another equal one another.

  5. The whole is greater than the parts.

• The Five Postulates:

  1. We can draw a straight line from any point to any point.

  2. We can produce a finite straight line continuously in a straight line.

  3. We can describe a circle with any center and radius.

  4. All right angles are equal to one another.

  5. The Fifth Postulate: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Non-Euclidean Geometry and Alternative Postulates

• Nature of the 5th Postulate: Euclid had no proof for it, and no proof is possible, yet he could not progress without it.

• Alternative Statements:

  • Poseidonius ($c. 135-51 BC$): Two parallel lines are equidistant.

  • Proclus ($c. 500 AD$): If a line intersects one of two parallel lines, it intersects the other.

  • Saccheri ($c. 1700$): The sum of the interior angles of a triangle is two right angles.

  • Legendre ($1752-1833$): A line through a point in the interior of an angle (other than a straight angle) intersects at least one arm of the angle.

  • Farkas B3lyai ($1775-1856$): There is a circle through every set of three non-collinear points.

  • Playfair’s Axiom: Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line. (Proposed $200$ years ago, but stated by Proclus $1300$ years prior).

• Non-Euclidean Revolution: Led by Gauss ($1777-1855$), Lobachevskii ($1792-1856$), and Jonas B3lyai ($1802-1850$).

  • Supposes the 5th postulate is false (e.g., sum of angles in a triangle is less than two right angles).

  • Later contributors include Beltrami, Hilbert, and Klein.

  • Fits Einstein’s Theory of Relativity.

Euclid's Number Theory and Algorithm

• Primes: A prime number is divisible only by $1$ and itself (e.g., $17$).

• Infinitude of Primes Theorem: There are infinitely many primes.

  • Proof Sketch: Suppose $2, 3, 5, \dots, p$ is a complete list of primes. Form $N = (2 \times 3 \times 5 \times \dots \times p) + 1$. $N$ is either prime or divisible by a prime q < N. In either case, there exists a prime greater than $p$.

• Prime Factorization: Every integer can be factored into primes (e.g., $100 = 2 \times 2 \times 5 \times 5$).

• Divisibility Observations:

  • If $n$ is divisible by $m$, then $n + 1$ is not divisible by $m$.

  • Two numbers are "relatively prime" if their only common factor is $1$.

• Euclidean Algorithm (Greatest Common Divisor):

  • Rule: If a number $n$ goes into $x$ and $y$, it goes into $x - y$.

  • Example 1: GCD of $30$ and $650$

    • $650 \div 30 = 21$ R $20$.

    • Compute GCD of $20$ and $30$, which is $10$.

  • Example 2: GCD of $48$ and $360$

    • $360 \div 48 = 7$ R $24$.

    • $48 \div 24 = 2$ R $0$.

    • Answer is the last divisor: $24$.

Additional Works of Euclid

• Surviving Works:

  • Data: $94$ propositions on deducing properties of figures.

  • On Divisions: Constructions to divide figures into areas of given ratios.

  • Optics: First Greek work on perspective.

  • Phaenomena: Introduction to mathematical astronomy (star rising/setting times).

• Lost Works: Surface Loci, Porisms ($171$ theorems), Conics (four books), Book of Fallacies, and Elements of Music.

Archimedes of Syracuse ($ca. 287-212 BC$)

• Overview: Considered one of the greatest mathematicians of all time. Native of Syracuse, Sicily; studied at the Museum in Alexandria.

• Legend of the Crown: Determined if a crown was pure gold or alloyed with silver using density and displacement of water. Shouted "Eureka!" after observing water overflow in his bath.

• Archimedes’ Law of Buoyancy: An object immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced.

• Archimedes’ Law of the Lever: $W \times D = w \times d$ (e.g., $140\text{ lb}$ boy $2\text{ ft}$ from fulcrum balances a $70\text{ lb}$ girl $4\text{ ft}$ from fulcrum). Famous quote: "Give me a place to stand and I will move the earth."

• Geometrical Achievements:

  • Volume of a sphere is $\frac{2}{3}$ the volume of a circumscribed cylinder.

  • Surface area of a sphere is four times the area of a great circle.

  • Area of a segment of a parabola is $\frac{4}{3}$ the area of the triangle with the same base and height.

Archimedes and the Measurement of the Circle

• Circle Formula: Discovered $A = \frac{1}{2} \times C \times r$.

  • Since $C = 2\pi r$, then $A = \pi r^2$.

• Approximation of $\pi$: Determined 3 \frac{10}{71} < \pi < 3 \frac{1}{7} by inscribing and circumscribing a circle with two $96$-sided polygons.

• Method of Exhaustion:

  • Area of an inscribed polygon is $A = \frac{1}{2} \times P \times h$ (where $P$ is perimeter and $h$ is distance from center to midpoint of sides).

  • By doubling sides ad infinitum, $P$ approaches $C$ and $h$ approaches $r$.

• Tombstone: Decorated with a sphere contained in a cylinder (ratio $3:2$). Archimedes considered this his greatest discovery.

Apollonius of Perga ($ca. 262-190 BC$)

• The Great Geometer: His book On Conics introduced the terms parabola, ellipse, and hyperbola.

• On Conics Structure:

  • Books I-IV: Elementary properties.

  • Books V-VII: Normals, tangents, similarity, and diameters.

  • Book VIII: Lost.

• The Problem of Apollonius: Constructing a circle tangent to three objects (points, lines, or circles). There are $10$ cases.

• Legacy: Without Conics, Kepler could not have discovered elliptical planetary orbits ($1609$), and Newton could not have formulated universal gravitation ($1687$).

Claudius Ptolemy ($ca. 85-165 AD$)

• Geography: Created a map with a coordinate system of latitude and longitude.

• Almagest: A $13$-book treatise detailing the motion of the Sun, Moon, and planets.

• Astronomical Model:

  • Geocentric Model: Earth at the center; unchallenged until the middle of the $16\text{th}$ century.

  • Epicycles: A circle whose center is carried around the circumference of another circle.

  • Eccentric Circles: Planets move in circles whose centers do not coincide with Earth.

• Trigonometry and $\pi$:

  • Calculated semi-chords (related to sine).

  • Calculated $\sin(\theta) = \frac{\text{Crd}(2\theta)}{120}$.

  • Using an inscribed $360$-gon, he approximated $\pi = 3 + \frac{17}{120} = 3.1417$.

  • Sexagesimal notation: $3;8,30$ ($3 + \frac{8}{60} + \frac{30}{3,600}$).