Greek Philosophy of Mathematics

Math 390 Lecture 5

Dr. Janssen

Introduction

Key figures

  • Pythagoras and his followers
  • Plato
  • Aristotle

The Pythagoreans

Pythagoras

  • 580-500 BCE
  • Born at Samos
  • Some say he studied under Thales, unlikely given age difference
  • Traveled to Egypt, Babylon, and possibly India
  • Contemporary of Buddha, Confucius, and Laozi
  • Said to have coined the words philosophy and mathematics

Pythagoras

Pythagoreans

  • Founded after travels on the island of Croton, SE coast of Italy
  • Knowledge and property held in common; attribution given to the group (but really the master)
  • Mathematical interest moved beyond the exigencies of daily life toward a love of wisdom
  • Motto: “All is number”

Number Mysticism

  • Odd numbers have male attributes, even numbers female
  • ‘There is divinity in odd numbers’
  • 1: The generator of numbers and the number of reason
  • 2: First even/female number; the number of opinion
  • 3: the first true male number, the number of harmony, composed of unity and diversity
  • 4: number of justice or retribution, indicating the squaring of accounts
  • 5: the number of marriage, the union of the first true male/female numbers
  • 6: the number of creation
  • 10, the tetractys: the holiest of numbers, representing the number of the universe, including the sum of all possible geometric dimensions

Figurate Numbers

  • 10, the holy tetractys, is an example of a triangular number
  • The pentagonal numbers are given by \(1+4+ 7 + \cdots + (3n-2) = \frac{n(3n-1)}{2}\)
  • The hexagonal numbers were derived from the sequence \(1+5+9+\cdots + (4n-3) = 2n^2 - n\)
  • And so on for larger polygonal numbers (and polyhedral numbers)

All things which can be known have number; for it is not possible that without number anything can be either conceived or known.

[The tetractys was] great, all-powerful and all-producing, the beginning and the guide of the divine as of the terrestrial life.

–Philolaus (died in approx.~390 BCE)

Arithmetic and Cosmology

What is a number?

  • Mesopotamia: number applied to spatial extension
  • Egypt: natural numbers and unit fractions
  • Babylonians: field of rational fractions
  • In Greece, “number” meant (positive) integer; fractions of integers described the relationship of ratio
  • When lengths of vibrating strings could be expressed as ratios of simple whole numbers, the results were harmonious
  • 2:1 is an octave, 2:3 is the fifth, 3:4 is the fourth
  • Extrapolated that the heavenly bodies emitted harmonious tones: the “harmony of the spheres”

Plato and Aristotle

Big Questions

  • Ontological question: what are mathematical objects?
  • Epistemological question: How do we know mathematics?
  • Aesthetic question: In what way(s) is mathematics beautiful?
  • Teleological question: Why should anyone do mathematics?

Plato (427(?)–347 BCE)

  • Born (and died) in Athens into an influential aristocratic family
  • A wrestler
  • Founded the Academy, at which many prominent philosophers studied, particularly Aristotle

Plato’s Ontology: Worlds of Being and Becoming

  • Identified a gap between the ideas we can conceive and the physical world around us
  • We have some understanding of perfect ideals, but we never find them
  • World of Being
  • World of Becoming

Plato’s Epistemology

How then can we know the Forms of things like Beauty and Goodness?

Through mental reflection:

Let me remind you of the distinction we drew earlier and have often drawn on other occasions, between the multiplicity of things that we call good or beautiful or whatever it may be and, on the other hand, Goodness itself or Beauty itself and so on. Corresponding to each of these sets of many things, we postulate a single Form or real essence as we call it … Further, the many things, we say, can be seen, but are not objects of rational thought; whereas the Forms are objects of thought, but invisible.

– The Republic, Book 6

Plato on Mathematics

  • Mathematics (especially geometry) is a helpful of example of Plato’s thinking
  • Rigorous definitions of circle, straight line?
  • Yet these do not exist in our physical world—so what do we study in geometry, and how do we study it?
  • Plato: the propositions of geometry are objectively true/false, independent of the mind, language, etc. of mathematicians, so where are the objects of study? How is geometry known?
  • Plato: Geometrical objects are like Forms and are in the world of Being

Plato’s Mathematical Epistemology

  • Geometry is not graspable via the senses
  • What, then, is the purpose of diagrams?
  • An aid to the mind
  • Platonists: geometrical knowledge is a priori

Aristotle (384-322 BCE)

  • Studied at Plato’s Academy
  • First codified principles of logical argument
  • Logical arguments built out of syllogisms
  • Distinguished between postulates and axioms
  • Principles of argument, e.g., law of the excluded middle

Aristotle’s Approach

  • Most of what Aristotle says about mathematics is a polemic against Plato
  • Rejected the world of Being
  • Accepted universals: Beauty is what all beautiful things have in common; likewise with Two-ness
  • Problem for Aristotle: If we reject the world of Being, then what reason is there to believe that mathematical objects exist? What is their nature (if they exist), and what do we need mathematical objects for?

Aristotle’s Ontology

  • Mathematical objects “exist in perceptible objects”
  • From Physics B: ``Physical bodies contain surfaces, volumes, lines, and points, and these are the subject matter of mathematics’’
  • One can then separate surfaces, lines, etc., from the physical object in thought
  • Abstraction!

Aristotle’s Epistemology

  • More empirical in nature: we learn of numbers by contemplating groups of objects and abstract from there
  • Then abstract from the abstraction, etc.
  • E.g: five people give rise to 5, which is a part of the natural numbers, which have certain arithmetical properties, etc.
  • Logical argument is the only certain way of gaining knowledge
  • Demonstration via syllogism is the only way to be sure of your results
  • One may draw any conclusions you want from any set of axioms, but truth comes from true axioms
  • Axiomatic method popularized by Aristotle still in use today, but in the form of propositions

What’s the point?

  • Heavily influential in all that follows
  • As Hersch argues, much of Plato and Aristotle were baptized by Augustine and Aquinas and so have influenced centuries of Christian thought

Discussion

  • Mathematicians often speak of their discoveries. Is this terminology more harmonious with Plato or Aristotle? Why?
  • Is mathematics discovered or invented?
  • Many (most?) modern Christian mathematicians have adapted Plato’s notion of a world of Being and replaced with the mind of God. So, a perfect circle exists in the mind of God. What do you think?
  • Are either of these views more resonant with your Christian faith than the other? Explain.

Coda: Aristotle and Zeno

Number vs. Magnitude

  • Pythagoreans: all is number
  • Aristotle divided the category of “quantity” into two classes: number and magnitude
  • Magnitude: continuous, infinitely divisible; e.g., lines
  • Number: discrete, indivisible
  • The idea that lines consist of an infinite number of points was nonsense to Aristotle
  • No conception of completed/actual infinity

Zeno’s Paradoxes

Aristotle wanted to refute the paradoxes of Zeno, which showed that current conceptions of motion, space, and time were insufficiently clear.

  • Dichotomy: “asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal.”
  • Achilles: “In a race, the quickest runner can never over­take the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”
  • Arrow: “If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion.”

Aristotle’s Response

  • Concedes that time, like distance is infinitely divisible
  • But this isn’t a problem: “while a thing in finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect to divisibility, for in this sense time itself is also infinite.”
  • Further noted, in refutation of the paradox of the arrow, that there are no such things as indivisible instants of time, but motion only defined over a period of time
  • Thus we describe instantaneous velocity as a limit of average velocities (which occur over a non-infinitesimal time unit)