73. The Continuous: Foundational to All Philosophy

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Lecture Notes

Main Topics #

The Centrality of the Continuous to All Philosophy #

• The continuous is absolutely fundamental for:
  • Natural Philosophy: Motion, place, distance, and time are all continuous
  • Mathematical Philosophy: Geometry deals with continuous quantity; arithmetic arises from division of the continuous
  • Metaphysics and Theology: Understanding immaterial substances (angels, God) requires negating the continuous
  • All Human Knowledge: Human reason always operates through imagination tied to the continuous and time

Why Philosophy of the Continuous Belongs to Natural Philosophy (Not Geometry) #

• Geometry assumes the continuous; Natural Philosophy must establish what the continuous fundamentally is
• Geometry treats only the continuous of lines, surfaces, and bodies
• Natural Philosophy must treat the continuous universally: magnitude, motion, and time together
• The continuous is broader than geometric objects—time and motion are continuous but not subjects of geometry
• Following Aristotle’s principle of demonstration: when multiple things share a common reason proportionally, they should be demonstrated together (like Euclid’s Proposition 47 demonstrating the Pythagorean theorem for all right triangles, not just isosceles ones)

The Continuous as Basic to All Fundamental Words #

• The first meaning of “beginning” (ἀρχή/principium) is the beginning of the continuous (e.g., the edge of a table)
• The first meaning of “end” or “limit” (τέλος/terminus) is the end or limit of the continuous
• The concept of “one” (ἕν/unum) is fundamentally connected to continuity: something is one when it remains continuous and undivided
• “Being” and “quantity” are intimately related through the continuous

Understanding Immaterial Things Through Negation of the Continuous #

• God and angels are not continuous
• In this life, we cannot know God or angels as they are in themselves
• Therefore, we understand them through negation of what is continuous
• This is also how we understand the immateriality of human reason and the soul
• Example from Aquinas: When showing that understanding creatures (angels) are not bodies, we demonstrate they are not continuous, establishing their immateriality

Key Arguments #

The Infinite Divisibility of the Continuous #

• The continuous is divisible forever and is not composed of indivisibles
• Number arises from the division of the continuous: dividing one line gives two, dividing again gives three, and this can continue infinitely
• The potential infinity of numbers corresponds to the infinite divisibility of the continuous
• Reference to Anaxagoras: “there is no smallest of the small” and “no greatest of the great”

The Relationship Between Proportional Demonstrations #

• When showing that a property belongs to something universally (as such), it is more perfect to demonstrate it for what is proportional to all instances
• Example: The Pythagorean theorem (Euclid I.47) demonstrates the property for any right-angled triangle proportionally, not just for isosceles right triangles
• Similarly, Socrates’ demonstration to the slave boy in the Meno shows a particular case (isosceles right triangle) before understanding the universal principle
• The same reasoning applies to the continuous: magnitude, motion, and time all share a common reason for infinite divisibility, so they should be demonstrated together in natural philosophy

Important Definitions #

The Continuous (ἀντεχές/continuus) #

• Philosophical definition: That which is divisible forever
• General principle: That whose parts meet at a common boundary, but the parts remain distinct

Key Distinctions Made #

• Continuous vs. Touching: Continuous things have boundaries that are one; touching things have boundaries together but distinct
• Beginning (ἀρχή): First, the beginning of the continuous (spatial/material); then extended to other meanings, all seen by likeness to the spatial beginning
• End/Limit (τέλος): First, the end of the continuous; then the end of motion; then purpose; finally definition
• One (ἕν): The continuous is fundamental to understanding unity—when something is broken, it ceases to be one (continuous) and becomes many

Examples & Illustrations #

The Plate Joke (Platonic Reference) #

• Socrates asks someone to define something but receives many examples instead of one definition
• He jokingly says: “I asked for one plate, and you’ve given it to me [broken]”
• Point: When the plate breaks (ceases to be continuous), it is no longer one but becomes many
• Illustrates the fundamental connection between continuity and unity

Socrates and the Slave Boy (Plato’s Meno) #

• Problem: Finding the side of a square twice as large as a given square
• False answer: The side would be twice as long (which produces a square four times as large)
• True answer: The diagonal of the original square is the side of the square twice as large
• Demonstration: Socrates constructs four equal squares, draws diagonals, and shows the inner square is composed of four halves of the four squares, hence twice the original
• Key insight: This particular demonstration is a special case of the Pythagorean theorem for an isosceles right triangle, but Socrates cannot assume the slave boy knows the universal theorem
• Philosophical point: The property (that the square on the hypotenuse equals the squares on the other sides) belongs to the triangle insofar as it is a right-angled triangle, not insofar as it is an isosceles right-angled triangle—the property is proportionally common to all right triangles

Euclid’s Propositions 5 and 47 (Book I) #

• Proposition 5: An isosceles triangle has equal angles at the base

  • The property belongs to isosceles triangles insofar as they have two equal sides
  • The theorem also applies to equilateral triangles (which have all angles equal)
  • An equilateral triangle can be shown to have all angles equal by applying Proposition 5
  • This shows Proposition 5 is not understood as belonging to the isosceles triangle as distinguished from equilateral, but rather as a more universal property

• Proposition 47: The Pythagorean theorem for right-angled triangles

  • Demonstrates that for any right-angled triangle (whether isosceles or not, like the 3-4-5 triangle), the square on the hypotenuse equals the squares on the other two sides
  • This is the more perfect demonstration because it assigns the property to what it is proportional to (right-angled triangles in general)
  • Whereas showing it for an isosceles right triangle would be less universal

The Cone Problem (Democritus) #

• If you bisect a cone parallel to the base, you get two circles
• Question: Are these circles equal or unequal?
• Paradox: If equal, the cone becomes a cylinder; if unequal, the cone’s side becomes jagged
• Referenced as an illustration of difficulties with treating the continuous as composed of indivisibles
• Implicit solution: There is no “next” circle in the continuous—an infinite number of circles lie between any two

Notable Quotes #

“The continuous and the discrete in modern science.” — Louis de Broglie (title of work cited as rare modern treatment of the continuous)

“You can’t think of a work on the continuous by the modern philosophers, can you?” — Berquist, noting the modern neglect of this fundamental philosophy

“The philosophy of the continuous is altogether fundamental for natural philosophy, because motion and place or distance and time… are all continuous.”

“Why does it belong to natural philosophy to determine, basically, that the continuous is, huh? That it’s divisible forever and it’s not divisible into indivisibles?” — Berquist’s central pedagogical question

“The highest kind of demonstration… is a demonstration poked or quit [demonstratio propter quid], giving you the reason… in the sense of the cause.”

“An opinion without a reason for it is an ugly thing.” — Plato (quoted via Berquist’s colleague, used to emphasize the necessity of reasoning)

“We understand nothing without the continuous and time, although time itself is continuous.” — Aristotle (via Thomas Aquinas commentary)

“We understand God by the negation of the continuous.” — Key principle for understanding immaterial substances

Questions Addressed #

Why is the Philosophy of the Continuous Fundamental for All Philosophy? #

Answer: The continuous is the foundation of:

• All motion, place, distance, and time (natural philosophy)
• Geometry and arithmetic (mathematical philosophy)
• Understanding immaterial substances through negation (metaphysics and theology)
• Human knowledge, since reason always operates through imagination tied to the continuous

Why Does This Inquiry Belong to Natural Philosophy Rather Than Geometry? #

Answer:

• Geometry assumes the continuous; natural philosophy must establish what it is
• The continuous is broader than geometry’s subject matter (includes time and motion)
• Natural philosophy demonstrates the continuous as divisible forever and not composed of indivisibles, which is the common reason (proportionally) for magnitude, motion, and time
• Euclid’s demonstrations in geometry depend on what natural philosophy establishes about the continuous

What is the Connection Between the Infinity of Numbers and the Infinite Divisibility of the Continuous? #

Answer:

• One line can be divided into two parts (first number: two)
• Division can continue indefinitely, producing three, four, five, etc.
• The potential infinity of numbers corresponds to the infinite divisibility of the continuous
• Both arise from the same fundamental principle: the continuous has no smallest part

How Do We Understand Immaterial Things (God, Angels) If They Are Not Continuous? #

Answer:

• We cannot know immaterial things as they are in themselves in this life
• We understand them through negation of the continuous
• By understanding what they are NOT (not extended, not divisible, not in time), we approach understanding what they are
• This is how we also understand the immateriality of the human soul and reason

Is the Continuous Equivocal (Having Multiple Unrelated Meanings)? #

Answer:

• The continuous is not equivocal by chance
• It is equivocal “by reason” (proportionally unified)
• “Beginning” is equivocally said of the beginning of magnitude, motion, and time, but these are unified by proportional relationship
• Melissus’ error: He confuses having no beginning in time with having no beginning in magnitude—a false equivocation (Aristotle’s critique in Physics I)

Pedagogical Method #

Berquist employs several key teaching strategies:

1. Socratic questioning: Asks students “which is better: philosophizing or breathing?” to elicit immediate intuitions before philosophical analysis
2. Comparison of demonstrations: Uses geometry (Euclid’s propositions) to illustrate degrees of universal vs. particular demonstration
3. Direct appeal to experience: References everyday objects (tables, plates, squares) and positions
4. Hierarchical construction: Builds from natural philosophy’s task to understand why it precedes geometry in the order of learning