215. Virtues, Passions, and the Geometry of Knowledge

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Lecture Notes

Main Topics #

Temperance and the Control of Passions #

• Mildness (Mansuetudo) is a virtue concerned with the regulation of anger, attached to the virtue of temperance
• Unlike courage, which requires individuals to be “pushed into” courageous action (overcoming fear), anger must be restrained rather than encouraged
• The strength of anger as a passion makes its restraint particularly important
• Example: Romeo and Juliet—Shakespeare uses the phrase “in the very wrath of love, clubs could not part them,” illustrating how powerful the passion of love can become, leading to the need for quick marriage to channel it appropriately

Envy as a Capital Vice #

• Envy is particularly insidious because people are ashamed to admit it
• It manifests in gossip and backbiting—when someone receives attention or acclaim, the envious respond by spreading negative things about that person
• This pattern is observable in academic and ecclesiastical contexts
• Unlike anger, which is openly passionate, envy operates covertly through social manipulation

Geometry and Recollection in Plato’s Meno #

• Socrates maintains that we naturally know geometry, but have forgotten it due to being “struck into this crazy body”
• Learning is recalling what we already know (μνήμη / mneme)
• The problem of doubling a square:
  • A student first incorrectly answers that you double the side of the square (which would make it four times as large)
  • Socrates guides the student by asking about smaller squares arranged together
  • The solution emerges: take the diagonal of the original square—this is the side of a square twice as large
  • This illustrates the Pythagorean theorem applied to an isosceles right triangle

The Wonder of Incommensurable Quantities #

• A square four times as large requires multiplying the side by 2 (a rational multiplier)
• A square twice as large requires multiplying by √2 (an irrational number)
• One cannot find any unit of measurement (inches, parts of inches, etc.) that would work: “72 is not a square number”
• This is an example of wonder (thaumazein) that Aristotle identifies as the beginning of philosophy
• The only way to solve this is geometrically, not arithmetically—you cannot multiply by anything to get exactly twice as large
• This demonstrates that geometry reveals truths that arithmetic cannot express

Natural Law and Natural Knowledge #

• When asked by a pro-life committee member “What is the natural law?” Berquist explains it as something naturally known
• The challenge: how do you lead someone to admit that natural law exists as something naturally known?
• The Meno provides a model: just as geometric truths are naturally known and must be recalled/developed through proper questioning, so too natural law is naturally known
• Not all knowledge requires explicit instruction—some truths are grasped through the use of reason on principles already possessed

Key Arguments #

The Slave Boy and Learning #

• Socrates does not teach the slave boy geometry; rather, the boy’s answers come from what he already knows
• Yet the boy’s initial answer (double the side) shows he was “even worse than not knowing—he was mistaken about it”
• The key insight: Learning is not the same as actually knowing
  • To know a new theorem requires recalling things already known
  • These recalled principles must be put together to see what follows
  • But one must begin somewhere with things naturally known (e.g., “the whole is greater than the part”)
• Socrates’ claim that we naturally know all geometry is exaggerated; what we naturally know are the very beginnings: basic principles of equality and proportion

The Geometric Solution and Mathematical Incommensurability #

• If a square is 1×1 (foot × foot), and we want one twice as large:
  • 1×1 = 144 square inches
  • Half of 144 = 72 square inches
  • But 72 is not a perfect square; no integer multiplied by itself gives 72
  • No matter how small a unit you use, you cannot express the side arithmetically
• The only solution is geometric: the diagonal of the original square is the side of the desired square
• This reveals something marvelous: some truths require geometry to express them; arithmetic alone is insufficient

Natural Law and the Order of Discovery #

• Aristotle proceeds in the opposite order from Thomas Aquinas
• Thomas (theological order): begins with God and sees other things in light of God; natural law is defined as “a partaking of the divine law”
• Aristotle (natural order): begins with the natural and visible, then ascends to higher principles
• Lawyers have difficulty understanding any law except positive law (law on the books)
• Sophocles’ Antigone is cited as an example of appeal to natural law: Creon passes a law forbidding burial, but this is against natural law, and Antigone appeals to this higher law

Important Definitions #

• Mildness (Mansuetudo): A virtue concerned with the proper regulation of anger, attached to temperance
• Temperance (Temperantia): The virtue governing appetitive desires (food, drink, sex); requires restraint
• Courage (Fortitudo): The virtue that overcomes fear; requires pushing oneself forward
• Envy (Invidia): A capital vice characterized by sadness at another’s good; operates covertly through gossip
• Natural Law (Lex Naturalis): Law according to natural reason; something naturally known, discoverable through the use of reason
• Positive Law (Lex Posita): Law explicitly enacted and written down
• Recollection (Ἀνάμνησις / Anamnesis): In Platonic theory, learning as the recovery of knowledge already possessed by the soul
• Marvelous/Wonder (Θαυμάζειν / Thaumazein): The experience of amazement at the discovery of something non-obvious; Aristotle identifies this as the beginning of philosophy

Examples & Illustrations #

The Doubling of the Square #

• Start with a square 2×2 feet (4 square feet total)
• To make a square 8 square feet:
  • Incorrect approach: double the sides to 4×4 = 16 square feet (four times as large)
  • Correct approach: take the diagonal of the 2×2 square; this becomes the side of the 8-square-foot square
  • Illustration: draw four small squares and arrange them around the diagonal to form a larger square

Romeo and Juliet #

• “In the very wrath of love, clubs could not part them”
• The passion of love is so strong that even violent force (clubs) cannot separate the lovers
• Parents must sometimes intervene to separate young lovers, just as one would use a club to part fighting dogs
• This necessitates quick marriage to properly channel such passionate inclinations

Academic Envy #

• When someone receives scholarly attention or acclaim, colleagues may respond by saying negative things about them
• This pattern is observable in universities and even in clergy
• The envious person does not want to admit their envy, so they mask it through gossip and criticism

The Girl on the Bike #

• A girl runs up and down the street on her bike while someone mows the lawn
• Others nearby try to stir up envy by suggesting “Don’t you wish you were doing that?”
• Yet the envious person may respond “That’s the last person I want to be with,” revealing their insinuating and defensive nature

Euclid’s Circle Inscription Problem #

• Given any triangle (equilateral or odd-shaped), can you always draw a circle through its three vertices?
• Intuition suggests yes for an equilateral triangle, but what about an irregular triangle?
• Euclid proves in Book IV that a circle can be inscribed through any three points of any triangle
• This exemplifies the marvelous simplicity of geometric truths once discovered

Notable Quotes #

“But in the case of anger, it’s something that needs to be, what, more restrained, right? So it’s like, you know, these strong emotions, right?”

“There’s nothing you could multiply it by [the side of the original square] that would give you one twice as big… There’s nothing you could multiply it by. I don’t believe that. How could that, you know? And that’s the thing that Aristotle takes as an example of wonder, right?”

“But you see that in the academic world, too, you know? They say bad things about somebody, you know? They take too much attention or something.”

“Learning is not… is really recalling what we already know, right? And he gives the appearance that the slave boy is recalling geometry because it’s out of the answers to the slave boy that comes the answer to how to double a square.”

“To put these things together and see what follows from them, right? Before you actually come to know, right? So he’s not really recalling how to, what, double a square, right? But he’s coming to know it by recalling things he already, what, knows.”

Questions Addressed #

1. How does one lead someone to admit that natural law exists as something naturally known?

   • Answer: Through the model provided by Plato’s Meno—by guiding them to see that some truths are grasped through reason applied to naturally known principles, not through explicit instruction

2. Does the slave boy actually recall how to double a square?

   • Answer: Not in the strict sense; he recalls the principles from which the solution follows, but learning the new theorem requires reasoning from these recalled principles

3. Can arithmetic express the relationship between a square and one twice its size?

   • Answer: No. There is no rational number that multiplies the side to produce exactly twice the area. Only geometry can express this through the diagonal

4. Why does geometry reveal truths that arithmetic cannot?

   • Answer: Because geometry works with continuous quantities and spatial relationships that transcend discrete numerical ratios. The diagonal of a square involves an incommensurable ratio (√2)

5. What is the difference between the theological order (Thomas) and the natural order (Aristotle) in understanding natural law?

   • Answer: Thomas proceeds from God downward to see natural law as participation in divine law; Aristotle proceeds from observable nature upward to natural law principles