8. The Natural Road, Rhetoric, and the Mathematics-Nature Distinction

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Lecture Notes

Main Topics #

The Four Reasons Philosophers Follow the Natural Road #

1. Man is rational - It is natural for humans as reasoning animals to follow this path
2. All knowledge passes through it - All knowledge must travel from senses to reason; there is no other route
3. Wisdom demands it - Following Heraclitus: wisdom is to speak truth and act in accord with nature (φύσις)
4. It leads to wisdom - This is the road to wisdom itself, as Aristotle demonstrates through the hierarchy of knowledge (sensation → memory → experience → art → science)

Rhetoric as a Mixed Science (Paraphysis) #

Rhetoric is defined as the art of persuasion but has a complex relationship to other disciplines:

• Offshoot of logic/dialectic: It uses argumentative structure and reasoning
• Offshoot of ethics and political philosophy: Persuasion depends on the ethos (character) of the speaker—appearing as a man of foresight, justice, and good character
• Context-dependent: Different political constitutions and audiences require different rhetorical approaches
  • Example: In ancient Athens, humble origins were a liability; in modern America, the self-made man is admired
  • Demosthenes would be ineffective in modern democratic contexts despite being the greatest Greek orator
• Historical significance: Great figures like St. Augustine and St. Jerome were trained in rhetoric; Augustine initially found the Bible unworthy compared to Cicero’s style
• Origins: Rhetoric developed in Greek cities of Sicily when tyrants were overthrown and power depended on persuading crowds

Modern Experimental Science as a Paraphysis #

Experimental science is not purely natural philosophy but a mixed science combining natural and technical science:

• Comes from the union of natural science (φύσις) and mechanical arts (mechanics)
• This union developed strongly in the 17th-18th centuries
• Two-way interplay: theoretical developments suggest new inventions; mechanical developments enable new experiments
• Important distinction: We know what we ourselves have made through instruments; we don’t know natural things as natural but as transformed by our artificial apparatus
• Difference from natural philosophy: Natural philosophy studies natural things as natural; experimental science shadows nature but doesn’t contain its real substance

The Problem of Mathematics and Natural Philosophy #

An apparent overlap creates a problem:

• The dilemma: Both natural philosophy and mathematics discuss surfaces, solids, lines, points, and shapes found in natural bodies
  • Do they study the same things, making them overlap?
  • Is mathematics part of natural science, or vice versa?
• Historical examples of the problem:
  • Pythagoras divided mathematical sciences into arithmetic, geometry, astronomy, and music (the quadrivium)
  • Astronomy appears to be both mathematical and natural
  • Newton’s Principia Mathematica Naturalis Philosophiae unites mathematics with natural philosophy

Aristotle’s Solution: Separation in Thought #

The mathematician and natural philosopher both discuss the same objects but in fundamentally different ways:

• The mathematician considers things (number, shape, surface, length, width, depth) in separation from matter and motion
  • Not insofar as they are limits of actual bodies
  • Does not consider what happens to these things when they exist in natural bodies
  • Example: A geometer studies circles but ignores how a tire flattens where it touches pavement
• The natural philosopher considers these things only insofar as they are in natural bodies
  • A surface as the surface of an actual body
  • A shape as it actually exists in matter
  • Example: The snub nose is a curved shape in flesh and cartilage; the definition must include the matter

This separation is in thought (mente), not in reality—things can be known in separation without existing in separation.

The Reason for the Solution: Definitions #

• Geometric definitions contain no reference to matter or motion
  • Definition of sphere: “a figure bounded by one surface such that all points on that surface are equidistant from an internal point”
  • No mention of what it is made of
• Natural philosophy definitions necessarily involve matter and motion
  • Example: “Snub” (turned up at the tip) in flesh
  • Bone must be defined with reference to its specific matter
• Since definitions are foundational to all reasoned knowledge, the difference in definitions proves the distinction between the sciences

The Sign for the Solution: Application #

• If mathematics were already in nature, no application would be necessary
• The fact that application is required proves prior separation
• Historical examples:
  • Kepler applied the ellipse (from Apollonius’s ancient conic sections) to planetary orbits; the Greeks knew nothing of this application
  • Einstein applied Riemann’s matrix mathematics to relativity theory; Riemann developed the mathematics with no idea of physical application
• Application is the sign that things are separated in thought but not in reality

Key Arguments #

The Logical Division Argument #

• Aristotle raises the problem through logical division: both mathematician and natural philosopher discuss the same sensible objects
• Either they overlap, or they are distinct
• If distinct, what makes them distinct?

The Definition Argument #

• Definitions reveal the essence of what is being studied
• Geometric definitions exclude matter and motion
• Natural definitions include matter and motion
• Therefore, the sciences studying these must be distinct

The Application Argument #

• Application of mathematics to nature is a historical fact (Kepler, Einstein, etc.)
• Application presupposes prior separation
• If mathematics already contained natural things, application would be unnecessary
• Therefore, mathematics and nature are separated in thought

Important Definitions #

Natural road (via naturalis): The path of knowledge from sensation through experience to art and finally to science; the natural way humans acquire knowledge

Rhetoric (ῥητορική): The art of persuasion; persuades through arguments, but more importantly through the character (ἦθος) and emotions of the speaker

Paraphysis: An offshoot or mixed science; a science that combines two distinct areas of knowledge (e.g., experimental science combining natural and technical science; rhetoric combining logic and ethics)

Separation in thought (mente): Considering something without reference to something else it is joined with in reality; knowing things in separation without falsity

Mathematical science: Knowledge of number, shape, surface, line, etc., in separation from matter and sensible motion

Natural science (natural philosophy): Knowledge of things insofar as they are composed of matter and exhibit motion or rest

Snub nose (σιμόν): Aristotle’s standard example of a natural definition: the curved shape (mathematical) must be understood as existing in flesh and cartilage (matter)

Examples & Illustrations #

Rhetoric and Demosthenes #

Demosthenes, the greatest Greek orator, made fun of his opponent for his humble origins, suggesting slavish birth. This was effective in aristocratic Athens. However, in modern democratic America, where the self-made man is admired, the same rhetoric would be ineffective or counterproductive. This shows that persuasion must adapt to the political constitution and customs of the audience.

The Tire in the Parking Lot #

A tire appears circular when viewed from the side, but when actually resting on pavement, it is flattened at the bottom due to gravity and pressure. The geometer studies the circle in separation and ignores this flattening. The natural philosopher must account for how the shape is actually affected by matter and forces. This illustrates the difference between mathematical and natural consideration.

Broccoli and Cauliflower Seeds #

With identical conditions (same sun, rain, soil, fertilizer), broccoli seeds produce broccoli and cauliflower seeds produce cauliflower. The difference must be due to something within the seeds themselves, not external conditions. When one cauliflower accidentally grew among broccoli, it was due to a cauliflower seed in the broccoli row, not a transformation of the broccoli seed. This shows nature is an internal cause.

The Tree and the Stone #

Given the same external conditions (sun, rain, soil), a tree grows while a stone remains the same. This obvious difference proves that there must be causes of change and rest within things themselves, which is what we call nature.

St. Augustine and Cicero #

When St. Augustine first read the Bible, he found it unworthy to be compared to Cicero’s beautiful style. This illustrates the power of rhetoric in great minds. St. Augustine and St. Jerome were both trained in the art of rhetoric before converting to Christianity.

Application in Physics: Kepler and the Ellipse #

Johannes Kepler discovered that planetary orbits follow ellipses by applying ancient conic sections (developed by Apollonius) to astronomical data. The Greeks who developed the mathematics of conic sections had no idea this mathematical form would apply to planetary motion. This shows that mathematics is worked out in separation from natural applications, and application is necessary to unite them.

Application in Relativity: Einstein and Riemann #

Albert Einstein applied Riemann’s matrix mathematics (developed in the 19th century) to relativity theory. Riemann had developed this mathematics with no knowledge of its physical application. This is another example of how mathematical development is separated from natural application, and application must be done by someone else.

Notable Quotes #

“Wisdom is to speak the truth and to act in accord with nature, giving ear thereto.” - Heraclitus (cited by Berquist on why philosophers follow the natural road)

“The mathematician then also treats of these things, but not insofar as each is the limit of an actual body.” - Aristotle (on the distinction between mathematics and natural philosophy)

“It would be laughable to try to prove that nature exists, for it is obvious that there are many such things among the things that are.” - Aristotle (on why nature’s existence requires no proof)

“To prove the obvious by what is not obvious belongs to one who can’t separate what is known to itself from what is not known to itself.” - Aristotle (on the error of trying to prove self-evident truths)

“Kick him in the ass.” - Berquist’s teacher Surrey (responding to those who deny material things exist, illustrating that sensory evidence is more certain than rational argumentation)

“The trouble with modern man is he’s trying to live entirely in a world of his own making.” - Cited by Berquist (on the consequences of limiting knowledge to what we manufacture)

Questions Addressed #

Why doesn’t Aristotle prove that nature exists? #

• Nature’s existence is self-evident from common experience
• To prove the obvious by appeal to what is not obvious would be absurd and circular
• A man blind from birth can syllogize about colors without understanding them; similarly, one who tries to prove nature exists confuses what is known to itself with what is not known to itself
• The obvious facts (tree grows, stone doesn’t; wood burns, stone doesn’t) are sufficient

How can mathematician and natural philosopher study the same things? #

• They study the same sensible objects but with different considerations
• The mathematician separates these objects from matter and motion in thought
• The natural philosopher considers them only as they exist in matter and motion
• This is not falsification but a legitimate difference in how one considers objects

What proves that mathematics is separated from nature? #

• Application is necessary to unite mathematics with nature
• If mathematics already contained natural things, application would be redundant
• Historical examples (Kepler, Einstein) show mathematicians often unaware of applications
• The fact that application must be done proves prior separation

Why does rhetoric depend on ethics and political philosophy? #

• Persuasion works through the character and credibility of the speaker, not arguments alone
• The virtues studied in ethics (justice, prudence, courage, temperance) create the ethos needed for persuasion
• Different political constitutions require different approaches to persuasion
• Therefore, the orator must know ethics and political philosophy