77. Natural Understanding and the Axiom of Limits

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Lecture Notes

Main Topics #

The Axiom: Nothing is an End of Itself #

• A fundamental axiom parallel to “nothing is a beginning of itself” from Physics Book I
• An end or limit is always other than that of which it is an end
• Example: the surface is the end of a body, but a surface is not itself a body
• This axiom is naturally known by all rational beings and cannot be avoided in thinking
• It underpins demonstrative reasoning in philosophy and natural science

The Four Senses of “End” (τέλος/peras/horos) #

1. End of magnitude: the boundary or surface (e.g., surface of a body)
2. End of motion: the terminus of change (e.g., hot and cold in a transformation)
3. End of intention: the purpose or goal (one of the four causes)
4. End as definition: that which separates a thing from everything else by drawing a limit around it

These senses are ordered hierarchically, each revealing a different aspect of the same fundamental principle.

Definition as a Limit #

• To define something is to draw a line around it, separating it from all other things
• Just as the city limits of Worcester exclude all non-Worcester and include all Worcester, a definition is “convertible” with the thing defined
• Example: “square” = equilateral and right-angled quadrilateral; every square is this, and every such figure is a square
• Division precedes definition: to define the square, divide figures → plain figures → rectilineal plane figures → quadrilaterals → equilateral quadrilaterals → right-angled equilateral quadrilaterals

The Hierarchy of Limits #

• Body’s limit is surface (length, width, depth → length, width)
• Surface’s limit is line (length, width → length)
• Line’s limit is point (length → no dimension)
• Point has no limit, because it is indivisible; to have a limit would require internal distinction, contradicting its nature

Natural Understanding and Philosophy’s Foundation #

• Natural understanding (νοῦς/intellectus): self-evident truths that reason naturally knows without reasoning
• Reasoned understanding: knowledge derived through demonstration from natural understanding
• Examples of naturally known statements: “A whole is greater than a part,” “Nothing is an end of itself,” “Statements exist”
• These are so natural that we cannot identify when we first learned them; they are universal in human experience
• The complete dependence of philosophy on the natural rests on three interconnected truths:
  1. There are some statements that reason naturally knows
  2. There is a natural desire to know what we do not naturally know (wonder/θαυμάζειν)
  3. We come to know what we do not naturally know through what we naturally know

The Problem of Equivocation #

• Modern confusion often arises from mixing different senses of the same word
• Example: “Man is an animal with reason” (animal is a propositional part); “Animal is the genus of man” (animal is a subjective part); “Animal includes cat, dog, horse” (more things than just man); therefore “a part is greater than the whole”
• This deceives because it equivocates on “part” (propositional vs. subjective)
• Students are easily deceived because they do not distinguish the senses of words

Key Arguments #

The Defense of Natural Knowledge #

• If all statements had to be known by reasoning, we would need statements to reason from—but then we would have no statements to begin with, and thus no knowledge
• Therefore, some statements must be naturally known
• These naturally known statements are common to all human reasoning; what is common to all is something natural

The Sophistical Argument Against “A Whole is Greater than a Part” #

• Premise 1: Everyone knows a whole is greater than a part
• Premise 2: Man is animal with reason; therefore animal is only part of man
• Premise 3: Animal includes cat, dog, horse, elephant—more things than man alone
• Conclusion: Therefore a part is greater than the whole
• Resolution: The fallacy equivocates on “part.” When animal is part of man’s definition, it is a propositional part. When we say animal includes more species, we mean the universal genus has more subjective parts (species) than any one species. These are different orders of whole and part.

Before and After Deception #

• Example: “Breathing is better than philosophizing because you cannot philosophize without breathing”
• This shows breathing is before philosophizing in the second sense (in existence/causation)
• But this does not show it is before in the fourth sense (better, more worthy, more good)
• One should not confuse temporal priority with priority in goodness or being

Important Definitions #

Key Greek and Latin Terms #

• τέλος (telos): end, purpose, goal; also can mean limit or boundary
• πέρας (peras): limit, boundary, end
• ὅρος (horos): limit, boundary; from this comes “horizon” (limit of the sky); also means “definition” (that which draws a limit)
• ἐσχάτον (eschaton): the last, the ultimate; also used for end or limit
• φύσις (physis): nature, natural being; foundational to all philosophical inquiry
• νοῦς (nous): mind, intellect, understanding; the faculty of natural understanding
• ἐπιστήμη (episteme): scientific knowledge, demonstrative knowledge derived through reasoning

Indivisible (ἄτομος/atomos) #

• That which has no parts
• A point is indivisible in magnitude
• Because it lacks internal distinction, it cannot have a limit or boundary
• Therefore two points cannot form a continuous line; points can only coincide

Examples & Illustrations #

The Point and Its Limit #

• A point has no parts, no edges, no boundaries
• Unlike a body (length, width, depth) or surface (length, width), a point has no dimension
• If a point had a limit, that would require distinction between the limit and the rest of the point—but a point has no “rest”
• Therefore, a point lacks a limit (properly speaking, it is not deprived of a limit, which would mean it should have one but doesn’t)

The Desk Analogy #

• The beginning of the desk is at one end; the end or limit is at the other
• Both the beginning and the end are other than the desk itself
• This illustrates the axiom: nothing is a beginning or end of itself

Defining the Square Through Division #

• Start with the universal: figure
• Divide: plain figure (excludes solid figures)
• Divide: rectilineal plane figure (excludes circles, ellipses)
• Divide: quadrilateral (excludes triangles, pentagons)
• Add difference: equilateral (excludes oblong, trapezium)
• Add difference: right-angled (excludes rhombus)
• Result: Equilateral, right-angled quadrilateral = square
• Each division separates the square from progressively fewer things until only the square remains

Personal Anecdotes on Nature vs. Social Construction #

• Berquist recounts his sister refusing to act like a boy despite their attempts, saying “girls are different than boys”
• He describes his niece saving money from babysitting to visit babies, showing natural maternal instinct
• He contrasts this with a modern parent who bought a daughter a truck instead of a doll, only to see the daughter cradle the truck like a baby
• He observed boys briefly playing with dolls before becoming bored—showing natural inclination differs from enforced social roles
• These examples illustrate how natural understanding and natural inclination cannot be entirely suppressed by contrary socialization

Shakespeare and Catholic Identity #

• Shakespeare purchased the Blackfriars gatehouse in London, a notorious place for hidden Catholics and secret Mass
• His daughter Susanna appeared on recusancy lists (those refusing Protestant communion)
• His godparents (who named his twins Hamnet and Judith) were known Catholics
• His father’s will was written in Catholic form
• These historical details illustrate how Shakespeare’s associations point to Catholic sympathies despite historical uncertainty

Nature and Crime #

• Berquist recounts a case where a boy sent threatening letters to a girl student
• The FBI handwriting analysis identified the boy; his father acknowledged guilt objectively
• The mother could not bring herself to accept it—showing the natural mercy of the mother vs. the natural justice of the father
• He notes that a father is more apt to disown a son who fails to live up to expectations, while the mother’s mercy is more constant

Notable Quotes #

“Nothing is an end of itself. There’s always some distinction between the end and that of which it is an end.”

“It’s hard to just hide the sparks of nature—it’s kind of natural, it’s sort of a thing.”

“The complete dependence of philosophy upon the natural.”

“If you revolt from the natural right, that would lead to the complete collapse.” (In reference to Shakespeare’s warning in Romeo and Juliet: “those who revolts from true birth…stumbling on abuse”)

“My rule of thumb is: Aristotle means what Thomas says he means.”

“Wonder is the beginning of philosophy.” (Attribution to Aristotle, via Plato)

Questions Addressed #

Can a Point Have a Limit? #

• Question: Since a point is an indivisible, can it have a boundary or limit?
• Answer: No. A point cannot have a limit because it lacks internal parts or distinction. For something to have a limit, there would need to be a distinction between the limit and the rest of the thing. A point, being indivisible, has no “rest.” Therefore, properly speaking, the point does not lack a limit (which would imply deprivation); rather, it simply has none.

How Does Definition Function as a Limit? #

• Question: In what way does defining something constitute drawing a limit around it?
• Answer: Definition separates a thing from all others by specifying what it is and is not. Like city limits that precisely include Worcester and exclude all surrounding areas, a definition is “convertible” with the thing defined—true of all instances and only instances of that thing. The definitions are achieved through progressive division, each step eliminating what the thing is not.

How Can We Know Anything if All Knowledge Requires Reasoning? #

• Question: If we must reason to all knowledge, don’t we face an infinite regress of needed premises?
• Answer: No, because some statements are naturally known without reasoning. These foundational truths (like “a whole is greater than a part” or “nothing is an end of itself”) are known by all rational beings and serve as starting points for all demonstrative reasoning. The natural desire to know drives us to reason from these natural understandings to discover what we do not naturally know.

Why Are Students Deceived by Equivocation? #

• Question: Why do students easily fall for arguments that equivocate on words like “part” or “end”?
• Answer: Because they fail to distinguish different senses of the same word. Words like “part,” “end,” and “before” have multiple related but distinct meanings. When two senses are used in the same argument without being distinguished, the mind naturally connects them as if they were the same sense. This is “the most common kind of mistake in thinking, according to the father of logic.”

What is the Difference Between “Lacking” and “Not Having”? #

• Question: Does the point “lack” a limit (in the sense of being deprived)?
• Answer: No. Strictly speaking, only something capable of having a property can be said to lack it. A stone is not blind because it cannot see; we or other animals would be blind if we couldn’t see. Similarly, the point does not lack a limit; it simply has no limit, because the nature of an indivisible precludes having one.

Pedagogical Insights #

On Teaching Natural Knowledge #

• Students naturally know certain truths but cannot articulate when they learned them
• The most effective deception comes from equivocation on familiar words, not entirely novel arguments
• Students must be trained to distinguish senses of words and recognize when a word is used in different ways within a single argument
• The axiom “nothing is an end of itself” is less immediately obvious than “a whole is greater than a part,” but can be demonstrated through the hierarchy of limits (body → surface → line → point)

On Modern Philosophy’s Failures #

• Modern philosophers often employ obscure language and resist clarity
• Hegel admitted writing obscurely because he thought profundity required obscurity
• This contrasts with the Aristotelian commitment to making things intelligible
• When natural understanding is rejected, philosophy becomes incoherent and contradicts itself