3. The Order of Sciences and Certainty

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Lecture Notes

Main Topics #

The Hierarchy of Sciences by Certainty #

Berquist presents an ordering of sciences based on how many factors must be considered:

1. Arithmetic (most certain) - only considers numbers, no material variation
2. Geometry - considers shape and size abstractly, but must account for some spatial properties
3. Natural Philosophy - adds physical matter and its variations
4. Practical Philosophy (least certain) - must consider customs, habits, choices, and human particularity

The principle: The more universal the knowledge, the fewer things one must take into account; therefore the more certain. As knowledge becomes more particular, more factors must be considered, making it less certain.

Matter as a Complicating Factor #

When matter is introduced into a discipline, certainty diminishes:

• Circle (geometry): The perfect mathematical circle
• Tire in the parking lot: The bottom is flattened (five and out), deviating from the geometric ideal because matter deforms the mathematical form
• Geometry does not need to account for this material deviation; it concerns only the abstract circle
• When we move from abstract principles to material instances, we must consider material imperfection

Application to Human Nature #

The same principle applies when moving from:

• What man is by nature (universal human essence)
• To 21st century American man (particular, culturally situated human)

The latter requires consideration of:

• Customs and traditions
• Personal choices
• Cultural circumstances
• Historical particularity

Progression and Addition of Considerations #

As one descends from the universal to the particular:

• Figure → Plane figure → Rectilineal plane figure → Quadrilateral → Square
• Each step adds more specifications to consider
• More specifications = less certainty
• This is why counting people in a room is more certain than counting people at Yankee Stadium

The Paradox of Universality and Certainty #

What is most universal is most certain because it requires the fewest considerations. What is said of all things requires the least to be taken into account, making it more certain than any particular instance.

Key Arguments #

Argument 1: The Inverse Relationship Between Universality and Complexity #

1. Knowledge of the universal requires fewer variables to be considered than knowledge of the particular
2. Fewer variables to consider means greater certainty
3. Therefore, knowledge of what is most universal is more certain than knowledge of particulars
4. Corollary: Arithmetic is more certain than geometry; geometry is more certain than natural philosophy; natural philosophy is more certain than practical philosophy

Argument 2: Matter Reduces Certainty #

1. Abstract principles (like the perfect geometric circle) are certain and invariable
2. Material instances deviate from abstract principles (like a tire’s flattened bottom)
3. When studying material instances, one must account for material imperfections
4. Therefore, the introduction of matter reduces certainty

Argument 3: Human Particularity Increases Complexity #

1. Universal human nature can be described with relative certainty
2. But particular humans (Israelites, Palestinians, 21st century Americans) involve customs, choices, and historical contingencies
3. These additional factors make knowledge of particular humans less certain
4. Therefore, practical philosophy applied to particular peoples is less certain than general philosophy

Important Definitions #

Universal (Universale) #

What is said of all instances of a kind; known with greater certainty because it requires consideration of fewer variables

Particular (Particulare) #

What applies to specific instances; known with less certainty because it requires consideration of many contextual variables

Matter (Materia) #

The physical substrate that receives form; its variations and imperfections cause abstract principles to deviate in material instances

Certainty (Certitudo) #

The degree of confidence with which a conclusion can be drawn; inversely proportional to the number of variables that must be considered

Examples & Illustrations #

The Tire and the Circle #

• A perfect mathematical circle has all points equidistant from center
• A tire in the parking lot has its bottom portion flattened (“five and out”)
• Geometry does not concern itself with this material deviation
• But if studying actual tires, one must account for the flattening caused by weight and material properties
• This illustrates how matter complicates abstract knowledge

Counting at Different Scales #

• It is easier and more certain to count the people in a room than in Yankee Stadium
• Fewer people = easier to track = greater certainty
• More people = more variables = more opportunity for error
• This demonstrates how quantity of particulars affects certainty

The Israeli-Palestinian Conflict #

• To understand Israelites and Palestinians, one cannot study them merely as “human beings” in the abstract
• One must consider:
  • Their customs and traditions
  • Their historical choices and conflicts
  • Their cultural practices
  • All the accidental particularities that distinguish them
• This makes practical knowledge of them more complicated and less certain than general philosophy

Questions Addressed #

Q1: Why is arithmetic more certain than geometry? #

A: Arithmetic considers only numbers without material variation. Geometry adds considerations of spatial properties. More variables to consider means less certainty. Therefore arithmetic, being more abstract and universal, is more certain.

Q2: How does the introduction of matter reduce certainty? #

A: Abstract principles (like geometric forms) are invariable. But when these principles are instantiated in matter, material imperfections cause deviations. One must then account for these material variations, complicating knowledge and reducing certainty.

Q3: Why is knowledge of universal human nature more certain than knowledge of particular peoples? #

A: Universal human nature concerns only what belongs to humans as such, requiring fewer considerations. But particular peoples (Americans, Palestinians) involve customs, choices, and historical contingencies that must all be taken into account. More factors to consider means less certainty.

Q4: What is the relationship between universality and certainty? #

A: The more universal the knowledge, the fewer things must be taken into account, and therefore the more certain it is. What applies to all requires consideration of only universal properties, making it more certain than knowledge of particulars.