MEANING OF THE DISTINCTION, A PRIORI’ CHARACTER OF MATHEMATICS

• In the theory of knowledge, a key distinction is made between two kinds of knowledge: a priori and empirical.

• Empirical knowledge is obtained by observation of the external world (sense-perception) and ourselves (introspection).

• A priori knowledge is gained by thinking alone, without requiring sensory experience.

• Examples of a priori knowledge are primarily found in logic and mathematics.

• For instance, to know that 5 + 7 = 12, we do not need to count physical objects; we know it just by thinking.

• A crucial difference: a priori knowledge shows that something must be true and explains why it is true, while empirical knowledge only shows that something is true in fact.

• Observing a flower is yellow tells us it is yellow but not why it must be yellow; it could have been red.

• Mathematical truths like 5 + 7 = 12 are necessarily true, and it is absurd to think otherwise.

• It is conceivable that objects might fuse or vanish, reducing the count, but it is inconceivable that 5 + 7 things could be less than 12 simultaneously.

• Saying something is inconceivable means a positive insight: it is logically impossible, not just difficult to imagine.

• Empirical propositions are contingent: they are true but might have been false.

• A priori propositions are necessary: they must be true and cannot be otherwise.

• For example, laws of motion might have been different, but arithmetic laws cannot be false for anything countable.

• It is conceivable the universe could be a homogeneous fluid without distinct objects, making arithmetic not applicable, but arithmetic must hold for anything that can be numbered.

• Although understanding arithmetic initially requires examples, once learned, we do not need examples to grasp or prove new arithmetic truths.

• For example, we know 3112 + 2467 = 5579 without ever physically counting that many objects.

• Examples serve only to help understand the concept of number, not to prove arithmetic truths.

• In geometry, examples are psychologically more needed; people often need drawn figures or mental images to understand proofs.

• However, geometric proofs are not based on the actual figures drawn; measurements are not part of Euclid’s proofs.

• We can follow proofs even if the illustrative figure is badly drawn or not accurate.

• If geometry were empirical, it would be risky to conclude that all triangles share a property based on a single example; it might be an exception.

• Recent developments in geometry, such as non-Euclidean geometries, have loosened the connection between geometric proofs and empirical figures.

• Some geometries can be developed without relying on actual physical figures.

THE ‘A PRIORI’ IN LOGIC

• Another important field of a priori knowledge is logic.

• The laws of logic must be known a priori; they cannot be derived from empirical observation.

• The purpose of logical argument is to provide conclusions not discovered by observation; if conclusions were already observed, argument would be unnecessary.

• Inference works because of a logical connection between premises and conclusion: if the premises are true, the conclusion must be true.

• Knowing the premises allows us to assert the conclusion without waiting for experience.

• Example story: Mr. X. is revealed as a murderer by inference, not observation — because he was the first penitent to a priest who heard a murder confession.

• This type of inference is called a syllogism.

• Syllogism example:

  • Premise 1: The first penitent was a murderer.

  • Premise 2: Mr. X was the first penitent.

  • Conclusion: Mr. X was a murderer.

• Syllogisms have sometimes been overemphasized, but they are as important as any form of inference.

• They allow us to gain information not previously observed or known directly.

• Inference depends on the special logical connections between propositions, not on observation.

• Logic’s chief function is to study these connections, with the syllogism being only one kind.

• A syllogism consists of three propositions: two premises and one conclusion.

• Each proposition has a subject and predicate connected by the verb “to be” (the copula).

• There are exactly three terms in a syllogism: one is the middle term common to both premises.

• The middle term connects the other two terms, which appear together only in the conclusion.

• Example:

  • All men are mortal (middle term: man)

  • Socrates is a man

  • Therefore, Socrates is mortal

• The middle term “man” links “Socrates” to “mortal,” enabling us to conclude Socrates is mortal even if we never observed his death.