Excerpt: Kant's Arguments for the Synthetic A Priori

Why did Kant believe that the propositions of geometry (and arithmetic) are synthetic a priori? Critically examine his view.

Introduction
A major piece of Kant’s epistemology is what he classed synthetic a priori statements and under this class of statements are mathematical statements within Euclidean geometry. It is my thesis to explicate this particular view within Kant’s transcendental philosophy.

Synthetic A Priori Statements of Euclidean Geometry
Assuming Kant’s distinction between analytic and synthetic statements, consider the following statement of Euclidean geometry:

P. “Parallel lines are such that they never intersect.”

For Kant, (P) is synthetic a priori for the following reasons:

1. (P) is a synthetic statement for the predicate is not contained within the concept of parallel lines. (B11-12)
2. The denial (P) does not produce a contradiction. (B14)
3. Experience shows parallel lines do not meet; since (P) has empirical content it is synthetic. (A714/B742; A734/B751)
4. Space does not exist in the world; space is mind-dependent because it is an a priori intuition in the mind. (A42, B59)
5. (P) is a priori because it is necessarily true and universal. (A48, B65)
6. Therefore (P) is a synthetic a priori statement.

Now I will explicate each premise in order to show how Kant’s conclusion following from them. First, premise (1); Kant made critical distinction between analytic and synthetic statements. (A) is an analytic statement and (S) is a synthetic statement:

A. “All bachelors are unmarried males.” (x) (Bx → Ux)
S. “All bodies are heavy.” (x) (Bx → Hx)

(A) is analytic because it is true by virtue of the meanings of its words and the predicate is contained within the subject, i.e., being an unmarried male is contained within the concept of a being a bachelor. Analytic statements also obey as Kant calls at (B14) the principle of contradiction for if one were to deny an analytic statement a contradiction would result. Considering (A), if one to deny it then one would be asserting the claim, “It is not the case that all bachelors are unmarried males” or in logical terms ~(x) (Bx → Ux); however, a contradiction arises because by definition a bachelor is an unmarried males and thus is impossible for one to be a bachelor and be a married male. (S) is synthetic because it is not true by virtue of the meanings of its words and its predicate is not contained within the subject and do not obey the principle of contradiction. The denial of (S) does not produce a contradiction because if one were to assert, “It is not the case that all bodies are heavy”, or in logical terms ~(x) (Bx → Hx), such a statement does not produce a contradiction because being heavy is not contained by the concept of being a body.

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