9. Philosophy of Mathematics, Metaphysics, and Philosophical Method

Kant on geometry

• Leibniz believes geometrical truths follow from basic axioms according with the laws of logic

• Kant rejects this, and doesn’t believe one can determine the relation of the angles of a triangle to that of a right angle by mere conceptual analysis

  • Some intuition is always required such that these truths are synthetic

    • Even mental constructions require the pure intuition of space for generality/necessity, so although intuition might not require experience, it still requires the forms of intuition

• Kant’s rejection of Leibnizian geometry has two characters — the logical and the substantive

The logical interpretation

• Developed especially by Michael Friedman (1985), Kant denies that ‘the angles in a triangle sum to 180 degrees’ is discoverable from axioms by logic alone

• Russell thinks Kant identifies a problem in Aristotelian logic which leads him to make mathematics non-logical (synthetic), but instead logic needed to be changed

  • This problem is syllogistic logic’s subject-predicate form, which requires that in an expression ‘5+7=12’, 5, 7, and 12 are separate subjects to which predicates can apply

    • In this form, relational properties are impossible to express without losing their logical content

    • Russell and Frege’s logic instead presents this expression as an identity relation: ‘5+7=12’ considers two expressions which have the same referent

• Friedman points out that modern formulations in logic are inexpressible in syllogistic logic without quantifier dependence

The substantive interpretation

• An alternative is to view Kant as believing all geometric truths are analytic, but the axioms aren’t

• That is, intuition is required to show metaphysical possibility, but not to cognise inferences (constructions in proofs are inessential heuristics)

• Furthermore, where Kant viewed geometry as the study of real space, non-euclidean geometry, where a space is defined according to axioms, opens the possibility that geometry need not accord with the space represented to us

Kant’s vision for metaphysics

• Unlike mathematics, which can proceed from axioms because the objects of these axioms can be exhibited in pure intuition by construction, metaphysics can’t proceed from similar axioms because concepts alone will never entail metaphysical possibility

• Causation can’t be derived axiomatically in the same way mathematical propositions can be

• The only possible place to begin with is to first discover the the conditions of experience, which are synthetic a priori

  • Thus, metaphysics effectively deals with the foundations of the empirical sciences — it shows how appearances are ordered

• Philosophical cognition is rational cognition from concepts whereas mathematical cognition is the construction of concepts a priori by exhibiting the intuitions belonging to them

  • In philosophy, definitions conclude arguments whereas in mathematics they precede them

German idealism and resistance to Kant

• Concerns philosophers like Reinhold, Fichte, Hegel, and Schelling who all considered themselves Kantian in some sense, although disagreed in important areas

• One famous problem for Kant is that of noumenal affection, which extends causality to the noumenal which contradicts his doctrine of noumenal ignorance

  • If noumena are unknowable, it is not clear that we have sufficient grounds to say that they exist

• Another major problem is his unjustified premises, such as the table of judgements, separation of cognitive faculties (into sensibility and understanding) and the transcendental argumentative strategy, which assumes certain types of knowledge and then attempts to prove them (question-begging against idealism)

  • These such premises led to a minor resurgence of rationalism in some form