Axiomatic Method (Peano, Hilbert)

You lay down a bunch of laws involving certain primitive terms, and the primitive terms are implicitly defined by the axioms. One big problem is that the definitions always fail: you don't get uniqueness.

Dedekind Abstraction

Dedekind uses something a lot like the Peano axioms to define the notion of a progression. He then says (I'll be careful later) that if we abstract what is common to all progressions, we obtain the numbers.

Cantorian Abstraction

Cantor in talking about cardinal numbers says that we get a cardinal number by taking a set and abstracting from all the particular properties of the elements except that they are different.

Context Principle

Example: Directions.

Frege rejects this because of the Julius Caesar problem.

Frege also introduces a principle of abstraction, one that Russell takes over in a slightly altered form, which is this: every equivalence relation induces an associated set of objects.

Frege is also of the view that every singular term denotes an object, where that is constitutive of what denotation is and what objects are.

Weierstrass's method

Construction

There was something important going on in 19th Century mathematics, and all of these ideas are, in part, attempts to make sense of it.

-- ShaughanLavine - 22 Aug 2006