Master History and Philosophy of knowledge

you can do a master of that at eth: link

Hilbert

finitization of calculus and Arithmetization of mathematics:

• replacement of infinitesimals by epsilon-delta limit definitions
• axiomatic approaches to numbers
• up until now a line was not a collection of points. Each of these points can be identified with a number (see numberline)
• acceptance of infinite sets
• functions are thought of mappings from set a to set b (before then it was lines) -> gives new problematic (non-analytic) functions: nowhere differentiable but continuous, nowhere continuous … (emergence of monsters)

Cantor’s theory

fourier expansion is not changed by change in finite number of places

Cantor tried to analyze which infinities would change fourier expansion -> there are different kinds of infinite.

This want to work with infinite set meant the analysation of infinite sets and overcoming the challenges and paradoxes

Paradoxes with infinite sets

• P(A) is always larger than A. What if A is the set of all sets?
• Russell’s paradox: B = {X : X ∉ X} ⇒ B ∈ B ≡ B ∉ B

The Kantian Matrix

Division of claim:

• Analytic: The claim is true by definition/logic (e.g. if Sokrates is a man and all men are mortal then Sokrates is mortal)
• Synthetic: The definition/logic are not enough to justify the claim (e.g. this bag is red)

as well as

• A-priori: The claim can be verified without external experience
• A-posteriori: th claim requires external experience for verification

Kant says a claim can be both Synthetic and A-priori: the sum of a triangles angles is 180 degrees. This is because a mathematical proof does not follow by definition or logic (in his sense) but by the construction of a mathematical proof. (This is not anymore a popular approach but it was a popular approach during Hilbert’s time and inspired him)

Hilbert’s goals

• infinite set theory
• certainty/no paradoxes

Finite arithmetic can be done by associating each numerical element on one side with exactly one numerical elements on the other side

Non-finite statements:

• ∀a, b ∈ ℕ : a + b = b + a (not allowed for Hilbert, since it is an infinite set it doesn’t exist, Hilbert allows potential infinite claims but not actually infinite claims)

Solution: use idealized concepts: they don’t (necessarily )exist but are useful. Mathematics already use idealized concepts such as imaginary numbers. However, we can’t just introduce any idealized concept: it can’t lead to any contradictions and needs to be successful. (He doesn’t really explain what he means by successful, probs along the lines of useful)

How do you prove an idealized element doesn’t lead to contradictions?

Need to avoid 1 ≠ 1

Three layers:

• finite layer:
  
  • refers to reality
  • The basis of mathematics for Hilbert is the givenness of human reason and correlating symbols. managed by immediate intuition concerning symbols
• ideal/formal layer:
  
  • managed by axioms and rules that aply to the ideal elements
  • does not refer to anything
  • a + b = b + a is still a meaningful statement when replacing a and b with a number.
• metamathematical layer:
  
  • think about this: a + b = b + a as: symbols from direct experience (a.k.a a number)
  • refers to proffs in layer 2
  • managed by immediate intuition concerning symbols

Goal: Prove in level 3 that layer 2 produces no contradiction

Gödel shows (using only the safe/finite layer which he needs to assume to be good and given) that infinite mathematics is inconsistent (Gensen shows consistency given some infinity, not super popular tho)