PY4813 Four topics on Formalism
FORMALISM 1. Hilbert’s Deductivism and the
Grundlagen der Geometrie
2. Frege-Hilbert controversy
Info for the (rest of the) course:
http://weka.ucdavis.edu/~ahwiki/bin/vi 3. Hilbert’s finitary mathematics
4. Gödel’s Incompleteness proof and
or easier go to: the end of Hilbert’s programme?
and then to “Teaching”.
I. Hilbert’s Deductivism and the
Three guiding issues: “Grundlagen der Geometrie” 1899
1) Metaphysical Issue
• Similar to the Logicist treatment of
arithmetic, Hilbert wants to avoid
What is the subject matter of any appeal to intuition.
• Axiomatic method:
2) Epistemological Issue ! |- "
(set of axiom) rules of Theorems
How do we know mathematics, i.e. inference
how do we know the subject matter?
and so Hilbert needs to explain how we
can establish the axioms and rules of
3) Application Issue inference in order to arrive at certain
theorems without requiring intuition.
Why is mathematics applicable in the
! What is the content and status of the
natural world? axioms and the inference rules, if
“nothing can be presupposed”?
Analogy: Consider the game “chess” (1) Metaphysical question?
Axioms: Positions of the various pieces • There is no specific subject matter
on the board, feature of the different
pieces, etc. • The pieces of the game of chess can
be any types of figures. If you want
Rules: How each figure can be moved, you can play with “tables, chairs
etc. and beer mugs, etc.” as the queen,
king, pawns, etc.
• We can just stipulate (fix) the
(1) Metaphysical issue: What is the axioms and rules and the game is up
subject matter? and running.
(2) Epistemological question: How (2) Epistemological Question?
do we know the rules.
• Understand the axioms and rules in
question, i.e. understand the role the
various pieces (however they are
instantiated) play in the game.
! Grasp the concepts involved.
Applied to the case of Geometry:
The constraint of consistency
• A set of axioms, containing “pieces”
such as the term “points”, “lines” • Hilbert’s only constraint/demand is
and “plane”. that the definitions are consistent.
• We specify the features of these • In the “Grundlagen der Geometrie”
pieces in relation to each other. he showed that his axioms are
(-> implicit, structural or relational consistent relative to Analysis.
• And that each axiom is independent
• We don’t need to appeal to intuition of each other.
(e.g. define “point” as extensionless)
– just like teaching chess, intuition is ! Issue: Is that sufficient?
a mere heuristic but not necessary.
! Issue of Existence:
! Provide a similar reply to the first Hilbert writes: “As long as I have been
two questions. What about the thinking […] about these things I have
third? been saying the exact reverse: if the
arbitrarily given axioms do not
Issue: Do we need constraints on the contradict each other, then they are true
axioms? What type? Analogy: When is a and the things defined by them exist.”
game not playable? (-> Analogy Games)
II. Frege-Hilbert controversy
III. Hilbert’s finitary mathematics
Three complaints by Frege:
The main idea is that mathematics is
1. From the truth of the axioms it follows that they divisble in the following areas:
do not contradict each other. There is no need to
show that they don't contradict each other.
• Finitary Mathematics: finitary part
2. Definitions and axioms have to be kept distinct. of mathematics (part of mathematics
An axiom can never be part of a definition. which is effectively decidable.)
3.Granted we accept Hilbert’s use of definitions ! Content of finitary mathematics
and axioms, how do we know that his system of seems to involve intuition: also it has
really is consistent?
a subject matter!!
• Ideal (Infinitary) Mathematics: Just
like a game!
Just reject (1), (2) and concerning (3)
• Metamathematics: Area which is
Hilbert suggested what is known as
concerned with the relation between
finitary and ideal mathematics.
IV. Gödel’s result and the end of
The only constraint on ideal mathematics
is that it can never contradict a statement • Gödel showed that even number
of finitary mathematics. theory is not a conservative
extension over finitary maths
! ideal mathematics is a conservative
extension of finitary mathematics. • He also showed that no consistent
theory (of sufficient strength) can
! If T is a conservative extension of S, prove its own consistency! Hence,
then T is consistent. neither are there such proofs
available for the finite part of
The aim of metamathematics is to secure mathematics nor for the ideal part.
that the CONSTRAINT obtains.
Hence, Hilbert’s demand to be able to
proof the conservativeness of ideal
mathematics over finitary mathematics is
! Gödel proved that there is no such