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FORMALISM

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					               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
          ew/Arche/PY4813
                                         4. Gödel’s Incompleteness proof and
           or easier go to:                 the end of Hilbert’s programme?

 http://www.st-andrews.ac.uk/~pae1/

       and then to “Teaching”.
                                        I.    Hilbert’s Deductivism and the
        Three guiding issues:                 “Grundlagen der Geometrie” 1899

                                       Features:
 1) Metaphysical Issue
                                        •    Similar to the Logicist treatment of
                                             arithmetic, Hilbert wants to avoid
What is the subject matter of                any appeal to intuition.
mathematics?
                                        •    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.
Ask yourself:
                                          •   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.
     definitions)
                                                •   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
Hilbert’s reply:
                                                            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.
            Hilbert’s Programme
            Metamathematics
                                               IV. Gödel’s result and the end of
                                                   Hilbert’s programme?
CONSTRAINT:
                                              Gödel’s Result:
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
                                              unavailable.
                                               ! Gödel proved that there is no such
                                                   proof

				
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posted:5/11/2011
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