Nov.11, 12, 2006, Graduate school of letters, Kyoto university.

Final Program

Nov 11th (Sat.)

Workshop on Hilbert part 1

13:00-14:00

W. Sieg (CMU), Existential Axiomatics

Full abstract (pdf)

Abridged version：

I sketch first the evolution to Hilbert's finitist consistency program,

point to the deep connections to Dedekind’s logicist views, and describe

the evolution of "existential axiomatics". Second, I describe the problematic

of articulating the epistemologically distinctive aspects of finitist mathematics.

An analysis of the informal ideas underlying this approach motivates, however,

the formulation of reductive structuralism. The goal in this third part is to

give an integrating perspective for philosophical work and to formulate focused

problems for mathematical investigation.

14:10-15:10

S. Hayashi (Kyoto Univ.), Hilbert's notebooks I

Abstract: Our investigations of Hilbert's mathematical notebooks are going to shed a new light on

Hilbert's thought on the foundations of mathematics. Some important new findings will be reported

in the couple of the talks.

(i) The original motivation of Hilbert's studies on the foundations of mathematics are his "Jungendraum"

conceived around 1889. The dream was to prove that every mathematical probelm can be solved in finite steps.

The motivation of the dream is very likely the non constructive proof of Hilbert's finite basis theorem.

It was discussed in the context of Kantian philosophy. Almost all of his studies of

mathematics are related to this dream. Even the study of the foundations of physics was related to this dream.

(ii)
his philosophical dream of solvability is deeply realted to the problem of computation in mathematics.

Hilbert had a strong "disgust" to computational way of mathematics. However, he planned a theory

of computation in the early 1890's. Hilbert's ideas on computations seemed led him to a bad direction in

his program of the foundations of mathematics in 1920's.

(iii) The idea of "consistency=existence"was seemingly conceived in the early 1890's. The seemingly oldest version

of the idea was not related to geometry by Hilbert but related to algebra/arithmetics.

Hilbert notebooks http://www.shayashi.jp/HNBP/index.html

15:20-16:20

R. Zach (Calgary univ.), Algorithms and decision problems in Hilbert's school

Abstract: Concurrent their work on proof theory in the 1920s, Hilbert and his

collaborators and students--in particular, Ackermann, Behmann, Bernays,

and Schönfinkel--did substatial work towards the decision problem for

first order logic. This begins with an unpublished talk by Behmann in

1921 in which the word "Entscheidungsproblem" first appears. In this

same talk, Behmann explicitly points to algebraic algorithms, e.g., for

solving equations, as the model according to which a solution to the

decision problem should be attempted. This kind of approach is

characteristic of approaches to elimination problems in algebraic logic,

and to quantifier elimination procedures. It is an important question

how this early work relates to other positive solutions to decision

problems based on exhaustive search over finite objects (e.g., by

establishing finite controllability of classes of formulas).

16:50-17:50

CC seminar part1

S. Hayashi (Kyoto univ.), Hilbert's basis theorem, Solvability, and LCM, Hilbert's finite basis theorem, and the law of excluded middle on Sigma^0_1-sentences are will be proved constructively equivalent.

Workshop Dinner： booking will be required.

Nov 12th (Sun.)

Workshop on Hilbert part 2

9:30-10:30

S. Hayashi, Hilbert's notebooks II

Abstract: the continuation of the talk "Hilbert's notebooks I".

10:30-11:00

K. Nakatogawa (Hokkaido univ,), Comments

Philosophical comments on the talks and research on Hilbert notebooks.

11:20-12:20

CC seminar part2

W. Sieg, Church without dogma: axioms for computability,

Full abstract (pdf)

abridged abstract：

I formulate finiteness and locality conditions for two types of calculators, human computing

agents and mechanical computing devices; the distinctive feature of the latter is that they can

operate in parallel.

The analysis leads to axioms for discrete dynamical systems (representing human and

machine computations) and allows the reduction of models of these axioms to Turing machines.

Cellular automata and a variety of artificial neural nets can be shown to satisfy the axioms for

machine computations.