|
The
year 2006 marks the centenary of Kurt Gödel's birth,
which is the reason
why the Philosophy Department at the University of Edinburgh
is organising a two-day conference that aims
to reflect on the foundations of mathematics, particularly
in the light of Gödel's work.
PROGRAMME WITH
ABSTRACTS
SATURDAY 25 MARCH
12.00
- 1.00pm Registration
1.00
- 2.30pm John
Dawson (Pennsylvania State University)
"Taking
Truth Seriously"
ABSTRACT:
The distinction between syntactic and semantic methods is fundamental
to modern logical studies, but
it was not always so.
Indeed, until surprisingly recently, the idea of formulating an objective
notion of truth was regarded with widespread suspicion. This talk
will trace the evolution of the notion of truth in propositional
and predicate logic, and the eventual acceptance of the legitimacy
of model-theoretic methods, through examination of the works of Post,
Bernays, Gödel, Tarski, Mal'cev, Robinson and others.
3.00
- 4.30pm Hannes
Leitgeb (University of Bristol)
"Type-Free
Necessity, Truth, and Informal Provability"
ABSTRACT:
We suggest a possible worlds semantics of type-free necessity
predicates
for sentences, where potentially paradoxical
instances of modal axiom schemes or rules are excluded by a principle
of groundedness. While the same approach yields a plausible semantics
of type-free truth predicates as a special case, it is not so clear
whether the corresponding semantics for type-free informal provability
predicates is satisfying or whether the latter needs a different
semantic treatment.
5.00
- 6.30pm Richard
Zach (University of Calgary)
"Gödel’s First Incompleteness Theorem and Mathematical Instrumentalism"
ABSTRACT:
According to instrumentalist views of mathematics, only certain
mathematical propositions
are straightforwardly meaningful, whereas much of mathematics is only a useful
tool which is not, strictly speaking, meaningful. David Hilbert’s foundational
program of the 1920s is often read as involving such a brand of instrumentalism:
only the finitary part of mathematics is meaningful, the rest—“ideal” mathematics—is
not, and its use was supposed to be legitimated by a finitary consistency proof.
Gödel’s second incompleteness theorem is generally considered to
show that this particular version of instrumentalism cannot be carried out.
Consensus on this question is not complete, however, and variants of Hilbert’s
Program have been proposed (in particular, by Detlefsen) which avoid this conclusion.
I will discuss an argument based on Gödel’s first incompleteness
theorem, which, I argue, is a stronger argument against Hilbert’s Program
than the usual argument invoking the second theorem, and which also raises
difficult issues about Detlefsen’s instrumentalism.
SUNDAY
26 MARCH
(British
Summer Time -- Clocks go forward one hour)
11.00am
- 12.30pm Philip
Welch (University of Bristol)
"Games
for supervaluation and dependency fixed points"
ABSTRACT: We
consider an epistemic variation on the construction of fixed points à la Kripke,
for finding possible partial extensions for a truth predicate. These
involve winning strategies for two person
perfect information open games. Now this has been done before by D.A.
Martin for Kripkean fixed points using Strong Kleene truth tables, but
our aim is to show how this can be effected for fixed points arising
from supervaluations, and thus to characterise Hannes Leitgeb’s
notion of dependency.
Lunch
2.00
- 3.30pm Stewart
Shapiro (Ohio State University / University
of St Andrews)
"We
hold these truths to be self-evident: But what do we mean by that?
The
rationalism of Frege and Zermelo."
ABSTRACT: At
the beginning of Die Grundlagen der Arithmetik (§2)
[1844], Frege observes that “it is in the nature of mathematics
to prefer proof, where proof is possible”. This, of course, is
true, but thinkers differ on why it is that mathematicians prefer proof.
And what of propositions for which no proof is possible? What of axioms?
This talk explores various notions of self-evidence, and the role they
play in various foundational systems.
4.00
- 5.30pm Panu
Raatikainen (University of Helsinki)
"Indefinite extensibility of mathematics and the powers of human mind"
ABSTRACT:
Several
alleged philosophical consequences of Gödel’s
incompleteness theorems are critically evaluated. These include anti-mechanist
conclusions of Lucas and Penrose, Benacerraf’s argument about
the limits of self-knowledge, Gödel’s own “disjunctive
conclusion” on
anti-mechanism and Platonism, and Dummett’s and Wright’s
views on indefinite extensibility, provability and proof. These claims
are discussed in light of relevant logical facts. Feferman’s
important work on progressions and reflective closures of theories
is also reviewed.
Please
complete the registration form (click on the link):
Registration
Form + Postgraduate Bursary Application (MS Word file)
Suggestions
for accommodation (HTML file)
Registration
fee: £15
(£5 student / unwaged)
(Postgraduate
Bursaries up to £50 are available to offset travel costs.)
For
further information, contact Jean-Louis Hudry: jl.hudry@ed.ac.uk
The
conference has been generously supported by:
Mind Association
London
Mathematical Society (LMS)
British Society
for the Philosophy of Science (BSPS)
British Logic
Colloquium (BLC)
Scots Philosophical Club (SPC)
Analysis
Trust
|