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| 7
December (Wednesday)
5:00
PM
Room
226 |
Gábor
Etesi
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Department
of Geometry,
Mathematical
Institute
Budapest
University of
Technology and
Economics
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Feynman integrals
and the current status
of the Continuum
Hypothesis
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It is
well-known that in infinite
dimensions there is no
analogue of the Lebesgue
measure. Mainly this is the
reason why Feynman
integration, a central
concept in quantum field
theory, lacks any rigorous
mathematical basis. On the
other hand recent advances
of set theory indicate that
the usual ZFC axioms of
mathematics should be
extended by further natural
ones such that within this
improved axiom system the
cardinality of the continuum
could be fixed. In the talk
we will speculate if these
new axioms could help to
define a meaningful measure
theory in infinite
dimensions. These
speculations will end
without any definitive
conclusion.
Finally, if time allows we
will also review two further
candidates for a meaningful
integration theory in
infinite dimensions: (i) A.
Connes' non-commutative
integration, (ii) an
integration based on
zeta-function
regularization. |
| 14
December (Wednesday)
5:00
PM
Room
226 |
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Institute
for Philosophical
Research, HAS, Budapest
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Dynamic Models of
the Growth of
Reflexive
Knowledge
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Motto:
“If
you have an idea and I
have an idea and we
exchange these ideas,
then will each of us
have two ideas…?”
(After G.B.
Show.)
1.
Introduction
The
‘growth
of knowledge’ was a
common assumption
lurking behind the
debates of the 60ies and
is still considered as
evidence for and
against epistemic views
in contemporary
philosophy of science,
let
alone in scientists’
opinions themselves. In
the intersection of
the various approaches
one finds the thesis
that knowledge
does grow as a result
of
collaboration and
information exchange.
The
willingness of
researchers, learners
and institutions to
exchange
information is usually
remarkable, but in
certain cases the
benefits
of such exchanges or the
evidence for the growth
of knowledge as
their result may not be
obvious. Simple
counterexamples to the thesis
and empirical studies of
research networks
underline the need for
more exact definitions
of ‘growth’ and logic
models of the
dynamics of information
exchange.
2.
Conceptions
of ‘growth’ and
reflexive
knowledge
Several
different
uses of the term of
‘growth’ can be
identified with
respect to knowledge,
including individual
level as well as collective
level
notions. The main
alternatives are
critically reviewed to
assess
their strengths and
weaknesses with respect
to the description of
the
dynamics of information
exchange. Leaving behind
various conceptions
of individual and
“objective” knowledge I
point to the importance
not only of shared views
(mutual knowledge), but
reflection on, and
awareness of what the
others know. Various
conceptions of social
and
group knowledge can be
defined and the
interpretation of growth
in
these contexts is a
challenging precondition
of exact models of the
growth of knowledge.
3. Dynamic logic
models of information
exchange
The
models
discussed introduce
conditions for common
knowledge in the
spirit of Logics
of Communication and
Change
(van Benthem et al.
2006). This framework is
capable of a
compositional analysis
of complex communication
scenarios such as
announcements to
subgroups, or private
and secret massaging.
Making
such models an integral
part of the assessment
of the growth of
knowledge in science or
research networks may
provide new assessment
paradigms for empirical
research. Based on
dynamic-models one can
analyze multi-agent
scenarios, such as
revealing individuals’
answers to a group in
order to promote
discussion, peer
assessment,
forming (sub)groups of
differing levels of
ability, or giving
particular roles to
different performers in
a group.
4.
Models
of distributed
knowledge in groups
and networks
The
process
of seeking and
interpreting assessment
of evidence where
researchers are to tell
how they need to go
further, naturally
introduced exchange of
information between
peers about
their
knowledge. In these
situations
distribution of
knowledge may depend on
rationality conditions
that
guarantee the existence
of a “wise man” who
represents the
cumulative knowledge of
the group. I present
some recent logic
results on the
preconditions of
no-wise-man
distributions. Crucial
issues depend on the
underlying protocols
that govern information
exchange in social
groups and networks. As
a result, the current
demand for social
software raise the
problem of the
comparison of the
efficiency of networking
scenarios based on
sharing and representing
information about
knowledge states.
5.
In conclusion
Analysis
of
conceptions of growth in
a composite framework of
dynamic
epistemic logic and
knowledge networks can
provide new models and
insights for the
development of knowledge
based communication
protocols and social
software. Making the
protocols dependent on
the
assessment of the
knowledge states of the
members can be shown to
have tremendous effects
to the performance of
the group that is
sharing information
according to different
models of information
exchange scenarios.
A
networking scenario
may determine not only
the efficiency of
information exchange but
the
very nature of group
knowledge that it
produces.
References
van
Benthem J., van Eijck, J,
and Kooi, B. (2006):
“Logics of
Communication and Change”
Information and
Computation, 204(11):
1620-1662.
Bird,
A.
(2007): ‘What is
Scientific Progress?’, Noûs
41:
64-89.
Bird,
A. (2008): ‘Scientific
Progress as Accumulation
of Knowledge—A
Reply to Rowbottom’, Studies
in History and
Philosophy of
Science 39,
279–281.
Fahrbach,
L.
(2011): How the Growth
of Science Ended Theory
Change. Synthese,
180(2):139-155.
J.Y.
Halpern
and Y.O. Moses. (1990):
Knowledge and common
knowledge in a
distributed environment.
Journal of the ACM, 37(3):549-587.
W.
van
der Hoek and J.-J. Ch
Meyer. (1992): Making
some issues of
implicit knowledge
explicit. International
Journal
on Foundations of
Computer Science,
3(2):193-224,
Pacuit.
E. and Simon, S. (2011):
Reasoning with Protocols
under Imperfect
Information. The
Review of Symbolic Logic,
4: 412-444.
Rowbottom,
D.
P. (2011): ‘Kuhn vs.
Popper on Criticism and
Dogmatism in
Science: A Resolution at
the Group Level’, Studies
in
History and Philosophy
of
Science
42
(1):117-124.
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