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Research |
AREAS OF RESEARCH
Philosophy of mathematics
One of the more important areas of my research is the philosophy of mathematics. The ontology and
epistemology of mathematics are areas in which not much agreement is found among researchers of them.
I find this to be one of the biggest scandals with in philosophy and philosophy of science. I attack the
problems
from three different
angles. Firstly, I find logic to be an important branch and tool of
mathematics - also as a meta-disciplin. Thus proof theory, computability
theory and modal logic play very important roles in the attempts to
characterise mathematics and our knowledge of it. Secondly, a philosophy of
mathematics must be based on a solid, robust and very general philosophy of
knowledge and science.
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In this connection I try to generalise several
important Kantian notions such as the notions of schema and regulative
idea - see for instance my thesis in this respect. In turn, my generalizations can, historically, be seen as an epistemological foundation
for the philosophy of mathematics which Hilbert and Bernays were advocating in
the 1920s and -30s (not to be identified with the so-called
program). Thirdly, I hold the opinion that a philosophy of
mathematics must square with contemporary mathematics as it is carried out by
actual mathematicians. This lead me - on the basis of a generalized
understanding of Kant's theory of schema - to define a very general notion of
constructibility in which different types of modern mathematics and methods
hereof is located.
Logic and theory of argumenation
One of the biggist mistakes in large areas of, especially, the so-called analytical philosophy
is the exclusive focus on deductive arguments leading us to true, justified beliefs. Way to
much energy has been put into the project of examining only those types of reasoning leading us to those kinds
of beliefs. And in this respect I very much agree with S. Toulmin when he writes in the preface to the second edition
of his influential book The Uses of Argument:
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"When I wrote it [The Uses of Argument], my aim was strictly
philosophical: to criticize the assumption, made by most Anglo-American
academic philosophers, that any significant argument can be put in formal
terms: not just as a syllogism, since for Aristotle himself any
inference can be called a 'syllogism' or 'linking of statements, but a
rigidly demonstrative deduction of the kind to be found in Euclidean
geometry. Thus was created the Platonic tradition that, some two millennia
later, was revived by René Descartes." (Toulmin, 2003, vii)
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I my recent work, however, I have discovered that deductive reasoning and reasoning in the style of Toulmin's
model is not so far from each other. In fact I have discovered a way to present to two types or resoning in which
Toulmin's model truly generalises first order logical reasoning.
Kant's theory of knowledge and philosophy of science
One of my big interests is the notion of schematism and its connection with and cognitive foundation
for human concepts.
The theory of schematism was initiated by I. Kant, who, however,
was never precise with respect to what he understood under this theory. In my thesis I give - based on
the theoretical works of Kant - an interpretation of the most important
aspects of Kant's theory of schematism. In doing this I show how schematism
can form a point of departure for a reinterpretation of Kant's theory of
knowledge. This can be done by letting the concept of schema be the central
concept. I show how strange passages in, say, the first Critique are in
fact understandable, when one takes schematism serious. Kant understood himself as a
philosopher in contact with science. It was science which he wanted to
provide a foundation for. I propose the view that one should take schematism to
be a very central feature of Kant's theory of knowledge and therefore it is also a
corner stone when examining his philosophy of science. I will do such an examination the
forthcoming years.
Mathematical logic
One of my fields of study is mathematical logic. I have written and published within proof theory, more
specfically, I have examined and related functional interpretations with different kinds of realisability interpretations as put
initiated by Kurt Gödel and Stephen S. Kleene, respectively. These
tools can be seen as reducing some ideal mathematics to some unproblematic
parts of mathematics, both with respect to consistency and computational
content. They also provide a frame work for the evalutation of mathematical
principles which at first sight look non-constructive but on a closer view in
a specific framework nevertheless are constructive, as for instance Markov's
principle, extensionality and restricted forms of independence-of-premise.
These
investigations lead to locally more precise descriptions of the logical operators
and this form a kind of critique of the Brouwer-Heyting-Kolmogorov
interpretation - if this is taken to be a global interpretation.
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A spin off from these investigations has been the search for a natural extension of the
recursive functions. Is there a unique, natural and philosophically satisfying generalization of the recursive
functions? In other words, is there a Church-Turing thesis for the computable functionals of higher types. This is
a question that take up important amounts of my time.
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© 2006 Klaus Frovin Jørgensen
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