1,377 research outputs found
Kriesel and Wittgenstein
Georg Kreisel (15 September 1923 - 1 March 2015) was a formidable mathematical
logician during a formative period when the subject was becoming
a sophisticated field at the crossing of mathematics and logic. Both with his
technical sophistication for his time and his dialectical engagement with mandates,
aspirations and goals, he inspired wide-ranging investigation in the metamathematics
of constructivity, proof theory and generalized recursion theory.
Kreisel's mathematics and interactions with colleagues and students have been
memorably described in Kreiseliana ([Odifreddi, 1996]). At a different level of
interpersonal conceptual interaction, Kreisel during his life time had extended
engagement with two celebrated logicians, the mathematical Kurt Gödel and
the philosophical Ludwig Wittgenstein. About Gödel, with modern mathematical
logic palpably emanating from his work, Kreisel has reflected and written
over a wide mathematical landscape. About Wittgenstein on the other hand,
with an early personal connection established Kreisel would return as if with
an anxiety of influence to their ways of thinking about logic and mathematics,
ever in a sort of dialectic interplay. In what follows we draw this out through
his published essays—and one letter—both to elicit aspects of influence in his
own terms and to set out a picture of Kreisel's evolving thinking about logic
and mathematics in comparative relief.Accepted manuscrip
Comparing hierarchies of total functionals
In this paper we consider two hierarchies of hereditarily total and
continuous functionals over the reals based on one extensional and one
intensional representation of real numbers, and we discuss under which
asumptions these hierarchies coincide. This coincidense problem is equivalent
to a statement about the topology of the Kleene-Kreisel continuous functionals.
As a tool of independent interest, we show that the Kleene-Kreisel functionals
may be embedded into both these hierarchies.Comment: 28 page
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
Predicativity and parametric polymorphism of Brouwerian implication
A common objection to the definition of intuitionistic implication in the
Proof Interpretation is that it is impredicative. I discuss the history of that
objection, argue that in Brouwer's writings predicativity of implication is
ensured through parametric polymorphism of functions on species, and compare
this construal with the alternative approaches to predicative implication of
Goodman, Dummett, Prawitz, and Martin-L\"of.Comment: Added further references (Pistone, Poincar\'e, Tabatabai, Van Atten
On the alleged simplicity of impure proof
Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim
Pi01 encodability and omniscient reductions
A set of integers is computably encodable if every infinite set of
integers has an infinite subset computing . By a result of Solovay, the
computably encodable sets are exactly the hyperarithmetic ones. In this paper,
we extend this notion of computable encodability to subsets of the Baire space
and we characterize the encodable compact sets as those who admit a
non-empty subset. Thanks to this equivalence, we prove that weak
weak K\"onig's lemma is not strongly computably reducible to Ramsey's theorem.
This answers a question of Hirschfeldt and Jockusch.Comment: 9 page
A Rice-like theorem for primitive recursive functions
We provide an explicit characterization of the properties of primitive
recursive functions that are decidable or semi-decidable, given a primitive
recursive index for the function. The result is much more general as it applies
to any c.e. class of total computable functions. This is an analog of Rice and
Rice-Shapiro theorem, for restricted classes of total computable functions
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