2,154 research outputs found
Computational reverse mathematics and foundational analysis
Reverse mathematics studies which subsystems of second order arithmetic are
equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main
philosophical application of reverse mathematics proposed thus far is
foundational analysis, which explores the limits of different foundations for
mathematics in a formally precise manner. This paper gives a detailed account
of the motivations and methodology of foundational analysis, which have
heretofore been largely left implicit in the practice. It then shows how this
account can be fruitfully applied in the evaluation of major foundational
approaches by a careful examination of two case studies: a partial realization
of Hilbert's program due to Simpson [1988], and predicativism in the extended
form due to Feferman and Sch\"{u}tte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be
proved in the same way as inequivalences, namely by considering only
-models of the systems in question. Shore refers to this approach as
computational reverse mathematics. This paper shows that despite some
attractive features, computational reverse mathematics is inappropriate for
foundational analysis, for two major reasons. Firstly, the computable
entailment relation employed in computational reverse mathematics does not
preserve justification for the foundational programs above. Secondly,
computable entailment is a complete relation, and hence employing it
commits one to theoretical resources which outstrip those available within any
foundational approach that is proof-theoretically weaker than
.Comment: Submitted. 41 page
On the alleged simplicity of impure proof
Roughly, a proof of a theorem, is âpureâ if it draws only on what is âcloseâ or âintrinsicâ to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is âextrinsic,â âextraneous,â âdistant,â âremote,â âalien,â or âforeignâ 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
On the Invariance of G\"odel's Second Theorem with regard to Numberings
The prevalent interpretation of G\"odel's Second Theorem states that a
sufficiently adequate and consistent theory does not prove its consistency. It
is however not entirely clear how to justify this informal reading, as the
formulation of the underlying mathematical theorem depends on several arbitrary
formalisation choices. In this paper I examine the theorem's dependency
regarding G\"odel numberings. I introduce deviant numberings, yielding
provability predicates satisfying L\"ob's conditions, which result in provable
consistency sentences. According to the main result of this paper however,
these "counterexamples" do not refute the theorem's prevalent interpretation,
since once a natural class of admissible numberings is singled out, invariance
is maintained.Comment: Forthcoming in The Review of Symbolic Logi
Introduction to clarithmetic I
"Clarithmetic" is a generic name for formal number theories similar to Peano
arithmetic, but based on computability logic (see
http://www.cis.upenn.edu/~giorgi/cl.html) instead of the more traditional
classical or intuitionistic logics. Formulas of clarithmetical theories
represent interactive computational problems, and their "truth" is understood
as existence of an algorithmic solution. Imposing various complexity
constraints on such solutions yields various versions of clarithmetic. The
present paper introduces a system of clarithmetic for polynomial time
computability, which is shown to be sound and complete. Sound in the sense that
every theorem T of the system represents an interactive number-theoretic
computational problem with a polynomial time solution and, furthermore, such a
solution can be efficiently extracted from a proof of T. And complete in the
sense that every interactive number-theoretic problem with a polynomial time
solution is represented by some theorem T of the system. The paper is written
in a semitutorial style and targets readers with no prior familiarity with
computability logic
Justifications-on-demand as a device to promote shifts of attention associated with relational thinking in elementary arithmetic
Student responses to arithmetical questions that can be solved by using arithmetical structure can serve to reveal the extent and nature of relational, as opposed to computational thinking. Here, student responses to probes which require them to justify-on-demand are analysed using a conceptual framework which highlights distinctions between different forms of attention. We analyse a number of actions observed in students in terms of forms of attention and shifts between them: in the short-term (in the moment), medium-term (over several tasks), and long-term (over a year). The main factors conditioning studentsÂŽ attention and its movement are identified and some didactical consequences are proposed
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