67 research outputs found

    Classical and Intuitionistic Arithmetic with Higher Order Comprehension Coincide on Inductive Well-Foundedness

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    Assume that we may prove in Classical Functional Analysis that a primitive recursive relation R is well-founded, using the inductive definition of well-founded. In this paper we prove that such a proof of well-foundation may be made intuitionistic. We conclude that if we are able to formulate any mathematical problem as the inductive well-foundation of some primitive recursive relation, then intuitionistic and classical provability coincide, and for such a statement of well-foundation we may always find an intuitionistic proof if we may find a proof at all. The core of intuitionism are the methods for computing out data with given properties from input data with given properties: these are the results we are looking for when we do constructive mathematics. Proving that a primitive recursive relation R is inductively well-founded is a more abstract kind of result, but it is crucial as well, because once we proved that R is inductively well-founded, then we may write programs by induction over R. This is the way inductive relation are currently used in intuitionism and in proof assistants based on intuitionism, like Coq. In the paper we introduce the comprehension axiom for Functional Analysis in the form of introduction and elimination rules for predicates of types Prop, Nat->Prop, ..., in order to use Girard\u27s method of candidates for impredicative arithmetic

    Hilbert's Program Then and Now

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    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

    Anti-exceptionalism and methodological pluralism in logic

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    According to methodological anti-exceptionalism, logic follows a scientific methodology. There has been some discussion about which methodology logic has. Authors such as Priest, Hjortland and Williamson have argued that logic can be characterized by an abductive methodology. We choose the logical theory that behaves better under a set of epistemic criteria (such as fit to data, simplicity, fruitfulness, or consistency). In this paper, I analyze some important discussions in the philosophy of logic (intuitionism versus classical logic, semantic paradoxes, and the meaning of conditionals), and I show that they presuppose different methodologies, involving different notions of evidence and different epistemic values. I argue that, rather than having a specific methodology such as abductivism, logic can be characterized by methodological pluralism. This position can also be seen as the application of scientific pluralism to the realm of logic

    Reverse mathematical bounds for the Termination Theorem

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    In 2004 Podelski and Rybalchenko expressed the termination of transition-based programs as a property of well-founded relations. The classical proof by Podelski and Rybalchenko requires Ramsey's Theorem for pairs which is a purely classical result, therefore extracting bounds from the original proof is non-trivial task. Our goal is to investigate the termination analysis from the point of view of Reverse Mathematics. By studying the strength of Podelski and Rybalchenko's Termination Theorem we can extract some information about termination bounds

    Sraffa's Mathematical Economics - A Constructive Interpretation

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    The claim in this paper is that Sraffa employed a rigorous logic of mathematical reasoning in his book, Production of Commodities by Means of Commodities (PCC), in such a way that the existence proofs were constructive. This is the kind of mathematics that was prevalent at the beginning of the 19th century, which was dominated by the concrete, the constructive and the algorithmic. It is, therefore, completely consistent with the economics of the 19th century, which was the fulcrum around which the economics of PCC was conceived.Existence Proofs, Constructive Mathematics, Algorithmic Mathematics, Mathematical Economics, Standard System.
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