54,123 research outputs found
Arithmetic, Set Theory, Reduction and Explanation
Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences
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
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
Classical Predicative Logic-Enriched Type Theories
A logic-enriched type theory (LTT) is a type theory extended with a primitive
mechanism for forming and proving propositions. We construct two LTTs, named
LTTO and LTTO*, which we claim correspond closely to the classical predicative
systems of second order arithmetic ACAO and ACA. We justify this claim by
translating each second-order system into the corresponding LTT, and proving
that these translations are conservative. This is part of an ongoing research
project to investigate how LTTs may be used to formalise different approaches
to the foundations of mathematics.
The two LTTs we construct are subsystems of the logic-enriched type theory
LTTW, which is intended to formalise the classical predicative foundation
presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has
also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA
as LTTs, we are able to compare them with LTTW. It is a consequence of the work
in this paper that LTTW is strictly stronger than ACAO.
The conservativity proof makes use of a novel technique for proving one LTT
conservative over another, involving defining an interpretation of the stronger
system out of the expressions of the weaker. This technique should be
applicable in a wide variety of different cases outside the present work.Comment: 49 pages. Accepted for publication in special edition of Annals of
Pure and Applied Logic on Computation in Classical Logic. v2: Minor mistakes
correcte
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
W-types in setoids
W-types and their categorical analogue, initial algebras for polynomial
endofunctors, are an important tool in predicative systems to replace
transfinite recursion on well-orderings. Current arguments to obtain W-types in
quotient completions rely on assumptions, like Uniqueness of Identity Proofs,
or on constructions that involve recursion into a universe, that limit their
applicability to a specific setting. We present an argument, verified in Coq,
that instead uses dependent W-types in the underlying type theory to construct
W-types in the setoid model. The immediate advantage is to have a proof more
type-theoretic in flavour, which directly uses recursion on the underlying
W-type to prove initiality. Furthermore, taking place in intensional type
theory and not requiring any recursion into a universe, it may be generalised
to various categorical quotient completions, with the aim of finding a uniform
construction of extensional W-types.Comment: 17 pages, formalised in Coq; v2: added reference to formalisatio
Displayed Categories
We introduce and develop the notion of *displayed categories*.
A displayed category over a category C is equivalent to "a category D and
functor F : D --> C", but instead of having a single collection of "objects of
D" with a map to the objects of C, the objects are given as a family indexed by
objects of C, and similarly for the morphisms. This encapsulates a common way
of building categories in practice, by starting with an existing category and
adding extra data/properties to the objects and morphisms.
The interest of this seemingly trivial reformulation is that various
properties of functors are more naturally defined as properties of the
corresponding displayed categories. Grothendieck fibrations, for example, when
defined as certain functors, use equality on objects in their definition. When
defined instead as certain displayed categories, no reference to equality on
objects is required. Moreover, almost all examples of fibrations in nature are,
in fact, categories whose standard construction can be seen as going via
displayed categories.
We therefore propose displayed categories as a basis for the development of
fibrations in the type-theoretic setting, and similarly for various other
notions whose classical definitions involve equality on objects.
Besides giving a conceptual clarification of such issues, displayed
categories also provide a powerful tool in computer formalisation, unifying and
abstracting common constructions and proof techniques of category theory, and
enabling modular reasoning about categories of multi-component structures. As
such, most of the material of this article has been formalised in Coq over the
UniMath library, with the aim of providing a practical library for use in
further developments.Comment: v3: Revised and slightly expanded for publication in LMCS. Theorem
numbering change
Domains via approximation operators
In this paper, we tailor-make new approximation operators inspired by rough
set theory and specially suited for domain theory. Our approximation operators
offer a fresh perspective to existing concepts and results in domain theory,
but also reveal ways to establishing novel domain-theoretic results. For
instance, (1) the well-known interpolation property of the way-below relation
on a continuous poset is equivalent to the idempotence of a certain
set-operator; (2) the continuity of a poset can be characterized by the
coincidence of the Scott closure operator and the upper approximation operator
induced by the way below relation; (3) meet-continuity can be established from
a certain property of the topological closure operator. Additionally, we show
how, to each approximating relation, an associated order-compatible topology
can be defined in such a way that for the case of a continuous poset the
topology associated to the way-below relation is exactly the Scott topology. A
preliminary investigation is carried out on this new topology.Comment: 17 pages; 1figure, Domains XII Worksho
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