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Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also ..."
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
Unfolding finitist arithmetic
, 2010
"... The concept of the (full) unfolding U(S) of a schematic system S is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? The program to determine U(S) for various systems S of foundational significan ..."
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The concept of the (full) unfolding U(S) of a schematic system S is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? The program to determine U(S) for various systems S of foundational significance was previously carried out for a system of nonfinitist arithmetic, NFA; it was shown that U(NFA) is prooftheoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system of finitist arithmetic, FA, and for an extension of that by a form BR of the socalled Bar Rule. It is shown that U(FA) and U(FA + BR) are prooftheoretically equivalent, respectively, to Primitive Recursive Arithmetic, PRA, and to Peano Arithmetic, PA.
Complexity Science for Simpletons
 Progress in Physics, Progress in Physics, 2006
"... In this article, we shall describe some of the most interesting topics in the subject of Complexity Science for a general audience. Anyone with a solid foundation in high school mathematics (with some calculus) and an elementary understanding of computer programming will be able to follow this artic ..."
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In this article, we shall describe some of the most interesting topics in the subject of Complexity Science for a general audience. Anyone with a solid foundation in high school mathematics (with some calculus) and an elementary understanding of computer programming will be able to follow this article. First, we shall explain the significance of the P versus NP problem and solve it. Next, we shall describe two other famous mathematics problems, the Collatz 3n+1 Conjecture and the Riemann Hypothesis, and show how both Chaitin’s incompleteness theorem and Wolfram’s notion of “computational irreducibility” are important for understanding why no one has, as of yet, solved these two problems. Disclaimer: This article was authored by Craig Alan Feinstein in his private capacity. No official support or endorsement by the U.S. Government is intended or should be inferred. Imagine that you have a collection of one billion lottery tickets scattered throughout your basement in no
Constructive representation of . . .
, 2015
"... The theory of nominal sets provide a mathematical analysis of names that is based upon symmetry. It formalizes the informal reasoning we employ while working with languages involving name binding operators. The central ideas of the theory are support, freshness and name abstraction, which respecti ..."
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The theory of nominal sets provide a mathematical analysis of names that is based upon symmetry. It formalizes the informal reasoning we employ while working with languages involving name binding operators. The central ideas of the theory are support, freshness and name abstraction, which respectively encapsulate the ideas of name dependence, name independence and alpha equivalence. This theory has been devel
Retrieving the Mathematical Mission of the Continuum Concept from the Transfinitely Reductionist Debris of Cantor’s Paradise (Extended Abstract)
, 2011
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"... ii The aim of this dissertation is to outline and defend the view here dubbed “antifoundational categorical structuralism ” (henceforth AFCS). The program put forth is intended to provide an answer the question “what is mathematics?”. The answer here on offer adopts the structuralist view of math ..."
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ii The aim of this dissertation is to outline and defend the view here dubbed “antifoundational categorical structuralism ” (henceforth AFCS). The program put forth is intended to provide an answer the question “what is mathematics?”. The answer here on offer adopts the structuralist view of mathematics, in that mathematics is taken to be “the science of structure ” expressed in the language of category theory, a language argued to accurately capture the notion of a structural property. In characterizing mathematical theorems as both conditional and schematic in form, the program is forced to give up claims to securing the truth of its theorems, as well as give up a semantics which involves reference to special, distinguished “mathematical objects”, or which involves quantification over a fixed domain of such objects. One who wishes—contrary to the AFCS view—to inject mathematics with a “standard ” semantics, and to provide a secure epistemic foundation for the theorems of mathematics, in short, one who wishes for a foundation for mathematics, will surely find this view lacking. However, I argue that a satisfactory development of the structuralist view, couched in the language of category theory, accurately represents our best understanding of the content of mathematical theorems and thereby obviates the need for any foundational program.
Reconstructing Hilbert to Construct Category Theoretic Structuralism
"... This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, its relation to the “algebraic”1 approach to mathematical structuralism. I first consider the FregeHilbert debate with the aim of distinguishing b ..."
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This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, its relation to the “algebraic”1 approach to mathematical structuralism. I first consider the FregeHilbert debate with the aim of distinguishing between axioms as assertions, i.e., as statements that are used to express or assert truths about a unique subject matter, and an axiom system as a schema that is used to provide “a system of conditions for what might be called a relational structure ” (Bernays [1967], p. 497) so that axioms, as implicit definitions, are about whatever satisfies the conditions set forth. I then use this inquiry to reevaluate arguments against using category theory to frame an algebraic structuralist philosophy of mathematics. Hellman has argued that category theory cannot stand on its own as a “foundation ” for a structuralist interpretation of mathematics because “the problem of the home address remains ” (Hellman [2003], pgs. 8 & 15). That is, since the axioms for a category “merely tell us what it is to be a structure of a certain kind ” and because “its axioms are not assertory ” (Ibid. 7), we need a background mathematical theory whose axioms are
In the Beginning Was the Verb: The Enigma of the Emergence and Evolution of Language and the Big Bang Epistemological Paradigm
, 2008
"... The enigma of the Emergence of Natural Languages, coupled or not with the closely related problem of their Evolution, E&ENL Problem for short, is perceived today as one of the most important scientific problems and even, according to the provocative title of [13], as the “hardest ” one: “Despite ..."
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The enigma of the Emergence of Natural Languages, coupled or not with the closely related problem of their Evolution, E&ENL Problem for short, is perceived today as one of the most important scientific problems and even, according to the provocative title of [13], as the “hardest ” one: “Despite a staggering growth in our scientific knowledge about the origin of life, the universe and (almost) everything else that we have seen fit to ponder, we know comparatively little about how our unique ability for language originated and evolved into the complex linguistic systems we use today.” All living beings are known to somehow communicate with their fellow creatures. It means that the language has been evolving over a very long stretch of time and, before becoming the language we learn, use, and enhance today, it has passed through a number of stages, or plateaux of relative stability, with each particular radical transition driven by proper
In the Beginning Was the Verb: The Emergence and Evolution of Language Problem in the Light of the Big Bang Epistemological Paradigm
, 2008
"... The enigma of the Emergence of Natural Languages, coupled or not with the closely related problem of their Evolution, E&ENL Problem for short, is perceived today as one of the most important scientific problems and even, according to the provocative title of [11], as the “hardest ” one: “Despite ..."
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The enigma of the Emergence of Natural Languages, coupled or not with the closely related problem of their Evolution, E&ENL Problem for short, is perceived today as one of the most important scientific problems and even, according to the provocative title of [11], as the “hardest ” one: “Despite a staggering growth in our scientific knowledge about the origin of life, the universe and (almost) everything else that we have seen fit to ponder, we know comparatively little about how our unique ability for language originated and evolved into the complex linguistic systems we use today.” Incredibly as it might appear in this pure linguistic, resolutely nonphysical and even less cosmological context, the case of the universe of the above quotation has not be taken by its authors in vain, whether knowledgeably or not. The purpose of the present study is actually to outline such a solution to our problem which is epistemologically con