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Mathematical method and proof
"... Abstract. On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that ..."
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Abstract. On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not wellequipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.
The Cognitive Foundations of Mathematics: The Role of Conceptual Metaphor Handbook of Mathematical Cognition New York: Psychology Press
"... analyze the biological foundations of human cognition. A crucial component of their arguments is a simple but profound aphorism: Everything said is said by someone. It follows from this that any concept, idea, belief, definition, drawing, poem, or piece of music, has to be produced by a living human ..."
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analyze the biological foundations of human cognition. A crucial component of their arguments is a simple but profound aphorism: Everything said is said by someone. It follows from this that any concept, idea, belief, definition, drawing, poem, or piece of music, has to be produced by a living human being, constrained by the peculiarities of his or her body and brain. The entailment is straightforward: without living human bodies with brains, there are no ideas — and that includes mathematical ideas. This chapter deals with the structure of mathematical ideas themselves, and with how their inferential organization is provided by everyday human cognitive mechanisms such as conceptual metaphor. The Cognitive Study of Ideas and their Inferential Organization The approach to Mathematical Cognition we take in this chapter is relatively new, and it differs in important ways from (but is complementary to) the ones taken by many of the authors in this Handbook. In order to avoid potential misunderstandings regarding the subject matter and goals of our piece, we believe that it is important to clarify these differences right upfront. The differences reside mainly on three fundamental aspects:
NonStandard Models of Arithmetic: a Philosophical and Historical perspective MSc Thesis (Afstudeerscriptie)
, 2010
"... 1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5 ..."
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1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
"... Ask a mathematician of the age of Gauss “What is mathematics? ” and you could expect a stock answer somewhat along the following lines: “Mathematics consists of arithmetic and geometry, arithmetic being the science of quantity, just as geometry is the science of space. ” A few philosophers (Berkeley ..."
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Ask a mathematician of the age of Gauss “What is mathematics? ” and you could expect a stock answer somewhat along the following lines: “Mathematics consists of arithmetic and geometry, arithmetic being the science of quantity, just as geometry is the science of space. ” A few philosophers (Berkeley or Kant, for instance) might raise questions about the nature and sources of mathematical knowledge or about the legitimacy of certain forms of mathematical reasoning, but those questions existed on the fringes of mathematics and seemed to have little to do with the core of the discipline. Then, over the course of the nineteenth century, under the pressure of developments within mathematics itself, the accepted answer dramatically broke down. In analysis, Bolzano, investigating the foundations of the calculus, gives his “purely analytic proof ” of the intermediate value theorem; Weierstrass and his students independently rediscover his results and attempt to put the calculus on a rigorous arithmetical foundation. In algebra, Gauss and Hamilton provide geometric interpretations of the complex numbers; Hamilton widens the number concept, introducing
Large Cardinals and Determinacy
, 2011
"... The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measura ..."
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The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measurable?”), cardinal arithmetic (“Does Cantor’s Continuum Hypothesis hold?”), combinatorics(“DoesSuslin’sHypotheseshold?”), andgrouptheory (“Is there a Whitehead group?”). These developments gave rise to two conflicting positions. The first position—which we shall call pluralism—maintains that the independence results largely undermine the enterprise of set theory as an objective enterprise. On this view, although there are practical reasons that one might give in favour of one set of axioms over another—say, that it is more useful for a given task—, there are no theoretical reasons that can be given; and, moreover, this either implies or is a consequence of the fact—depending on the variant of the view, in particular, whether it places realism before reason,
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, 2012
"... This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Pattern matching through Chaos Game Representation: bridging numerical and discrete data structures for biological sequence analysis ..."
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This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Pattern matching through Chaos Game Representation: bridging numerical and discrete data structures for biological sequence analysis
Journal of Pragmatics Creating Mathematical Infinities: Metaphor, Blending, and the Beauty of Transfinite Cardinals
"... The Infinite is one of the most intriguing ideas in which the human mind has ever engaged. Full of paradoxes and controversies, it has raised fundamental issues in domains as diverse and profound as theology, physics, and philosophy. The infinite, an elusive and counterintuitive idea, has even playe ..."
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The Infinite is one of the most intriguing ideas in which the human mind has ever engaged. Full of paradoxes and controversies, it has raised fundamental issues in domains as diverse and profound as theology, physics, and philosophy. The infinite, an elusive and counterintuitive idea, has even played a central role in defining mathematics, a fundamental field of human intellectual inquiry characterized by precision, certainty, objectivity, and effectiveness in modeling our real finite world. Particularly rich is the notion of actual infinity, that is, infinity seen as a “completed, ” “realized ” entity. This powerful notion has become so pervasive and fruitful in mathematics that if we decide to abolish it, most of mathematics as we know it would simply disappear, from infinitesimal calculus, to projective geometry, to set theory, to mention only a few. From the point of view of cognitive science, conceptual analysis, and cognitive semantics the study of mathematics, and of infinity in particular, raises several intriguing questions: How do we grasp the infinite if, after all, our bodies are finite, and so are our experiences and everything we encounter with our bodies? Where does then the infinite come from? What cognitive mechanisms make it possible? How an elusive and