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Output Devices, Computation, and the Future of Mathematical Crafts
 International Journal of Computers in Mathematical Learning
, 2002
"... As I write this sentence, I am glancing over at the color printer sitting beside my screen. In the popular jargon of the computer industry, that printer is called a "peripheral"—which, upon reflection, is a rather odd way to describe it. What, precisely, is it peripheral to? If the ultimat ..."
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As I write this sentence, I am glancing over at the color printer sitting beside my screen. In the popular jargon of the computer industry, that printer is called a "peripheral"—which, upon reflection, is a rather odd way to describe it. What, precisely, is it peripheral to? If the ultimate
On the theoretical foundations of LePUS3 and its application to objectoriented design verification
, 2011
"... Software systems are the most complex artefacts ever produced by humans, and managing their complexity is one of the central challenges of software engineering. One major source of complexity arises from maintaining consistency between a program and its design documentation. Design verification redu ..."
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Software systems are the most complex artefacts ever produced by humans, and managing their complexity is one of the central challenges of software engineering. One major source of complexity arises from maintaining consistency between a program and its design documentation. Design verification reduces such complexity by improving consistency between design and implementation. Fully automated tools are of paramount importance so that the budget, time and quality tradeoff is minimized. This thesis combines aspects of both the theory and practise of design verification. We begin by defining a theory of classes abstracted and refined from the theoretical foundations of LePUS3, and developed in the Typed Predicate Logic. LePUS3 is a promising formal and visual design description language for the decidable aspects of objectoriented design. Our theory fixes many of the criticisms of LePUS3 and is also more expressive, rigorous, and extensible. We then move onto the practise of design verification, which we define for a restricted subset of our theory. We demonstrate our design verification method in three case studies: verifying an
Loss of vision: How mathematics turned blind while it learned to see more clearly
 In B. Löwe and T. Müller (Eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice
, 2010
"... The aim of this paper is to provide a framework for the discussion of mathematical ontology that is rooted in actual mathematical practice, i.e., the way in which mathematicians have introduced and dealt with mathematical ..."
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The aim of this paper is to provide a framework for the discussion of mathematical ontology that is rooted in actual mathematical practice, i.e., the way in which mathematicians have introduced and dealt with mathematical
ARISTOTELIAN REALISM
"... Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as rat ..."
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Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity,
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granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. It is a publisher's requirement to display the following notice: The documents distributed by this server have been provided by the contributing authors as a means to ensure timely dissemination of scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may not be reposted without the explicit permission of the copyright holder. In the case of Springer, it is the publisher’s requirement that the following note be added: “An author may selfarchive an authorcreated version of his/her article on his/her own website and his/her institution’s repository, including his/her final version; however he/she may not use the publisher’s PDF version which is posted on www.springerlink.com. Furthermore, the author may only post his/her version provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer’s website. The link must be accompanied by the following text: “The original publication is available at www.springerlink.com. ” The Philosophy of Information as a Conceptual Framework
Mathematics and Statistics in the Social Sciences
"... Over the years, mathematics and statistics have become increasingly important in the social sciences 1. A look at history quickly confirms this claim. At the beginning of the 20th century most theories in the social sciences were formulated in qualitative terms while quantitative methods did not pla ..."
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Over the years, mathematics and statistics have become increasingly important in the social sciences 1. A look at history quickly confirms this claim. At the beginning of the 20th century most theories in the social sciences were formulated in qualitative terms while quantitative methods did not play a substantial role in their formulation and establishment. Moreover, many practitioners considered mathematical methods to be inappropriate and simply unsuited to foster our understanding of the social domain. Notably, the famous Methodenstreit also concerned the role of mathematics in the social sciences. Here, mathematics was considered to be the method of the natural sciences from which the social sciences had to be separated during the period of maturation of these disciplines. All this changed by the end of the century. By then, mathematical, and especially statistical, methods were standardly used, and their value in the social sciences became relatively uncontested. The use of mathematical and statistical methods is now ubiquitous: Almost all social sciences rely on statistical methods to analyze data and form hypotheses, and almost all of them use (to a greater or lesser extent) a range of mathematical methods to help us understand the social world. Additional indication for the increasing importance of mathematical and statistical methods in the social sciences is the formation of new subdisciplines, and the establishment of specialized journals and societies. Indeed, subdisciplines such as
The cognitive basis of arithmetic
"... Arithmetic is the theory of the natural numbers and one of the oldest areas of mathematics. Since almost all other mathematical theories make use of numbers in some way or other, arithmetic is also one of the most fundamental theories of mathematics. But numbers are not just abstract entities ..."
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Arithmetic is the theory of the natural numbers and one of the oldest areas of mathematics. Since almost all other mathematical theories make use of numbers in some way or other, arithmetic is also one of the most fundamental theories of mathematics. But numbers are not just abstract entities
Discussion Do children learn the integers by induction?
, 2007
"... www.elsevier.com/locate/COGNIT ..."
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"... Methodology and metaphysics in the development of Dedekind’s theory of ideals ..."
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Methodology and metaphysics in the development of Dedekind’s theory of ideals
The Internal Relatedness of All Things
"... The argument from internal relatedness was one of the major nineteenth century neoHegelian arguments for monism. This argument has been misunderstood, and may even be sound. The argument, as I reconstruct it, proceeds in two stages: first, it is argued that all things are internally related in ways ..."
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The argument from internal relatedness was one of the major nineteenth century neoHegelian arguments for monism. This argument has been misunderstood, and may even be sound. The argument, as I reconstruct it, proceeds in two stages: first, it is argued that all things are internally related in ways that render them interdependent; second, the substantial unity of the whole universe is inferred from the interdependence of all of its parts. The guiding idea behind the argument is that failure of free recombination is the modal signature of an integrated monistic cosmos. Frequently consider the connection of all things in the universe and their relation to one another. For in a manner all things are implicated with one another … (Marcus Aurelius, Meditations, p.43) Many of us were raised to believe the following story. By the end of the nineteenth century, darkness was over the surface of the deep. Philosophy was dominated by neoHegelian monistic idealism, and plunged into obscurity and confusion. And then there was light. As the twentieth century dawned, Russell and Moore separated from the neoHegelians by defending external relations, pluralism, realism, clarity, and all that is Good. The creation myth of analytic philosophy — like many founding myths — contains some traces of truth. By the end of the nineteenth century, philosophy was indeed dominated by neoHegelian monistic idealism. Russell and Moore did indeed separate from the neoHegelians by defending external relations, pluralism, and realism. Russell and Moore were much clearer than their predecessors, and this was Good. But the neoHegelians actually had an argument for their monism — the argument from internal relatedness — which went misunderstood, and which may even be sound. Or so I will argue. Overview of the argument: The argument from internal relatedness, as I will reconstruct it, proceeds in two stages. First, it is argued that all