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Geometry as a universal mental construction
 In E. Brannon & S. Dehaene (Eds.), Space, Time and Number in the Brain: Searching for the Foundations of Mathematical Thought (pp. 319332): Attention & Performance XXIV
, 2011
"... Geometry, etymologically the “science of measuring the Earth”, is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. I ..."
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Geometry, etymologically the “science of measuring the Earth”, is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. Is the spatial content of formal Euclidean geometry universally present in the way humans perceive space, or is Euclidean geometry a mental construction, specific to those who have received appropriate instruction? The spatial content of the formal theories of geometry may depart from spatial perception for two reasons: first, because in geometry, only some of the features of spatial figures are theoretically relevant; and second, because some geometric concepts go beyond any possible perceptual experience. Focusing in turn on these two aspects of geometry, we will present several lines of research on US adults and children from the age of three years, and participants from an Amazonian culture, the Mundurucu. Almost all the aspects of geometry tested proved to be shared between these two cultures. Nevertheless, some aspects involve a process of mental construction where explicit instruction seem to play a role in the US, but that can still take place in the absence of instruction in geometry.
Orihedra: Mathematical Sculptures in Paper
 International Journal of Computers for Mathematical Learning
, 1997
"... Mathematics, as a subject dealing with abstract concepts, poses a special challenge for educators. In students ' experience, the subject is often associated with (potentially) unflattering adjectives—"austere", "remote", "depersonalized", and so forth. This paper descri ..."
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Mathematics, as a subject dealing with abstract concepts, poses a special challenge for educators. In students ' experience, the subject is often associated with (potentially) unflattering adjectives—"austere", "remote", "depersonalized", and so forth. This paper describes a computer program named HyperGami whose purpose is to alleviate this harsh portrait of the mathematical enterprise. HyperGami is a system for the construction of decorated paper polyhedral shapes; these shapes may be combined into larger polyhedral sculptures, which we have dubbed "orihedra. " In this paper, we illustrate the methods by which orihedra may be created from HyperGami solids (using the construction of a particular sculpture as an example); we describe our experiences with elementary and middleschool students using HyperGami to create orihedra; we discuss the current limitations of HyperGami as a sculptural medium; and we outline potential directions for future research and software development.
A New Look At Euclid's Second Proposition
 The Mathematical Intelligencer
, 1993
"... There has been considerable interest during the past 2300 years in comparing different models of geometric computation in terms of their computing power. One of the most well known results is Mohr's proof in 1672 that all constructions that can be executed with straightedge and compass can be carri ..."
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There has been considerable interest during the past 2300 years in comparing different models of geometric computation in terms of their computing power. One of the most well known results is Mohr's proof in 1672 that all constructions that can be executed with straightedge and compass can be carried out with compass alone. The earliest such proof of the equivalence of models of computation is due to Euclid in his second proposition of Book I of the Elements in which he establishes that the collapsing compass is equivalent in power to the modern compass. Therefore in the theory of equivalence of models of computation Euclid's second proposition enjoys a singular place. However, like much of Euclid's work and particularly his constructions involving cases, his second proposition has received a great deal of criticism over the centuries. Here it is argued that it is Euclid's early Greek commentators and more recent expositors and translators that are at fault and that Euclid's original...
Abstract Computerizing Mathematical Text with
"... Mathematical texts can be computerized in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which c ..."
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Mathematical texts can be computerized in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which capture the full mathematical meaning and have proofs expressed in a formal foundation of mathematics. In between, there are computer typesetting systems (e.g., LATEX and Presentation MathML) and semantically oriented systems (e.g., Content MathML, OpenMath, OMDoc, etc.). The MathLang project was initiated in 2000 by Fairouz Kamareddine and Joe Wells with the aim of developing an approach for computerizing mathematical texts and knowledge which is flexible enough to connect the different approaches to computerization, which allows various degrees of formalization, and which is compatible with different logical frameworks (e.g., set theory, category theory, type theory, etc.) and proof systems. The approach is embodied in a computer representation, which we call MathLang, and associated software tools, which are being developed by ongoing work. Three Ph.D. students (Manuel Maarek (2002/2007), Krzysztof Retel (since 2004), and Robert Lamar (since 2006)) and over a dozen master’s degree and undergraduate students have worked on MathLang. The project’s progress and design choices are driven by the needs for computerizing real representative mathematical texts chosen from various
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
Blending and Other Conceptual Operations in the Interpretation of Mathematical Proofs
"... this paper I benefited from discussions with Gilles Fauconnier. ..."
5 WordVectors and Search Engines
"... So far in this book we have discussed symmetric and antisymmetric relationships between particular words in a graph or a hierarchy, described one way to learn symmetric relationships from text, and shown how to use ideas such as similarity measures and transitivity to find ‘nearest neighbours ’ of a ..."
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So far in this book we have discussed symmetric and antisymmetric relationships between particular words in a graph or a hierarchy, described one way to learn symmetric relationships from text, and shown how to use ideas such as similarity measures and transitivity to find ‘nearest neighbours ’ of a particular word. But ideally we should be able to measure the similarity or distance between any pair of words or concepts. To some extent, this is possible in graphs and taxonomies by finding the lengths of paths between concepts, but there are problems with this. First of all, finding shortest paths is often computationally expensive and may take a long time. Secondly, we might not have a reliable taxonomy, and as we’ve seen already, that there is a short path between two words in a graph doesn’t necessarily mean that they’re very similar, because the links in this short path may have arisen from very different contexts. Thirdly, the meanings of words we encounter in documents and corpora may be very different from those given by a general taxonomy such as WordNet — for example, WordNet 2.0 only gives the fruit and tree meanings for the word apple, which is a stark contrast with the top 10 pages returned by Google when doing an internet search with the query apple, which are all about Apple Computers. Another limitation of our methods so far is that we have focussed our attention purely on individual concepts, mainly single words. Ideally, we should be able to find the similarity between two arbitrary collections of words, and quickly. For this, we need some process for semantic composition — working out how to represent the meaning of a sentence or document based on the meaning of the words it contains.
Under supervision of:
, 2006
"... MathLang is a language for mathematics on computers. It allows computerisation of existing and new mathematical texts written in the Common Mathematical Language, and checking the grammatical correctness of this computerisation. The framework also allows the user to take incremental steps towards th ..."
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MathLang is a language for mathematics on computers. It allows computerisation of existing and new mathematical texts written in the Common Mathematical Language, and checking the grammatical correctness of this computerisation. The framework also allows the user to take incremental steps towards the generation of a fully formalised document in such a way that the result can be checked by a proof checker. This report describes the language itself, its grammar, elements, characteristics and one of the concrete syntaxes: the plain syntax. An encoding of a large example is presented and explained. The main focus of the report lies on the implementation of the heart of the framework: MathLangCore. While the architecture has changed to a more XMLcentred design during the implementation, both the old and proposed new architectures are discussed. Four components can be distinguished in the framework: the parser, the abstract syntax tree, the checker and the printer. The latter has many different instances for many formats.
Computerising Mathematical Text with MathLang
"... Mathematical texts can be computerised in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which c ..."
Abstract
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Mathematical texts can be computerised in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which capture the full mathematical meaning and have proofs expressed in a formal foundation of mathematics. In between, there are computer typesetting systems (e.g., LATEX and Presentation MathML) and semantically oriented systems (e.g., Content MathML, OpenMath, OMDoc, etc.). The MathLang project was initiated in 2000 by Fairouz Kamareddine and Joe Wells with the aim of developing an approach for computerising mathematical texts which is flexible enough to connect the different approaches to computerisation, which allows various degrees of formalisation, and which is compatible with different logical frameworks (e.g., set theory, category theory, type theory, etc.) and proof systems. The approach is embodied in a computer representation, which we call MathLang, and associated software tools, which are being developed by ongoing work. Four Ph.D. students (Manuel Maarek (2002/2007), Krzysztof Retel (since 2004), Robert Lamar (since 2006)), and Christoph Zengler (since 2008) and over a dozen master’s degree and undergraduate