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Towards Selfverification of HOL Light
 In International Joint Conference on Automated Reasoning
, 2006
"... Abstract. The HOL Light prover is based on a logical kernel consisting of about 400 lines of mostly functional OCaml, whose complete formal verification seems to be quite feasible. We would like to formally verify (i) that the abstract HOL logic is indeed correct, and (ii) that the OCaml code does c ..."
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Abstract. The HOL Light prover is based on a logical kernel consisting of about 400 lines of mostly functional OCaml, whose complete formal verification seems to be quite feasible. We would like to formally verify (i) that the abstract HOL logic is indeed correct, and (ii) that the OCaml code does correctly implement this logic. We have performed a full verification of an imperfect but quite detailed model of the basic HOL Light core, without definitional mechanisms, and this verification is entirely conducted with respect to a settheoretic semantics within HOL Light itself. We will duly explain why the obvious logical and pragmatic difficulties do not vitiate this approach, even though it looks impossible or useless at first sight. Extension to include definitional mechanisms seems straightforward enough, and the results so far allay most of our practical worries. 1 Introduction: quis custodiet ipsos custodes? Mathematical proofs are subjected to peer review before publication, but there
Understanding Mathematical Discourse
 Dialogue. Amsterdam University
, 1999
"... Discourse Understanding is hard. This seems to be especially true for mathematical discourse, that is proofs. Restricting discourse to mathematical discourse allow us, however, to study the subject matter in its purest form. This domain of discourse is rich and welldefined, highly structured, offers ..."
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Discourse Understanding is hard. This seems to be especially true for mathematical discourse, that is proofs. Restricting discourse to mathematical discourse allow us, however, to study the subject matter in its purest form. This domain of discourse is rich and welldefined, highly structured, offers a welldefined set of discourse relations and forces/allows us to apply mathematical reasoning. We give a brief discussion on selected linguistic phenomena of mathematical discourse, and an analysis from the mathematician’s point of view. Requirements for a theory of discourse representation are given, followed by a discussion of proofs plans that provide necessary context and structure. A large part of semantics construction is defined in terms of proof plan recognition and instantiation by matching and attaching. 1
The journey of the four colour theorem through time
 The NZ Math. Magazine
"... This is a historical survey of the Four Colour Theorem and a discussion of the philosophical implications of its proof. The problem, first stated as far back as 1850s, still causes controversy today. Its computeraided proof has forced mathematicians to question the notions of proofs and mathematical ..."
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This is a historical survey of the Four Colour Theorem and a discussion of the philosophical implications of its proof. The problem, first stated as far back as 1850s, still causes controversy today. Its computeraided proof has forced mathematicians to question the notions of proofs and mathematical truth.
Reflections on Quantum Computing
, 2000
"... In this rather speculative note three problems pertaining to the power and limits of quantum computing are posed and partially answered: a) when are quantum speedups possible?, b) is fixedpoint computing a better model for quantum computing?, c) can quantum computing trespass the Turing barrier? 1 ..."
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In this rather speculative note three problems pertaining to the power and limits of quantum computing are posed and partially answered: a) when are quantum speedups possible?, b) is fixedpoint computing a better model for quantum computing?, c) can quantum computing trespass the Turing barrier? 1 When are quantum speedups possible? This section discusses the possibility that speedups in quantum computing can be achieved only for problems which have a few or even unique solutions [12]. For instance, this includes the computational complexity class UP [15]. Typical examples are Shor's quantum algorithm for prime factoring [18] and Grover's database search algorithm [13] for a single item satisfying a given condition in an unsorted database (see also Gruska [14]). In quantum complexity, one popular class of problems is BQP,whichisthe set of decision problems that can be solved in polynomial time (on a quantum computer) so that the correct answer is obtained with probability at l...
Foolproof
"... Mathematical proof is foolproof, it seems, only in the absence of fools I was a teenage angle trisector. In my first fulltime job, fresh out of high school, I trisected angles all day long for $1.75 an hour. My employer was a maker of voltmeters, ammeters and other electrical instruments. This was ..."
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Mathematical proof is foolproof, it seems, only in the absence of fools I was a teenage angle trisector. In my first fulltime job, fresh out of high school, I trisected angles all day long for $1.75 an hour. My employer was a maker of voltmeters, ammeters and other electrical instruments. This was back in the analog age, when a meter had a slender pointer swinging in an arc across a scale. My job was drawing the scale. A technician would calibrate
WHAT IS THE PHILOSOPHY OF MATHEMATICS EDUCATION?
"... This question (what is the philosophy of mathematics education?) provokes a number of reactions, even before one tries to answer it. Is it a philosophy of mathematics education, or is it the philosophy of mathematics education? Use of the preposition ‘a ’ suggests that what is being offered is one o ..."
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This question (what is the philosophy of mathematics education?) provokes a number of reactions, even before one tries to answer it. Is it a philosophy of mathematics education, or is it the philosophy of mathematics education? Use of the preposition ‘a ’ suggests that what is being offered is one of several such perspectives, practices or areas of study. Use of the definite article ‘the ’ suggests to some the arrogation of definitiveness to the account given. 1 In other words, it is the dominant or otherwise unique account of philosophy of mathematics education. However, an alternative reading is that ‘the ’ refers to a definite area of enquiry, a specific domain, within which one account is offered. So the philosophy of mathematics education need not be a dominant interpretation so much as an area of study, an area of investigation, and hence something with this title can be an exploratory assay into this area. This is what I intend here. Moving beyond the first word, there is the more substantive question of the reference of the term ‘philosophy of mathematics education’. There is a narrow sense that can be applied in interpreting the words ‘philosophy ’ and ‘mathematics education’. The philosophy of some area or activity can be understood as its aims or rationale. Mathematics education understood
HOW TO COUNT
"... Counting is something we learn so early in life that we tend to dismiss it as a trivial skill, beneath the notice of mathematics. The recent U.S. presidential election suggests otherwise. Although most of the votecounting controversies last fall concerned what to count rather than how to count, the ..."
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Counting is something we learn so early in life that we tend to dismiss it as a trivial skill, beneath the notice of mathematics. The recent U.S. presidential election suggests otherwise. Although most of the votecounting controversies last fall concerned what to count rather than how to count, the counting process itself also proved to be imprecise and unreliable. Counting and recounting the same batch of ballots seldom gave the same total twice. Evidently, counting is not the utterly deterministic procedure we take it to be. There is some wiggle and wobble in it. And ballots are not the only things we can lose count of. The Census Bureau has reported the U.S. population as 281,421,906, but no one believes