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Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
Formal proof—theory and practice
 Notices AMS
, 2008
"... Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are nume ..."
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Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are numerous computer programs known as proof assistants that can check, or even partially construct, formal proofs written in their preferred proof language. These can be considered as practical, computerbased realizations of the traditional systems of formal symbolic logic and set theory proposed as foundations for mathematics. Why should we wish to create formal proofs?
Spherical harmonic analysis on affine buildings
 Mathematische Zeitschrift
, 2006
"... Abstract. Let X be a locally finite regular affine building with root system R. There is a commutative algebra A spanned by averaging operators Aλ, λ ∈ P +, acting on the space of all functions f: VP → C, where VP is in most cases the set of all special vertices of X, and P + is a set of dominant co ..."
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Cited by 8 (3 self)
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Abstract. Let X be a locally finite regular affine building with root system R. There is a commutative algebra A spanned by averaging operators Aλ, λ ∈ P +, acting on the space of all functions f: VP → C, where VP is in most cases the set of all special vertices of X, and P + is a set of dominant coweights of R. This algebra is studied in [6] and [7] for Ãn buildings, and the general case is treated in [15]. In this paper we show that all algebra homomorphisms h: A → C may be expressed in terms of the Macdonald spherical functions. We also provide a second formula for these homomorphisms in terms of an integral over the boundary of X. We may regard A as a subalgebra of the C ∗algebra of bounded linear operators on ℓ 2 (VP), and we write A2 for the closure of A in this algebra. We study the Gelfand map A2 → C(M2), where M2 = Hom(A2, C), and we compute M2 and the Plancherel measure of A2. We also compute the ℓ 2operator norms of the operators Aλ, λ ∈ P +, in terms of the Macdonald spherical functions.
A short survey of automated reasoning
"... Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so f ..."
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Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for
Contents
, 2011
"... In contemporary theoretical physics, the powerful notion of symmetry stands for a web of intricate meanings among which I identify four clusters associated with the notion of transformation, comprehension, invariance and projection. While their interrelations are examined closely, these four facets ..."
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In contemporary theoretical physics, the powerful notion of symmetry stands for a web of intricate meanings among which I identify four clusters associated with the notion of transformation, comprehension, invariance and projection. While their interrelations are examined closely, these four facets of symmetry are scrutinised one after the other in great detail. This decomposition allows us to examine closely the multiple different roles symmetry plays in many places in physics. Furthermore, some connections with others disciplines like neurobiology, epistemology, cognitive sciences and, not least, philosophy are proposed in an attempt to show that symmetry can be an organising principle also in these fields. pacs: 11.30.j, 11.30.Qc, 01.70.+w,
Communities of Practice in Mathematical ELearning
, 2008
"... With the globalization in education, bridging cultural differences by making course material more accessible and adaptable to individual user needs becomes an important goal. In this paper we attack this goal for the field of mathematics where knowledge is abstract, highly structured, and extraordin ..."
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With the globalization in education, bridging cultural differences by making course material more accessible and adaptable to individual user needs becomes an important goal. In this paper we attack this goal for the field of mathematics where knowledge is abstract, highly structured, and extraordinary interlinked. Modern representation formats like our OMDOC format allow us to capture, model, relate, and represent mathematical learning objects and thus make them contextaware and machineadaptable to the respective learning contexts. But to make mathematical knowledge accessible to learners of diverse cultural backgrounds we also need to model mathematical practice. In this paper, we show that many practices of mathematical communities can already be modeled in OMDOC and outline extensions to support further ones. We have implemented a collection of services that allow applications to interpret and manage OMDOC and its practice representations as well as to adapt OMDOC for users and communities. These services have been integrated into our prototype ELearning platform panta rhei to demonstrate how systems can improve the accessibility of mathematical ELearning materials.
Formal Proof: Reconciling Correctness and Understanding
, 2009
"... Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical th ..."
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Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved. Formal proof is practically achievable. With formal proof, correctness reaches a standard that no penandpaper proof can match, but an essential component of mathematics — the insight and understanding — seems to be in short supply. So, what makes a proof understandable? To answer this question we first suggest a list of symptoms of understanding. We then propose a vision of an environment in which users can write and check formal proofs as well as query them with reference to the symptoms of understanding. In this way, the environment reconciles the main features of proof: correctness and understanding.
Pure Mathematics in a Mechanized Logic
"... : It is widely believed that in principle it's possible to reduce most of presentday mathematics to reasoning in a formal logical system. The technical difficulty of actually doing so is quite formidable. However, the arrival of the computer is changing this situation, since computers are good ..."
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: It is widely believed that in principle it's possible to reduce most of presentday mathematics to reasoning in a formal logical system. The technical difficulty of actually doing so is quite formidable. However, the arrival of the computer is changing this situation, since computers are good at helping with such tedious symbolic manipulation. The computer formalization of mathematics is now a popular research topic. Here we report on our own development of mathematical analysis starting just from the axioms of simple type theory and reducing all reasoning (with the aid of the computer) to a formal deductive calculus of great simplicity. 1 Introduction The idea of reducing reasoning to computation in some kind of mechanical calculus is quite an old one. For example, Hobbes [1] remarked that `reasoning . . . is but reckoning', and even noted the difficulty of ensuring that logical reasoning is correct: For as Arithmeticians teach to adde and subtract in numbers [...] The Logicians te...