Results 1  10
of
82
The Impact of the Lambda Calculus in Logic and Computer Science
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the represent ..."
Abstract

Cited by 28 (1 self)
 Add to MetaCart
One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand.
Flexible encoding of mathematics on the computer
 In MKM 2004, volume 3119 of LNCS
, 2004
"... Abstract. This paper reports on refinements and extensions to the MathLang framework that add substantial support for natural language text. We show how the extended framework supports multiple views of mathematical texts, including natural language views using the exact text that the mathematician ..."
Abstract

Cited by 24 (13 self)
 Add to MetaCart
(Show Context)
Abstract. This paper reports on refinements and extensions to the MathLang framework that add substantial support for natural language text. We show how the extended framework supports multiple views of mathematical texts, including natural language views using the exact text that the mathematician wants to use. Thus, MathLang now supports the ability to capture the essential mathematical structure of mathematics written using natural language text. We show examples of how arbitrary mathematical text can be encoded in MathLang without needing to change any of the words or symbols of the texts or their order. In particular, we show the encoding of a theorem and its proof that has been used by Wiedijk for comparing many theorem prover representations of mathematics, namely the irrationality of √ 2 (originally due to Pythagoras). We encode a 1960 version by Hardy and Wright, and a more recent version by Barendregt. 1 On the way to a mathematical vernacular for computers Mathematicians now use computer software for a variety of tasks: typing mathematical texts, performing calculation, analyzing theories, verifying proofs. Software tools like
Toward an objectoriented structure for mathematical text
 MATHEMATICAL KNOWLEDGE MANAGEMENT, 4TH INT’L CONF., PROCEEDINGS. VOLUME 3863 OF LECTURE NOTES IN ARTIFICIAL INTELLIGENCE
, 2006
"... Computerizing mathematical texts to allow software access to some or all of the texts ’ semantic content is a long and tedious process that currently requires much expertise. We believe it is useful to support computerization that adds some structural and semantic information, but does not require j ..."
Abstract

Cited by 19 (12 self)
 Add to MetaCart
(Show Context)
Computerizing mathematical texts to allow software access to some or all of the texts ’ semantic content is a long and tedious process that currently requires much expertise. We believe it is useful to support computerization that adds some structural and semantic information, but does not require jumping directly from the wordprocessing level (e.g., L ATEX) to full formalization (e.g., Mizar, Coq, etc.). Although some existing mathematical languages are aimed at this middle ground (e.g., MathML, OpenMath, OMDoc), we believe they miss features needed to capture some important aspects of mathematical texts, especially the portion written with natural language. For this reason, we have been developing MathLang, a language for representing mathematical texts that has weak type checking and support for the special mathematical use of natural language. MathLang is currently aimed at only capturing the essential grammatical and binding structure of mathematical text without requiring full formalization. The development of MathLang is directly driven by experience encoding real mathematical texts. Based on this experience, this paper presents the changes that yield our latest version of MathLang. We have restructured and simplified the core of the language, replaced our old notion of “context” by a new system of blocks and local scoping, and made other changes. Furthermore, we have enhanced our support for the mathematical use of nouns and adjectives with objectoriented features so that nouns now correspond to classes, and adjectives to mixins.
Computerizing Mathematical Text with MathLang
"... 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 ..."
Abstract

Cited by 18 (4 self)
 Add to MetaCart
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
NONEUCLIDEAN PYTHAGOREAN TRIPLES, A PROBLEM OF EULER, AND RATIONAL POINTS ON K3 SURFACES
, 2006
"... Abstract. We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a nonEuclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an alg ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
Abstract. We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a nonEuclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler’s work on the second problem with a modern point of view. 1. Problem I: Pythagorean triples An ordinary Pythagorean triple is a triple (a, b, c) of positive integers satisfying a 2 + b 2 = c 2. Finding these is equivalent, by the Pythagorean theorem, to finding right triangles with integral sides. Since the equation is homogeneous, the problem for rational numbers is the same, up to a scale factor. Some Pythagorean triples, such as (3, 4, 5), have been known since antiquity. Euclid [6, X.28, Lemma 1] gives a method for finding such triples, which leads to a complete solution of the problem. The primitive Pythagorean triples are exactly the triples of integers (m 2 − n 2, 2mn, m 2 + n 2) for various choices of m, n (up to
Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λcalculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced
The notion of dimension in geometry and algebra
 ArXiv:math.AG/0502016 v1 1 Feb
, 2005
"... Abstract. This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are invoked and compared. Glenn Gould disapproved of his ow ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
Abstract. This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are invoked and compared. Glenn Gould disapproved of his own recording of Goldberg variations. “There is a lot of piano playing going on there, and I mean that as the most disparaging comment possible.” NYRB, Oct. 7, 2004, p. 10 0.1. Some history. The notion of dimension belongs to the most fundamental mathematical ideas. In Western civilization and our school system, we become exposed in early life to the assertion that the dimension of our physical space is three (and somewhat later, that time furnishes the fourth dimension).
The Constructed Objectivity of Mathematics and the Cognitive Subject
, 2001
"... Introduction This essay concerns the nature and the foundation of mathematical knowledge, broadly construed. The main idea is that mathematics is a human construction, but a very peculiar one, as it is grounded on forms of "invariance" and "conceptual stability" that single out ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Introduction This essay concerns the nature and the foundation of mathematical knowledge, broadly construed. The main idea is that mathematics is a human construction, but a very peculiar one, as it is grounded on forms of "invariance" and "conceptual stability" that single out the mathematical conceptualization from any other form of knowledge, and give unity to it. Yet, this very conceptualization is deeply rooted in our "acts of experience", as Weyl says, beginning with our presence in the world, first in space and time as living beings, up to the most complex attempts we make by language to give an account of it. I will try to sketch the origin of some key steps in organizing perception and knowledge by "mathematical tools", as mathematics is one of the many practical and conceptual instruments by which we categorize, organise and "give a structure" to the world. It is conceived on the "interface" between us and the world, or, to put it in husserlian terminology, it is "de