Results 1  10
of
14
A proof of the Kepler conjecture
 Math. Intelligencer
, 1994
"... This section describes the structure of the proof of ..."
Abstract

Cited by 209 (13 self)
 Add to MetaCart
(Show Context)
This section describes the structure of the proof of
Some considerations on the usability of Interactive Provers
"... Abstract. In spite of the remarkable achievements recently obtained in the field of mechanization of formal reasoning, the overall usability of interactive provers does not seem to be sensibly improved since the advent of the “second generation ” of systems, in the mid of the eighties. We try to ana ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
Abstract. In spite of the remarkable achievements recently obtained in the field of mechanization of formal reasoning, the overall usability of interactive provers does not seem to be sensibly improved since the advent of the “second generation ” of systems, in the mid of the eighties. We try to analyze the reasons of such a slow progress, pointing out the main problems and suggesting some possible research directions. 1
Stateless HOL
"... Dedicated to Roel de Vrijer, in the tradition of Automath. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far h ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Dedicated to Roel de Vrijer, in the tradition of Automath. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far has been moved out of the kernel. This means that the kernel now is purely functional. The changes to the system are small. All existing HOL Light developments can be run by the stateless system with only minor changes. The basic principle behind the system is not to name constants by strings, but by pairs consisting of a string and a definition. This means that the data structures for the terms are all merged into one big graph. OCaml – the implementation language of the system – can use pointer equality to establish equality of data structures fast. This allows the system to run at acceptable speeds. Our system runs at about 85 % of the speed of the stateful version of HOL Light.
Stateless HOL Dedicated to Roel de Vrijer, in the tradition of Automath
"... Abstract. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far has been moved out of the kernel. This means that ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far has been moved out of the kernel. This means that the kernel now is purely functional. The changes to the system are small. All existing HOL Light developments can be run by the stateless system with only minor changes. The basic principle behind the system is not to name constants by strings, but by pairs consisting of a string and a definition. This means that the data structures for the terms are all merged into one big graph. OCaml – the implementation language of the system – can use pointer equality to establish equality of data structures fast. This allows the system to run at acceptable speeds. Our system is about 1 6 version of HOL Light. th slower than the stateful
unknown title
"... Abstract: According to the most popular non‐skeptical views about intuition, intuitions justify beliefs because they are based on understanding. More precisely: if intuiting that p justifies you in believing that p it does so because your intuition is based on your understanding of the proposition ..."
Abstract
 Add to MetaCart
Abstract: According to the most popular non‐skeptical views about intuition, intuitions justify beliefs because they are based on understanding. More precisely: if intuiting that p justifies you in believing that p it does so because your intuition is based on your understanding of the proposition that p. The aim of this paper is to raise some challenges for accounts of intuitive justification along these lines. I pursue this project from a non‐skeptical perspective. I argue that there are cases in which intuiting that p justifies you in believing that p, but such that there is no compelling reason to think this is because your intuition is based on your understanding of the proposition that p. A common idea about intuition is that it is based on understanding. 2 According to one way of developing the idea, in intuiting that p you simply draw on
Chapter 1 of Kelley’s famous book General Topology introduces the most
"... fundamental concepts of a topological space. One such notion is defined as follows [1]: A topological space (X, τ) is connected if and ony if X is not the union of two nonvoid separated subsets, where A and B are separated in X if and only if A ∩ B = � and A ∩ B = �. As usual, Y denotes the closure ..."
Abstract
 Add to MetaCart
(Show Context)
fundamental concepts of a topological space. One such notion is defined as follows [1]: A topological space (X, τ) is connected if and ony if X is not the union of two nonvoid separated subsets, where A and B are separated in X if and only if A ∩ B = � and A ∩ B = �. As usual, Y denotes the closure of a subset Y
THE FUNDAMENTAL THEOREM OF ALGEBRA MADE EFFECTIVE: AN ELEMENTARY REALALGEBRAIC PROOF VIA STURM CHAINS
, 2009
"... Sturm’s famous theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any given real polynomial. In his residue calculus of complex functions, Cauchy (1831/37) extended this to an algebraic method to count and locate the complex roots of any given complex polynomial. ..."
Abstract
 Add to MetaCart
Sturm’s famous theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any given real polynomial. In his residue calculus of complex functions, Cauchy (1831/37) extended this to an algebraic method to count and locate the complex roots of any given complex polynomial. We give a realalgebraic proof of Cauchy’s theorem starting from the axioms of a real closed field, without appeal to analysis. This allows us to algebraically formalize Gauss ’ geometric argument (1799) and thus to derive a realalgebraic proof of the Fundamental Theorem of Algebra, stating that every complex polynomial of degree n has n complex roots. The proof is elementary inasmuch as it uses only the intermediate value theorem and arithmetic of real polynomials. It can thus be formulated in the firstorder language of real closed fields. Moreover, the proof is constructive and immediately translates to an algebraic rootfinding algorithm. The latter is sufficiently efficient for moderately sized polynomials, but in its present form it still lags behind Schönhage’s nearly optimal numerical algorithm (1982).
ii FORMAL COMPUTATIONS AND METHODS
, 2012
"... This dissertation was presented by ..."
(Show Context)
Understanding, formal verification, and the philosophy of mathematics
, 2010
"... The philosophy of mathematics has long been concerned with determining the means that are appropriate for justifying claims of mathematical knowledge, and the metaphysical considerations that render them so. But, as of late, many philosophers have called attention to the fact that a much broader ra ..."
Abstract
 Add to MetaCart
(Show Context)
The philosophy of mathematics has long been concerned with determining the means that are appropriate for justifying claims of mathematical knowledge, and the metaphysical considerations that render them so. But, as of late, many philosophers have called attention to the fact that a much broader range of normative judgments arise in ordinary mathematical practice; for example, questions can be interesting, theorems important, proofs explanatory, concepts powerful, and so on. The associated values are often loosely classified as aspects of “mathematical understanding.” Meanwhile, in a branch of computer science known as “formal verification,” the practice of interactive theorem proving has given rise to software tools and systems designed to support the development of complex formal axiomatic proofs. Such efforts require one to develop models of mathematical language and inference that are more robust than the the simple foundational models of the last century. This essay explores some of the insights that emerge from this work, and some of the ways that these insights can inform, and be informed by, philosophical theories of mathematical understanding.