18,996 research outputs found
Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth
Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation
A Survey of Languages for Specifying Dynamics: A Knowledge Engineering Perspective
A number of formal specification languages for knowledge-based systems has been developed. Characteristics for knowledge-based systems are a complex knowledge base and an inference engine which uses this knowledge to solve a given problem. Specification languages for knowledge-based systems have to cover both aspects. They have to provide the means to specify a complex and large amount of knowledge and they have to provide the means to specify the dynamic reasoning behavior of a knowledge-based system. We focus on the second aspect. For this purpose, we survey existing approaches for specifying dynamic behavior in related areas of research. In fact, we have taken approaches for the specification of information systems (Language for Conceptual Modeling and TROLL), approaches for the specification of database updates and logic programming (Transaction Logic and Dynamic Database Logic) and the generic specification framework of abstract state machine
A decompilation of the pi-calculus and its application to termination
We study the correspondence between a concurrent lambda-calculus in
administrative, continuation passing style and a pi-calculus and we derive a
termination result for the latter
Bootstrapping Lexical Choice via Multiple-Sequence Alignment
An important component of any generation system is the mapping dictionary, a
lexicon of elementary semantic expressions and corresponding natural language
realizations. Typically, labor-intensive knowledge-based methods are used to
construct the dictionary. We instead propose to acquire it automatically via a
novel multiple-pass algorithm employing multiple-sequence alignment, a
technique commonly used in bioinformatics. Crucially, our method leverages
latent information contained in multi-parallel corpora -- datasets that supply
several verbalizations of the corresponding semantics rather than just one.
We used our techniques to generate natural language versions of
computer-generated mathematical proofs, with good results on both a
per-component and overall-output basis. For example, in evaluations involving a
dozen human judges, our system produced output whose readability and
faithfulness to the semantic input rivaled that of a traditional generation
system.Comment: 8 pages; to appear in the proceedings of EMNLP-200
On Constructive Axiomatic Method
In this last version of the paper one may find a critical overview of some
recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure
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