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It is the immodest hope of the authors that this Dictionary will not only prove valuable as a reference book for students of mathematics at all levels from secondary school to master's degree, but also offer much to interest a more general readership. We have no illusions that a reference book can supplant authoritative text-books, but we nonetheless believe that the Dictionary will serve a distinct, complementary, and useful purpose. We have thus attempted to include anything we consider of interest or mnemonic value to the reader: there are formal accounts of certain terms of elementary arithmetic (in terms unavoidably reminiscent of Tom Lehrer's "New Math"), informal accounts of a number of common logical paradoxes, and less than hagiographic biographies of the luminaries in the history of mathematics.
Our more serious aspiration was the inclusion of any term that an undergraduate might encounter not only within, but also in reading around, any course at any college or university, and we have also deliberately set out to tailor the explanation of each term to the mathematical knowledge of the reader who is likely to consult it. But this ideal overreaches the practicalities, since syllabuses vary greatlyin scope, level, and order of presentation, and there are also various demarcation disputes, particularly at the intersection of applied mathematics and physics. In resolving these conflicts, we gave some weight to the existence of the other volumes of the present series; for that reason there is little computing or economics, but a considerable amount of logic. However, our prime consideration remained the likelihood of a mathematics student encountering a given term, so that our comparative liberality towards mechanics and statistics is a consequence of the prominence of these subjects in many undergraduate syllabuses. Thus in general we believe we have erred on the side of inclusiveness, albeit that some lacunae undoubtedly remain, however inadvertantly, and that for some advanced terms we have taken refuge in mere indications of the context in which the term occurs, and limitations of space and the demands of readability have dictated that some definitions are relatively informal.
In accordance with our principle of relative inclusiveness, we have attempted to define every term used in our definitions. However, just as we have tried to set the level of each entry for the reader who is likely to have recourse to it, so we have only explicitly signalled a CROSS-REFERENCE where we judge it likely to assist that reader, and for synonymous terms we have attempted to cross-refer the less common to the more common, although a degree of arbitrariness (to say nothing of subjectivity) is inevitable in such judgements. We have likewise attempted, in addition to the substantive biographical entries, to give biographical information under every headword that includes a personal name, but we have not always succeeded, and the absence of such information should not be taken to reflect on the standing of the individual; whenever our research was successful, the biographical matter appears as a subsidiary entry following the first substantive eponym. More generally, italics denote a term which has no entry of its own, other than (unless the two entries would in any event be adjacent in the lexical ordering) a cross-reference. Cross-references are always to the identical form, except that singular and plural are not distinguished; in particular, a sequence of words in this style always corresponds to a single entry except in a very few cases in which it was impossible to avoid ambiguity without violence to syntax or sense.
A major difficulty faced by a reference book of this kind is the existence of distinct -- and inconsistent -- usages. We have sought to record the contradictions and ambiguities between recognizable mathematical "dialects", as in the terminology of orderings and metric spaces, or with respect to the commutativity of certain algebraic objects, but the reader must remain on guard against the unbounded possibility of idiosyncrasy. Use of this handbook should thus be subject to the caveat that authors are authoritative about their own usage, and we should not be taken to be promulgating a unique definitively correct usage where none exists.
The Dictionary originated in 1984-85, when I was asked successively to review the logic, philosophy, and mathematics entries in the Collins English Dictionary, and it therefore affords me an opportunity to express my appreciation for the encouragement of its editors, Patrick Hanks and Bill McLeod. It soon became apparent that the only English language dictionaries of postschool mathematics in print were out of date, poorly written, prohibitively expensive, or, in the classic case, all three. Jon Borwein readily agreed to fulfil my immediate need for a consultant's consultant, and this work has grown out of that collaboration, following us both around the world.
Many of our friends, students, and colleagues were sucked into the project, and we are happy to acknowledge their unstinting assistance; Margaret Jones in particular was always ready to lend a hand. The entire text was read and commented on at various stages by Professor Robin Knops of Heriot-Watt University and Dr. John Bowers of Leeds, on behalf of the publishers, and by Glasgow students Michael McQuillan, Andrew Robertson, Martin Hendry, and John Lamb; Karen Chandler and Todd Cardno in Halifax also assisted us with research. By virtue of the addictive nature of lexicography, John Bowers, Andrew Robertson (now at Oxford), and Michael McQuillan (now at Harvard) became increasingly involved in the project, and we are grateful for their many suggestions which we have included in the present text. In addition, Peter Breeze of Glasgow University was particularly helpful in steering us away from statistical solecism.
We must also acknowledge material help -- substantial in both senses -- from our respective families, from whom I must single out my mother for her help with the early clerical chores. Our respective universities and departments enabled us to consult not only by granting us leave of absence but also by more technological means; I must also...
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