The language of mathematics. A linguistic and philosophical investigation
From a linguistic perspective the author analyses several sentences used in mathematical texts and papers (written in English) as those of a certain particular natural language, namely, the language of mathematics, one of whose objectives is to provide a parser for it. There are very few and limited previous studies of this kind: the works of Ranta deal with those of certain specific theories (such as arithmetic, for example) by a linguistic approach, and those of de Bruijn are motivated by an interest coming from computer science more strongly. On the other hand, the subject of this book is a language supposed to cover all mathematical theories. Referring to the observations due to Ranta, the analysis is carried out based on a context-free generative grammar syntactically and discourse representation theory semantically. Some particular characteristics of the language are featured first; for example, the mixture of textual and symbolic expressions. It is also emphasized that for the language to cover all mathematical theories, its grammar needs to satisfy a requirement called `adaptivity', or roughly, to enjoy a mechanism in itself to produce a new term or notation according to a given definition (so that its lexicon cannot be given manually but needs to be open). Then, the main part of the discussion is devoted to the disambiguation problem, for which on symbolic statements in particular it is necessary to categorize the symbols to be used in certain types. But the problem is more complicated than in the case of formal languages because the ambiguity of a sentence containing both textual and symbolic expressions comes from both aspects depending on each other. Therefore, the mechanism for disambiguation is required so as to cover both aspects uniformly. Thus, taking account of the above-mentioned adaptivity as well, the author introduces a new system of type inference, and proposes a typed parsing system based on it. It is also pointed out that the analysis brings forward some philosophical discussions on the foundation of mathematics, that is, the set-theoretical reconstruction of mathematical theories, the argument on which (on whose ontological problem particularly) is another main issue of the book. This is the book version of the author's PhD thesis, which was awarded the E. W. Beth Dissertation Prize for outstanding dissertations in the fields of logic, language, and information. Reviewer's remarks: In the description there are some mistakes and ambiguities which may happen to leave certain opaqueness in the argument. Among others, for example, on p. 126 (in the middle) ``Axiom of Extensionality'' should be read instead of ``Axiom of Foundation''; and on p. 152 (at the top) the type inference rule (with respect to the set intersection) must be applied correctly under the restriction to exclude the disjoint case (depending on a semantical aspect) because the empty set seems to be required not to enjoy any inferential type (cf. p. 148), while this kind of remark is not specified in other similar cases either.
scientific article; zbMATH DE number 6142694