Mathematics. A minimal introduction (Q2863027)

According to the author, the book under review ``can serve as a course bridging the gap between `procedural' mathematics (that emphasizes calculation) and `conceptual' mathematics (that emphasizes ideas); it can also serve as a rigorous introduction to the basic concepts of logic and mathematics.'' In the preface, two features of existing texts which introduce logic and mathematics are pointed out. Firstly, ``mathematics is being used as an illustration (and even as a tool) in the construction of the logical apparatus while, on the other hand, logic is being used to construct mathematical proofs. This circularity is generally presented as acceptable, although, for excellent reasons, students are often uncomfortable accepting it.'' Secondly, ``most books on the subject implicitly subscribe to a correspondence theory of truth; according to such a theory mathematical sentences have a meaning, they are supposed to refer to an ``infinite universe of mathematical reality,'' and they are supposed to be either true or false depending on their meaning and on the ``state of affairs'' in the mathematical universe. This leads to an implicit acceptance of a ``maximal ontology,'' such as that of Cantor set theory, where ``actual infinities'' are being treated as ``reality'' and statements about them are being treated as referring to ``empirical facts''.''NEWLINENEWLINEProfessor Buium goes on by explaining his vision as follows. ``The present book will adopt, however, a radically different position: we will declare that the ``infinite universe of mathematical reality'' is a fiction; and that mathematics does not need this ``fairy tale universe'' in order to exist. All that mathematics needs are the mathematical words themselves which should be regulated by a ``pre-mathematical logic'' of language. For instance we will agree that empirical ``infinity'' is not available but that we can use the word ``infinity'' as a mere word, devoid of any reference, provided that strict rules are observed in its use. Consequently, in contrast with most available books, we will begin here by asking the students to ``forget,'' for a while, about all the mathematical objects/concepts that they were ever exposed to; one will have to act as if one does not know the ``meaning'' of the symbols \(1, 2, 3, +, \times, \mathbb{Z}\), \(\sin\), \(\cos\), etc. Then the course will introduce each of these words/symbols in a non-circular manner. In this way we will build, from scratch, first a pre-mathematical logic, then mathematics itself, and, finally, a mathematical logic.''NEWLINENEWLINEDesigned as an undergraduate introduction to pure mathematics and promoting a non-standard formalist approach to mathematics, labeled by the author as ``rather extreme,'' this text is an interesting offer for instructors teaching mathematical logic and pure mathematics and looking for innovative ideas. Professor's Buium statement that ``extreme formalism can be easily grasped by students, who generally seem to enjoy playing the game of reconstructing mathematics from language itself'' sounds quite optimistic and reassuring.