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Structures and Algorithms

Jens Erik Fenstad
Publication Date: 
Number of Pages: 
Logic. Argumentation and Reasoning 15
[Reviewed by
Salim Salem
, on

Structures and Algorithms is a collection of nine essays that discuss the role of mathematics in knowledge. In the first essay the author explains how the mathematical approach is important in other subjects, such as physics, biology and language. He discusses structures, algorithms, and how they are linked to other sciences.

The second essay he describes the relation between structures and science by looking at Leonardo da Vinci scientific works. The third essay is concerned with relationships between social and natural sciences. Here Fenstad discusses anthropological models and the taxonomy of mathematical models. He introduces notions like probability, chaos and catastrophes, and give economics as his example. He tries to find common ground between social and natural sciences.

In the fourth chapter, the author reviews recent trends in natural and biomedical science and the impact of their development on the future of research training. The fifth chapter contains a study of the relationship between language and logic and the development of grammar through the history.

Mathematical ideas are the topic of the sixth essay. A large part of it is concerned with L. E. J. Brouwer and David Hilbert’s ideas and their disagreement. It concludes the first part of the book

The second part contains three essays. The seventh chapter traces Tarski’s idea of truth and its relation to natural language through history. Logic and grammar are given in matrix and geometrical forms. The eighth chapter discusses formal semantics, geometry and the mind. The author recalls the matrix structure of grammar, then he study the ontology of formal semantics, explains the model theory of geometry, and ends with a good discussion of the relation between geometry and human mind.

The last chapter is a review of the notion of non-standard analysis. Fenstad tries to convince the reader of the importance of infinitesimals and the extreme applicability of this notion.

The book’s subject is important, but it is never clear what exactly the author means by structure. Is it what we see? Or is it the model theory’s structure? In addition, algorithms do not appear to be really important in the book’s argument: we hardly see their role in constructing or in learning knowledge.

One lovely thing about the book is that it gives good historical reviews of the development of logic, grammar, geometry, intuitionism and non-standard analysis. But the nice things are obscured by the large number of typos and the repetition of the same stories in different essays.

Salim Salem is Professor of Mathematics at the Saint-Joseph University of Beirut.

See the table of contents in the publisher's webpage.