The European Mathematical Society

As the title suggests this book discusses the subject of logic. It begins with the basic elements but it also introduces new elements such as a logic of exceptions. Hence the last part of the book requires a more advanced level. It also includes part of the exciting history of philosophy, mathematics and logic, that usually doesn’t appear in other logic texts.
The examples are written in Mathematica which is a language or system for doing mathematics on the computer, but you can still read and understand this book even if you are not familiar with Mathematica and you decide not to run the programs. In any case the basics are supplied.
The book is organised in three parts:
Part I – The basic propositional operators are introduced.
Part II – This part deals with the basic notions of inference: inferences, syllogisms, axiomatics and proof theory. It is first developed for propositional logic and then for predicate logic. For the first time, the existence of sentences that require some three-valued logic is discussed. (There are propositions that are either true or false (two-valued) and sentences that may be true, false or indeterminate.)
It also includes a short history of the Liar paradox , which is the root of many other paradoxes and its different solution approaches such as the theory of types, proof theory and three-valued logic. It analyses the problems which arise in the first two approaches, so that the author defends that only three-valued logic allows for a consistent development.
The historical review of the Liar shows how various cases propose the introduction of an exception as part of the proposed solution. Hence a logic of exceptions is developed.
Part III - This part deals with three-valued logic. First of all, the author discusses the need to introduce a third value different to true or false. Once again the basic operators are considered and its definitions modified because of this third value.
Finally three-valued logic is applied to the Liar paradox, which is now solved. Gödel’s theorems, and Brouwer’s intuitionism are revisited.

The author begins with a brief description of Hungary at the end of the XIX century and Lanczos’ family. Then she goes into the early works of Lanczos on Relativity and Quantum Mechanics. Then she writes about the american tour of Lanczos from Purdue University (Indiana) to Boeing in Seattle and the interest Lanczos paid to numerical and Fourier analysis. Finally, we learn about his come back to Europe at the Dublin Institute for Advanced Studies, where he continued his research on Relativity and the Unified Field Theory.

The reader will find many curious things about Lanczos' private life, which are very useful to grasp that Lanczos was a humble person, involved with pedagogical questions on the teaching and learning of science and so we can read on the preface of his book “The variational principles of mechanics” the following:
“Many of the scientific treatises of today are formulated in a half-mystical language, as though to impress the reader with the uncomfortable feeling that he is in the permanent presence of a superman. The present book is conceived in a humble spirit and is written for humble people.”
Also H. Goldstein refers to that book in a manner which reflects the character of Lanczos. He say in his famous book “Classical Mechanics”:
“Lanczos has a different point of view (from the rest of the authors of mechanical treatises); he talks a lot and writes few equations”

On the other side, I miss some chapter dedicated to show and explain a little bit further the mathematical and physical work of Lanczos. Perhaps, it would be nice to have some formulas for trained readers, for instance, his description of the electromagnetic field with quaternions, or a much more deep explanation of his operator of quantum mechanics in contrast to the Schrodinger equation. Also some examples of his use of the Fast Fourier Transformation in numerical examples would be fine.
Nonetheless, she gives detailed references of the cites she uses throughout the book.

Ultimately, it is very pleasant to read as it is written in a very concise manner.

This volume has been edited in commemoration of the 100th anniversary of the birth of the eminent mathematician and physicist Stanislaw Ulam (April 3, 1909 to May 13, 1984). The editors of the volume are Themistocles M. Rassias (National Technical University of Athens, Greece) and Janusz Brzdek (Pedagogical University of Krakow, Poland).

The volume consists of 47 articles written by experts who present research works in the field
of Mathematical Analysis and related subjects, in particular, in Functional Equations and
Inequalities. These works provide an insight in the study of various problems of nonlinear
analysis. Several of the results have been influenced by the work of S. Ulam. In particular,
a special emphasis is given to one of his questions concerning approximate homomorphisms.

The book is divided in two parts. Part I focuses on several aspects of the Ulam stability
theory. The original stability problem was posed by S. M. Ulam in 1940 and concerned
approximate homomorphisms. The pursuit of solutions to this problem, but also to its
generalizations and modifications for various classes of (difference, functional, differential, and integral) equations and inequalities, is an expanding area of research and has led to the development of what is now known as the Hyers-Ulam stability theory.

Part II consists of papers on various subjects of Mathematical Analysis, mainly
Functional Analysis, Inequalities, and Geometric Analysis.

All together, it is a very nice volume collecting a lot of material of high level, which can be of interest to researchers in different areas related to Ulam's work.