Playing with the infinite. Mathematics for outsiders (Hungarian) (1944), by Rózsa Péter.**1.1. Review by: John George Kemeny.**

*The Journal of Symbolic Logic***13**(3) (1948), 141-142.This book is a popular account of modern mathematical ideas. The author states that her purpose is to reach that very large section of the population which always wanted to find out what modern mathematics was like, but thought that it was too difficult to understand. She attempts to give a clear picture of as many advanced concepts as possible without sacrificing rigour. The book is divided into three parts. The first two parts develop some of the fundamental ideas; the third part is devoted to mathematical logic. ... Mathematical ideas are illustrated by examples from a wide variety of fields. In the second part the emphasis is on the concept of infinity. The author makes many interesting remarks about the nature of mathematics throughout these two parts. The third part opens with a discussion of geometry. The development of non-Euclidean geometry is described, and this is followed by a few remarks on the fourth dimension. Set-theory is used as an introduction to the paradoxes. Russell's paradox in particular is singled out. The different attempts at a solution are discussed. This is followed by an introduction to formal logic, the propositional connectives, and quantifiers. These elementary ideas lead to a discussion of symbolic systems in general and of the whole axiomatic method. The author then makes a very interesting attempt at a popularization of some advanced ideas. She discusses consistency, the Hilbert program, Gödel's result about the impossibility of proving consistency, Gentzen's consistency proof, the continuum hypothesis, and the unsolvability of the general decision problem. The last chapter contains a detailed account of Gödel's incompleteness theorem. The book is remarkable in the clarity of presentation, even of the most complex ideas. The illustrations are simple and interesting. The author's humour makes every page enjoyable. The most interesting part of the book is the popular account of Gödel's very difficult theorem. The author seems to have found a perfect compromise between rigour and clarity.

Playing with infinity: Mathematical Explorations and Excursions (1961), by Rózsa Péter.**2.1. Review by: Reuben Louis Goodstein.**

*The Mathematical Gazette***46**(356) (1962), 157.This is easily the best book on mathematics for everyman that I have ever seen. The author is both a highly creative mathematician and an experienced teacher of young children, and this happy combination, allied to a gift for lucid exposition has produced a delightful book, which has been successfully translated from the Hungarian by Z P Dienes. Starting with counting, the chapters range over integers and rationals, with a passing glance at topology and a close look at the Calculus. From there we pass to non-Euclidean geometry and mathematical logic and end with an account of Gödel's construction of undecidable sentences; and all this is accomplished with a minimum of mathematical symbolism, A remarkable achievement.

**Playing with Infinity:**Mathematical Explorations and Excursions (1976), by Rózsa Péter.**3.1. Review by: Philip Peak.**

*The Mathematics Teacher***70**(3) (1977), 282.This is a translation of the author's first publication in 1943. Written for nonmathematicians, it is designed to help the reader enjoy and appreciate some mathematics without being involved in formulas and algorithms. The author succeeds fairly well. Section 1, "Playing with Fingers," emphasizes the key place of counting as a basis for mathematical thought. As concepts are developed, the author shows how they are related and how hard-to-define notions sometimes help resolve situations. The last section, "What Is Mathematics Not Capable Of?" clearly shows, as the author states, that "there is no system of axioms that can grasp quite tightly exactly what it is intended to circumscribe." This is a delightful book, and the mathematician as well as the layperson could profit from reading it.

**3.2. Review by: Michael Holt.**

*Mathematics in School***6**(3) (1977), 35.Written for the cultivated non-mathematician the book makes the sort of reading that could keep a solitary mind from going to pieces. Perhaps a copy, like the Gideon bibles in hotel rooms, could be placed in every modern prison cell! Professor Péter offers the reader a thoughtful panorama of the mathematical jungle without ever letting him trip over the undergrowth of chains of logical reasoning; she comes within an ace of offering instant insight. The book is amply illustrated and written in an engagingly fancy-free style that reflects, I suspect, something of the pedagogical methods we have come to expect of Hungarian educators. The book is written in three parts. Part 1 is called fancifully "The Sorcerer's Apprentice". It opens with finger counting, covers graphs, divisibility of whole numbers before daring, as the writer puts it, "to pluck up courage" to face divisions with remainders. In the next chapter the writer shows the link between geometry and arithmetic in the celebrated feat of the 9-year-old Gauss summing the numbers from 1 to 100. ... Part II, "The Creative Role of Form" looks deeply at the role of inverse operations, subtraction and directed numbers and that teachers' bane "minus times minus", here treated as the speeds (positive and negative) of a walker along a number line. Next follows fractions and how they do not fill the number line and a beautifully simple sketch of Cantor's proof that, contrary to appearances, there are as many positive integers as there are rational numbers (fractions). ... Part III, "The Self-Critique of Pure Reason", deals with the many different kinds of geometries, the fourth dimension, and mathematical logic. ... This is certainly a book for teachers to dip into if only to sample the Hungarian approach to mathematics education compared to the British, pragmatic approach.

Rekursive Funktionen (1951), by Rózsa Péter.**4.1. Review by: Raphael Mitchel Robinson.**

*The Journal of Symbolic Logic***16**(4) (1951), 280-282.The development of the theory of recursive functions may perhaps be said to have begun in 1923 with the publication of Skolem. The 1930's were especially fruitful in the development of this domain, seeing in particular the introduction of the concept of general recursive function, which gave for the first time a satisfactory formalization of the concept of computable function. But only now has the first book on recursive functions appeared. The book under review is a summary of known results in the field, many of which are due to the author herself. It treats all kinds of recursive definitions, from primitive to general, with considerable attention to the intermediate types. The question of which types of recursion are equivalent and which are more general is studied. Thus the book is particularly suited to the reader who wishes to explore all types of recursive definitions, rather than just primitive and general recursive functions. The book is entirely elementary, and no knowledge of recursive functions or of mathematical logic is presupposed. Adequate proofs are given, though the arguments are often simplified by showing how to make various reductions in typical special cases, rather than attempting to give a formal proof in the most general case.

**4.2. Review by: David Nelson.**

Mathematical reviews, MR0044467**(13,421l)**.This is the first book to appear on the theory of recursive functions, a topic of central interest to the foundations of mathematics and one which aroused general interest in its application to the incompleteness of arithmetic systems in the work of Gödel. While the author devotes one section to the history and applications of the theory and the final three sections to topics primarily of foundational interest, her major concern is the arithmetic aspect of the subject, especially the classification of types of recursive functions and theorems on the reduction of the schemata required to generate various classes of functions. In this field the book will be a valuable and authoritative reference work. The study proceeds through the consideration of primitive recursive functions to general recursive functions, discussing in detail the scale of k-fold recursive functions. A large part of the theory is made up of the author's own work which has appeared in a series of papers since 1932. ... The book is clearly and concisely written and will be useful as an introduction to the subject as well as a reference work.

**4.3. Review by: Stephen Kleene.**

*Bull. Amer. Math. Soc.***58**(1952), 270-272.Beginning in 1932, Rósza Péter has published a series of papers, examining the relationship of various special forms of recursion, and showing the definability of new functions by successively higher types of recursion, which establish her as the leading contributor to the special theory of recursive functions. ... In line with Ms Péter's research interests, the special theory occupies about two-thirds of the present book ... The aim of the book is primarily to give an elementary exposition of the existing theory, rather than to push into new territory. But the part dealing with the special theory is especially complete, and the latter chapters of this part contain material not covered or covered only summarily in the literature. It is of great value to have in this book for the first time a connected account of the special theory. ... In writing this book Ms Péter has carried out a considerable undertaking; and to go further would have constituted a still greater one, and required either a much larger book or a more compact style. Only a minimum of knowledge of elementary number theory, analysis, and set theory including transfinite ordinals is presupposed and none of mathematical logic. Ms Péter aims to make the subject intelligible to the beginner by working out the treatment of many topics (particularly in the special theory) on an example, whence the reader can surmise how the treatment would go in general (or consult the literature). This method has both advantages and disadvantages. No student can complain that he has lost contact with the reality for want of concrete examples; but an unwary reader may be oppressed by the immense amount of detail involved in working out the examples and proofs.

Rekursive Funktionen (2nd edition) (1957), by Rózsa Péter.**5.1. Review by: Raphael Mitchel Robinson.**

*The Journal of Symbolic Logic***23**(3) (1958), 362-363.The second edition of this book differs but little from the first edition, the increase from 206 pages to 278 pages being due mainly to the fact that there are fewer words per page in the new edition. However, the author has made some corrections, the bibliography is considerably expanded, and at least ten pages of new material have been added to the text.

Rekursive Funktionen in der Komputer-Theorie (1976), by Rózsa Péter.**6.1. Review by: Ivan Hal Sudborough**

Mathematical reviews, MR0433956**(55 #6926).**The objective of this book is principally to show that many of the definitional structures used in programming languages can be expressed formally as partial recursive functions and that these partial recursive functions can be implemented by straightforward techniques on a random access machine with a simple assembly language instruction set. The author shows that many of these definitional structures are primitive recursive and that others can be expressed as an unbounded minimalization of a primitive recursive predicate.

Recursive functions in computer theory (1981), by Rózsa Péter.English translation of the German text.