What is Mathematical Logic?

by John Newsome Crossley, C.J. Brickhill, J.C. Stillwell

Although mathematical logic can be a formidably abstruse topic, even for mathematicians, this concise book presents the subject in a lively and approachable fashion. It deals with the very important ideas in modern mathematical logic without the detailed mathematical work required of those with a professional interest in logic.
The book begins with a historical survey of the development of mathematical logic from two parallel formal deduction, which originated with Aristotle, Euclid, and others; and mathematical analysis, which dates back to Archimedes in the same era. The streams began to converge in the seventeenth century with the invention of the calculus, which ultimately brought mathematics and logic together. The authors then briefly indicate how such relatively modern concepts as set theory, Gödel's incompleteness theorems, the continuum hypothesis, the Löwenheim-Skolem theorem, and other ideas influenced mathematical logic.
The ideas are set forth simply and clearly in a pleasant style, and despite the book's relative brevity, there is much covered on these pages. Nonmathematicians can read the book as a general survey; students of the subject will find it a stimulating introduction. Readers will also find suggestions for further reading in this lively and exciting area of modern mathematics.

Genres: Logic, Mathematics, Philosophy

96 pages, Paperback

First published December 28, 1972

About the author

John Newsome Crossley, DPhil, MA (Oxon), (born 1937, Yorkshire, England) is a British-Australian mathematician and logician who writes in the field of logic in computer science, history of mathematics and medieval history. He is involved in the field of mathematical logic in Australia and South East Asia.

As of 2010, Crossley is Emeritus Professor of Logic at Monash University, Australia where he has been connected since 1968. Crossley studied at Oxford University where he received his DPhil and MA (Mathematics) in 1963. His early career was spent at Oxford where he was the first university lecturer in mathematical logic and was a Fellow of All Souls College, Oxford. He is still a Quondam Fellow there. He was offered a Readership position and following a lecturing visit to Monash University in 1968, he was elected to a Chair in Pure Mathematics. He accepted this position and as of 2010, Crossley continues to be active at Monash University where he serves through its Faculty of Information Technology.

Crossley has written books in logic, mathematics and computer science. He is known as the lead author of the book What is Mathematical Logic. Co-written by some of his students, the book popularised the subject to the interested layman. Many of Crossley's doctoral students have gone on to be professors themselves and have written books in the field of mathematics or computing, including Peter Aczel, Wilfrid Hodges, John Lane Bell and Rod Downey.

Crossley is also an avid photographer. In 1974 he first exhibited his photographs in Melbourne and again 2005 he exhibited Composition and Context, a collection of photographs shot by Crossley around the world that illustrates the title and theme of the exhibition. A number of these photographs since have appeared in publications in Australia, Britain and the Philippines.

Reviews

[Philip Naw · Rating 4 out of 5]

A review of this book on Amazon said something like this:

'This book is a tad bit advanced for the beginner, yet too elementary for anyone who could read it.'

This is one partial downfall. However, for those who know just a little, this book is extremely stimulating. So rather than a downfall, to the right student(1 semester of logic), this book could be very helpful. The author spans upon several key points of 20th century logic, in a quick, dense, but very informal fashion.

The introduction of the book acts as if the book is nice and easy reading, but the book is tough! Do not underestimate this read. The first chapter does not require any prerequisite knowledge, but the other chapters are difficult and quick.

I thought about giving this book only a 3 star, but I remember the reading experience being so enjoyable. With all of its oddities, this book brings back good memories to when I first read the phrase 'Cantor's Continuum Problem.' So extra star for nostalgia!

[James Lancelot · Rating 2 out of 5]

My main criticism of this book is that it seemed to be too terse. For a book with an informal title like, "What is Mathematical Logic", I thought this would be a light read. The structure of the book seemed to be: idea introduction; idea formalization in a predicate calculus; proof of idea; repeat. It wasn't always obvious to me why the particular topic was chosen, how they led into one another or even why the proof was in fact a proof.

The authors' stated aim was "to introduce the very important ideas in modern mathematical logic without the detailed mathematical work which is required of those with a professional interest in logic", which also made this book seem simpler that it was.

I'm certain that I just don't understand mathematical logic well enough to judge this book fairly, but that's really my main criticism: for what seemed like it was supposed to be an introduction to the topic, it's an awfully tough read.

[Jamal Rahman · Rating 3 out of 5]

3.5

I am trying to delve into other subjects to widen my knowledge and whilst this book does explain mathematical logic pretty good, it is not explained the best.

Of course this is an advanced topic, but I am referring to how they presented the information. Einstein once said if you are truly smart you can present advanced topics in a simple manner.

[Latchezar Tomov · Rating 4 out of 5]

The chapter about set theory is the best written, except for the forcing part, which could be a little more expanded as explanations. Overall, this is very good way to trigger interest in logic for the student how knows a little bit of it.

[Yvan Moses Levy · Rating 5 out of 5]

Hard, clear and concentrate

A very valuable lecture. There are a lot of typo in kindle version. The chapter on Turing's machines is not easy to understand.

[Maurizio Codogno · Rating 4 out of 5]

Non è semplicissimo catalogare questo libro. La parte centrale del testo è molto tecnica, ahimè con qualche refuso che non so se sia poi stato corretto nelle edizioni successive alla prima che è entrata in mio possesso: ma il capitolo introduttivo fa una storia della logica con un linguaggio colloquiale e allo stesso tempo chiaro - un bravo alla traduttrice Teresa Pallucchini - e il capitolo finale dà un'ottima ideadi come si sia arrivati agli assiomi ZF per la teoria degli insiemi e cerca persino di gettare qualche luce sulle tecniche di forcing che dieci anni prima avevano permesso a Cohen di dimostrare l'indipendenza di assioma della scelta e ipotesi del continuo. Direi che per chi non è un patito della logica matematica queste due parti sono sufficienti per apprezzare il testo.