Mathematics & Logic

by Mark Kac

What is mathematics? How was it created and who were and are the people creating and practicing it? Can one describe its development and role in the history of scientific thinking and can one predict the future? This book is a thought provoking attempt to answer such questions and to suggest the scope and depth of the subject. The volume begins with a discussion of problems involving integers in which ideas of infinity appear and proceeds through the evolution of more abstract idea about numbers and geometrical objects. The authors show how mathematicians came to consider groups of general transformations and then, looking upon the sets of such subjects as spaces, how they attempted to build theories of structures in general. Also considered here are the relations between mathematics and the empirical disciplines, the profound effect of high-speed computers on the scope of mathematical experimentation and the question of how much mathematical progress depends on "invention" and how much on "discovery". For mathematicians, physicists or any student of the evolution of mathematical thought, this highly regarded study offers a stimulating investigation of the essential nature of mathematics.

Genres: Mathematics, Logic, Philosophy, Science

Mass Market Paperback

First published January 1, 1968

Reviews

[J. · Rating 2 out of 5]

This had some good stuff in it, but I'm not sure who the intended target audience was. It suffered from deep-and-wide syndrome, where it's too broad for a specialist and too deep for an introduction.

[Mark · Rating 4 out of 5]

Parts were fascinating, but the whole did not hold together. It was a disjointed read.

[Anthony O'Connor · Rating 3 out of 5]

A couple of mathematicians writing a brief essay on 'What is Mathematics?' We are promised a look into the role of logic, the use of infinity, and a look at how it is applied to describing the physical world. Sounds all right.
But. NO! We get very little of the above. In fact the authors explicitly refuse to define/describe what mathematics is under the glib assertion that any such definition is too limiting. Instead we are treated to a scattered and superficial treatment of a range of mini-topics they find interesting. And some of them are. But that's not the point. And I will kick (figuratively speaking only of course ) the next person who brings up the tired old repetition of Eugene Wigner's "the unreasonable effectiveness of mathematics". Which is just so stupid. If mathematics is the generalization, abstraction and idealization of our innate notions of classification and ordering et al evolved in brains fighting to adapt to the world then how could it possibly FAIL to be effective. Or putting it a completely (?) different way if mathematics is the abstract science of possible forms/patterns/structures and the physical world is an instantiation of some of the these structures then once again how could mathematics fail to be effective. I think people who repeat this tired old trope are just impressed by the unexpected efficacy of arcane notations. But that is to miss the real point.
Anyway as a quick and scattered survey of some elementary and interesting mathematical results this book is OK. As anything else it isn't.

[Lucille Nguyen · Rating 3 out of 5]

This book suffers from the problem that many technical books face: they seek to make it readable, so they don't go into detail (perhaps exemplified by a rather pitiful non-attempt at handwaving Godel's incompleteness theorem in a couple pages); but simultaneously they get too into the weeds of the topics they are most familiar. Nonetheless, there is a rather fascinating history of mathematics and a wide sweep of concepts covered that one does walk away from this rather disjointed read with some additional insight into this great field.