The Search for Certainty: A Philosophical Account of Foundations of Mathematics [Paperback]

Marcus Giaquinto (Author)

Book Description

Publication Date: July 29, 2004 | ISBN-10: 0198752458 | ISBN-13: 978-0198752455 | Edition: First Paperback Edition

Marcus Giaquinto tells the compelling story of one of the great intellectual adventures of the modern era: the attempt to find firm foundations for mathematics. From the late nineteenth century to the present day, this project has stimulated some of the most original and influential work in logic and philosophy.

Editorial Reviews

Review

`Review from previous edition Giaquinto has provided a careful and judicious discussion and analysis of these matters, supplying needed technical background for readers who are not mathematicians ... Readers of this book will be well prepared to follow the current literature on foundations of mathematics.' Martin Davis, American Scientist

About the Author

Marcus Giaquinto is in the Department of Philosophy, University College London.

Product Details

• Paperback: 304 pages
• Publisher: Oxford University Press, USA; First Paperback Edition edition (July 29, 2004)
• Language: English
• ISBN-10: 0198752458
• ISBN-13: 978-0198752455
• Product Dimensions: 9.2 x 6.1 x 0.8 inches
• Shipping Weight: 15.2 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #2,451,882 in Books (See Top 100 in Books)

More About the Author

Discover books, learn about writers, read author blogs, and more.

Customer Reviews

Most Helpful Customer Reviews

7 of 9 people found the following review helpful:

5.0 out of 5 stars Exceptionally well-written, November 22, 2004

By 

Christopher Grant (Salem, UT USA) - See all my reviews
(REAL NAME)   

Amazon Verified Purchase

This review is from: The Search for Certainty: A Philosophical Account of Foundations of Mathematics (Paperback)

It is not unusual for professional mathematicians to make it through their entire schooling without much in the way of formal instruction in the foundations of math. If, like me, you fall into this category, or if you just enjoy reading well-written books, I recommend Giaquinto's book most strongly.

The author has performed an impressive balancing act. He manages to treat details precisely without being pedantic. He doesn't shortchange history, but he also doesn't permit the pursuit of historical authenticity to interfere with clarity of exposition.

I am grateful to the author for having written this book, and I think he deserves to be proud of his accomplishment.

6 of 8 people found the following review helpful:

5.0 out of 5 stars Good history of Foundations Search In mathematics, February 7, 2005

By 

P. Nagy "revreader" (Chapel Hill, NC USA) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: The Search for Certainty: A Philosophical Account of Foundations of Mathematics (Paperback)

The Search for Certainty: A Philosophical Account of Foundations of Mathematics by Marcus Giaquinto (Oxford University Press) (Hardcover) [...]In the early decades of the twentieth century, mathematicians showed an unprecedented concern for the foundations of their subject, not just in expres¬sions of disquiet but also in attempts to find a secure basis. This search for cer¬tainty and the crisis that sparked it off is the central subject of this book. First Giaquinto shows mathematical setting of this story to see how the foundational accomplishments grew out of the nineteenth-century quest for clarity and rigor in mathematics.
The clarification of basic properties and relations of analysis set the tone for the search for certainity. The objects of analysis-real numbers and more generally points, classes of points, and functions on classes of points-were taken for granted. But in the later decades of the nineteenth century, according to Giaquinto, mathematicians came to feel that an explicit account of real numbers was needed. also an account of the transfinite numbers discovered-or, some would say, invented by Cantor is explored.
Giaquinto first explains the two best-known accounts of real numbers. Next, he presents a sketch of the way in which the ideas for the transfinite ordinal and cardinal number systems grew out of the study of classes of points, and the rudiments of those number systems are presented. After which he looks at accounts of the natural numbers.
Towards the end of the nineteenth century, the drive for clarity and rigor seemed to be reaching a successful conclusion. Among its fruits were precise accounts of the real and natural numbers, the first general theory of transfinite classes and numbers, and a first account of quantifier logic-no meager harvest. But celebrations had barely begun when certain paradoxes were found in the general theory of classes, which was the basis for all supposedly rigorous accounts of the number systems. This defeat in the hour of triumph made foundational research a major area of concern for mathematicians.
Deeper excavation was needed, and the younger mathematicians who took up the task intended to reach bedrock. So the drive to find sure foundations for mathematics issued largely from problems internal to mathematics, together with the conviction that, if certainty is to be found anywhere, it is to be found in mathematics. In this way, the mathematical concern was tied to a philosophical one: how can we be certain that the theorems of mathematics are trustworthy? The bulk this book examines the attempts to meet this challenge.
The central concern of this book is the epistemic status of non-finitary mathematics. Epistemology is not the only concern in foundational studies, though it has been dominant. The nature and intrinsic organization of math¬ematics has also been a major concern. Later developments in mathematics show that set theory is not the only basis for this kind of inquiry. Of course, those who think that true mathematics must be constructed will reject not only classical set theory but also the nineteenth-century mathematics out of which it grew. In this regard the development of constructive analysis can be regarded as partial fulfillment of an alternative foundational program. Giaquinto is not able to evaluate the success and significance of this program, and perhaps we are too close to see all of what needs to be seen. For those who accept classical mathematics, category theory has been offered as an alternative to set theory for its catholic reach. Mathematics is definitely not just logic, not just higher-order logic, not just set theory. The old picture of a single fundamental theory to which all else must be reduced has faded. If pure mathematics is the study of abstract structures, set theory is just one framework among others for thinking about that subject matter, and it may not be the best. Universes of sets are themselves structures, and these may be instances of something more general, as is suggested by topos theory. In addition, topos theory sheds new light on the intrinsic organization of mathematics, revealing a surprising unity between apparently disparate fields, topology and algebraic geometry on the one hand, and logic and set theory on the other.
The initial impulse for foundational study was the need to clarify our understanding of the continuum and the basis of infinitesimal calculus. The standard set-theoretic account is an explication that has served well - wit¬ness the use made of it in classic textbooks on analysis. But now there are other explications of the basic intuitions. Robinson's non-standard analysis rehabilitated infinitesimals. Non-classical accounts include intuitionistic analysis and Bishop's constructive analysis. The development of synthetic differential geometry gives yet another perspective on the continuum and a novel theory of infinitesimals. Thus we now have a plurality of mathemati¬cal ways of refining and abstracting from what are originally spatial intu¬itions. This too is a way in which foundational study has spread out and away from the monolithic view.
If new developments within mathematics advance our understanding of the nature and intrinsic organization of mathematics, epistemological advances are likely to come from developments outside. In the period covered by Giaquinto in this book, the central epistemological concern has been to justify a body of mathematics. Another concern is to explain how it is possible for an individ¬ual to have mathematical knowledge and understanding. This inquiry needs fine-grained information about how we actually acquire our mathematical beliefs and abilities; then we can investigate how best to evaluate those modes of cognitive growth in epistemic terms. The empirical input must come primarily from cognitive sciences. Investigations of simple numerical abilities have already proved fruitful, aided by a recent confluence of evidence from different sources: experiments on healthy adults, children, and even infants, clinical tests on brain-damaged patients, brain imaging studies, and animal studies. There is still a long way to go. The history of mathematics is anoth¬er source of information.