《The Foundations of Mathematics》PDF电子书下载
《The Foundations of Mathematics》内容简介
Ian Stewart / Oxford University Press / 2015-5-1 / GBP 14.99 -
PREFACE TO THE FIRST EDITION
This book is intended for readers in transition from school mathematics to the fully-fledged type of thinking used by professional mathematicians. It should prove useful to first-year students in universities and colleges, and to advanced students in school contemplating further study in pure mathematics. It should also be of interest to a wider class of reader with a grounding in elementary mathematics seeking an insight into the foundational ideas and thought processes of mathematics.
The word ‘foundations’, as used in this book, has a broader meaning than it does in the building trade. Not only do we base our mathematics on these foundations: they make themselves felt at all levels, as a kind of cement which holds the structure together, and out of which it is fabricated. The foundations of mathematics, in this sense, are often presented to students as an extended exercise in mathematical formalism: formal mathematical logic, formal set theory, axiomatic descriptions of number systems, and technical constructions of them; all carried out in an exotic and elaborate symbolism. Sometimes the ideas are presented ‘informally’ on the grounds that complete formalism is too difficult for the delicate flowering student. This is usually true, but for an entirely different reason.
A purely formal approach, even with a smattering of informality, is psychologically inappropriate for the beginner, because it fails to take account of the realities of the learning process. By concentrating on the technicalities, at the expense of the manner in which the ideas are conceived, it presents only one side of the coin. The practising mathematician does not think purely in a dry and stereotyped symbolism: on the contrary, his thoughts tend to concentrate on those parts of a problem which his experience tells him are the main sources of difficulty. While he is grappling with them, logical rig- our takes a secondary place: it is only after a problem h
PREFACE TO THE FIRST EDITION
This book is intended for readers in transition from school mathematics to the fully-fledged type of thinking used by professional mathematicians. It should prove useful to first-year students in universities and colleges, and to advanced students in school contemplating further study in pure mathematics. It should also be of interest to a wider class of reader with a grounding in elementary mathematics seeking an insight into the foundational ideas and thought processes of mathematics.
The word ‘foundations’, as used in this book, has a broader meaning than it does in the building trade. Not only do we base our mathematics on these foundations: they make themselves felt at all levels, as a kind of cement which holds the structure together, and out of which it is fabricated. The foundations of mathematics, in this sense, are often presented to students as an extended exercise in mathematical formalism: formal mathematical logic, formal set theory, axiomatic descriptions of number systems, and technical constructions of them; all carried out in an exotic and elaborate symbolism. Sometimes the ideas are presented ‘informally’ on the grounds that complete formalism is too difficult for the delicate flowering student. This is usually true, but for an entirely different reason.
A purely formal approach, even with a smattering of informality, is psychologically inappropriate for the beginner, because it fails to take account of the realities of the learning process. By concentrating on the technicalities, at the expense of the manner in which the ideas are conceived, it presents only one side of the coin. The practising mathematician does not think purely in a dry and stereotyped symbolism: on the contrary, his thoughts tend to concentrate on those parts of a problem which his experience tells him are the main sources of difficulty. While he is grappling with them, logical rig- our takes a secondary place: it is only after a problem h