Truth In Mathematics
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Truth in Mathematics
The nature of truth in mathematics is a problem which has exercised the minds of thinkers from at least the time of the ancient Greeks. The great advances in mathematics and philosophy in the twentieth centuryand in particular the proof of Gödel's theorem and the development of the notion of independence in mathematicshave led to new viewpoints on this question in our era. This book is the result of the interaction of a number of outstanding mathematicians and philosophersincluding Yurii Manin, Vaughan Jones, and Per Martin-Löfand their discussions of this problem. It provides an overview of the forefront of current thinking, and is a valuable introduction and reference for researchers in the area.
How Mathematicians Think
Author: William Byers
language: en
Publisher: Princeton University Press
Release Date: 2010-05-02
To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself.
Another Sort of Mathematics
Years ago, James V. Schall wrote Another Sort of Learning, a book listing those things you should read but probably were never required to read. It is not a curriculum, except maybe one “for life.” This book you have in your hands is something of a mathematical tribute to Schall’s basic idea and is aptly titled Another Sort of Mathematics. Like Schall’s book, it is not a curriculum. It is, however, a list of some things from mathematics you should experience but probably were never required to experience. The theorems and proofs in this book represent, in a small way, some of the best that has been said within the discipline of mathematics. There is something unique in the human soul that can only be satisfied by wondering about mathematics. And that means, regardless of your background, this book is for you. Reclaim your mathematical inheritance. Embrace the mathematician within you. Choose to wonder.