# Welcome

**Klaus Truemper** is Professor Emeritus of Computer Science at the University of Texas at Dallas. He started out at that university in Mathematics and Computer Science, but now also focuses on questions of philosophy. Below is a recent book on the foundation of mathematics. See **More Books** and **Software** for additional material.

## The Construction of Mathematics: The Human Mind’s Greatest Achievement

Is mathematics created or discovered? The answer has been debated for centuries. This book answers the question clearly and decisively by applying the concept of language games, invented by the philosopher Wittgenstein to solve difficult philosophical issues.

Using the results of modern brain science, the book also explains how it is possible that eminent mathematicians and scientists offer diametrically opposed answers to the question of creation vs. discovery.

Interested in the topic but intimidated by mathematics? Not to worry. If you are familiar with the elementary operations of addition, subtraction, multiplication, and division, you can follow the arguments of this book.

## Reviews

“The question whether mathematics is discovered or created divides the mathematical community into two camps. Some mathematicians—like me—are in one camp in the morning and in the other one in the afternoon. My opinion, I confess, depends on the type of work being done at the moment. But deep in my mind, I am fully convinced that, based on some very elementary and not yet understood endowment of our brain, the fantastic mathematical universe is human-made.

This can’t be proved mathematically. The best one can hope for are compelling arguments and strong empirical evidence. This is what Klaus Truemper’s book *The Construction of Mathematics: The Human Mind’s Greatest Achievement* delivers. It sheds surprising and fascinating new light on the issue. Powerful arguments are provided by using the method of language games invented by the philosopher Ludwig Wittgenstein.

Employing results of modern brain science about human cognition, the book also explains how it is possible that eminent mathematicians and scientists arrive at diametrically opposed answers for the creation vs. discovery question.

Truemper’s findings are consistent with my ultimate conviction.”

** —Martin Grötschel, mathematician and President, Berlin-Brandenburg Academy of Sciences and Humanities, Germany**

“Klaus Truemper has made an original and daring attack on the foundations of mathematics. Readers will enjoy his forthright and unswerving analysis. His ideas should become recognized and influential.”

** —Reuben Hersh, mathematician and award-winning author of a number of books on the nature, practice, and social impact of mathematics**

“As computational methods become increasingly powerful, the engineer of today often resorts to simulation methods and is acutely aware of the limitations of contrived analytical mathematical methods. Truemper’s exposition puts into focus the debate as to whether mathematics is really intrinsic to the physical world or is in fact made up as we go along.

The elucidation the book delivers on this topic is of significance, as science often moves along faster when we are released from anachronistic notions that imprison freedom of thought. Klaus Truemper permits us to be more daring with our mathematics.”

** —Derek Abbott, physicist and engineer, University of Adelaide, Australia**

“Is mathematics the product of human creativity and ingenuity or does it exist independently of mankind and only waits to be discovered? Is the latter the reason for its incredible success or is mathematics not so indispensable in our world after all?

After a fascinating tour through the history of mathematics and computation that provides astonishing new perspectives, Klaus Truemper’s book addresses the philosophical question of creation versus discovery from many different directions ranging from Wittgenstein’s philosophy to brain science. Pros and cons are collected and discussed in a comprehensive and—given the difficult subject—remarkably light and entertaining way and: the book comes to a conclusion.

Is the issue finally settled? No! Can it possibly be? Most certainly not! The given circumstantial evidence does, however, stimulate the reader to rethink his or her point of view, question his or her own arguments, rethink the given reasoning and sharpen the issue. The book is a pleasure to read!”

** —Peter Gritzmann, mathematician, author of mathematics and popular science books**

“This wonderful book addresses the oldest and thorniest question in the philosophy of mathematics: Is mathematics discovery or invention? The key contribution of this book is to use Ludwig Wittgenstein’s technique of language games to shed light on this deep philosophical question. Overall this is the most insightful and compelling contribution to the debate that I have read and I am inclined to agree with the conclusions.

But the book is much more than an important contribution to a fundamental debate. It is beautifully and lucidly written and, in contrast to most texts on philosophical matters, is enjoyable and easy to read. Mathematicians know that a picture is worth a thousand words, and the rich use of beautifully presented illustrations genuinely enriches the readers experience.

The book begins with a short history of the major themes of mathematics. It is, by a country mile, the most insightful such history that I have read. An intelligent reader with no interest whatsoever in philosophy would still learn much from this book. I cannot recommend it too highly.”

** —Geoffrey Whittle, mathematician, Victoria University of Wellington, New Zealand**

“Klaus Truemper’s *The Construction of Mathematics: The Human Mind’s Greatest Achievement* is a unique blend of history and penetrating philosophical analysis. After taking the reader on a dizzying journey through the history of mathematics and computation, Truemper arrives at his central question: Is mathematics discovered or invented?

Relying on a range of philosophical approaches and some brilliant argumentation, he arrives at what seems like the only possible answer. Mathematics, he shows, is not “out there,” waiting to be discovered; it is, rather, the highest creation of the human mind.

Truemper’s book is not only insightful and original but also fast-paced and gripping. Whether you are a mathematician, historian, philosopher, or layman, you will find it thought-provoking as well as highly enjoyable.”

** —Amir Alexander, historian of science and award-winning author of books on the interconnection of mathematics and its social, cultural, and political settings**