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Krantz takes the reader on a journey around the globe and through centuries of history , exploring the many transformations that mathematical proof has undergone from its inception at the time of Euclid and Pythagoras to its versatile, present-day use . The author elaborates on the beauty, challenges and metamorphisms of thought that have accompanied the search for truth through proof.

The first two chapters examine the early beginnings of concept of proof and the creation of its elegant structure and language, touching on some of the logic and philosophy behind these developments. The history then unfolds as the author explains the changing face of proofs. The more well-known proofs , the mathematicians behind them, and the world that surrounded them are all highlighted . Each story has its own unique past; there was often a philosophical, sociological, technological or competitive edge that restricted or promoted progress. But the author's commentary and insights create a seamless thread throughout the many vignettes.

Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to date. This is shown in noting some of the more prominent discussions currently underway, such as Gorenstein's effort to classify finance groups, Thomas Hale's resolution of the Kepler sphere-packing problem, and other modern tales. Most of the proofs are discussed in detail with figures and

some equations accompanying them, allowing both the professional mathematician and those less familiar with mathematics to derive the same joy from reading this book.

Inhaltsverzeichnis

Frontmatter

1. What Is a Proof and Why?

Abstract
A well-meaning mother was once heard telling her child that a mathematician is someone who does “scientific arithmetic.” Others think that a mathematician is someone who spends all day hacking away at a computer.
Steven G. Krantz

2. The Ancients

Abstract
Perhaps the first mathematical proof in recorded history is due to the Babylonians. They seem (along with the Chinese) to have been aware of the Pythagorean theorem (discussed in detail below) well before Pythagoras.29 The Babylonians had certain diagrams that indicate why the Pythagorean theorem is true, and tablets have been found to validate this fact.30
Steven G. Krantz

3. The Middle Ages and An Emphasis on Calculation

Abstract
The history of the Near East and the Muslims is complex, and there are many gaps. The Islamic religion was born in Arabia, and lives on there today. What some people call “Arabic mathematics” others would call “Islamic” or “Muslim” mathematics. One of the principal figures in our discussion here is Muhammad ibn Musa al-Khwarizmi. As far as we know, hewas born in Baghdad around 780 CE. Thus it is not out of place to call him an Arab; but he is more commonly referred to as an “Islamic mathematician,” or a “Persian mathematician.” It appears that the Muslim culture was a driving force in many of the developments of the Middle Ages. So we shall adopt the policy of referring to “Islamic mathematics.”
Steven G. Krantz

4. The Dawn of the Modern Age

Abstract
Leonhard Euler (1707–1783)was one of the greatest mathematicians who ever lived. Hewas also one of the most prolific. His collected works comprise more than 70 volumes, and they are still being edited today. Euler worked in all parts of mathematics, as well as mechanics, physics, and many other parts of science. He did mathematics almost effortlessly, often while dangling a grandchild on his knee. Late in life he became partially blind, but he declared that this would help him to concentrate more effectively, and his scientific output actually increased.
Steven G. Krantz

5. Hilbert and the Twentieth Century

Abstract
Along with Henri Poincar’e (1854–1912) of France, David Hilbert (1862–1943) of Germany was the spokesman for early twentieth century mathematics. Hilbert is said to have been one of the last mathematicians to be conversant with the entire subject—from differential equations to analysis to geometry to logic to algebra. He exerted considerable influence over all parts of mathematics, and he wrote seminal texts in many of them. Hilbert had an important and profound vision for the rigorization of mathematics (one that was later dashed by the work of Bertrand Russell, Kurt Gödel, and others), and he set the tone for the way that mathematics was to be practiced and recorded in our time.
Steven G. Krantz

6. The Tantalizing Four-Color Theorem

Abstract
In 1852 Francis W. Guthrie, a graduate of University College London, posed the following question to his brother Frederick: Imagine a geographic map of the earth (i.e., a sphere) consisting of countries only—no oceans, lakes, rivers, or other bodies of water. The only rule is that a country must be a single contiguous mass—in one piece, and with no holes [see Figure 6.1]. As cartographers, we wish to color the map so that no two adjacent countries will be of the same color [Figure 6.2—note that R, G, B, Y stand for red, green, blue, and yellow].
Steven G. Krantz

7. Computer-Generated Proofs

Abstract
The first counting devices were counting boards. The most primitive of these may be as old as 1200 BCE and consisted of a board or stone tablet with mounds of sand in which impressions or marks could be made. Later counting boards had grooves or metal disks that could be used to mark a position. The oldest extant counting board is the Salamis tablet, dating to 300 BCE. See Figure 7.1.
Steven G. Krantz

8. The Computer as an Aid to Teaching and a Substitute for Proof

Abstract
Geometer’s Sketchpad is a (software) learning tool, marketed by Key Curriculum Press, designed for teaching Euclidean geometry to high school students. There has been a great trend in the past twenty-five years to reverse-engineer high school geometry so that proofs are deemphasized and empiricism and speculation more highly developed. Geometer’s Sketchpad fits in very nicely with this program.
Steven G. Krantz

9. Aspects of Modern Mathematical Life

Abstract
As this book has endeavored to describe, prior to two hundred years ago the mathematician was of necessity a dedicated, independent, largely self-sufficient lone wolf. There were few academic positions and no granting agencies (of the government or otherwise). If a mathematician was extraordinarily lucky, a patron could be found who would provide financial and other support. Euler and Descartes had patrons. Galois and Riemann did not.
Steven G. Krantz

10. Beyond Computers: The Sociology of Mathematical Proof

Abstract
The idea of “group” came about in the early nineteenth century. A product of the work of Évariste Galois (1812–1832) and Augustin-Louis Cauchy, this was one of the first cornerstones of what we now think of as abstract algebra.
Steven G. Krantz

11. A Legacy of Elusive Proofs

Abstract
Bernhard Riemann (1826–1866) was one of the true geniuses of nineteenth century mathematics. He lived only until the age of 39, ultimately defeated by poverty and ill health. He struggled all his life, and finally landed a regular professorship only when he was on his deathbed. But the legacy of profound mathematics that Riemann left continues to have a major influence in the subject.
Steven G. Krantz

12. John Horgan and “The Death of Proof?”

Abstract
In 1993 John Horgan, a staff writer for Scientific American, published an article called The Death of Proof? [HOR1]. In this piece the author claimed that mathematical proof no longer had a valid role in modern thinking. There were several components of his argument, and they are well worth considering here.
Steven G. Krantz

13. Closing Thoughts

Abstract
Before proofs, about 2600 years ago, mathematics was a heuristic and phenomenological subject. Spurred largely (though not entirely) by practical considerations of land surveying, commerce, and counting, there seemed to be no real need for any kind of theory or rigor. It was only with the advent of abstract mathematics—or mathematics for its own sake—that it began to become clear why proofs are important. Indeed, proofs are central to the way we now view our discipline.
Steven G. Krantz

Backmatter

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