Wilbur Knorr
Encyclopedia
Wilbur Richard Knorr was an American historian of mathematics
and a professor in the departments of philosophy and classics at Stanford University
. He has been called "one of the most profound and certainly the most provocative historian of Greek mathematics" of the 20th century.
from 1963 to 1966 and stayed there for his Ph.D., which he received in 1973 under the supervision of John Emery Murdoch and G. E. L. Owen
. After postdoctoral studies at Cambridge University, he taught at Brooklyn College
, but lost his position when the college's Downtown Brooklyn campus was closed as part of New York's mid-1970's fiscal crisis. After taking a temporary position at the Institute for Advanced Study
, he joined the Stanford faculty as an assistant professor in 1979, was tenured there in 1983, and was promoted to full professor in 1990.
He died March 18, 1997 in Palo Alto, California
, of melanoma
.
Knorr was a talented violinist, and played first violin in the Harvard Orchestra, but he gave up his music when he came to Stanford, as the pressures of the tenure process did not allow him adequate practice time.
s from their first discovery (in Thebes
between 430 and 410 BC, Knorr speculates), through the work of Theodorus of Cyrene
, who showed the irrationality of the square roots of the integers up to 17, and Theodorus' student Theaetetus
, who showed that all non-square integers have irrational square roots. Knorr reconstructs an argument based on Pythagorean triple
s and parity that matches the story in Plato
's Theaetetus
of Theodorus' difficulties with the number 17, and shows that switching from parity to a different dichotomy in terms of whether a number is square or not was the key to Theaetetus' success. Theaetetus classified the known irrational numbers into three types, based on analogies to the geometric mean
, arithmetic mean
, and harmonic mean
, and this classification was then greatly extended by Eudoxus of Cnidus
; Knorr speculates that this extension stemmed out of Eudoxus' studies of the golden section.
Ancient Sources of the Medieval Tradition of Mechanics: Greek, Arabic, and Latin studies of the balance.
The Ancient Tradition of Geometric Problems.:This book, aimed at a general audience, examines the history of three classical problems from Greek mathematics
: doubling the cube
, squaring the circle
, and angle trisection
. It is now known that none of these problems can be solved by compass and straightedge
, but Knorr argues that emphasizing these impossibility results is an anachronism due in part to the foundational crisis in 1930s mathematics. Instead, Knorr argues, the Greek mathematicians were primarily interested in how to solve these problems by whatever means they could, and viewed theorem and proofs as tools for problem-solving more than as ends in their own right.
Textual Studies in Ancient and Medieval Geometry.:This is a longer and more technical "appendix" to The Ancient Tradition of Geometric Problems in which Knorr examines the similarities and differences between ancient mathematical texts carefully in order to determine how they influenced each other and untangle their editorial history. One of Knorr's more provocative speculations in this work is that Hypatia may have played a role in editing Archimedes
' Dimension of a Circle.
History of mathematics
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....
and a professor in the departments of philosophy and classics at Stanford University
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
. He has been called "one of the most profound and certainly the most provocative historian of Greek mathematics" of the 20th century.
Biography
Knorr was born August 29, 1945, in Richmond Hill, New York. He did his undergraduate studies at Harvard UniversityHarvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
from 1963 to 1966 and stayed there for his Ph.D., which he received in 1973 under the supervision of John Emery Murdoch and G. E. L. Owen
Gwilym Ellis Lane Owen
Gwilym Ellis Lane Owen was a Welsh philosopher, concerned with the history of Ancient Greek philosophy. From 1973 until his death he was the fourth Laurence Professor of Ancient Philosophy at the University of Cambridge...
. After postdoctoral studies at Cambridge University, he taught at Brooklyn College
Brooklyn College
Brooklyn College is a senior college of the City University of New York, located in Brooklyn, New York, United States.Established in 1930 by the New York City Board of Higher Education, the College had its beginnings as the Downtown Brooklyn branches of Hunter College and the City College of New...
, but lost his position when the college's Downtown Brooklyn campus was closed as part of New York's mid-1970's fiscal crisis. After taking a temporary position at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
, he joined the Stanford faculty as an assistant professor in 1979, was tenured there in 1983, and was promoted to full professor in 1990.
He died March 18, 1997 in Palo Alto, California
Palo Alto, California
Palo Alto is a California charter city located in the northwest corner of Santa Clara County, in the San Francisco Bay Area of California, United States. The city shares its borders with East Palo Alto, Mountain View, Los Altos, Los Altos Hills, Stanford, Portola Valley, and Menlo Park. It is...
, of melanoma
Melanoma
Melanoma is a malignant tumor of melanocytes. Melanocytes are cells that produce the dark pigment, melanin, which is responsible for the color of skin. They predominantly occur in skin, but are also found in other parts of the body, including the bowel and the eye...
.
Knorr was a talented violinist, and played first violin in the Harvard Orchestra, but he gave up his music when he came to Stanford, as the pressures of the tenure process did not allow him adequate practice time.
Books
The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry.:This work incorporates Knorr's Ph.D. thesis. It traces the early history of irrational numberIrrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
s from their first discovery (in Thebes
Thebes, Greece
Thebes is a city in Greece, situated to the north of the Cithaeron range, which divides Boeotia from Attica, and on the southern edge of the Boeotian plain. It played an important role in Greek myth, as the site of the stories of Cadmus, Oedipus, Dionysus and others...
between 430 and 410 BC, Knorr speculates), through the work of Theodorus of Cyrene
Theodorus of Cyrene
Theodorus of Cyrene was a Greek mathematician of the 5th century BC. The only first-hand accounts of him that we have are in two of Plato's dialogues: the Theaetetus and the Sophist...
, who showed the irrationality of the square roots of the integers up to 17, and Theodorus' student Theaetetus
Theaetetus (mathematician)
Theaetetus, Theaitētos, of Athens, possibly son of Euphronius, of the Athenian deme Sunium, was a classical Greek mathematician...
, who showed that all non-square integers have irrational square roots. Knorr reconstructs an argument based on Pythagorean triple
Pythagorean triple
A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...
s and parity that matches the story in Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
's Theaetetus
Theaetetus (dialogue)
The Theaetetus is one of Plato's dialogues concerning the nature of knowledge. The framing of the dialogue begins when Euclides tells his friend Terpsion that he had written a book many years ago based on what Socrates had told him of a conversation he'd had with Theaetetus when Theaetetus was...
of Theodorus' difficulties with the number 17, and shows that switching from parity to a different dichotomy in terms of whether a number is square or not was the key to Theaetetus' success. Theaetetus classified the known irrational numbers into three types, based on analogies to the geometric mean
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
, arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...
, and harmonic mean
Harmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
, and this classification was then greatly extended by Eudoxus of Cnidus
Eudoxus of Cnidus
Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy...
; Knorr speculates that this extension stemmed out of Eudoxus' studies of the golden section.
- Along with this history of irrational numbers, Knorr reaches several conclusions about the history of EuclidEuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's ElementsEuclid's ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
and of other related mathematical documents; in particular, he ascribes the origin of the material in Books 1, 3, and 6 of the Elements to the time of Hippocrates of ChiosHippocrates of ChiosHippocrates of Chios was an ancient Greek mathematician, , and astronomer, who lived c. 470 – c. 410 BCE.He was born on the isle of Chios, where he originally was a merchant. After some misadventures he went to Athens, possibly for litigation...
, and of the material in books 2, 4, 10, and 13 to the later period of Theodorus, Theaetetus, and Eudoxos. However, this suggested history has been criticized by van der WaerdenBartel Leendert van der WaerdenBartel Leendert van der Waerden was a Dutch mathematician and historian of mathematics....
, who believed that books 1 through 4 were largely due to the much earlier Pythagorean schoolPythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
.
Ancient Sources of the Medieval Tradition of Mechanics: Greek, Arabic, and Latin studies of the balance.
The Ancient Tradition of Geometric Problems.:This book, aimed at a general audience, examines the history of three classical problems from Greek mathematics
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...
: doubling the cube
Doubling the cube
Doubling the cube is one of the three most famous geometric problems unsolvable by compass and straightedge construction...
, squaring the circle
Squaring the circle
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...
, and angle trisection
Angle trisection
Angle trisection is a classic problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one-third of a given arbitrary angle, using only two tools: an un-marked straightedge, and a compass....
. It is now known that none of these problems can be solved by compass and straightedge
Compass and straightedge
Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....
, but Knorr argues that emphasizing these impossibility results is an anachronism due in part to the foundational crisis in 1930s mathematics. Instead, Knorr argues, the Greek mathematicians were primarily interested in how to solve these problems by whatever means they could, and viewed theorem and proofs as tools for problem-solving more than as ends in their own right.
Textual Studies in Ancient and Medieval Geometry.:This is a longer and more technical "appendix" to The Ancient Tradition of Geometric Problems in which Knorr examines the similarities and differences between ancient mathematical texts carefully in order to determine how they influenced each other and untangle their editorial history. One of Knorr's more provocative speculations in this work is that Hypatia may have played a role in editing Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
' Dimension of a Circle.