The Principles of Mathematics
Encyclopedia
The Principles of Mathematics is a book written by Bertrand Russell
in 1903. In it he presented his famous paradox
and argued his thesis that mathematics
and logic
are identical.
The book presents a view of the foundations of mathematics
and has become a classic reference. It reported on developments by Giuseppe Peano
, Mario Pieri
, Richard Dedekind
, Georg Cantor
, and others. In 1937 Russell prepared a new introduction saying, "Such interest as the book now possesses is historical, and consists in the fact that it represents a certain stage in the development of its subject." Further editions were printed in 1938, 1951, 1996, and 2009.
and appeared in Times Literary Supplement (Issue #88, 18 September 1903). Hardy titles his review "The Philosophy of Mathematics" and expects the book to appeal more to philosophers than mathematicians. But he says
In 1904 another review appeared in Bulletin of the American Mathematical Society
(11(2):74–93) written by Edwin Bidwell Wilson
.He says "The delicacy of the question is such that even the greatest mathematicians and philosophers of to-day have made what seem to be substantial slips of judgement and have shown on occasions an astounding ignorance of the essence of the problem which they were discussing. ... all too frequently it has been the result of a wholly unpardonable disregard of the work already accomplished by others." Wilson recounts the developments of Peano that Russell reports, and takes the occasion to correct Henri Poincaré
who had ascribed them to David Hilbert
. In praise of Russell, Wilson says "Surely the present work is a monument to patience, perseverance, and thoroughness."(page 88)
in which he recalled the impetus to write the Principles:
Recalling the book after his later work, he provides this evaluation:
Such self-deprecation from the author after half a century of philosophical growth is understandable. On the other hand, Jules Vuillemin
wrote in 1968
Russel's Principles of 1903 also looms large in Ivor Grattan-Guinness
' study of the roots of modern logic.
physics in the final part as the last three chapters consider Newton's laws of motion, absolute and relative motion, and Hertz's dynamics. However, Russell rejects what he calls "the relational theory", and says on page 489
In his review, Hardy (1903) says "Mr. Russell is a firm believer in absolute position in space and time, a view as much out of fashion nowadays that Chapter [58: Absolute and Relative Motion] will be read with peculiar interest."
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
in 1903. In it he presented his famous paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
and argued his thesis that mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
are identical.
The book presents a view of the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
and has become a classic reference. It reported on developments by Giuseppe Peano
Giuseppe Peano
Giuseppe Peano was an Italian mathematician, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in...
, Mario Pieri
Mario Pieri
Mario Pieri was an Italian mathematician who is known for his work on foundations of geometry.Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pieri began his higher education at University of Bologna where he drew the attention of Salvatore...
, Richard Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
, Georg Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
, and others. In 1937 Russell prepared a new introduction saying, "Such interest as the book now possesses is historical, and consists in the fact that it represents a certain stage in the development of its subject." Further editions were printed in 1938, 1951, 1996, and 2009.
Early reviews
Reviews were prepared by G. E. Moore and Charles Sanders Peirce, but Moore's was never published and that of Peirce was, as one author described it, "so brief and cursory that I am convinced that he never read the book." However, a long and generally favorable review was written by G. H. HardyG. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
and appeared in Times Literary Supplement (Issue #88, 18 September 1903). Hardy titles his review "The Philosophy of Mathematics" and expects the book to appeal more to philosophers than mathematicians. But he says
- [I]n spite of its five hundred pages the book is much too short. Many chapters dealing with important questions are compressed into five or six pages, and in some places, especially in the most avowedly controversial parts, the argument is almost too condensed to follow. And the philosopher who attempts to read the book will be especially puzzled by the constant presupposition of a whole philosophical system utterly unlike any of those usually accepted.
In 1904 another review appeared in Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society...
(11(2):74–93) written by Edwin Bidwell Wilson
Edwin Bidwell Wilson
Edwin Bidwell Wilson was an American mathematician and polymath. He was the sole protégé of Yale's physicist Josiah Willard Gibbs and was mentor to MIT economist Paul Samuelson. He received his AB from Harvard College in 1899 and his PhD from Yale University in 1901, working under Gibbs.E.B...
.He says "The delicacy of the question is such that even the greatest mathematicians and philosophers of to-day have made what seem to be substantial slips of judgement and have shown on occasions an astounding ignorance of the essence of the problem which they were discussing. ... all too frequently it has been the result of a wholly unpardonable disregard of the work already accomplished by others." Wilson recounts the developments of Peano that Russell reports, and takes the occasion to correct Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
who had ascribed them to David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
. In praise of Russell, Wilson says "Surely the present work is a monument to patience, perseverance, and thoroughness."(page 88)
Later reviews
In 1959 Russell wrote My Philosophical DevelopmentMy Philosophical Development
My Philosophical Development is a book written by Bertrand Russell where he is summing up his philosophical beliefs and how they have changed during his life....
in which he recalled the impetus to write the Principles:
- It was at the International Congress of Philosophy in Paris in the year 1900 that I became aware of the importance of logical reform for the philosophy of mathematics. ... in every discussion [Peano] showed more precision and more logical rigor than was shown by anybody else. ... It was [Peano's works] that gave the impetus to my own views on the principles of mathematics.
Recalling the book after his later work, he provides this evaluation:
- The Principles of Mathematics, which I finished on May 23, 1902, turned out to be a crude and rather immature draft of the subsequent work (Principia MathematicaPrincipia MathematicaThe Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
), from which, however, it differed in containing controversy with other philosophies of mathematics..
Such self-deprecation from the author after half a century of philosophical growth is understandable. On the other hand, Jules Vuillemin
Jules Vuillemin
Jules Vuillemin was a French philosopher, succeeding to Maurice Merleau-Ponty at the Collège de France from 1962 to his death. A friend of Michel Foucault, he supported his election at the College, and was also close to Michel Serres...
wrote in 1968
- The Principles inaugurated contemporary philosophy. Other works have won and lost the title. Such is not the case with this one. It is serious, and its wealth perseveres. Furthermore, in relation to it, in a deliberate fashion or not, it locates itself again today in the eyes of all those that believe that contemporary science has modified our representation of the universe and through this representation, our relation to ourselves and to others.
Russel's Principles of 1903 also looms large in Ivor Grattan-Guinness
Ivor Grattan-Guinness
Ivor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...
' study of the roots of modern logic.
Contents
The Principles of Mathematics consists of 59 chapters divided into seven parts: indefinables in mathematics, number, quantity, order, infinity and continuity, space, matter and motion. There is an anticipation of relativityTheory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
physics in the final part as the last three chapters consider Newton's laws of motion, absolute and relative motion, and Hertz's dynamics. However, Russell rejects what he calls "the relational theory", and says on page 489
- For us, since absolute space and time have been admitted, there is no need to avoid absolute motion, and indeed no possibility of doing so.
In his review, Hardy (1903) says "Mr. Russell is a firm believer in absolute position in space and time, a view as much out of fashion nowadays that Chapter [58: Absolute and Relative Motion] will be read with peculiar interest."
External links
- The Principles of Mathematics Online text
- The Principles of Mathematics Full Text at the Internet ArchiveInternet ArchiveThe Internet Archive is a non-profit digital library with the stated mission of "universal access to all knowledge". It offers permanent storage and access to collections of digitized materials, including websites, music, moving images, and nearly 3 million public domain books. The Internet Archive...