The Principles Of Mathematics
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''The Principles of Mathematics'' (''PoM'') is a 1903 book by
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
, in which the author presented his famous
paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
and argued his thesis that mathematics and
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...
are identical. The book presents a view of the
foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
and Meinongianism and has become a classic reference. It reported on developments by
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. 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 sta ...
, Mario Pieri,
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
,
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
, and others. In 1905
Louis Couturat Louis Couturat (; 17 January 1868 – 3 August 1914) was a French logician, mathematician, philosopher, and linguist. Couturat was a pioneer of the constructed language Ido. Life and education Born in Ris-Orangis, Essonne, France. In 1887 h ...
published a partial French translation that expanded the book's readership. 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.


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. In chapter one, "Definition of Pure Mathematics", Russell asserts that :
The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.
Russell deconstructs pure mathematics with relations, by positing them, their converses and complements as
primitive notions Primitive may refer to: Mathematics * Primitive element (field theory) * Primitive element (finite field) * Primitive cell (crystallography) * Primitive notion, axiomatic systems * Primitive polynomial (disambiguation), one of two concepts * P ...
. Combining the calculus of relations of DeMorgan, Pierce and Schroder, with the
symbolic logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal s ...
of Peano, he analyses
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
s using
serial relation In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation. Bert ...
s, and writes that the theorems of measurement have been generalized to
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
. He notes that Peano distinguished a term from the set containing it: the
set membership In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. Sets Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are subset ...
relation versus
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
. Epsilon (ε) is used to show set membership, but Russell indicates trouble when x \epsilon x.
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contain ...
is mentioned 15 times and chapter 10 "The Contradiction" explains it. Russell had written previously on foundations of geometry, denoting, and relativism of space and time, so those topics are recounted.
Elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
according to Clifford, and the Cayley-Klein metric are mentioned to illustrate
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ...
. There is an anticipation of relativity 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 Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame. Before Newton A version of the concept of absolute space (in the sense of a prefer ...
have been admitted, there is no need to avoid absolute motion, and indeed no possibility of doing so. In his review, G. H. Hardy says "Mr. Russell is a firm believer in absolute position in space and time, a view as much out of fashion nowadays that Chapter 8: Absolute and Relative Motionwill be read with peculiar interest."


Early reviews

Reviews were prepared by G. E. Moore and
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...
, but Moore's was never published and that of Peirce was brief and somewhat dismissive. He indicated that he thought it unoriginal, saying that the book "can hardly be called literature" and "Whoever wishes a convenient introduction to the remarkable researches into the logic of mathematics that have been made during the last sixty years ..will do well to take up this book." G. H. Hardy wrote a favorable review G. H. Hardy (18 September 1903) "The Philosophy of Mathematics",
Times Literary Supplement ''The Times Literary Supplement'' (''TLS'') is a weekly literary review published in London by News UK, a subsidiary of News Corp. History The ''TLS'' first appeared in 1902 as a supplement to ''The Times'' but became a separate publication ...
#88
expecting the book to appeal more to philosophers than mathematicians. But he says : : 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 The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...
'' (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 Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
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)


Second edition

In 1938 the book was re-issued with a new preface by Russell. This preface was interpreted as a retreat from the realism of the first edition and a turn toward nominalist philosophy of
symbolic logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal s ...
. James Feibleman, an admirer of the book, thought Russell’s new preface went too far into nominalism so he wrote a rebuttal to this introduction. Feibleman says, "It is the first comprehensive treatise on symbolic logic to be written in English; and it gives to that system of logic a realistic interpretation."


Later reviews

In 1959 Russell wrote ''
My Philosophical Development ''My Philosophical Development'' is a 1959 book by the philosopher Bertrand Russell, in which the author summarizes his philosophical beliefs and explains how they changed during his life.. Summary Russell gives an account of his philosophical ...
'', 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. ... I was impressed by the fact that, in every discussion, eanoshowed more precision and more logical rigour than was shown by anybody else. ... It was eano's worksthat 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 23 May 1902, turned out to be a crude and rather immature draft of the subsequent work /nowiki>''Principia Mathematica''">Principia_Mathematica.html" ;"title="/nowiki>''Principia Mathematica">/nowiki>''Principia Mathematica''/nowiki>, 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 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. When W. V. O. Quine penned his autobiography, he wrote: :Peano's symbolic notation took Russell by storm in 1900, but Russell’s ''Principles'' was still in unrelieved prose. I was inspired by its profundity
n 1928 N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History ...
and baffled by its frequent opacity. In part it was rough going because of the cumbersomeness of ordinary language as compared with the suppleness of a notation especially devised for these intricate themes. Rereading it years later, I discovered that it had been rough going also because matters were unclear in Russell's own mind in those pioneer days. ''The Principles'' was an early expression of
analytic philosophy Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United ...
and thus has come under close examination.Peter Hylton (1990) ''Russell, Idealism, and the Emergence of Analytic Philosophy'', chapter 5: Russell’s ''Principles of Mathematics'', pp 167 to 236,
Clarendon Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
,
Peter Hylton wrote, "The book has an air of excitement and novelty to it ... The salient characteristic of ''Principles'' is ... the way in which the technical work is integrated into metaphysical argument."
Ivor Grattan-Guinness Ivor Owen Grattan-Guinness (23 June 1941 – 12 December 2014) was a historian of mathematics and logic. Life Grattan-Guinness was born in Bakewell, England; his father was a mathematics teacher and educational administrator. He gained his ...
made an in-depth study of ''Principles''. First he published ''Dear Russell – Dear Jourdain'' (1977), which included correspondence with Philip Jourdain who promulgated some of the book’s ideas. Then in 2000 Grattan-Guinness published ''The Search for Mathematical Roots 1870 – 1940'', which considered the author’s circumstances, the book’s composition and its shortcomings. In 2006, Philip Ehrlich challenged the validity of Russell's analysis of infinitesimals in the Leibniz tradition. A recent study documents the non-sequiturs in Russell's critique of the infinitesimals of
Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...
and
Hermann Cohen Hermann Cohen (4 July 1842 – 4 April 1918) was a German Jewish philosopher, one of the founders of the Marburg school of neo-Kantianism, and he is often held to be "probably the most important Jewish philosopher of the nineteenth cent ...
..


See also

* ''
Introduction to Mathematical Philosophy ''Introduction to Mathematical Philosophy'' is a book (1919 first edition) by philosopher Bertrand Russell, in which the author seeks to create an accessible introduction to various topics within the foundations of mathematics. According to the pr ...
'' *
Russellian change The B-theory of time, also called the "tenseless theory of time", is one of two positions regarding the temporal ordering of events in the philosophy of time. B-theorists argue that the flow of time is only a subjective illusion of human consciousn ...


Notes


References

* Stefan Andersson (1994). ''In Quest of Certainty: Bertrand Russell's Search for Certainty in Religion and Mathematics Up to'' The Principles of Mathematics. Stockholm: Almquist & Wiksell. .


External links


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{{DEFAULTSORT:Principles of Mathematics 1903 non-fiction books Books by Bertrand Russell English-language books Logic literature Mathematics books Methodology Philosophy books Cambridge University Press books