Philosophy of Arithmetic
Encyclopedia
The Philosophy of Arithmetic is the English language title of Edmund Husserl
's first published book. It was first published, in the German language, under the full title, Philosophie der Arithmetik. Psychologische und logische untersuchungen. The work is a synthesis of his studies in mathematics, under Karl Weierstrass
, with his studies in philosophy and psychology, under Franz Brentano
and Carl Stumpf
.
The first volume is divided in two parts, the first of which purports to analyse the "Proper concepts of multiplicity, unity and amount" (Die eigentliche Begriffe von Vielheit, Einheit und Anzahl) and the second "The symbolic amount-concepts and the logical sources of amount-arithmetic" (Die symbolischen Anzahlbegrife und die logischen Quellen der Anzahlen-Arithmetik).
analysis of the concept of number
, which is the most basic concept on which the entire edifice of arithmetic
and mathematics
can be founded. In order to proceed with this analysis, Husserl, following Brentano and Stumpf, uses the tools of psychology
to look for the "origin and content" of the concept of number. He begins with the classical definition, already given by Euclid
, Hobbes and Leibniz, that "number is a multiplicity of unities" and then asks himself: what is multiplicity and what is unity? Anything that we can think of, anything we can present, can be considered at its most basic level to be "something". Multiplicity is then the "collective connection" of "something and something and something etc." In order to get a number instead of a mere quantity, we can also think of these featureless, abstract "somethings" as "ones" and then get "one and one and one etc." as basic definition of number in abstracto. However, these are just the proper numbers, i.e. number which we can conceive of properly, without the help of instruments or symbols. Psychologically we are limited to just the very first few numbers if we want to conceive of them properly, with higher numbers our short term memory is not enough to think of them all together, but still as identical to themselves and different from all others. Hence, Husserl says, we have to move on to the analysis of symbolically conceived numbers, which are effectively those that are used in mathematics.
(in Vienna) on the mathematical side and his studies with Brentano (in Vienna) and Stumpf (in Halle) on the psychological/philosophical side. The book is mostly based on his habilitationsschrift
of 1887 "On the Concept of Number" (Über den Begriff der Zahl). Husserl also lectured on the concept of number between 1889 and 1891, much in the same vein. He continued working on the second volume up to at least 1894.
Philosophie der Arithmetik
Psychologische und logische untersuchungen,
von Dr. E. G. Husserl. 1. bd.
1891, xvi, 324 p. 23cm.
LC Classification: QA9 .H8
Dewey Class No.: 510.1
Other System No.: (OCoLC)4787664
Husserliana edition:
Philosophie der Arithmetik
Psychologische und logische untersuchungen - mit erguanzenden Texten (1890-1901)
Series: Husserliana - Edmund Husserl Gesammelte werke, Vol. XII
Husserl, Edmund
Ed. Eley, Lothar
1970, 585 p., Hardcover
ISBN: 90-247-0230-5
Official English translation of the Husserliana edition:
Philosophy of Arithmetic
Psychological and Logical Investigations - with Supplementary Texts from 1887-1901
Series: Edmund Husserl Collected Works, Vol. X
Husserl, Edmund
Tr. Willard, Dallas
2003, 580 p., Hardcover
ISBN: 1-4020-1546-1
Edmund Husserl
Edmund Gustav Albrecht Husserl was a philosopher and mathematician and the founder of the 20th century philosophical school of phenomenology. He broke with the positivist orientation of the science and philosophy of his day, yet he elaborated critiques of historicism and of psychologism in logic...
's first published book. It was first published, in the German language, under the full title, Philosophie der Arithmetik. Psychologische und logische untersuchungen. The work is a synthesis of his studies in mathematics, under Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
, with his studies in philosophy and psychology, under Franz Brentano
Franz Brentano
Franz Clemens Honoratus Hermann Brentano was an influential German philosopher and psychologist whose influence was felt by other such luminaries as Sigmund Freud, Edmund Husserl, Kazimierz Twardowski and Alexius Meinong, who followed and adapted his views.-Life:Brentano was born at Marienberg am...
and Carl Stumpf
Carl Stumpf
Carl Stumpf was a German philosopher and psychologist.Born in Wiesentheid, he studied with Franz Brentano and Hermann Lotze...
.
Structure
The Philosophy of Arithmetic (PA) consititues the first volume of a work which Husserl intended to comprise two volumes, of which the second was never published. Comprehensively it would have encompassed four parts and an Appendix.The first volume is divided in two parts, the first of which purports to analyse the "Proper concepts of multiplicity, unity and amount" (Die eigentliche Begriffe von Vielheit, Einheit und Anzahl) and the second "The symbolic amount-concepts and the logical sources of amount-arithmetic" (Die symbolischen Anzahlbegrife und die logischen Quellen der Anzahlen-Arithmetik).
Content
The basic issue of the book is a philosophicalPhilosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
analysis of the concept of number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
, which is the most basic concept on which the entire edifice of arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
and 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...
can be founded. In order to proceed with this analysis, Husserl, following Brentano and Stumpf, uses the tools of psychology
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...
to look for the "origin and content" of the concept of number. He begins with the classical definition, already given by Euclid
Euclid
Euclid , 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...
, Hobbes and Leibniz, that "number is a multiplicity of unities" and then asks himself: what is multiplicity and what is unity? Anything that we can think of, anything we can present, can be considered at its most basic level to be "something". Multiplicity is then the "collective connection" of "something and something and something etc." In order to get a number instead of a mere quantity, we can also think of these featureless, abstract "somethings" as "ones" and then get "one and one and one etc." as basic definition of number in abstracto. However, these are just the proper numbers, i.e. number which we can conceive of properly, without the help of instruments or symbols. Psychologically we are limited to just the very first few numbers if we want to conceive of them properly, with higher numbers our short term memory is not enough to think of them all together, but still as identical to themselves and different from all others. Hence, Husserl says, we have to move on to the analysis of symbolically conceived numbers, which are effectively those that are used in mathematics.
History
The book is a product of his years of study with Weierstrass (in Berlin) and his student Leo KönigsbergerLeo Königsberger
Leo Königsberger was a German mathematician, and historian of science. He is best known for his three-volume biography of Hermann von Helmholtz, which remains the standard reference on the subject.-Biography:...
(in Vienna) on the mathematical side and his studies with Brentano (in Vienna) and Stumpf (in Halle) on the psychological/philosophical side. The book is mostly based on his habilitationsschrift
Habilitation
Habilitation is the highest academic qualification a scholar can achieve by his or her own pursuit in several European and Asian countries. Earned after obtaining a research doctorate, such as a PhD, habilitation requires the candidate to write a professorial thesis based on independent...
of 1887 "On the Concept of Number" (Über den Begriff der Zahl). Husserl also lectured on the concept of number between 1889 and 1891, much in the same vein. He continued working on the second volume up to at least 1894.
Editions
The original edition:Philosophie der Arithmetik
Psychologische und logische untersuchungen,
von Dr. E. G. Husserl. 1. bd.
1891, xvi, 324 p. 23cm.
LC Classification: QA9 .H8
Dewey Class No.: 510.1
Other System No.: (OCoLC)4787664
Husserliana edition:
Philosophie der Arithmetik
Psychologische und logische untersuchungen - mit erguanzenden Texten (1890-1901)
Series: Husserliana - Edmund Husserl Gesammelte werke, Vol. XII
Husserl, Edmund
Ed. Eley, Lothar
1970, 585 p., Hardcover
ISBN: 90-247-0230-5
Official English translation of the Husserliana edition:
Philosophy of Arithmetic
Psychological and Logical Investigations - with Supplementary Texts from 1887-1901
Series: Edmund Husserl Collected Works, Vol. X
Husserl, Edmund
Tr. Willard, Dallas
2003, 580 p., Hardcover
ISBN: 1-4020-1546-1