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Philosophy of Arithmetic: Psychological and Logical Investigations with Supplementary Texts from 1887–1901
This volume is a window on a period of rich and illuminating philosophical activity that has been rendered generally inaccessible by the supposed "revolution" attributed to "Analytic Philosophy" so-called. Careful exposition and critique is given to every serious alternative account of number and number relations available at the time.
579 pages, Hardcover
First published September 30, 2003
About the author
Edmund Husserl
541 books575 followersEdmund Gustav Albrecht Husserl (Dr. phil. hab., University of Halle-Wittenberg, 1887; Ph.D., Mathematics, University of Vienna, 1883) was a philosopher who is deemed the founder of phenomenology. He broke with the positivist orientation of the science and philosophy of his day, believing that experience is the source of all knowledge, while at the same time he elaborated critiques of psychologism and historicism.
Born into a Moravian Jewish family, he was baptized as a Lutheran in 1887. Husserl studied mathematics under Karl Weierstrass, completing a Ph.D. under Leo Königsberger, and studied philosophy under Franz Brentano and Carl Stumpf. Husserl taught philosophy, as a Privatdozent at Halle from 1887, then as professor, first at Göttingen from 1901, then at Freiburg im Breisgau from 1916 until his 1928 retirement.
Born into a Moravian Jewish family, he was baptized as a Lutheran in 1887. Husserl studied mathematics under Karl Weierstrass, completing a Ph.D. under Leo Königsberger, and studied philosophy under Franz Brentano and Carl Stumpf. Husserl taught philosophy, as a Privatdozent at Halle from 1887, then as professor, first at Göttingen from 1901, then at Freiburg im Breisgau from 1916 until his 1928 retirement.
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Displaying 1 - 2 of 2 reviews
July 12, 2019
A quintessential 19th century work of philosophy of mathematics, focusing on two sides of mathematics: authentic representations, understood as the conceptualization that underlies mathematical concepts, and symbolic representations, the way of representing mathematics with signs that correctly capture the authentic representation without forcing us to check it against the authentic representation.
The first part, two thirds of the book, is about the authentic representation. The authentic representation of a number is something like an equivalence class of totalities, multiplicities, groups, or (non-technically understood) sets. Against other authors, he doesn't identify this with a certain category of sets, such as unit intervals on a continuum, unlike certain other authors. He also doesn't try to wash out the psychological content of these notions, unlike his most memorable opponent, Frege.
Husserl deals with Frege on multiple occasions but most thoroughly near the end of the first part, he has many objections to Frege: (i) that his reference theory of names is obviously false (which Frege realized as well, hence developing his sense-reference theory), (ii) that we can arrive at a judgement of equinumerosity in numerous ways, not just by describing a bijection between two sets, and (iii) that a definition of numbers simply needs to give a 'feel' for what numbers are & most of the work Frege does (e.g. demonstrating in his model of the natural numbers that there is only one successor to 0) is frankly redundant. I think a lot of what Husserl says is very damaging to Frege's project, and both of them needed time to recoup after their critiques of each other.
The second part of the book, I'm sorry, is a bit boring so reader beware. This section as I said is about symbolic representation. The main focus for this is the development of a formalism for doing arithmetic, and especially the development of efficient algorithms for representing and calculating numbers. Husserl aims to show that our system, the "Indic" system because of its Indian roots, is a work of genius because base-10 counting is fairly efficient, natural, and a very compressed way of representing very larger numbers. For the most part he's right, although we get long digressions about the counting in bases which can feel a bit basic.
However as this part goes on, he sets out a mathematical programme which is interesting: he wants to have us represent all languages and systems for representing the natural numbers (base-10 counting, the "sequential" way (1+1 = 2, 2+1=3...), etc) and prove that they're equivalent. To him this would justify using any of these systems when faced with a particular arithmetical problem that might be better suited for one of those systems than another. In some ways I think this does a better job of predicting the direction of mathematical logic than Frege did, since Husserl is focusing on the constraints of our formal language, a favourite topic of mathematical logicians.
Finally he attempts to give a distinction at the end for dividing algebra from arithmetic, although it is unfortunately much more aimed at the elementary algebra of the time than abstract algebra, which might be more interesting to mathematicians.
The first part, two thirds of the book, is about the authentic representation. The authentic representation of a number is something like an equivalence class of totalities, multiplicities, groups, or (non-technically understood) sets. Against other authors, he doesn't identify this with a certain category of sets, such as unit intervals on a continuum, unlike certain other authors. He also doesn't try to wash out the psychological content of these notions, unlike his most memorable opponent, Frege.
Husserl deals with Frege on multiple occasions but most thoroughly near the end of the first part, he has many objections to Frege: (i) that his reference theory of names is obviously false (which Frege realized as well, hence developing his sense-reference theory), (ii) that we can arrive at a judgement of equinumerosity in numerous ways, not just by describing a bijection between two sets, and (iii) that a definition of numbers simply needs to give a 'feel' for what numbers are & most of the work Frege does (e.g. demonstrating in his model of the natural numbers that there is only one successor to 0) is frankly redundant. I think a lot of what Husserl says is very damaging to Frege's project, and both of them needed time to recoup after their critiques of each other.
The second part of the book, I'm sorry, is a bit boring so reader beware. This section as I said is about symbolic representation. The main focus for this is the development of a formalism for doing arithmetic, and especially the development of efficient algorithms for representing and calculating numbers. Husserl aims to show that our system, the "Indic" system because of its Indian roots, is a work of genius because base-10 counting is fairly efficient, natural, and a very compressed way of representing very larger numbers. For the most part he's right, although we get long digressions about the counting in bases which can feel a bit basic.
However as this part goes on, he sets out a mathematical programme which is interesting: he wants to have us represent all languages and systems for representing the natural numbers (base-10 counting, the "sequential" way (1+1 = 2, 2+1=3...), etc) and prove that they're equivalent. To him this would justify using any of these systems when faced with a particular arithmetical problem that might be better suited for one of those systems than another. In some ways I think this does a better job of predicting the direction of mathematical logic than Frege did, since Husserl is focusing on the constraints of our formal language, a favourite topic of mathematical logicians.
Finally he attempts to give a distinction at the end for dividing algebra from arithmetic, although it is unfortunately much more aimed at the elementary algebra of the time than abstract algebra, which might be more interesting to mathematicians.
November 18, 2024
This is a book about uniting. The number is a symbol, and it stands for the authentic contents which are noticed. What is noticed is noticed out of our special interest, whatever that may be.
The work is narrow minded. The psychological border is not to be trespassed. What makes a number a symbol, what justifies the unification? The answer is left to progress.
To me, a thing is part of an encompassing whole, a composite state of consciousness, an acceptance of an order. A box is not only, or mostly not, a means to protect your delivery, but part of a symbol in your mind for the trustworthyness of the deliverer.
The work is narrow minded. The psychological border is not to be trespassed. What makes a number a symbol, what justifies the unification? The answer is left to progress.
To me, a thing is part of an encompassing whole, a composite state of consciousness, an acceptance of an order. A box is not only, or mostly not, a means to protect your delivery, but part of a symbol in your mind for the trustworthyness of the deliverer.
Displaying 1 - 2 of 2 reviews