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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
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About the author
Dallas Willard
120 books1,144 followersDallas Willard was a widely respected American philosopher and Christian thinker, best known for his work on spiritual formation and his expertise in phenomenology, particularly the philosophy of Edmund Husserl. He taught philosophy at the University of Southern California from 1965 until his death in 2013, where he also served as department chair in the early 1980s. Willard held degrees in psychology, philosophy, and religion, earning his PhD in philosophy from the University of Wisconsin–Madison with a focus on the history of science. He was recognized as a leading translator and interpreter of Husserl's thought, making foundational texts available in English and contributing significantly to the fields of epistemology, philosophy of mind, and logic.
Though a serious academic, Willard became even more widely known for his books on Christian living, including The Divine Conspiracy and Renovation of the Heart, both of which earned major awards and helped shape the modern spiritual formation movement. He believed that discipleship to Jesus was an intentional process involving not only belief but transformation through spiritual disciplines like prayer, study, solitude, and service. For Willard, spiritual growth was not about earning God’s favor but about participating in the divine life through active cooperation with grace.
His teachings emphasized the concept of apprenticeship to Jesus—being with him, learning to be like him—and his influence extended to ministries such as Renovaré, the Apprentice Institute, and the Dallas Willard Center for Spiritual Formation. He served on the boards of organizations like the C.S. Lewis Foundation and Biola University, and his intellectual and spiritual legacy continues through Dallas Willard Ministries and academic institutions inspired by his work.
Willard was also a deeply personal writer who shared candidly about the challenges of balancing academic life with family. Despite his own admitted shortcomings, those closest to him regarded him as a man of deep love, humility, and grace. His enduring impact can be seen in the lives and works of many contemporary Christian thinkers and writers, including Richard J. Foster, James Bryan Smith, and John Mark Comer. As both philosopher and pastor to the mind, Dallas Willard remains a towering figure in the dialogue between rigorous thought and transformative Christian practice.
Though a serious academic, Willard became even more widely known for his books on Christian living, including The Divine Conspiracy and Renovation of the Heart, both of which earned major awards and helped shape the modern spiritual formation movement. He believed that discipleship to Jesus was an intentional process involving not only belief but transformation through spiritual disciplines like prayer, study, solitude, and service. For Willard, spiritual growth was not about earning God’s favor but about participating in the divine life through active cooperation with grace.
His teachings emphasized the concept of apprenticeship to Jesus—being with him, learning to be like him—and his influence extended to ministries such as Renovaré, the Apprentice Institute, and the Dallas Willard Center for Spiritual Formation. He served on the boards of organizations like the C.S. Lewis Foundation and Biola University, and his intellectual and spiritual legacy continues through Dallas Willard Ministries and academic institutions inspired by his work.
Willard was also a deeply personal writer who shared candidly about the challenges of balancing academic life with family. Despite his own admitted shortcomings, those closest to him regarded him as a man of deep love, humility, and grace. His enduring impact can be seen in the lives and works of many contemporary Christian thinkers and writers, including Richard J. Foster, James Bryan Smith, and John Mark Comer. As both philosopher and pastor to the mind, Dallas Willard remains a towering figure in the dialogue between rigorous thought and transformative Christian practice.
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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