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The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number
The Foundations of Arithmetic is undoubtedly the best introduction to Frege's thought; it is here that Frege expounds the central notions of his philosophy, subjecting the views of his predecessors and contemporaries to devastating analysis. The book represents the first philosophically sound discussion of the concept of number in Western civilization. It profoundly influenced developments in the philosophy of mathematics and in general ontology.
119 pages, Paperback
First published January 1, 1884
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About the author
Gottlob Frege
126 books170 followersFriedrich Ludwig Gottlob Frege (German: [ˈɡɔtloːp ˈfreːɡə]) was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and mathematics. While he was mainly ignored by the intellectual world when he published his writings, Giuseppe Peano (1858–1932) and Bertrand Russell (1872–1970) introduced his work to later generations of logicians and philosophers.
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Displaying 1 - 30 of 68 reviews
April 13, 2007
Yeah, yeah, his theory of numbers doesn't really make any sense. But, whatever. You try to define a number.
Read
June 3, 2011I have no recollection of having read this book but I have it on good authority that I did.
June 15, 2019
Bloody weird to slap a star rating on this, but there you go, welcome to where nothing is not rateable.
September 29, 2024
“The laws of numbers, therefore, are not really applicable to external things; they are not laws of nature. They are, however, applicable to judgements holding good of things in the external world: they are laws of the laws of nature” (99)
A deep dive into the concept of Number and an attempt to “prove” arithmetic functions to the same degree of rigor the Euclid brought to geometry. Frege spends the early part of the book examining various philosophical treatments of how we fix our idea of Number. Following Kant, Frege asks whether numbers are parts of analytic judgements where the meaning is contained in a definitions of itself or if they are parts of synthetic judgements where their meaning is grounded in the world. And are numbers known a priori (as a matter of reason) or a posteriori (known by experience). Eventually he decides that numbers are analytic judgements that we understand a priori (118).
Frege reaches this conclusion, interestingly, by focusing on language use and how numbers are deployed in how we talk about the world. In a statement like “There are three books on the table” the number “three” is used to add meaning to the concept “books on the table” where “books on the table” is something that we can conceive of conceptually, even in the absence of a context where this is true and there are three actual books on the table in front of you. Furthermore, it is a concept that allows for use of a number … in this case three. We would not say of the same of “There are lazy books on the table” because the idea of “laziness” does not belong to the concept of books in the same way. Numbers appearing in sentences with concepts make propositions about those concepts and allow us to formulate judgements about the world that then support claims, intuitions, and applications of those in the world.
In principle, following from Frege’s arguments, it seems possible to work from language and expressions that are allowed and sensible (as described in the Tractatus) to derive a sense of the rules of reason that are at work. However, as someone who works in language, I would say that what we say is at best an unreliable source from which to understand applicable analytic judgements or a priori acts of knowledge. People say things they don’t mean or say things imprecisely all the time. Our choices of grammar, which Frege is taking as an accurate and rule-governed way of determining the edges of the concepts at play, is more driven by intuitive internalization of rules rather than linguistic expressions of a priori rationality.
Still, I like the idea of looking at our choices in language to determine how they define the edges of concepts (via determiners, articles, and pronouns), identify properties about them (via adjectives), locate them in contexts (via prepositions), position interlocutors to them (via metadiscourse), and provide guidance for their uptake and understanding (via verbs).
A deep dive into the concept of Number and an attempt to “prove” arithmetic functions to the same degree of rigor the Euclid brought to geometry. Frege spends the early part of the book examining various philosophical treatments of how we fix our idea of Number. Following Kant, Frege asks whether numbers are parts of analytic judgements where the meaning is contained in a definitions of itself or if they are parts of synthetic judgements where their meaning is grounded in the world. And are numbers known a priori (as a matter of reason) or a posteriori (known by experience). Eventually he decides that numbers are analytic judgements that we understand a priori (118).
Frege reaches this conclusion, interestingly, by focusing on language use and how numbers are deployed in how we talk about the world. In a statement like “There are three books on the table” the number “three” is used to add meaning to the concept “books on the table” where “books on the table” is something that we can conceive of conceptually, even in the absence of a context where this is true and there are three actual books on the table in front of you. Furthermore, it is a concept that allows for use of a number … in this case three. We would not say of the same of “There are lazy books on the table” because the idea of “laziness” does not belong to the concept of books in the same way. Numbers appearing in sentences with concepts make propositions about those concepts and allow us to formulate judgements about the world that then support claims, intuitions, and applications of those in the world.
In principle, following from Frege’s arguments, it seems possible to work from language and expressions that are allowed and sensible (as described in the Tractatus) to derive a sense of the rules of reason that are at work. However, as someone who works in language, I would say that what we say is at best an unreliable source from which to understand applicable analytic judgements or a priori acts of knowledge. People say things they don’t mean or say things imprecisely all the time. Our choices of grammar, which Frege is taking as an accurate and rule-governed way of determining the edges of the concepts at play, is more driven by intuitive internalization of rules rather than linguistic expressions of a priori rationality.
Still, I like the idea of looking at our choices in language to determine how they define the edges of concepts (via determiners, articles, and pronouns), identify properties about them (via adjectives), locate them in contexts (via prepositions), position interlocutors to them (via metadiscourse), and provide guidance for their uptake and understanding (via verbs).
July 18, 2008
Russell's paradox be damned!
August 5, 2025
pea valutab
paha on olla
tahaks oksendada
üsna psühhootiline olid "head-ööd lugemiseks" sellised värsid:
"Kui iga objekt y, millega x on suhtes φ, kuulub mõiste F alla, ja kui sellest, et d kuulub mõiste F alla, üldiselt, mis d ka oleks, järeldub, et iga objekt, millega d on suhtes φ, kuulub mõiste F alla, siis kuulub y mõiste F alla sõltumata sellest, mis mõiste F ka olla võiks."
või
"Koguarv, mis omistub mõistele F, on mõiste "mõiste, mis on samaarvuline mõistega F" maht, kus me nimetasime mõistet F samaarvuliseks mõistega G, kui on olemas too üksühese vastavuse võimalus."
ju vist siin on teatav omalaadi muusika
paha on olla
tahaks oksendada
üsna psühhootiline olid "head-ööd lugemiseks" sellised värsid:
"Kui iga objekt y, millega x on suhtes φ, kuulub mõiste F alla, ja kui sellest, et d kuulub mõiste F alla, üldiselt, mis d ka oleks, järeldub, et iga objekt, millega d on suhtes φ, kuulub mõiste F alla, siis kuulub y mõiste F alla sõltumata sellest, mis mõiste F ka olla võiks."
või
"Koguarv, mis omistub mõistele F, on mõiste "mõiste, mis on samaarvuline mõistega F" maht, kus me nimetasime mõistet F samaarvuliseks mõistega G, kui on olemas too üksühese vastavuse võimalus."
ju vist siin on teatav omalaadi muusika
July 9, 2023
Áhugaverð bók. Ekki fyrir alla, en fyrir mig.
Ég datt samt eitthvað út á nokkrum stöðum (nennti ekki að einbeita mér alveg).
Mér finnst oft erfiðara að skilja gamlar bækur, ekki hægt að kenna bókinni um það.
Gaman og fræðandi að lesa vitandi að hún er röng að einhverju leiti.
Ég datt samt eitthvað út á nokkrum stöðum (nennti ekki að einbeita mér alveg).
Mér finnst oft erfiðara að skilja gamlar bækur, ekki hægt að kenna bókinni um það.
Gaman og fræðandi að lesa vitandi að hún er röng að einhverju leiti.
February 6, 2025
Fins i tot tenint en compte que Russel va demostrar aquest intent no ser més que això, la seva influència en el món de la matemàtica és inmensa. Com a lector, es un plaer (intentar) seguir els raonaments exposats i pensar en conceptes tant abstractes com atractius. Després de llegir-lo, tinc la sensació d'haver desbloquejat una part del meu raonament (potser només una il·lusió efímera).
September 28, 2022
1. Fun Memories:
Often, I read in Bagelz in Kingston, Rhode Island.
I discussed about this with an Older American.
We were discussing about cultures, Europeans, Africa.
I remember, I shared how works like this are from Europeans.
2. Background of this work:
Lutz Hamel was my Theoretical Computer Science Professor.
Lutz has a PhD from Oxford University.
We started the class with Peano's Axioms.
I scratched my head at that moment, what was it?
I bumped into this work, during my reading of Russell, wanting to survey about Computing.
How did Computer Science come into existence?
Lutz Hamel classes were mostly abstract concepts.
Mostly, from theoretical computer science.
If you were to go in any computer-science department.
Almost all would take a class of Theoretical Computer Science.
3. So, What is in this work?
a. Start of Analytical Philosophy
b. Frege goes into language & concept of number
c. Russell developed this work, further
d. Although, from what I call, Russell's work went into brick-wall
e. If I recall, they were refuted, especially logicism
f. Where Russell & Whitehead wanted to ground everything in basic logic
g. Frege in Chapter four gives meat of the work
I read philosophy of mathematics after this, which is not used anywhere, now that I contemplate about it.
This work might be relevant to Math majors & theoretical computer scientists.
Deus Vult,
Gottfried
Often, I read in Bagelz in Kingston, Rhode Island.
I discussed about this with an Older American.
We were discussing about cultures, Europeans, Africa.
I remember, I shared how works like this are from Europeans.
2. Background of this work:
Lutz Hamel was my Theoretical Computer Science Professor.
Lutz has a PhD from Oxford University.
We started the class with Peano's Axioms.
I scratched my head at that moment, what was it?
I bumped into this work, during my reading of Russell, wanting to survey about Computing.
How did Computer Science come into existence?
Lutz Hamel classes were mostly abstract concepts.
Mostly, from theoretical computer science.
If you were to go in any computer-science department.
Almost all would take a class of Theoretical Computer Science.
3. So, What is in this work?
a. Start of Analytical Philosophy
b. Frege goes into language & concept of number
c. Russell developed this work, further
d. Although, from what I call, Russell's work went into brick-wall
e. If I recall, they were refuted, especially logicism
f. Where Russell & Whitehead wanted to ground everything in basic logic
g. Frege in Chapter four gives meat of the work
I read philosophy of mathematics after this, which is not used anywhere, now that I contemplate about it.
This work might be relevant to Math majors & theoretical computer scientists.
Deus Vult,
Gottfried
February 13, 2014
Read this for my philosophy of maths course and I really enjoyed it. Not just in an, for an academic book it was pretty readable and not too slow, kinda-way, but in an I'm a massive geek for all things mathematically philosophical or philosophically mathematical and genuinely liked reading it. As the book that led to most modern discussions within the area it is well worth reading, even if the system presented is as wrong as can be, there are still some salient points to be found.
August 13, 2012
Originally read many years ago. Michael Dummett cites its #62 as the "most pregnant philosophical paragraph ever written"!
January 13, 2022
Nerd heaven! The most curl-up-with friendly, piquant, and provocative philosophy of math I’ve read so far - though perhaps that’s not saying much. I’m convinced of his theory of Number and of arithmetic as analytic a priori. Gonna re-read when I’m smarter. Some reservations and questions:
1. He distinguishes between one’s doxastic method for arriving at a statement and its “ultimate ground for justification”; the first is necessarily a posteriori whereas the second can be a priori. Given his aversion to the first, his argument could benefit from clarification of what counts as the “ultimate ground for justification”, and if there is even a question as to whether it is a priori or otherwise.
2. What’s with the constant, kinda gratuitous shitting on psychology and experimental philosophy? Was Frege hurt by daddy Freud?
3. Unclear what he means by the demand for definitions which are justified per se, eliminating the possibility that contradiction arises from them. This seems an either unachievable or trivially achievable and hence overall meaningless criterion. Worse, I’m missing how his definition of Number meets it. Sure, it is sufficient for our incumbent propositions. But this is only an “empirical” defence which does not altogether foreclose the possibility of contradiction. Is his more conclusive defence that he exhibits concrete concepts under which something falls, earning their consistency? This might work for 1, 2, and so on, but how do we know there is a *Number* in particular, and not just some arbitrary object, belonging to the concepts he defines? And how does this accommodate 0 (on which all following numbers rest), given the concept is deliberately contradictory?
4. I was lost in the section on phi-series. Might be the translator’s slight, but he moves freely between “concept” as concept versus “concept” as extension of concept, which addles me more (yeah I’m a pedant sue me). A worked example would be lovely.
5. A deeper question that this books dances around: is it purely contingent fact that laws of arithmetic,
governing aphysical, atemporal objects, apply so faithfully to the physical, temporal world of which they float free?
6. I love his discussion of Kant’s synthetic a priori judgments, such as “self-evident” steps in a geometric proof which are not purely grounded in axioms of logic (although perhaps they differ in their synthetic character from these axioms not in kind but only in degree). If these judgments are necessary, wither automated theorem proving?
1. He distinguishes between one’s doxastic method for arriving at a statement and its “ultimate ground for justification”; the first is necessarily a posteriori whereas the second can be a priori. Given his aversion to the first, his argument could benefit from clarification of what counts as the “ultimate ground for justification”, and if there is even a question as to whether it is a priori or otherwise.
2. What’s with the constant, kinda gratuitous shitting on psychology and experimental philosophy? Was Frege hurt by daddy Freud?
3. Unclear what he means by the demand for definitions which are justified per se, eliminating the possibility that contradiction arises from them. This seems an either unachievable or trivially achievable and hence overall meaningless criterion. Worse, I’m missing how his definition of Number meets it. Sure, it is sufficient for our incumbent propositions. But this is only an “empirical” defence which does not altogether foreclose the possibility of contradiction. Is his more conclusive defence that he exhibits concrete concepts under which something falls, earning their consistency? This might work for 1, 2, and so on, but how do we know there is a *Number* in particular, and not just some arbitrary object, belonging to the concepts he defines? And how does this accommodate 0 (on which all following numbers rest), given the concept is deliberately contradictory?
4. I was lost in the section on phi-series. Might be the translator’s slight, but he moves freely between “concept” as concept versus “concept” as extension of concept, which addles me more (yeah I’m a pedant sue me). A worked example would be lovely.
5. A deeper question that this books dances around: is it purely contingent fact that laws of arithmetic,
governing aphysical, atemporal objects, apply so faithfully to the physical, temporal world of which they float free?
6. I love his discussion of Kant’s synthetic a priori judgments, such as “self-evident” steps in a geometric proof which are not purely grounded in axioms of logic (although perhaps they differ in their synthetic character from these axioms not in kind but only in degree). If these judgments are necessary, wither automated theorem proving?
April 12, 2020
Frege'nin bu eserinin kendisine ve modern felsefedeki önemine dair laf kalabalığı yaratmak lüzumsuz. Bu eserin de nadide bir parçası olduğu Frege külliyatının günümüz felsefesinin önemli bir kısmını anlamada mihenk taşları olduğunu belirtip geçebiliriz okura vakit kaybettirmemek ve okurun asıl kaynaklara yönelik değerli vaktini harcamamak için.
Benim bu kısa yorumum, daha ziyade Gözkân'ın bu özenli çevirisine ama daha da önemlisi, nerede ise Frege'nin eseri kadar uzun olan 'Sunuş'una dair. Pek çok değerli felsefe eserinde olduğu gibi, bu eserde de zorluk sadece dilden kaynaklanmıyor; derin ve geniş bir tarihi bağlamda, diğer keskin düşünürlerin bazı anlaşılması zor düşünceleri ile de bir hesaplaşma söz konusu. Dolayısı ile böyle bir sunuşu sayfa sayısını belli bir sınırın altında tutarak ve bu kısıt koşul altında hakkını olabildiğince vererek yapmak da hiç kolay değil! Hele de söz konusu eser, okurun sadece Kant, Leibniz ve diğer büyük düşünürlerin ilgili fikirlerini bildiğini var saymakla kalmayıp, aynı zamanda matematik ve mantık temelleri gibi alanlarda da bir birikim öngörüyorsa. Gözkân olabildiğince düzgün bir Türkçe çeviri sunmakla kalmayıp, böyle zorlu bir metinle uğraşacak okura da bir hayli yardımcı oluyor yaptığı açıklamalar ve yorumlarla. Tam da bu yüzden bana "iyi ki özellikle bu Türkçe çeviriden okumuşum da, böyle bir sunuştan mahrum şekilde bazı terimleri daha kolay anlayabileceğim bir başka dilde, mesela İngilizce okumamışım," dedirtmeyi başardı.
Sunuşun ve çevirinin eleştirilecek yanları yok mudur? Eminim vardır, mesela kendi adıma sunuşun sonlarına doğru Cantor, Dedekind ve Peano ile ilgili bir şeyler beklerdim ama Frege'nin Kant'a yönelik dediği gibi, böyle küçük tartışmaları olduklarından önemli şeylermiş gibi gösterip çevirinin düzgünlüğüne ve önemine gölge düşürür gibi bir yanılgıya yol açmamalıyım asla.
Bunun ötesinde, belki son bir kişisel not olarak şunu düşmeliyim: benim açımdan böyle bir eserin değeri sadece içeriği değil, aynı zamanda Frege'nin ele aldığı içeriği analiz etmek için izlediği yöntem, asla taviz vermediğini sık sık vurguladığı prensipler yani tabiri caizse biraz da "meta-içerik". Böyle hisseden sadece ben miyim bilmiyorum ama bazı kısımlarda aritmetiğin temellerinden ve "sayal sayı nedir, 0 nasıl tanımlanabilir, 1 nasıl tanımlanabilir", vb. sorular dışında dilbilime, zihin felsefesine ve hatta bilgisayar programlamaya, semantik web'e kadar pek çok düşünce tetiklendi zihnimde ve bunlar da başka sorulara yol açtı. Belki böyle eski bir eserin günümüzde halen yaşamasının sebeplerinden biri de, az da olsa, budur.
Benim bu kısa yorumum, daha ziyade Gözkân'ın bu özenli çevirisine ama daha da önemlisi, nerede ise Frege'nin eseri kadar uzun olan 'Sunuş'una dair. Pek çok değerli felsefe eserinde olduğu gibi, bu eserde de zorluk sadece dilden kaynaklanmıyor; derin ve geniş bir tarihi bağlamda, diğer keskin düşünürlerin bazı anlaşılması zor düşünceleri ile de bir hesaplaşma söz konusu. Dolayısı ile böyle bir sunuşu sayfa sayısını belli bir sınırın altında tutarak ve bu kısıt koşul altında hakkını olabildiğince vererek yapmak da hiç kolay değil! Hele de söz konusu eser, okurun sadece Kant, Leibniz ve diğer büyük düşünürlerin ilgili fikirlerini bildiğini var saymakla kalmayıp, aynı zamanda matematik ve mantık temelleri gibi alanlarda da bir birikim öngörüyorsa. Gözkân olabildiğince düzgün bir Türkçe çeviri sunmakla kalmayıp, böyle zorlu bir metinle uğraşacak okura da bir hayli yardımcı oluyor yaptığı açıklamalar ve yorumlarla. Tam da bu yüzden bana "iyi ki özellikle bu Türkçe çeviriden okumuşum da, böyle bir sunuştan mahrum şekilde bazı terimleri daha kolay anlayabileceğim bir başka dilde, mesela İngilizce okumamışım," dedirtmeyi başardı.
Sunuşun ve çevirinin eleştirilecek yanları yok mudur? Eminim vardır, mesela kendi adıma sunuşun sonlarına doğru Cantor, Dedekind ve Peano ile ilgili bir şeyler beklerdim ama Frege'nin Kant'a yönelik dediği gibi, böyle küçük tartışmaları olduklarından önemli şeylermiş gibi gösterip çevirinin düzgünlüğüne ve önemine gölge düşürür gibi bir yanılgıya yol açmamalıyım asla.
Bunun ötesinde, belki son bir kişisel not olarak şunu düşmeliyim: benim açımdan böyle bir eserin değeri sadece içeriği değil, aynı zamanda Frege'nin ele aldığı içeriği analiz etmek için izlediği yöntem, asla taviz vermediğini sık sık vurguladığı prensipler yani tabiri caizse biraz da "meta-içerik". Böyle hisseden sadece ben miyim bilmiyorum ama bazı kısımlarda aritmetiğin temellerinden ve "sayal sayı nedir, 0 nasıl tanımlanabilir, 1 nasıl tanımlanabilir", vb. sorular dışında dilbilime, zihin felsefesine ve hatta bilgisayar programlamaya, semantik web'e kadar pek çok düşünce tetiklendi zihnimde ve bunlar da başka sorulara yol açtı. Belki böyle eski bir eserin günümüzde halen yaşamasının sebeplerinden biri de, az da olsa, budur.
March 14, 2023
He's grasping at the idea that the natural numbers are given by the sequence of sets 0 = {}, 1 = {{}}, 2 = {{}, {{}}}, ...
This is rendered confusing to the modern reader, due to his logical/metaphysical framework where the primary notion is Concept (essentially statements of the form "property P holds for Objects x"), while Object is left essentially undefined/arbitrary, ranging over both physical and immaterial objects, including Concepts themselves. He defines "the Number belonging to a Concept F" to be "the extension of the Concept 'the extension of Concept F is in one-to-one correspondence with the extension of the Concept G'". He then builds the "finite numbers" as special sorts of numbers, built as a series starting with "0 = the number of the Concept 'x is not identical to x'".
This seems to really be some form of modern Platonism, and I'm curious to what extent it was criticized as such by contemporaries. Wittgenstein's repudiation of Frege-style philosophy in his "Philosophical Investigations" suggests that Russell and the Logical Positivists were insufficiently critical or aware of the metaphysics they had accepted.
In the history of philosophy, this text seems relevant for its attack on Kant's idea that arithmetic judgments like "2+5=7" are "synthetic a priori" judgments. Frege, despite having essentially made up new logical forms to support his proofs, insists that he has shown that proofs of arithmetic statements always had been secretly reducible to logical deductions, and hence are "analytic a priori".
Curiously, he still accepted Kant's claim that statements in Euclidean geometry are synthetic a priori, even though Descartes had, in some sense, reduced Euclidean geometry to number centuries before! (This seems like a much greater threat than the discovery of non-Euclidean geometries.) This makes me wonder about how widespread the use of Cartesian coordinates for doing geometry was at the time.
This is rendered confusing to the modern reader, due to his logical/metaphysical framework where the primary notion is Concept (essentially statements of the form "property P holds for Objects x"), while Object is left essentially undefined/arbitrary, ranging over both physical and immaterial objects, including Concepts themselves. He defines "the Number belonging to a Concept F" to be "the extension of the Concept 'the extension of Concept F is in one-to-one correspondence with the extension of the Concept G'". He then builds the "finite numbers" as special sorts of numbers, built as a series starting with "0 = the number of the Concept 'x is not identical to x'".
This seems to really be some form of modern Platonism, and I'm curious to what extent it was criticized as such by contemporaries. Wittgenstein's repudiation of Frege-style philosophy in his "Philosophical Investigations" suggests that Russell and the Logical Positivists were insufficiently critical or aware of the metaphysics they had accepted.
In the history of philosophy, this text seems relevant for its attack on Kant's idea that arithmetic judgments like "2+5=7" are "synthetic a priori" judgments. Frege, despite having essentially made up new logical forms to support his proofs, insists that he has shown that proofs of arithmetic statements always had been secretly reducible to logical deductions, and hence are "analytic a priori".
Curiously, he still accepted Kant's claim that statements in Euclidean geometry are synthetic a priori, even though Descartes had, in some sense, reduced Euclidean geometry to number centuries before! (This seems like a much greater threat than the discovery of non-Euclidean geometries.) This makes me wonder about how widespread the use of Cartesian coordinates for doing geometry was at the time.
August 10, 2022
Frege attempts a reduction of the general laws of arithmetic to the elementary rules of logic in an effort to secure a more robust and a priori foundation for the discipline. This involves, among other things, purging all reference to intuition and sensibility whenever the question of the legitimacy of the concepts of arithmetic (quid juris) is concerned. Hence the lesson--Numbers do not belong to objects but to concepts. What does "0" in the "Venus has 0 moons" belong to? Certainly not to the nonexistent moon. According to Frege, we assign a property "including nothing under it" to the concept "moon of Venus" (p 59). Numbers are creatures of the mind that nonetheless wield a certain sort of objectivity due to the fact that they belong to the rules by which we make sense of the world. Struggling to intuit 10000 African Elephants into a unitary representation? Just bring them all under the concept "Largest land animal on Earth"! The excited Frege then proceeds to claim that the concept has far superior "power of collecting together than than the synthetic [unity of] apperception" (p 61). I guess this is technically true, but on the Kantian account the synthetic apperception [a priori synthesis] also happens the very basis of the faculty that generates rules viz concepts in the first place. Then we come to defining numbers 0 and 1 in purely logical terms. For example Frege defines 0 as the Number which falls under the concept "not identical with itself". This is where Frege departs from Kant.
Here Frege's logical reduction, if successful, would demonstrate that at least in the case of arithmetic, by simply unpacking the consequences of the definitions and proofs (a thoroughly analytic enterprise) we can indeed extend our knowledge (extension in the sense of synthetic as Kant intended it) but without appealing to anything given in intuition.
Here Frege's logical reduction, if successful, would demonstrate that at least in the case of arithmetic, by simply unpacking the consequences of the definitions and proofs (a thoroughly analytic enterprise) we can indeed extend our knowledge (extension in the sense of synthetic as Kant intended it) but without appealing to anything given in intuition.
October 17, 2014
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Read this and reviews of other classics in Western Philosophy on the History page of www.BestPhilosophyBooks.org (a thinkPhilosophy Production).
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Discovered and popularized posthumously by Bertrand Russell (see Russell and Alfred North Whitehead'’s Principia Mathematica ), Ludwig Wittgenstein, and others, Gottlob Frege is now known as the father of modern logic and of analytic philosophy, and The Foundations of Arithmetic is his most important work.
Frege examines the philosophical foundations for mathematics, destroying some earlier theories (i.e. Psychologism, the idea that knowledge of numbers is subjective) and proposing a rigorous system that establishes objectivity for mathematics. In short, he proved how mathematical truths and numbers have properties that are independent of the subjects who think them.
This annotated edition of this, Frege’s most important work, makes the work more accessible and meaningful for the general reader. It includes an introduction to Frege’s life and works, outlining his influences and some key biographical details.
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Read this and reviews of other classics in Western Philosophy on the History page of www.BestPhilosophyBooks.org (a thinkPhilosophy Production).
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Read this and reviews of other classics in Western Philosophy on the History page of www.BestPhilosophyBooks.org (a thinkPhilosophy Production).
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Discovered and popularized posthumously by Bertrand Russell (see Russell and Alfred North Whitehead'’s Principia Mathematica ), Ludwig Wittgenstein, and others, Gottlob Frege is now known as the father of modern logic and of analytic philosophy, and The Foundations of Arithmetic is his most important work.
Frege examines the philosophical foundations for mathematics, destroying some earlier theories (i.e. Psychologism, the idea that knowledge of numbers is subjective) and proposing a rigorous system that establishes objectivity for mathematics. In short, he proved how mathematical truths and numbers have properties that are independent of the subjects who think them.
This annotated edition of this, Frege’s most important work, makes the work more accessible and meaningful for the general reader. It includes an introduction to Frege’s life and works, outlining his influences and some key biographical details.
---
Read this and reviews of other classics in Western Philosophy on the History page of www.BestPhilosophyBooks.org (a thinkPhilosophy Production).
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December 19, 2020
به عنوان یکی از متون قدیمی و مبادی فلسفه تحلیلی خواندش جالب بود و پر از ایده، اما نارسا هم بود (نمی دونم از ترجمه بود یا متن اصلی هم همینقدر نارسا بود) تا جایی که من فهمیدم فرگه اول دیدگاه های روانشناسی یا تجربی عدد رو مناسب نمی دونه، بعد تعریف عدد رو انجام میده و نشون میده با قوانین منطق میشه از روی تعریف عدد، نظریه حساب رو ساخت بنا بر این نظریه حساب تحلیلیه و صرفا از منطق در میاد، و چون منطق عینی و همیشه صادقه پس نظریه حساب هم عینی و همیشه صادقه (فرگه به نوعی در مورد منطق افلاطونی فکر می کنه و در مورد ریاضی هم نتیجه منطق میدونه و به نظرش تقریبا ریاضی کشف میشه نه ابداع) با این همه من نفهمیدم تکلیف فرگه با اعداد مختلط و منفی و کسری و حقیقی چیه؟ از طرفی میگه تعاریف صوری تناقض ندارن اما همچنان گیر میده که عدم وجود تناقض به معنی وجود مدلولهای این تعاریفِ صوری نیست و همون طور که در مورد اعداد طبیعی و حسابشون، معنی اعداد کاملا تعریف شده، در مورد بقیه هم باید این اتفاق بیافته اما آخرش جملاتی میگه که به نظر میرسه مشکلی هم با اینها نداره.
در نهایت برای کسی که با مفهوم عدد مشکل داشته می تونه متن بسیار جذابی باشه.
در نهایت برای کسی که با مفهوم عدد مشکل داشته می تونه متن بسیار جذابی باشه.
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July 19, 2019A close study of Kant's first critique certainly helps understanding this text (which I lack and consequently do not understand some parts fully). If you come from modern mathematics you'll feel as if you are studying its very inception.
February 11, 2023
Gottlob Frege owes his reputation as perhaps the greatest logician since Aristotle to his invention of the predicate calculus, first published in his Begriffsschrift, eine der Arithmetischen nachgebildete Formelsprache des reinen Denkens (1879). As one can tell from its title, Frege’s motivation stems from a desire to frame arithmetic as an ideal model of all thought which, for him, rests upon a strictly logical foundation. Acting upon the advice of his colleagues, though, he decided to gather his reflections on arithmetic itself and to write them up in an informal manner, independent of the technical refinements of the then still little known predicate calculus and his inconvenient mode of presentation of it, which even to this day most everyone shies away from (we prefer a simpler but equivalent notation that goes back to Peano and Russell). This very thing he undertakes in his Grundlagen der Arithmetik (1884), a classic text that has proved instrumental to the development of modern analytic philosophy.
Frege’s goal is not to advance pure mathematics or number theory in and of itself but to disclose the source of its knowledge [Erkenntnisquelle]. This invites polemics against psychologism, as represented for instance by J.S. Mill who wants somewhat ineptly to ascribe mathematical knowledge, even in basic arithmetic, to an empirical origin. For Mill, so to speak, the truths of arithmetic constitute observable laws of nature! Frege has little trouble putting paid to views such as this (a number cannot simply be a property of an aggregate of objects, though it may well be applied to knowledge of the natural world), though Leibniz’ concept of innate principles of necessary truth from which arithmetic flows and Kant’s idea of the synthetic a priori call for a more extensive discussion in part i. For Frege, mathematics has to have an objective ground (although he will concede that subjective experience could be necessary for us to learn about mathematics, it merely leads us to recognize propositions that require logical justification apart from the conditions of space and time).
In part ii, Frege seeks to define rigorously the concept of counting number [Anzahl] on which he intends to base his theory. Two basic premises are announced that will guide his reflections and that are pregnant for the further development of analytical philosophy: first, that the meaning of a sentence can be drawn only from the connection among its terms [nach der Bedeutung der Wörter muss in Zusammenhange, nicht in ihrer Vereinzelung gefragt werden] and second, one must distinguish between a concept and an object [der Untershied zwischen Begriff und Gegenstand ist im Auge zu behalten]. An object can be an instance of a concept, or fall under it, but the two terms fulfill categorically different functions in our thought; it is always a mistake to try to make a concept into an object. Moreover, a concept [Begriff] is to be differentiated from its extension [Umfang], or perhaps scope would be a better translation. Roughly speaking, the extension of a concept comprises everything that enjoys its defining properties. So, to employ a typical example due to Frege, the direction of a line is by definition the extension of the concept of parallelism between lines, that is to say, every line has a direction which amounts to nothing but the concept of all other lines parallel to it.
Now, how are these ideas to be applied to arithmetic? Frege appeals to what may be called Hume’s principle: he wants number [Anzahl] to be the extension of the concept of equinumerosity. Thus, two sets are to be regarded as having the same number if and only if there exists a one-to-one correspondence between them. Frege’s task in the remainder of parts iii and iv is to legitimate the claim that this analytical definition incorporates the essence of what we mean by number and to show that he has thereby placed arithmetic upon a strictly logical foundation. To this end, he discusses the concept of a successor and supplies logical definitions of the numbers zero and one: for instance, zero is the number [Anzahl] characterized by the concept of ‘being unequal to itself’ and one is the number [Anzahl] characterized by the concept of ‘being equal to zero’. It follows fairly immediately that zero and one correspond to different numbers. In all this, two things elicit the present reviewer’s attention: first, the treatment of the distinction between unity [Einheit] and one [Eins], in which Frege reviews the status quaestionis going back to Aristotle and pursuing it through Leibniz, Hobbes, Hume, Locke, Thomae, Descartes up to several of his contemporaries, such as Baumann, St. Jevons, Köpp, Schröder, Hankel before giving his own solution [pp. 44-65] – clearly, a profound subject independent of whatever Frege may conclude about it; and second, the treatment of Cantor’s transfinitum [pp. 89-91] – at a juncture when Cantor had yet to hone his inchoate thoughts into systematic form (which was not to arrive until the latter’s Beiträge zur Begründung der transfiniten Mengenlehre of 1895/97; cf. our review of Cantor’s collected works here). Frege justifies the first uncountable ordinal as a genuine number [Anzahl] but finds it hard to see how the concept of successor would apply to it or to any higher infinities. The root of his difficulty lies in a failure on his part to observe a clear distinction between ordinality and cardinality – for which one can scarcely blame him, given the primitive state of development of set theory at the moment, even in Cantor’s mind.
A few reservations as to the stringency of the argumentation: it is not so evident from Frege’s definition of 1 that it must be the successor of 0; hence, it remains unclear why his logical definitions do in fact capture what we really mean by the numbers 0, 1, etc. For one might well wonder, can the concept of successor mean anything without the inner intuition of time? Also, in the present work Frege never defines anywhere the arithmetic operations of addition and multiplication nor does he justify the principle of induction. Lastly, though he declares himself forcefully against psychologism, he enters into little sustained discussion of it. Certainly, one suspects there must be rather more cogent defenders of the doctrine than J.S. Mill, say for instance the latter’s contemporary in Germany, the psychologist and post-Kantian Friedrich Eduard Beneke. Yet Beneke does not even so much as merit mention by Frege, here at least.
All the same, despite its flaws Frege’s foundation of arithmetic herein presented deserves attentive study as harbinger of what was to come, not just in the Grundgesetze der Arithmetik of 1893 but also in the pioneering reflections on the philosophy of language to which Frege was led by his early fixation on arithmetic – for instance, to his celebrated distinction between sense [Sinn] and reference [Bedeutung]. But all of this at the proper occasion! For now, contemplate the intellectual audacity of Frege’s project and look forward to seeing whether he makes good on its promise in the Grundgesetze in which he will deploy all of the newly-won formalism of the predicate calculus so as to render rigorous the informal argumentation of the present work (and for which we must therefore promise a review to come in due time).
Frege’s goal is not to advance pure mathematics or number theory in and of itself but to disclose the source of its knowledge [Erkenntnisquelle]. This invites polemics against psychologism, as represented for instance by J.S. Mill who wants somewhat ineptly to ascribe mathematical knowledge, even in basic arithmetic, to an empirical origin. For Mill, so to speak, the truths of arithmetic constitute observable laws of nature! Frege has little trouble putting paid to views such as this (a number cannot simply be a property of an aggregate of objects, though it may well be applied to knowledge of the natural world), though Leibniz’ concept of innate principles of necessary truth from which arithmetic flows and Kant’s idea of the synthetic a priori call for a more extensive discussion in part i. For Frege, mathematics has to have an objective ground (although he will concede that subjective experience could be necessary for us to learn about mathematics, it merely leads us to recognize propositions that require logical justification apart from the conditions of space and time).
In part ii, Frege seeks to define rigorously the concept of counting number [Anzahl] on which he intends to base his theory. Two basic premises are announced that will guide his reflections and that are pregnant for the further development of analytical philosophy: first, that the meaning of a sentence can be drawn only from the connection among its terms [nach der Bedeutung der Wörter muss in Zusammenhange, nicht in ihrer Vereinzelung gefragt werden] and second, one must distinguish between a concept and an object [der Untershied zwischen Begriff und Gegenstand ist im Auge zu behalten]. An object can be an instance of a concept, or fall under it, but the two terms fulfill categorically different functions in our thought; it is always a mistake to try to make a concept into an object. Moreover, a concept [Begriff] is to be differentiated from its extension [Umfang], or perhaps scope would be a better translation. Roughly speaking, the extension of a concept comprises everything that enjoys its defining properties. So, to employ a typical example due to Frege, the direction of a line is by definition the extension of the concept of parallelism between lines, that is to say, every line has a direction which amounts to nothing but the concept of all other lines parallel to it.
Now, how are these ideas to be applied to arithmetic? Frege appeals to what may be called Hume’s principle: he wants number [Anzahl] to be the extension of the concept of equinumerosity. Thus, two sets are to be regarded as having the same number if and only if there exists a one-to-one correspondence between them. Frege’s task in the remainder of parts iii and iv is to legitimate the claim that this analytical definition incorporates the essence of what we mean by number and to show that he has thereby placed arithmetic upon a strictly logical foundation. To this end, he discusses the concept of a successor and supplies logical definitions of the numbers zero and one: for instance, zero is the number [Anzahl] characterized by the concept of ‘being unequal to itself’ and one is the number [Anzahl] characterized by the concept of ‘being equal to zero’. It follows fairly immediately that zero and one correspond to different numbers. In all this, two things elicit the present reviewer’s attention: first, the treatment of the distinction between unity [Einheit] and one [Eins], in which Frege reviews the status quaestionis going back to Aristotle and pursuing it through Leibniz, Hobbes, Hume, Locke, Thomae, Descartes up to several of his contemporaries, such as Baumann, St. Jevons, Köpp, Schröder, Hankel before giving his own solution [pp. 44-65] – clearly, a profound subject independent of whatever Frege may conclude about it; and second, the treatment of Cantor’s transfinitum [pp. 89-91] – at a juncture when Cantor had yet to hone his inchoate thoughts into systematic form (which was not to arrive until the latter’s Beiträge zur Begründung der transfiniten Mengenlehre of 1895/97; cf. our review of Cantor’s collected works here). Frege justifies the first uncountable ordinal as a genuine number [Anzahl] but finds it hard to see how the concept of successor would apply to it or to any higher infinities. The root of his difficulty lies in a failure on his part to observe a clear distinction between ordinality and cardinality – for which one can scarcely blame him, given the primitive state of development of set theory at the moment, even in Cantor’s mind.
A few reservations as to the stringency of the argumentation: it is not so evident from Frege’s definition of 1 that it must be the successor of 0; hence, it remains unclear why his logical definitions do in fact capture what we really mean by the numbers 0, 1, etc. For one might well wonder, can the concept of successor mean anything without the inner intuition of time? Also, in the present work Frege never defines anywhere the arithmetic operations of addition and multiplication nor does he justify the principle of induction. Lastly, though he declares himself forcefully against psychologism, he enters into little sustained discussion of it. Certainly, one suspects there must be rather more cogent defenders of the doctrine than J.S. Mill, say for instance the latter’s contemporary in Germany, the psychologist and post-Kantian Friedrich Eduard Beneke. Yet Beneke does not even so much as merit mention by Frege, here at least.
All the same, despite its flaws Frege’s foundation of arithmetic herein presented deserves attentive study as harbinger of what was to come, not just in the Grundgesetze der Arithmetik of 1893 but also in the pioneering reflections on the philosophy of language to which Frege was led by his early fixation on arithmetic – for instance, to his celebrated distinction between sense [Sinn] and reference [Bedeutung]. But all of this at the proper occasion! For now, contemplate the intellectual audacity of Frege’s project and look forward to seeing whether he makes good on its promise in the Grundgesetze in which he will deploy all of the newly-won formalism of the predicate calculus so as to render rigorous the informal argumentation of the present work (and for which we must therefore promise a review to come in due time).
July 1, 2024
I would be lying if I said that this book was not hard to get through. It requires a lot of attention to every single line to be able to grasp the entire picture. While I had very little knowledge of the in-depth concept of numbers and arithmetic, I believe that I was able to grasp most of what Frege was trying to convey.
As a philosophy giant who inspired the likes of Bertrand Russell and Ludwig Wittgenstein, it is surprising to see how little attention to paid to Gottlob Frege. When I was recommended to read this book by a man I met in a used bookstore, I was a bit hesitant to try and fully delve into this book. But once I read the first couple of pages, I eased into it.
Here are the main points that I was able to pick up:
1. Number is not abstract nor a property.
2. Number is not physical nor subjective.
3. Number doesn't come from the combination of things (composite).
4. Multitude, set, and plurality are unfit to define number.
5. Undivided, incapable of dissection, and or being isolated are not what we mean when we express "one".
6. Calling numbers "numerical units" and then asserting that they're identical is a falsity. It is necessary for the things to be numbered to be different to go beyond 1.
7. Units are then containing two contradicting elements, identity and distinguishability.
8. Numbers are self-sufficient objective objects that do not occupy space.
9. Numbers only form an element in the assertion of a concept.
Overall, this book made me more interested in the concept of number and has started to push me into the rabbit hole of analytic philosophy. Would recommend.
As a philosophy giant who inspired the likes of Bertrand Russell and Ludwig Wittgenstein, it is surprising to see how little attention to paid to Gottlob Frege. When I was recommended to read this book by a man I met in a used bookstore, I was a bit hesitant to try and fully delve into this book. But once I read the first couple of pages, I eased into it.
Here are the main points that I was able to pick up:
1. Number is not abstract nor a property.
2. Number is not physical nor subjective.
3. Number doesn't come from the combination of things (composite).
4. Multitude, set, and plurality are unfit to define number.
5. Undivided, incapable of dissection, and or being isolated are not what we mean when we express "one".
6. Calling numbers "numerical units" and then asserting that they're identical is a falsity. It is necessary for the things to be numbered to be different to go beyond 1.
7. Units are then containing two contradicting elements, identity and distinguishability.
8. Numbers are self-sufficient objective objects that do not occupy space.
9. Numbers only form an element in the assertion of a concept.
Overall, this book made me more interested in the concept of number and has started to push me into the rabbit hole of analytic philosophy. Would recommend.
January 6, 2021
Wrong, but brilliantly so!
June 4, 2025
A classic in the Philosophy of Mathematics, revolutionary in its time though nonsensical in application. Well worth reading for seeing the difficulty of finding foundations.
March 3, 2021
Okuduğum en zor kitaplardan biri, bu puan da anlamamama. :(
June 14, 2013
This was a fun read.
Unfortunately, Frege does some hand-waving throughout the book. He is attempting to show that arithmetic is an extension of logic. Critical to his argument is the definition of number as a concept expresssable by second-order classical logic. In showing his definition of number is in fact correct, he isolates number away from the realm of psychology. There are some minor points throughout the text in which he relies on the reader's intuition and some popular beliefs to 'wave away' a point. Maybe he's done so because his treatment indirectly expresses how stupid the point in question was. Whatever the reason, the result is a break from rigorous definition.
My wish for the content of the book is to see it written in logical formalism.
Unfortunately, Frege does some hand-waving throughout the book. He is attempting to show that arithmetic is an extension of logic. Critical to his argument is the definition of number as a concept expresssable by second-order classical logic. In showing his definition of number is in fact correct, he isolates number away from the realm of psychology. There are some minor points throughout the text in which he relies on the reader's intuition and some popular beliefs to 'wave away' a point. Maybe he's done so because his treatment indirectly expresses how stupid the point in question was. Whatever the reason, the result is a break from rigorous definition.
My wish for the content of the book is to see it written in logical formalism.
March 25, 2017
Agonistic philosophy is rare enough after the Greeks. Frege's bashing of J.S. Mill ranks right up there with Schopenhauer's sarcastic hostility toward Fichte and Hegel, and Nietzsche making fun of everyone (but especially Kant).
December 26, 2007
Perhaps some the clearest writing I've ever encountered.
September 21, 2008
Written like a knife, this concise exposition of Frege's philosophy of mathematics still stands up well to scrutiny today.
June 9, 2021
1 ay süresince yan okumalarla beraber anca bitirebildim. Frege’den önceki düşünürlerin sayı kuramları hakkında ön bilgi gerektirdiğini düşünüyorum. Frege kendisi bunları özet şeklinde verip bunların neden aritmetiğin kuramı olamayacağını açıklamış ancak belirli bir hazırlık olmadan okuma süresi uzayabiliyor. Yine de anladığımı düşünüyorum. Oluşturduğu kuramın sonradan aslında paradokslar içerdiğinin anlaşılması kitabın değerini azaltmıyor. Aksine matematiksel gelişim sürecinde ne gibi değişimlerin olduğunu görmek, şu an kabul edilen kurama ulaşmadaki çabaları ve zorlukları anlamaya yardımcı oluyor. Dil felsefesiyle ilgilenenlere ve matematikçilere öneririm.
June 9, 2022
Este libro está dividido en dos partes, la primera los Fundamentos de la aritmética de Frege y la segunda un análisis de la obra realizado por Claude Imbert.
Frege se dedica a investigar el origen de los números, el 0, el 1 y el concepto de número siguiente a otro. Esto, que parece tan sencillo, al tratar de hacerlo de forma totalmente rigurosa se convierte en un problema filosófico muy complejo. De hecho ninguno de sus predecesores, filósofos o matemáticos, lo consigue de manera satisfactoria. Más aún, a pesar del esfuerzo del autor su solución no es completa y así se lo hizo ver más tarde Bertrand Russell.
El libro intenta que los conceptos sean lo más sencillos posible. Las profundidades filosóficas a veces se vuelven complicadas, a pesar de la aparente simplicidad del problema.
En la segunda mitad Imbert se dedica a explicar el trabajo de Frege con la perspectiva que da el tiempo y la evolución del problema con posterioridad al libro. El lenguaje resulta más cercano y más fácil de seguir.
Se trata de un libro de filosofía de las matemáticas, interesante para conocer un paso muy concreto de la evolución del concepto de número.
Frege se dedica a investigar el origen de los números, el 0, el 1 y el concepto de número siguiente a otro. Esto, que parece tan sencillo, al tratar de hacerlo de forma totalmente rigurosa se convierte en un problema filosófico muy complejo. De hecho ninguno de sus predecesores, filósofos o matemáticos, lo consigue de manera satisfactoria. Más aún, a pesar del esfuerzo del autor su solución no es completa y así se lo hizo ver más tarde Bertrand Russell.
El libro intenta que los conceptos sean lo más sencillos posible. Las profundidades filosóficas a veces se vuelven complicadas, a pesar de la aparente simplicidad del problema.
En la segunda mitad Imbert se dedica a explicar el trabajo de Frege con la perspectiva que da el tiempo y la evolución del problema con posterioridad al libro. El lenguaje resulta más cercano y más fácil de seguir.
Se trata de un libro de filosofía de las matemáticas, interesante para conocer un paso muy concreto de la evolución del concepto de número.
April 10, 2023
Simple but not simplistic presentation of a logical framework for arithmetic. The extent to which such a basis is needed is debatable as he notes Indians among others have carried out complex and sophisticated mathematics with having laid the foundation in specifically classical logic (I don’t quite agree non Greek math is less rigorous/logical). I wish more detail on extension properties were dealt with and more could have been written about predecessor functions (Conway discusses this) but still very rigorous text.
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