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Cambridge Introductions to Philosophy
An Introduction to the Philosophy of Mathematics
This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics.
200 pages, Hardcover
First published May 31, 2012
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Displaying 1 - 10 of 10 reviews
July 23, 2023
Reading Stella Maris spurred an interest in philosophy of mathematics, so I found a couple of recommendations for where to start and here I am. This book provides a somewhat brief overview of relevant topics mostly concentrating on current (as of 2012) issues of debate in the field, weighted towards those the author is particularly interested in and wants to push attention towards (paraconsistent logic, theories of mathematical explanation, mathematical notation). Thus it’s not a great source for getting up to speed on the central historical debates between realist (Platonist) and anti-realist (nominalist) interpretations of mathematics and mathematical entities, although it does go over those in a few earlier chapters. Those chapters were my favorite.
The discussion questions at the end of each chapter I found valuable for leading to more thought and development of the ideas presented, even though this necessarily took place just in my head and not in a classroom including other students, as the book was created for. A brief presentation and remarks about sources for further reading on those specific topics in the chapter is included at the end of each chapter as well, and these would seem to be valuable for the student.
Some understanding of formal logic and complex math would certainly be helpful, though I managed to muddle through ok, sometimes consulting other sources like the online Stanford Encyclopedia of Philosophy for assistance.
Includes some nice quotes:
“Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” - Bertrand Russell, in “Recent Work on the Principles of Mathematics”
“In mathematics you don’t understand things. You just get used to them.” - John von Neumann
“Beauty is the first test: there is no permanent place in the world for ugly mathematics.” - G.H. Hardy in “A Mathematician’s Apology”
The discussion questions at the end of each chapter I found valuable for leading to more thought and development of the ideas presented, even though this necessarily took place just in my head and not in a classroom including other students, as the book was created for. A brief presentation and remarks about sources for further reading on those specific topics in the chapter is included at the end of each chapter as well, and these would seem to be valuable for the student.
Some understanding of formal logic and complex math would certainly be helpful, though I managed to muddle through ok, sometimes consulting other sources like the online Stanford Encyclopedia of Philosophy for assistance.
Includes some nice quotes:
“Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” - Bertrand Russell, in “Recent Work on the Principles of Mathematics”
“In mathematics you don’t understand things. You just get used to them.” - John von Neumann
“Beauty is the first test: there is no permanent place in the world for ugly mathematics.” - G.H. Hardy in “A Mathematician’s Apology”
April 2, 2026
Bien escrito, en un tono ameno, y bien explicado. Muy buen resumen introductorio. Lectura somera. A continuación comento algunas cosas que me han llamado la atención:
===== La paradoja de Skolem =====
Se llama así al aparente choque entre el teorema de Cantor y el teorema de Löwenheim–Skolem
> In 1915 the German mathematician Leopold Löwenheim (1878–1957) proved a remarkable result. He proved that if a first-order sentence has a model at all, it has a countable model. 4 In 1922 the Norwegian mathematician Thoralf Skolem (1887–1963) generalised this result to systems of first-order sentences. 5 What is remarkable about these results is that they appear to fly in the face of Cantor’s Theorem. The Löwenheim–Skolem Theorem seems to be telling us that we do not need to entertain infinities beyond the countable. In particular, it seems to be telling us that there are countable models of the real numbers and of set theory itself. This apparent conflict between Cantor’s Theorem and the Löwenheim–Skolem Theorem became known as Skolem’s paradox.
En efecto, podemos encontrar un modelo con contables elementos que satisfaga los axiomas de, por ejemplo, los números reales. La resolución aceptada de la paradoja es que ese modelo seguramente no tenga nada que ver con los números reales que nos interesan.
> Despite the hint of paradox here, there is in fact no paradox. The generally agreed-upon solution to the apparent paradox is that although under some interpretations of the mathematical terms in question (set membership, successor, subset, and the like) there will be uncountable models, under different interpretations of the terms in question there will be countable models. What is crucial is the failure of the theory to absolutely fix the reference of the mathematical terms.
Junto con los teoremas de la incompletitud de Godel, suponen un buen ejemplo de los límites de las matemáticas.
Putnam lo emplea contra el realismo:
> Hilary Putnam (1926–) argues that the Löwenheim–Skolem Theorem undermines common-sense realism, not just about mathematics but elsewhere as well (Putnam 1980). The idea, very roughly, is that if we were to formulate our best scientific theories in a first-order language, we’d find the same relativity. We find that there is no fact of the matter about the reference of our theoretical terms. Moreover, the indeterminacy in question undermines any confidence we have in the ontology of our physical and even common-sense theories. This, in turn, invites a turn to anti-realism.
===== Los problemas de Benacerraf =====
Paul Benacerraf estableció en las d.60-80 las preguntas que desde entonces estructuran el debate en la filosofía de las matemáticas.
Ver: problemas de Benacerraf
> The agenda for contemporary philosophy of mathematics was shaped by Paul Benacerraf in a couple of landmark papers. In the first of the papers (Benacerraf 1983a, originally published in 1965), Benacerraf outlines an underdetermination problem for the project of reducing all of mathematics to set theory. Such underdetermination or non-uniqueness problems had been around for some time, but Benacerraf’s presentation was compelling, and its relevance to a popular position in philosophy of mathematics was firmly established. The second and third problems (Benacerraf 1983b, originally published in 1973) are presented as a challenge that any credible philosophy of mathematics must meet: (i) allow for a semantics that is uniform across both mathematical and non-mathematical discourse and (ii) provide a plausible epistemology for mathematics. As Benacerraf went on to show, it is difficult to satisfy both parts of this challenge simultaneously. Any philosophy of mathematics that meets one part of the challenge typically has serious difficulties meeting the other part.
Resumen: ¿Como ofrecer al tiempo de la //semántica// (de qué hablan los enunciados matemáticos) y una //epistemología// (cómo conocemos los enunciados matemáticos) de las matemáticas que sean //uniformes// (que sean las mismas que para otros ámbitos no matemáticos)?\\
El realismo/platonismo ofrece la semántica pero no la epistemología. El nominalismo ofrece la epistemología pero no la semántica.
===== Metafísica de las matemáticas =====
Recorre varias propuestas:
* platonismo - filosofía de las matemáticas
* nominalismo - filosofía de las matemáticas
* ficcionalismo - filosofía de las matemáticas
* formalismo - filosofía de las matemáticas
* matemáticas como metáfora - filosofía de las matemáticas
* estructuralismo - filosofía de las matemáticas
===== Explicación =====
Analiza varias teorías de cómo funciona la explicación en matemáticas.
===== Argumentos de indispensabilidad =====
Referir a las entidades matemáticas (o cuantificar sobre ellas) es indispensable para el funcionamiento de nuestras mejores teorías científicas. Sin matemáticas no habría ciencia.
Este es el argumento de indispensabilidad para el realismo matemático - Quine, Putnam. Se le suele considerar el mejor argumento a favor del platonismo/realismo.
Esto lo niega Hartry Field, que para mostrar que se puede hacer ciencia sin matemáticas, reformuló buena parte de la teoría gravitatoria de Newton (comparando puntos espaciotemporales en base a su potencial gravitacional, etc. [mico] ¿Acaso no son estas realciones abstractas, no numéricas pero sí estructurales, también matemáticas?). Su tesis es que la matemática no añade nada nuevo a la ciencia, es una herramienta útil pero dispensable, un atajo.
Otra vía para negar el argumento de indispensabilidad es negar, simplemente, que el uso de las matemáticas en la ciencia implique un compromiso metafísico con la existencia
===== La paradoja de Skolem =====
Se llama así al aparente choque entre el teorema de Cantor y el teorema de Löwenheim–Skolem
> In 1915 the German mathematician Leopold Löwenheim (1878–1957) proved a remarkable result. He proved that if a first-order sentence has a model at all, it has a countable model. 4 In 1922 the Norwegian mathematician Thoralf Skolem (1887–1963) generalised this result to systems of first-order sentences. 5 What is remarkable about these results is that they appear to fly in the face of Cantor’s Theorem. The Löwenheim–Skolem Theorem seems to be telling us that we do not need to entertain infinities beyond the countable. In particular, it seems to be telling us that there are countable models of the real numbers and of set theory itself. This apparent conflict between Cantor’s Theorem and the Löwenheim–Skolem Theorem became known as Skolem’s paradox.
En efecto, podemos encontrar un modelo con contables elementos que satisfaga los axiomas de, por ejemplo, los números reales. La resolución aceptada de la paradoja es que ese modelo seguramente no tenga nada que ver con los números reales que nos interesan.
> Despite the hint of paradox here, there is in fact no paradox. The generally agreed-upon solution to the apparent paradox is that although under some interpretations of the mathematical terms in question (set membership, successor, subset, and the like) there will be uncountable models, under different interpretations of the terms in question there will be countable models. What is crucial is the failure of the theory to absolutely fix the reference of the mathematical terms.
Junto con los teoremas de la incompletitud de Godel, suponen un buen ejemplo de los límites de las matemáticas.
Putnam lo emplea contra el realismo:
> Hilary Putnam (1926–) argues that the Löwenheim–Skolem Theorem undermines common-sense realism, not just about mathematics but elsewhere as well (Putnam 1980). The idea, very roughly, is that if we were to formulate our best scientific theories in a first-order language, we’d find the same relativity. We find that there is no fact of the matter about the reference of our theoretical terms. Moreover, the indeterminacy in question undermines any confidence we have in the ontology of our physical and even common-sense theories. This, in turn, invites a turn to anti-realism.
===== Los problemas de Benacerraf =====
Paul Benacerraf estableció en las d.60-80 las preguntas que desde entonces estructuran el debate en la filosofía de las matemáticas.
Ver: problemas de Benacerraf
> The agenda for contemporary philosophy of mathematics was shaped by Paul Benacerraf in a couple of landmark papers. In the first of the papers (Benacerraf 1983a, originally published in 1965), Benacerraf outlines an underdetermination problem for the project of reducing all of mathematics to set theory. Such underdetermination or non-uniqueness problems had been around for some time, but Benacerraf’s presentation was compelling, and its relevance to a popular position in philosophy of mathematics was firmly established. The second and third problems (Benacerraf 1983b, originally published in 1973) are presented as a challenge that any credible philosophy of mathematics must meet: (i) allow for a semantics that is uniform across both mathematical and non-mathematical discourse and (ii) provide a plausible epistemology for mathematics. As Benacerraf went on to show, it is difficult to satisfy both parts of this challenge simultaneously. Any philosophy of mathematics that meets one part of the challenge typically has serious difficulties meeting the other part.
Resumen: ¿Como ofrecer al tiempo de la //semántica// (de qué hablan los enunciados matemáticos) y una //epistemología// (cómo conocemos los enunciados matemáticos) de las matemáticas que sean //uniformes// (que sean las mismas que para otros ámbitos no matemáticos)?\\
El realismo/platonismo ofrece la semántica pero no la epistemología. El nominalismo ofrece la epistemología pero no la semántica.
===== Metafísica de las matemáticas =====
Recorre varias propuestas:
* platonismo - filosofía de las matemáticas
* nominalismo - filosofía de las matemáticas
* ficcionalismo - filosofía de las matemáticas
* formalismo - filosofía de las matemáticas
* matemáticas como metáfora - filosofía de las matemáticas
* estructuralismo - filosofía de las matemáticas
===== Explicación =====
Analiza varias teorías de cómo funciona la explicación en matemáticas.
===== Argumentos de indispensabilidad =====
Referir a las entidades matemáticas (o cuantificar sobre ellas) es indispensable para el funcionamiento de nuestras mejores teorías científicas. Sin matemáticas no habría ciencia.
Este es el argumento de indispensabilidad para el realismo matemático - Quine, Putnam. Se le suele considerar el mejor argumento a favor del platonismo/realismo.
Esto lo niega Hartry Field, que para mostrar que se puede hacer ciencia sin matemáticas, reformuló buena parte de la teoría gravitatoria de Newton (comparando puntos espaciotemporales en base a su potencial gravitacional, etc. [mico] ¿Acaso no son estas realciones abstractas, no numéricas pero sí estructurales, también matemáticas?). Su tesis es que la matemática no añade nada nuevo a la ciencia, es una herramienta útil pero dispensable, un atajo.
Otra vía para negar el argumento de indispensabilidad es negar, simplemente, que el uso de las matemáticas en la ciencia implique un compromiso metafísico con la existencia
January 13, 2021
Eğer bir şeyi bildiğinizden eminseniz felsefe okuyun. Kitapta matematiğe nasıl şüpheyle bakılabileceğini ve bu şüphelere de nasıl şüpheyle yaklaşılabileceğini basitçe anlatılmış. Basit dediysem herkes bir kerede anlar demek istemiyorum ama karmaşık da değil. Giriş seviyesi bir kitap olduğundan ana akımlara ve bu akımların önemli temsilcilerine yer verilmiş. Bence biraz matematik temeli olan ve felsefeye ilgi duyan herkes okuyabilir.
February 24, 2018
Clearly written and sensibly organised textbook which gave a brief overview of several areas of debate. As somebody with almost no knowledge of algebra or set theory I was able to skip past the proofs, as Colyvan does a good job of explaining their significance.
August 10, 2020
Monella tapaa erinomainen johdatus matematiikan filosofiaan. Colyvan käsittelee tieteiden kuningatarta sekä ansaitulla kunnioituksella, että tarpeellisella kritiikillä.
Matematiikan erityisluonne tieteiden joukossa tekee siitä filosofisesti hyvin kiinnostavan. Kirjan ytimessä ovat kysymykset matemaattisten olioiden luonteesta, matematiikan sovellettavuudesta empiirisissä tieteissä, todistuksista matematiikassa ja notaation merkityksestä. Näistä erityisesti kaksi ensimmäistä olivat mielestäni sangen kiinnostavia ja varmasti sovellettavissa esimerkinomaisesti myös tämän kapean alueen ulkopuolelle.
Ylipäätänsä tämä sopi hyvin matematiikasta viehättyvälle, mutta taidoiltaan alimittaiselle uskonnonfilosofille. Ainoastaan yhdessä kohdassa kun Colyvan käsittelee Löwenheim-Skolem teoreeman ja Cantorin teoreeman ristiriitaisuutta koskien eri kokoisia äärettömyyksiä totesin, että tätä pitäisi ehkä ymmärtää aika paljon paremmin, jotta voisi hahmottaa näiden pohjalle rakentuvan filosofisen argumentin oikeasti. Kirjassa on toki lukuisia matemaattisia esimerkkejä, jotka menevät täysin oman taitotason yli, mutta Colyvan avaa niiden merkitystä siinä määrin, että itse esimerkkien syvällinen ymmärtäminen ei ole tarpeen. Kyseessä oli siis oikeasti hyvin kirjoitettu johdantoteos melko haastavaan aiheeseen.
Matematiikan erityisluonne tieteiden joukossa tekee siitä filosofisesti hyvin kiinnostavan. Kirjan ytimessä ovat kysymykset matemaattisten olioiden luonteesta, matematiikan sovellettavuudesta empiirisissä tieteissä, todistuksista matematiikassa ja notaation merkityksestä. Näistä erityisesti kaksi ensimmäistä olivat mielestäni sangen kiinnostavia ja varmasti sovellettavissa esimerkinomaisesti myös tämän kapean alueen ulkopuolelle.
Ylipäätänsä tämä sopi hyvin matematiikasta viehättyvälle, mutta taidoiltaan alimittaiselle uskonnonfilosofille. Ainoastaan yhdessä kohdassa kun Colyvan käsittelee Löwenheim-Skolem teoreeman ja Cantorin teoreeman ristiriitaisuutta koskien eri kokoisia äärettömyyksiä totesin, että tätä pitäisi ehkä ymmärtää aika paljon paremmin, jotta voisi hahmottaa näiden pohjalle rakentuvan filosofisen argumentin oikeasti. Kirjassa on toki lukuisia matemaattisia esimerkkejä, jotka menevät täysin oman taitotason yli, mutta Colyvan avaa niiden merkitystä siinä määrin, että itse esimerkkien syvällinen ymmärtäminen ei ole tarpeen. Kyseessä oli siis oikeasti hyvin kirjoitettu johdantoteos melko haastavaan aiheeseen.
June 14, 2021
Overall, this is a good introduction to the philosophy of mathematics. I found the discussion of the Löwenheim-Skolem Theorem in chapter 2 to be unnecessarily confusing with key definitions relegated to footnotes. Otherwise, I found this book to reasonably easy to read. It was nice to read about the big “isms” in PoM: realism, fictionalism, formalism, logicism, etc. I particularly enjoyed the section on paraconsistent logic.
August 2, 2025
The limits to math, what is math, what does math tell us about be world… all so interesting
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September 20, 2013510.1 C7279 2012
April 4, 2019
A splendid and clear presentation of many of mathematic's concerns... a helpful introduction to help navigate with a wealth of resources for further reading.
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