An Introduction to the Philosophy of Mathematics

Cambridge Introductions to Philosophy

by Mark Colyvan

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.

Genres: Philosophy, Mathematics, Science, Logic, Nonfiction

199 pages, Kindle Edition

First published May 31, 2012

Reviews

[Lee · Rating 3 out of 5]

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”

[Yilmaz Aksoy · Rating 5 out of 5]

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.

[Nick · Rating 4 out of 5]

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.

[Rasmus Tillander · Rating 4 out of 5]

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.

[Aaron · Rating 4 out of 5]

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.

[nyaru · Rating 4 out of 5]

hmmm

[Alfred Timperley · Rating 4 out of 5]

The limits to math, what is math, what does math tell us about be world… all so interesting

[Mills College Library]

510.1 C7279 2012

[An Te · Rating 4 out of 5]

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.