What do you think?
Rate this book
The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise
This volume offers a guided tour of modern mathematics' Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor's transfinite paradise; axiomatic set theory, and more. 1989 edition. Includes 32 figures.
272 pages, Paperback
First published January 1, 1989
33 people are currently reading
172 people want to read
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 9 of 9 reviews
February 24, 2021
Nice presentation of Set theory for a Dover book. It is one of the more accessible areas of math and came out of the foundational crisis of mathematics in the early twentieth century because its ties to foundational thinking in mathematics overlap with the philosophy of mathematics. If you know a little symbolic logic it helps but you can skim over it and not miss a lot of the argument. A fun subject for me which is why the book was enjoyable.
February 25, 2017
This book is an exploration of the philosophic consequences of the infinite, both potential and actual. It starts out looking at Zeno’s paradoxes. From there Aristotle’s view is look at. Then, limits are examine and what they mean for the finitist. Next the continuum is introduce and Cantor’s continuum hypothesis that was posited under his study of transfinite numbers. Axiomatic set theory comes on the scene, and after this the logicists’ program is discussed. Finally, the attempts to solve the continuum hypothesis and the independence results and the axiom of choice’s roll is presented. The last chapter attempts to provide some sense of a resolution of the status of the infinite–its necessity from both a realist and a constructivist point of view.
Here are some of the my comments I thought interesting. (Pagination is from the Kindle edition)
[page 3] In discussing the opposing viewpoints whether mathematics is discovered or invented, Mary Tiles writes: “It may be found that one of these alternatives is to be preferred, or that different alternatives are useful for different purposes.” Upon reading this I thought that pragmatism is alive and well even in mathematics.
[page 8] After discussing the view that mathematics needs to be linked to some empirical content, she states: “Following this line of argument, some empiricists have been led to conclude that there is no sense to be given to such talk [of the actual infinite].” I thought what would the pure mathematicians have to say of such a view of mathematics.
[page 9] “But what case can the realist make which might persuade the finitist (an anti-realist about the infinite), motivated by empiricism, of the error of his ways?” My answer is that one does not need realism or empiricism if mathematics is created by the human mind/brain whether or not it has any connection to the world. Once created mathematics takes on an objective status or at the very least, a inter-subjective one. Further on “. . . for space and time are presumed to be continuous.” From my understanding of quantum mechanics space and time may actually be discrete. Lastly, “. . . since neither space or time can coherently be thought to have a boundary . . .” Again science intervenes. Under general relativity the universe has a boundary, even though we may not be able to see it, being outside our location’s light-cone. This is so even for the ever expanding universe, which we very well may be living in. Nothing lies outside this expanding boundary of the universe.
[page 20] In discussing Aristotle’s response to Zeno’s paradoxes, somewhat technically, she states: “Where is M at t? It seems that either M does not have a position at t or that it is in two positions at t, which violates the assumption that material objects occupy exactly one place at one time and occupy some place at all times.” These are no violations according to quantum mechanics description of atomic particles and events.
[page 22] She asks: “Can that all-embracing whole, the physical world, or the universe, be anything short of infinite?” Most definitely yes (see previous comments).
[page 47] According to the nominalists: “All classification is an imposition by the human mind (whether this is as a product of human nature, as in Ockham . . . or of human convention in defining words, as in Hobbes . . .)” This fits in with my notion that it is us humans who give meaning to the world.¹ I guess this makes me a nominalist. Platonic realism is a shadow on the wall in the cave, not the Sun casting the shadows.
[page 129] Another example of possible pragmatism in mathematics: “The question regarding the adoption of further axioms may be complicated, but is basically a question of what will be useful or what is required by other areas of mathematics.”
[page 195] After asking whether the continuum hypothesis should be considered true or false, she asks further: “Should one not perhaps conclude that there are several set-theoretic structures, each of which can legitimately be explored by the mathematician?” If this attitude should be taken that leaves a big hole in foundationalism, at least in connection with the hope that set theory was to provide such a foundation.
[page 208-9] “One might find some definitions more useful than others, but ultimately utility will be judged by reference to non-mathematical applications, not by application strictly within mathematics.” Once again she brings in pragmatism, though she never mentions the term in the book. I disagree with her here (not her pragmatism). If mathematics is a creative discipline, than any system created is valid as long as it is consistent. Think of art for art’s sake.
I must admit right off the bat that I got lost in the sauce of technicalities making my way through the book, but there was just enough philosophy to keep me from bagging it. I probably skimmed through half of the more technical parts of the book. I did enjoy the philosophy parts of the book, and a bit of the easier technical material. I found that Mary Tiles might have ignore some of the necessary physics in her discussion of space and time. And while I am not certain, she seems to favor a pragmatic approach to set theory and possibly the rest of mathematics, which in my limited experience in philosophy of mathematics I have not found all that much of. I find such an approach to be reasonable. This maybe connected with my anti-realism (Plato’s kind), but I will spare the reader my critique of this.
If, you are up to the challenge of technical and philosophical exploration of the infinite, the continuum hypothesis, and set theory, you may find the book interesting. If you are naive to any of these topics, it is definitely not a book for you. If, you lay somewhere in between like me, it may be a fifty-fifty proposition.
¹ For more on my take on meaning see my blog: “What Do You Mean?” at https://aquestionersjourney.wordpress....
Here are some of the my comments I thought interesting. (Pagination is from the Kindle edition)
[page 3] In discussing the opposing viewpoints whether mathematics is discovered or invented, Mary Tiles writes: “It may be found that one of these alternatives is to be preferred, or that different alternatives are useful for different purposes.” Upon reading this I thought that pragmatism is alive and well even in mathematics.
[page 8] After discussing the view that mathematics needs to be linked to some empirical content, she states: “Following this line of argument, some empiricists have been led to conclude that there is no sense to be given to such talk [of the actual infinite].” I thought what would the pure mathematicians have to say of such a view of mathematics.
[page 9] “But what case can the realist make which might persuade the finitist (an anti-realist about the infinite), motivated by empiricism, of the error of his ways?” My answer is that one does not need realism or empiricism if mathematics is created by the human mind/brain whether or not it has any connection to the world. Once created mathematics takes on an objective status or at the very least, a inter-subjective one. Further on “. . . for space and time are presumed to be continuous.” From my understanding of quantum mechanics space and time may actually be discrete. Lastly, “. . . since neither space or time can coherently be thought to have a boundary . . .” Again science intervenes. Under general relativity the universe has a boundary, even though we may not be able to see it, being outside our location’s light-cone. This is so even for the ever expanding universe, which we very well may be living in. Nothing lies outside this expanding boundary of the universe.
[page 20] In discussing Aristotle’s response to Zeno’s paradoxes, somewhat technically, she states: “Where is M at t? It seems that either M does not have a position at t or that it is in two positions at t, which violates the assumption that material objects occupy exactly one place at one time and occupy some place at all times.” These are no violations according to quantum mechanics description of atomic particles and events.
[page 22] She asks: “Can that all-embracing whole, the physical world, or the universe, be anything short of infinite?” Most definitely yes (see previous comments).
[page 47] According to the nominalists: “All classification is an imposition by the human mind (whether this is as a product of human nature, as in Ockham . . . or of human convention in defining words, as in Hobbes . . .)” This fits in with my notion that it is us humans who give meaning to the world.¹ I guess this makes me a nominalist. Platonic realism is a shadow on the wall in the cave, not the Sun casting the shadows.
[page 129] Another example of possible pragmatism in mathematics: “The question regarding the adoption of further axioms may be complicated, but is basically a question of what will be useful or what is required by other areas of mathematics.”
[page 195] After asking whether the continuum hypothesis should be considered true or false, she asks further: “Should one not perhaps conclude that there are several set-theoretic structures, each of which can legitimately be explored by the mathematician?” If this attitude should be taken that leaves a big hole in foundationalism, at least in connection with the hope that set theory was to provide such a foundation.
[page 208-9] “One might find some definitions more useful than others, but ultimately utility will be judged by reference to non-mathematical applications, not by application strictly within mathematics.” Once again she brings in pragmatism, though she never mentions the term in the book. I disagree with her here (not her pragmatism). If mathematics is a creative discipline, than any system created is valid as long as it is consistent. Think of art for art’s sake.
I must admit right off the bat that I got lost in the sauce of technicalities making my way through the book, but there was just enough philosophy to keep me from bagging it. I probably skimmed through half of the more technical parts of the book. I did enjoy the philosophy parts of the book, and a bit of the easier technical material. I found that Mary Tiles might have ignore some of the necessary physics in her discussion of space and time. And while I am not certain, she seems to favor a pragmatic approach to set theory and possibly the rest of mathematics, which in my limited experience in philosophy of mathematics I have not found all that much of. I find such an approach to be reasonable. This maybe connected with my anti-realism (Plato’s kind), but I will spare the reader my critique of this.
If, you are up to the challenge of technical and philosophical exploration of the infinite, the continuum hypothesis, and set theory, you may find the book interesting. If you are naive to any of these topics, it is definitely not a book for you. If, you lay somewhere in between like me, it may be a fifty-fifty proposition.
¹ For more on my take on meaning see my blog: “What Do You Mean?” at https://aquestionersjourney.wordpress....
July 3, 2016
A very helpful introduction to set theory, the internal feuds between mathematicians and philosophers, and the reprecussions of much of modern math on our view of rationality. A very helpful book...
June 7, 2023
Not too bad a survey of the origins of Set Theory. (Aristotelian) logic and classes, combinatorics, and Analysis ( from the need to understand the implications of Fourier Analysis and the drive towards rigor and arithmetization as an answer to the imprecision and growing doubts about geometric intuition). A good enough though skimpy discussion of Cantors early 'naive' theory and the resulting paradoxes. Ironically enough throwing doubts in turn on these new set theoretic intuitions. ( By the way to call it 'naive' is really to do it a disservice. One of the great discoveries of all time. Beginning to map out the nature of infinity which turns out to be far more subtle than was initially thought. ) A turgid and confusing description of Frege's and Russel's logicism but that's OK because the subject itself is turgid and confused. I wouldn't waste too much time on it. A skimpy and inadequate treatment of the axiomatic theories ( Zermelo/Fraenkel mainly) but I suppose that can be left to Set Theory Texts.
The author seems to be a bit obsessed with the old question, 'is mathematics discovered or invented'. It's a dumb question. The obvious answer is Both. To discover something is in part to invent a way of seeing it. But to invent anything is to discover a pre-existing possibility. What does pre-existing mean in this context. Ah who knows. That's something for philosophers to babble about.
Then we get onto the good stuff. Independence results. Axiom of Choice and Continuum Hypothesis. And higher Infinity Axioms ( including Large Cardinals ). Plotting out the nature of the higher infinite as Akihiro Kanamori so memorably described it. But only to be disappointed. The last couple of chapters are brief, sketchy and relatively useless.
Apparently one can use higher order infinity axioms to prove first order properties of numbers than cant be proved in first order logic ( or second ... etc ). This blows my mind. It says something deep about the nature of reality. High on my bucket list. After decades of being distracted by trivia I have just gotta know how this works.
The author seems to be a bit obsessed with the old question, 'is mathematics discovered or invented'. It's a dumb question. The obvious answer is Both. To discover something is in part to invent a way of seeing it. But to invent anything is to discover a pre-existing possibility. What does pre-existing mean in this context. Ah who knows. That's something for philosophers to babble about.
Then we get onto the good stuff. Independence results. Axiom of Choice and Continuum Hypothesis. And higher Infinity Axioms ( including Large Cardinals ). Plotting out the nature of the higher infinite as Akihiro Kanamori so memorably described it. But only to be disappointed. The last couple of chapters are brief, sketchy and relatively useless.
Apparently one can use higher order infinity axioms to prove first order properties of numbers than cant be proved in first order logic ( or second ... etc ). This blows my mind. It says something deep about the nature of reality. High on my bucket list. After decades of being distracted by trivia I have just gotta know how this works.
November 30, 2023
The intended audience of both mathematicians and philosophers limits the depth of material so I can't hold that against the author.
Lovely book that serves as a good introduction to the philosophy of mathematics with some classic examples that have been clearly communicated.
Kind of exactly what you'd be expecting which I think is a good thing, considering the intentions of the book.
Lovely book that serves as a good introduction to the philosophy of mathematics with some classic examples that have been clearly communicated.
Kind of exactly what you'd be expecting which I think is a good thing, considering the intentions of the book.
May 10, 2024
Definitely not an introduction. This is one of the most thorough accounts of set theory and the continuum hypothesis from a philosophy perspective. I did enjoy how it maintained a high level of rigor at the same time as being very readable.
August 27, 2024
An accessible overview of the crisis in mathematics in the 20th century, and the debate between finitivism and its alternatives. A good overview of concepts of the transfinite and how they affect the philosophy of mathematics.
January 3, 2019
2.5/5.0
February 11, 2013
Good overview but at times unnecessarily verbose.
Displaying 1 - 9 of 9 reviews