The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise (Dover Books on Mathematics);An;Dover Books on Mathematics

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. 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.

Excellent book! Having some background in mathematics and philosophy will definitely help. I have been interested in these topics since the seventies and have never found a book as informative and interesting as this! I have found good philosophical books that were mathematically deficient and good mathematical books that were philosophically naïve. This book is a rare find!

My interest in set theory and the foundations of Mathematics is already established around here, just look at my reviews, heheheh... The advantage of this text is the nice presentation of the logical foundations, even telling the history of concepts development. A nice reading for those who like the subject.

Incredible book, very thoughtful

It was for my husband. He really liked it.

I was a little surprised that this book dealt with more than just set theory, and it does this to set the stage for discussing Cantor's ideas and beyond. The book delves into Zeno's paradox, Aristotelian logic, and other subjects, and contrasts differing views (finitist/infinitist, nominalist/realist). This is the part of the book I enjoyed the most, as it was accessible to the amateur philosopher or mathematician. Of the nine chapters in the book, four of them are part of this programme of setting the background for the problems discussed. One chapter explicitly focuses on Cantor's ideas. The remaining four chapters delve into ZF Axioms and later reflections on set theory by great minds such as Kurt Godel. The remaining four chapters of the book were extremely difficult for me to comprehend.The only reason I give this book four stars is that the back cover claims I only need introductory philosophy or mathematics to get through this small book, but the last four chapters prove this isn't the case. I've taken an introductory philosophy course and differential and integral calculus classes and I was still overwhelmed toward the end of the book. Maybe I'm just getting old and can't handle the material? I still highly recommend the first half of the book, which was exactly as advertised.

This book is excellent for philosophy students who wish, like me, to improve their understanding of set theory.

No significant comments.