Defending the Axioms: On the Philosophical Foundations of Set Theory

by Penelope Maddy

Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account of the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments.

Genres: Philosophy, Mathematics, Nonfiction

150 pages, Paperback

First published January 27, 2011

About the author

Penelope Maddy is a UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well-known for her influential work in the philosophy of mathematics where she has worked on realism and naturalism.
Maddy received her Ph.D. from Princeton University in 1979.
Her early work, culminating in Realism in Mathematics, tried to defend Kurt Gödel's position that mathematics is a true description of a mind-independent realm that we can access through our intuition. However, she suggested that some mathematical entities are in fact concrete, unlike, notably, Gödel, who assumed all mathematical objects are abstract. She suggested that sets can be causally efficacious, and in fact share all the causal and spatiotemporal properties of their elements. Thus, when I see the three cups on the table in front of me, I also see the set as well. She used recent work in cognitive science and psychology to support this position, pointing out that just as at a certain age we begin to see objects rather than mere sense perceptions, there is also a certain age at which we begin to see sets rather than just objects.
In the 1990s, she moved away from this position, towards a position described in Naturalism in Mathematics. Her "naturalist" position, like Quine's, suggests that since science is our most successful project so far for knowing about the world, philosophers should adopt the methods of science in their own discipline, and especially when discussing science. However, rather than a unified picture of the sciences like Quine's, she has a picture on which mathematics is separate. This way, mathematics is neither supported nor undermined by the needs and goals of science, but is allowed to obey its own criteria. This means that traditional metaphysical and epistemological concerns of the philosophy of mathematics are misplaced. Like Wittgenstein, she suggests that many of these puzzles arise merely because of the application of language outside its proper domain of significance.
Throughout her career, she has been dedicated to understanding and explaining the methods that set theorists use in agreeing on axioms, especially those that go beyond ZFC.

http://en.wikipedia.org/wiki/Penelope...

Reviews

[Mark Moon · Rating 2 out of 5]

Read this with some friends. We didn't care for it.

[Viv · Rating 2 out of 5]

I have so many problems with this book. I’m not going to list them all but… Yikes.

I’m going to do a compliment sandwich thing and try to open and close with a compliment because I need to lower my blood pressure.

Compliment: there was a valuable insight in defining abstracta as those objects that are not doubt-apt. But this feels more like a Putnam insight than a Maddy insight. And is not particularly relevant to the project at hand.

Now onto what I actually thought:

The first 20 pages had me thinking that I might as well be reading Kline 1972. Maddy, I get that you’re in love with this source, but an absence of a mono-citation wasteland really can make the heart grow fonder. I believe in you.

As for the content of the book… If the second philosopher takes no view on what is metaphysically real but merely aims for the most parsimonious explanation of set-theoretic methods, should she prefer the finitely (provably co-extensive with ZFC) axiomatizable NBG? Or does the second-philosopher have a different criterion prohibiting the introduction of classes? What could this be? Are they less metaphysically pure, and if so, doesn't that imply that she really does care about metaphysics deep down despite her protestations? I don’t think Maddy has an answer to this question.

The "idealised inquirer" taking no particular philosophical view on what science is but working from "inside" science sounds a lot to me like letting the dominating ideologies of the scientific inquirer go unchallenged. I'm worried about (the ideal inquirer) being controlled by power structures (they) are not cognisant of and again Maddy’s solution seems to be “just don’t think about it.”

Maddy’s solution to the whole problem of how to justify the axioms seems to be “don’t think about it” and this really constitutes my biggest objection to the book. The book "Defending the Axioms" does not once defend any particular axioms!

And to end the sandwich with another compliment. It really wasn’t entirely bad, the last chapter was actually alright. She finally concedes that mathematicians might do something wrong and it looks like she’s caught in the trap between acknowledging Cohen’s forcing was mathematically “fruitful” and saying that independence results are not. Her discussion of Borel determinacy feels mostly like a red herring but it does feel conspicuous that she leaves out Wadge hierarchy. There is at least an attempt here to defend some axioms. And of course it's not an actual defence, she doesn't do that, but there is some description of what a defence of axioms might look like at least. This should have been the first chapter and the rest of the book should have followed from here. That might’ve made a good inquiry into the philosophical foundations of set theory.

[Paige McLoughlin · Rating 5 out of 5]

excellent guide to a beautiful Axiomatic Subject of Set Theory. I love the borderlands of Philosophy Mathematics and Philosophy and Physics and Maddy is an excellent guide to the former while others like Tegmark, Carroll, or Wallace are guides to the latter. Maddy has written a really good book on set theory and philosophy here.

[Akhil · Rating 4 out of 5]

Well written, with a breezy first half and dense second half. More focused and limited in scope than a generic “philosophy of math” book, which I think is good. I’m going to try learning more about the actual set theory now. Also, “second philosophy” is great! I’m going to use that in conversation a lot.

[Logan · Rating 3 out of 5]

I appreciate her disdain for metaphysics and blatant transcendentalism. You should be well versed in mathematical foundations and proof theory before reading.

[meadow · Rating 4 out of 5]

Maddy looks into how we decide what axioms to add to set theory. In doing so she briefly recounts the history of math and its gradual tendency towards abstraction and rigor. Then she gives an overview of two "natural" philosophies of math (one slightly Platonistic (she calls it "Thin Realism"), and one a bit nominalistic ("Arealism"), before concluding that on some level it doesn't matter which of the two you choose, because you will more or less be saying the same thing. The final chapter looks at what we've learned from all this and this is where Maddy really comes up with something new (everything before kind of proceeds kind of very simply and shouldn't strike anyone as too crazy or extreme). Her idea is that extrinsic justification for new axioms is more important than intrinsic justification. She makes a very good case for this by more or less looking at why we've developed set theory, and what set theory studies (from both the Thin Realist perspective and Arealist perspective), along with lots of "evidence" from the interrelations of proposed new axioms. She does a good job of being realistic and getting at what math is actually about, rather than getting too caught up in ontological or epistemological details. There was also a very good section in which she outlined other philosopher's ontological/epistemological views and how they relate to her conclusions.