Naturalism in Mathematics

by Penelope Maddy

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Genres: Philosophy, Mathematics, Nonfiction

266 pages, Paperback

First published November 13, 1997

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

[Hobart Mariner · Rating 4 out of 5]

So you're a philosopher and you want to weigh in on pressing "methodological" questions of set theory: are there measurable cardinals, what is the status of the continuum hypothesis, etc. Yet you are a modest Quinean with some Carnap and Wittgenstein thrown in for good measure and you don't want to assess these questions by way of an "external" standard -- you want to be a good naturalist and take for granted that the general consensus arrived at by practitioners is the good one, and that philosophical questions shouldn't drive the resolution of these methodological questions. How can you do this? It seems impossible: you want to advise or at least narrate some of the developments in mathematics *as a philosopher*... but you don't want to philosophize or externalize any criticism. The best possible outcome is to Maddy suggests that you can accomplish it via "mathematical naturalism," which she outlines in the last part of this book via an extremely thorough, persuasive "naturalist" argument against ZFC + V=L.


Great book that includes huge swaths of history of math and science in order to present certain analogies. Maddy, to all our benefit, considers the rehearsal of historical analogies to be a vital step in a naturalist philosophy of math, so we get fun and eminently readable accounts of Frege, Godel, some electrodynamics, relativity, chemistry, etc.

Some of her arguments are, I think, deliberately weak, because they hinge on presenting things in historical analogy with previously established scientific principles. "ZFC+V=L" is associated with "Definabilism" which is shown to play a role similar to Mechanism in physics, and Mechanism wound up getting discarded because of how it limited electromagnetics, hence we should discard V=L. Now, this is her rehearsing some of the earlier, realist arguments, and it's allowed for it to be weak. Just don't chuck the book because you think this is bad.

One of the strongest parts of the book is when she kicks Quinean indispensability arguments for mathematics to the curb. This requires her to go fairly deep into how math is actually used in science, and I think she nails it: physicists will use any mathematics that enables them to carry out experimentally interesting measurements, without granting any ontological status or privilege to these concepts. Also, in the newer theories of physics, the infinite-divisibility of the time and position axes ceases to be necessary or even plausible: therefore, we can't affirm the existence of the real number line based on these applications.

The last part of the book, where she presents a case study in the application of mathematical naturalism, is certainly the most challenging and will buck the most readers. For her, a mathematical naturalist is a philosopher who adopts mathematical methods in order to address methodological questions; if this seems like the philosopher is becoming a mathematician, it is, and Maddy is fine with that. The ideal naturalist method consists of building an idealized model of some practice (say, ZFC or one of its extensions); enhancing or amplifying that model (ZFC -> ZFC + MC, the measurable cardinal axiom); testing the model against others; and evaluating the rationality of methods. The second step basically requires the philosopher to become a mathematician, and the third does as well as you see when she gets into the EXTREMELY nitty-gritty of how to compare different extensions of ZFC. Much of this will be completely unreadable to people who don't have any set theory under their belt; once we hit Woodin cardinals I was skimming. However, I do think that she has taken great pains to present a fair and reasonable argument, entirely within mathematics, that adopting the constructibility axiom V=L is "too restrictive" in the sense of limiting future math development.

Great book for anyone interested in the relationship between math and science, or math and philosophy. Loved her paraphrasing of Wittgenstein, who she evidently admires as an avowed practitioner of anti-philosophy but whose own claims on how mathematics should be trimmed belie that. (He basically says its wrong for philosophers to tell mathematicians what to do, but then tells mathematicians that they will voluntarily give up set theory once he's done rapping at them, lmao.)

[Alexi Parizeau · Rating 5 out of 5]

Reading this is probably what it feels like to drink a Pan Galactic Gargle Blaster (i.e. it's like having your brains smashed out by a slice of lemon wrapped around a large gold brick). That is to say, this book is hard as hell for a non-mathematician to digest, but it is possible. And though Maddy writes in a crisp concise style, the book will only be appreciated by someone willing to put in the effort.

[emily · Rating 5 out of 5]

bit of a different read this time around. i wanted to read about the philosophy of maths and was recommended to penelope maddy's writing by peter smith's fantastic blog. lo and behold, while i was scrolling the math section of my library, i saw her name and decided to give it a go. what a great decision that was!

maddy writes with astounding clarity, every point was well-articulated to the point that i think even non-mathematicians could possibly give this a read if they really wanted to. the independent questions have always piqued my interest, so the subject matter was right up my alley as well. as for her thesis, i'm not sure i buy it just yet—this is my first foray into mathematical philosophy, after all. i haven't gotten to hear any other ideas! but she sure does make a compelling stance. in some ways, i think her approach reflects the main principles of phenomenology. perhaps that's not much of a surprise, given she addresses that in the opening remarks of one of her chapters, but it just felt very familiar to me since i've taken a class in phenomenology.

anyway, this definitely inspires me to go check out more books about the philosophy of maths. next i'm eyeing the seminal "thinking about mathematics" by stewart shapiro. i really want to read something philosophical about computer science that isn't godel, escher, bach as well, but time will tell if i can find a book that fulfills that.