Realism in Mathematics

by Penelope Maddy

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism
about mathematics raises serious What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. She
answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics.

Genres: Philosophy, Mathematics, History

216 pages, Hardcover

First published August 16, 1990

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

[Peter]

The book is a clear and careful treatment of a technical subject that defends a particular brand of mathematical realism that has been developed from certain understandings found primarily in Quine, Putnam and Godel. For the most part I would imagine the book would be read almost exclusively by other academics as I am not sure the problems themselves would be of interest or perhaps even credible to the non-professional. The naturalizing of epistemology (indeed epistemology itself) seems ultimately to depends on a rather dogmatic believe in the progress of man and the presupposition that science is obviously our best understanding of the world. I am always somewhat amuse by the out of hand dismissal of say the Homeric gods or Plato as being a kind of proto-science and the certainty with which the book understands philosophy as a body of knowledge that moves forward leaving behind, no matter how respectfully, the work that came before.

There is something quite startling about human nature that shows in our stubborn and ruthless believe in human progress, as if somehow we had left dogma behind. This assumption is nowhere clearer than in the work of the modern professional philosopher whose always well documented regard for and discussion of the work "on which they build" is grounded seems to be there mostly to assure us that progress has been made.

Notwithstanding, I often get a lot out of such careful work as in the end, the history of philosophy, its very construction, seems to point me towards more careful understandings of what this human love of wisdom, or what was once called original sin, actually or possibly is.

[Akhil · Rating 5 out of 5]

A spirited defense of Platonism in terms of set theoretic realism. I thought the move to naturalizing mathematical intuition (as perceptual and even visual) was interesting, and the engagement with Quine-Putnam well developed. Chapter 4 was the most interesting IMO and helped clarify for me what intrinsic and extrinsic justification of axioms might look like. This continues to be, in my opinion, the most crucial question, and Maddy justifies its relevance not just for the Platonist but also the nominalist and structuralist.

[Lucille Nguyen · Rating 4 out of 5]

A clear and thoughtful argument along the lines of Quine & Gödel that the foundations of mathematics lie in sets, that numbers are properties of sets, and that mathematics itself is describing a Platonic realm that we access through our intuition of mathematical truth. Nonetheless, I must admit that I find the non-existence of number and the primacy of sets as what is being conceived by someone seeing "three eggs" (to use an example from the set) as part of a set and not the number itself to be unconvincing, at best. But Maddy provides a nuanced and considered argument why it would be so, which is worth reading to better one's understanding of numbers and, more broadly, the nature of mathematics itself.

[John Reis · Rating 3 out of 5]

I bought this many years ago. I have read it off and on over the past three years. She offers the notion that sets are real. And that would be the foundation of arithmetic. I am not convinced because most of humanity does not know what a mathematical set is. Am I being too pedantic?