After a couple dozen interviews with bloggers, teachers, and popular math authors, I realized one area I'd not heard from yet was "philosophy of math," so that omission is corrected today!
Penelope Maddy is Distinguished Professor of logic and philosophy-of-science at the University of California Irvine, and well known in philosophy circles for her explorations of the foundations of logic and math.
A brief Wikipedia page covers a little more of her background and publications here:
Without further adieu, here are her responses to some questions I posed:
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What first caught my attention was the rigorous proofs in high school geometry -- the idea that pure reasoning could squeeze so much information out of such simple assumptions. Algebra, word problems, had something of the same charm: you were given some meager bits of information, and by manipulating equations, you could answer questions you wouldn't have thought answerable. But I really caught the bug in one of those great NSF summer programs for high school kids, where I learned that 2+2=4 could be proved in naive set theory! I soon read about transfinite numbers (what?! more than one kind of infinity!), but the topper was Cantor's Continuum Hypothesis. This is Cantor's answer to the first question that arises in the exponentiation table for transfinite numbers, a question that involves the very structures used scientifically to model space and time -- and it turns out not to be provable or disprovable from the accepted axioms that found all the rest of classical mathematics! What could a solution to that kind of problem even look like?1) My understanding is that your interest in mathematics was initiated by set theory, at an age when I think most young people are more interested in computations, recreational math, or methods (not theory).
Can you describe what it was about set theory that appealed to you at a young age?
Well, soon I was off to UC Berkeley to study set theory.
Since the Continuum Hypothesis remains unsettled, I'd have to list it as a significant unresolved problem. What I've been trying to do on the more philosophical end of things is to develop a cogent picture of what would count as a resolution and why.2) What would you say are a couple of the most significant unresolved problems in the philosophy of mathematics these days?
And what questions/issues (if different from above) do you spend most of your own time working on these days?
[...I asked Dr. Maddy to flesh this out a bit more and explain how she approaches such a resolution, given that within Zermelo–Fraenkel set theory (with the axiom of choice, ZFC) mathematicians generally find CH UNdecidable.]:
Everyone pretty much agrees that CH can't be settled from ZFC. You have to buy that ZFC is consistent (otherwise it proves both CH and not-CH!) and that our formal model of 'proof' captures what mathematicians actually do well enough for these purposes -- but most people accept these things.
So, as you suggest, the obvious way to try to resolve CH is to look for a new axiom to add to ZFC. Some mathematicians resist this idea, taking ZFC to be somehow sacrosanct, so that independence from ZFC is the end of the story, but when you examine the reasons why the axioms of ZFC are accepted, it doesn't seem so far-fetched to think that there might be others that could pass similar tests. There are various candidates up for debate, and for a philosopher like me, there's the fascinating problem of understanding what kinds of considerations should count one way or the other.
[...I further inquired if she had her own hunch regarding the truth/falsity of CH?]:
Years ago, when I was a young math major, the consensus among set theorists willing to take the question seriously (that is, among those who didn't take independence from ZFC to be the end of the story) was that CH is not only false, but badly false. (If you think about the real line, CH says there's no set of points with size in-between that of the set of points corresponding to natural numbers only and that of the set of all points on the line. For CH to be 'badly false' is for there to be sets of lots of different sizes between these two.) Since then, it's come to seem more plausible that it's false but not badly so (say only one intermediate size) or even true. There are potential axioms that settle it various ways, but no clear winner.
So the truth is that I don't have a hunch. It could even turn out that two conflicting new axiom candidates might have all the attractions one could ask for, and that set theory would end up bifurcating. This would require a major rethinking of what set theory is supposed to be, but stranger things have happened. For what it's worth, on this one I do have a hunch -- I don't think set theory will bifurcate.
In the times of Newton and Euler, mathematics was just the study of the mathematical features of the world; as we might put it, all math was applied. Over the course of the 19th century, this changed, for a number of reasons: mathematicians began to pursue questions of purely mathematical interest, the rise of non-Euclidean geometries implied that at least some geometries were independent of the world, the role of atomic theory in physical chemistry and kinetic theory gradually revealed that differential equations, formerly regarded as the language of 'the great book of Nature' (Galileo), are actually 'a smoothed-out imitation of a really much more complicated microscopic world' (Feynman). The idea that mathematics shouldn't be tied down to physical applications became the orthodoxy of modern pure mathematics. It's certainly true that a surprising variety of these pure mathematical theories have ended up being useful in natural science after all, but I don't see any reason to assume that this will be true for every pure mathematical theory.3) I recently read a quote from a mathematician saying, "There are two kinds of mathematics: applied mathematics and mathematics that is not yet applied."
How do you feel about that? Is there any such thing as "pure mathematics" that will never have any practical application?
[...Dr. Maddy has a paper, which she cites further down (question #6) and I link to, that addresses this topic in greater detail.]
I guess I was more fascinated by questions about what mathematics is and how it manages to do what it does than by particular mathematical or logical puzzles.4) For many professional mathematicians, Martin Gardner played a significant role in steering them toward math at a young age. Did Gardner (or recreational math more generally) have much influence on you?Alternatively, any significant influence from logician Raymond Smullyan?
(I might note that Gardner's undergraduate degree from U. of Chicago was actually in philosophy, and he was a vocal mathematical Platonist.)
[...I'll just note that Gardner did write philosophical essays as well, and was heavily influenced by his own professor, Rudolf Carnap and logical positivism, though indeed his main influence on other mathematicians was through his "Recreational Games" column for Scientific American.]
In ancient times, the Pythagoreans thought that 'all is number', and they were stunned when it turned out that the square root of two isn't even a ratio of numbers. Nowadays most philosophers think of mathematical things as abstract -- without location in space and time, without causal powers -- and in this way quite different from ordinary objects. But there are those in the philosophy of science who take the position that 'everything is structure', and I suppose that might be a version of Tegmark's position.5) As you probably know, physicist Max Tegmark promotes an ultra-Platonist view that mathematics is ultimately ALL there is in the Universe; that fundamentally everything is reducible to mathematical description or relationships...
How do you view Tegmark's outlook? Do you know of any philosophers who share Max's view?
Let's see ...6) For readers who may be inclined toward mathematics, but are unfamiliar with "philosophy of mathematics" could you recommend a couple of "primer" books that are a good introduction?
And do you have any Web-accessible pieces of your own on math philosophy that you can recommend to non-specialists?
Gottlob Frege, Foundations of Arithmetic (1884) is a classic text.
A couple of nice contemporary introductions:
Stewart Shapiro, Thinking about Mathematics
Mark Colyvan, An Introduction to the Philosophy of Mathematics
People might enjoy a paper of mine called 'How Applied Mathematics Became Pure'.
This will come as no surprise: I'd want to know what ultimately became of the Continuum Hypothesis!7) Finally, David Hilbert famously said that if he ever woke up from a 1000-year nap, the first question he'd immediately ask would be, "Has the Riemann hypothesis been solved?"
1000 years is too far off to even imagine, but if YOU were to come back, say 200 years from now, what mathematical or philosophical question would you most like to find settled?
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Thanks for taking part here Dr. Maddy. Nice hearing from someone with a little different approach to mathematics than folks I usually interact with on this blog (wish I knew enough philosophy of mathematics to ask better/deeper questions!) And good luck with that Continuum Hypothesis!... just don't let it drive you batty like it did a previous fellow! ;-)
Another, longer, more detailed, transcribed interview with Penelope (from earlier this year) is also online:
http://www.3ammagazine.com/3am/the-stuff-of-proof/
Finally, this is the fourth female interviewee in a row that I've had here folks (and I think another is lined up for late June?), so are there any males out there who like math and are any good at it, who'd care to be interviewed? ;-)) Let me know....
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