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Discussing Russell’s *Introduction to Mathematical Philosophy *(1919), ch. 1-3 and 13-18.

How do mathematical concepts like number relate to the real world? Russell wants to derive math from logic, and identifies a number as a set of similar sets of objects, e.g. “3″ just IS the set of all trios. Hilarity then ensues.

This book is a shortened and much easier to read version of Russell and Whitehead’s much more famous *Principia Mathematica*, and given that we can’t exactly walk through the specific steps of lots of proofs on a purely audio podcast (nor would we want to put you through that), we spend some of the discussion comparing analytic (with its tendency to over-logicize) and continental (with its tendency towards obscurity) philosophy.

Featuring guest podcaster and number guy Josh Pelton, filling in for Seth.

Read with us online or buy the book.

End song: “Words and Numbers,” by Madison Lint (read more about this tune).

Great episode. I laughed out loud at the “New Math” reference. Here’s a primer:

http://youtu.be/UIKGV2cTgqA

I did too! Good episode– loved the critique of both camps of philosophy at the end.

The truth is analytic “philosophy” is to linguistics, mathematics, natuarl science, and genuine philosophy what a cargo cult is to an industiralized economy.

Analytic “philosophy”‘s problems aren’t problems. Analytic “philosophers” are intellectual flyweights.

Hi, Jorge,

I’m thinking we don’t have the same thing in mind by the term “analytic philosophy.” 80% of what’s been done in England and America since 1910 or so falls into this category, and its adherents range from behaviorist types like Ryle to those much more sympathetic to the humanist project like Danto and Nelson Goodman. Though analytic philosophers generally have different strategies for handling problems, different debates about what constitutes a problem (e.g. Quine and Rorty rule out a lot of traditional problems as legitimate ones), and certainly different conventions re. writing style, there’s a great overlap in their concerns with continental philosophers (and pragmatists, if you want to put those in a third category).

For a quick idea re. what analytic philosophy is now, listen to this Chalmers lecture: http://www.partiallyexaminedlife.com/2010/05/25/david-chalmers-on-merely-verbal-disputes/ or pick up Thomas Nagel’s book of essays “Mortal Thoughts.” Those two guys are good examples, I think, of analytic philosophers who are not shy about taking on big problems, but they write clearly and try to cut the problems down into manageable bits (thus the “analytic” designation). Being opaque so people have to guess what the hell you’re talking about is not a virtue, and often covers up shoddy thinking.

Just thought that I’d mention a couple of good semi-technical secondary readings that might go well with this episode:

For Goedel, try “Incompleteness: The Proof and Paradox of Kurt Goedel” by Rebecca Goldstein.

For infinity, you can’t go wrong with “Everything and More: A Compact History of [infinity symbol]” by David Foster Wallace. In my opinion this is one of the all-time great pieces of science writing. And if your first (perhaps only) foray into DFW was his “Infinite Jest”, don’t worry. I think that his essays are far better than his fiction. But I can’t say too much about DFW: He’s the only person that I did not personally know whose death actually brought me to tears.

Joe

Nice call, Joe — I’ve seen DFW’s “Everything and More” but haven’t yet got round to picking it up. I totally agree that his essays are more interesting and engaging than his fiction. The Anglophone world suffered a major loss with his death.

Thanks Joe, will check those out. I couldn’t bear Infinite Jest, but I’ve heard his essays are very good (and incidentally ordered “Consider the Lobster” the other day.

Wes, if you want a quick teaser/spoiler, here’s “Consider the Lobster” online:

http://www.gourmet.com/magazine/2000s/2004/08/consider_the_lobster

Wes –

“Consider the Lobster” contains some very good essays, including what is quite possibly my favorite by DFW: “Authority and American Usage”. I hope that you enjoy it.

Joe

Joe,

I think a better choice for an intro to the study of infinity is A.W. Moore’s “The Infinite”. Moore is professor of philosophy at Oxford, specializing in the philosophy of mathematics, among other topics. In a review titled, “How To Catch A Tortoise”, http://www.lrb.co.uk/v25/n24/aw-moore/how-to-catch-a-tortoise/ Moore criticizes Wallace’s work, in “Everything and More”, as full of mistakes and misconceptions. (see the last paragraph of the article).

I saw an interview with DFW where he acknowledged his ineptitude with the subject and didn’t understand why the publisher’s wanted him to write about it. That being said, his work’s in “Consider The Lobster” and other’s are excellent, as you and others have noted.

The Goldstein book on Godel is okay, but in my opinion, could’ve been better; Notices of the American Mathematical Society did a pretty good review of it:

http://www.ams.org/notices/200604/rev-kennedy.pdf

Is there any logic in the American habit of abbreviating the word “mathematics” to “math”? The rest of the English speaking world says “maths”, which, because the word being abbreviated is in the plural, is logical.

It’s getting to be a long time since this podcast was released, but I only just got to hear it last week. So if there’s still interest in discussion on this topic, I offer some ideas on why it is a more important topic than the interlocutors concluded.

I think the discussion missed an important epistemological angle. There is a tendency to think that maths and logic are well-grounded and a “safe” source of knowledge. That mathematical knowledge is “provable”. Russell’s work shows that even if a mathematical claim *is* provable, it is not *easily* provable.

But the challenges discussed in the podcast, such as Russell’s paradox (the set of all sets etc) and Godel’s theorem, show that even claims about provability are on shaky ground. The common axioms of set theory lead to the troubles such as the Banach-Tarski Paradox (http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox). And if that is so, why should we trust other conclusion derived from set theory?

In summary, I think a key question to address is why we trust the conclusions of maths.

A good read on this (OK, I’ll admit it — the *only* work I have read on the subject) is “Mathematics: The Loss of Certainty” by Morris Kline.

I think it should be important to note here (and it’s mentioned in the wikipedia article) that the Banach-Tarski Paradox isn’t much of a paradox at all, it’s considered a paradox only because of the counter-intuitive nature of its results; The Banach-Tarski result does nothing to diminish or doubt the results of set theory.

Have not listed to the PEL episode yet, but love this subject. Have you ever read Timothy Gower’s intro to Math in the Oxford Intro series or any of Douglas Hofstader, e.g. Godel Escher Bach and I am a Strange Loop. I think it is not so much that “provability is on shaky ground” as that things are provable only within systems. Its pretty clear (probably perfectly clear from Godel) that there is no one big system that captures everything we consider “true” in arithmetic (and which is also internally consistent). So “provability” can only make sense within a system, not as a way of showing the correctness of a choice of systems. That certainly does seem to challenge the notion of Math having some greater claim to reality than other disciplinces; although Godel himself thought he had showed the opposite and was apparently a Platonist. What I think is really neat is the possibilty that all other claims about truth and falisty / reality (i.e. outside Math), are like this too: that they only make sense after you have chosen a system within which to evaluate them — and your choice of system can’t be guided by the same logic-driven principles.

Have now listened to this. Have loved most every PEL episode. But this one kind of struck me as missing the point. There is something really weird from a philosphic standpoint that the patterns of nature are so suceptible to articulation by arithmetic tools. It didn’t have to be that way. Doesn’t that give math a status as a’thing in itself’ / a part of the basic structure of reality / whatever you want to call it, that is at least a little bit troubling for the Continetnal anti-realists? And then isn’t it more troubling that this tool (arithmetic)– which seems as you do it to be all symbol manipulation — turns out, according to Godel, not to be. If it isn’t all computable, what is it and how are we — with our finite brains — apparently doing it?

Those aren’t trivial philosophic concerns and some of the possible answers actually shed light on issues the PEL episodes spend a lot of time discussing.

We’ll definitely have more episodes on math at some point, with different guest participants. We realized as we were doing this that this particular reading didn’t really address the ontological issue much. For me, this one was about getting my brain back in the right mode to handle the analytic philosophers who use a lot of formal logic. What we read was fun, I think, but only tangentially philosophical. So, yes, I agree with you, I think.

Thanks. To add to your no doubt endless list of possible future episodes — one on the ontologocial issues surrounding numbers would be great. The seemingly mind-independent nature of math might be the best challenge to Continetnal thinking (a good way to get you guys out of the comfort zone, perhaps?), but then Godel (although he probably would have hated to hear it) can actually be strangely confirming of Continental thinking given that the weirdness he discovered all comes from issues involving interpretation — specifically that there are just too many interpretations of airthmetic langauge to fit into a fully computable langauge, so maybe truth in an arithmetic sense is also only available after you commit to a specific interpretation of things. Doug Hoffstader is great on this, and Stanislas Dehaene, although a nero-cognition guy rather than a philosopher, has some incredible stuff on how varied human understanding of Math really is. Thanks again.