We are philosophy professors Agustin Rayo (MIT) and Susanna Rinard (Harvard). Agustin is currently teaching a free online course “Paradox & Infinity” which covers time travel, infinity, Gödel’s Theorem. Susanna just finished teaching a class o...

I've always wanted to know; why are philosophy and maths so closely linked? Is it just because higher-level maths can get very abstract or what? Is there a lot of maths in a typical philosophy course?

Interesting question... I think they're linked for (at least!) three different reasons:

1. There is a certain kind of beauty that philosophy and math both share.

2. Mathematics gives rise to interesting philosophical questions. (How should we develop set theory after Russell's Paradox? How can we know about numbers if they are abstract?)

3. Philosophers -- like economists -- sometimes use mathematical tools to develop their ideas. (Much of my own work on the philosophy of mathematics, for example, is about thinking of how best to respond to a theorem that shows that there is a certain sense in which it is impossible to reduce mathematics to logic.)

I was always torn about whether to become a philosopher or a mathematician, and now that I'm a philosopher I often miss mathematics...

Wittgenstein, in his 1939 Lectures on the Foundations of Mathematics is critical of mathematics done to produce a "pleasant feeling of paradox":

There is a kind of misunderstanding which has a kind of charm:

[picture of a circle and a line not intersecting]

"The line cuts the circle but in imaginary points." This has a certain charm, now only for schoolboys and not for those whose whole work is mathematical...

The kind of misunderstanding arising from this assimilation is not important. The proof has a certain charm if you like that kind of thing; but that is irrelevant. The fact that it has this charm is a very minor point and is not the reason why those calculations were made.—That is is colossally important. The calculations here have their use not in charm but in their practical consequences.

It is quite different if the main or sole interest is this charm—if the whole interest is showing that a line does cut when it doesn't, which sets the whole mind in a whirl, and gives the pleasant feeling of paradox. If you can show there are numbers bigger than the infinite, your head whirls. This may be the chief reason this was invented. (page 16, emphasis mine)

Now it's quite certainly false that Cantor's work on 'multiple infinities' was done with this sort of motive. His work has application to real, serious mathematics. (Indeed, in his 1874 paper where he introduced his famous theorem that the reals and the integers are not equipotent, he used this fact to prove a new proof of Liouville's theorem that every interval contains transcendental numbers.) However, what is true is that his work is often presented as having as its main interest this charm of a pleasant feeling of paradox. The incompleteness theorems are often subjected to a similar presentation.

In the short description of the Paradox and Infinity course on the edX page, it seems like it might be presenting these results in this way.

Learn about how some infinities are bigger than others, and explore the mind-boggling hierarchy of bigger and bigger infinities.

My question to Prof. Rayo is: Do you think that we should shy away from presenting work on the higher infinite and on incompleteness as a sort of mind-boggling thing, invoking a 'pleasant feeling of paradox'? If so, how do you aim to avoid students in the edX course getting that impression about these results?

Dear completely-ineffable,

I have much sympathy for what you're saying. Ideas that are motivated because of their "pleasant feeling of paradox" are usually obscure, and trading with obscurity is no way to do philosophy.

(Oxford philosopher Timothy Williamson has a memorable quote about this sort of thing. I can't remember exactly how it goes, but it's to do with philosophers who think the muddy river is deeper than the open pond because they're unable to see the bottom...)

The nice thing about Infinity and Gödel's Theorem is that they involve both beautiful mathematics and fertile philosophical terrain. And both the mathematics and the philosophy stand on their own: they don't have to be motivated by pleasant obscurity.

The mathematics is so beautiful that it doesn't have to be buttressed by philosophy. And the philosophy is interesting not because it gives rise to a "pleasant feeling of paradox", but because it sheds light on the foundations of our thinking about mathematics.

Gödel's Theorem, for example, is philosophically important because it teaches us that we cannot aspire to absolute certainty in mathematics. And that's important because it changes our conception of what mathematical knowledge is all about, not because it's pleasantly obscure.

The bulk of the philosophy of mathematics and logic appears to me to focus on the subjects you focus on in your class: infinity, completeness/incompleteness, (in physics:) relativity, and quantum mechanics.

As a student focusing on math and the theory of computation I'm curious about whether you think that the majority of work in math, physics and theory of CS is relevant to philosophy? Maybe I can clarify my question by relating it to a more frequently brought up issue: many areas of math appear to be unrelated to any physical or practical question, but then decades or centuries later they are applied (ex: group theory to physics, category theory in CS etc.). Do you think a similar pattern could connect math and philosophy?

Is there any research in this direction? Or otherwise do you know of any textbooks or professors who teach in this way, i.e. with a focus on looking to apply their findings (math/physics etc.) to philosophy? The closest thing to what I'm looking for that I've found are Scott Aaronson's writings such as "Why philosophers should care about computational complexity".

Dear tsarnicky,

You're not going to believe this, but as I started reading your post I was thinking "I'll respond by recommending Scott Aaronson's "Why philosophers should care about computational complexity"." But you know it already, and the sad truth is that I don't know of many other texts that do what you want.

My guess -- but it's only a guess -- is that we will continue to discover deep connections between math (including computer science) and philosophy. But I think it's hard to predict where they will come from.

A recent example of discipline jumping involves philosophy and linguistics. Ideas in pragmatics which were largely the work of philosophers suddenly hit the point in which they could be developed systematically, and have led to incredibly interesting work in linguistics.

What books would you recommend to those who have interest in philosophy, but are new to the field?

Hi holiday_bandit!

I fell in love with philosophy reading two classics: Plato's Apology and Descartes' Meditations.

Those aren't really about the sorts of things that Susanna and I teach, though. A wonderful book that is about the sorts of things I cover in Paradox and Infinity is Hofstadter's Gödel, Escher, Bach.

I also think that Logicomix is pretty cool...

This might seem like a hostile question, so let me preface it by saying I also work in philosophy. I'm always interested in how philosophers justify what they do. If someone is doing research in, say, biology, there's an easy answer to the question "What's the value of what you're doing?": you could say something like "My research might one day be part of curing some deadly disease". For example. Even if someone is doing esoteric research in basic physics, I guess one could say "I'm learning about the basic structure of the world.". I guess I always find these justifications more convincing than the justifications I've heard for research in the humanities.

When someone asks you why your research in philosophy is valuable, what do you say?

This is a question that's close to my heart, since I'd like a better answer to it to use myself!

Dear scmbradley,

Your question is also close to my heart. It's an issue I've thought about a lot, and that has sometimes made me uncomfortable.

I think it's a mistake to think that ideas are only valuable insofar as they have practical applications. Finding a cure for a deadly disease is valuable, but so is creating a wonderful piece of music or proving a beautiful theorem.

We live in a world which is filled with horrible problems, and it makes sense to work hard to find solutions. But that doesn't mean that the only thing we should do with our lives is work hard to find solutions to our problems. That would be a bit like spending one's entire life making money -- for one's self and for one's children -- without ever finding time for other kinds of fulfillment.

Imagine a world in which most of our practical problems have disappeared. What would you want to spend your life doing in that world? If you're like me, you'd like to have meaningful relationships, and you'd like to surround yourself by art, and mathematics and philosophy. So there is a certain sense in which those things are what matters most of all. If they don't seem urgent to us now it's because we have so many problems. But, then again, someone who is worried about money might easily forget that making money is not the only thing there is to life.

I'll be addressing Professor Rayo.

I'm new to rigorous philosophy and have already signed up for your Paradox and Infinity class on edX because I'm very interested in the topics to be discussed. But isn't philosophy fundamentally a discussion? You describe in your first video that there are fruits to be had after a lot of work that we don't have time for so we'll be skipping the labor in order to get a taste of the fruit. But the topics covered so far aren't conclusively answered (time travel and free will) and without a proper discussion we largely get a single viewpoint that can make us feel like we're being mislead, as if you're hiding important counter points (not illustrated by the Frankfurt case) to the case you are making. I know we're only one week in, but this sours the fruit for me.

Do you feel that the online format does justice to the field of philosophy when the students aren't able to properly participate in discussion of the topics?

Dear SnakeDevil,

That's a very good point!

I think you're totally right to think that it's hard to make progress in philosophy unless one considers different points of view. (I think it's partly to do with the fact philosophical problems are not very clearly defined, so it's hard to be sure whether one's ideas are right.)

The kind of teaching I like best involves no lecturing: it's just a big discussion. But I've found that in order for this to work you need a small group of people, and it really helps if everyone in the group has a bit of background in philosophy: otherwise the discussion gets derailed, and becomes uninteresting. So I think it'd be hard to set up a MOOC with a lot of discussion.

Hello! What advice would you give to someone applying to the Philosophy graduate program at MIT? ps: Love the Paradox & Infinity course (specially the extra/bonus parts)!

Hi cpittella!

I think what matters most is originality. (We're looking for people who will grow up to make a real contribution to the field, and originality is very hard to teach.)

We get lots of really good applicants, though, so it's hard to make it past the first cut unless one also has strong letters of recommendation from professional philosophers.

This is for Prof. Rayo. Thank you for your work with WiPhi. I'm friends with those guys, and I think they fill an important gap between laypeople and professional philosophers (as do you by getting on Reddit).

I read the abstract on your website of your and Elga's paper. You seem to espouse a functionalist way of specifying mental states, and one that serves primarily explanatory purposes. I'm sympathetic to this kind of view of the mind, but I've run into some problems. What do you do about the problem of misrepresentation? Specifically how can a creature be said to misrepresent its environment on a theory where mental states are provided as explanations for action? (I tend to think that misrepresentation is an epistemic issue, rather than one for phil of mind, and I know this is a minority view, so I'd like help being convinced of it.) Do you see the problem of misrepresentation as a normative issue; and, in turn, does a theory of mind need to accommodate it?

Hi have_a_word!

I kind of like Bob Stalnaker's theory of representation. To a first approximation, his view is this:

To have beliefs B and desires D is to act in ways that would tend to bring about D in a world in which B is the case.

Now suppose that I have a strong desire for chocolate, and that I head for the pantry. (In other words: I act in ways that would tend to bring it about that I eat chocolate on the assumption that there is chocolate in the pantry.)

Then -- simplifying things a lot -- Stalnaker's theory delivers the result that I have the false belief that there is chocolate in the pantry.

I think this is a really deep issue, though, and that there's much more to be said about it...

Hello, professors. How do you explain the term "philosophy" to a 12 year old child?

One thing to say is that whereas scientists tend to tackle questions that are relatively well-defined, philosophers tend to focus on issues that are interesting, but in which the rules of the game aren't really clear.

(So a big part of the work of a philosopher is trying to figure out what the rules of the game are!)

First, I'd like to say that I've enrolled in Paradox and Infinity and I'm excited to get started on that this evening. I actually have a few questions, so I hope you'll take the time to answer them.

• If someone were interested in studying philosophy in their own time, how would you recommend they go about doing it?

I ask this because I'm an undergraduate student studying Mathematics & Computer Science, and although philosophy is interesting and something I considered, I sadly don't have many opportunities for more formal study (I can only take so many classes!).

• What is research actually like?

Since starting university last fall, obviously I've noticed that my professors are involved in research of various sorts. What I don't really understand yet is what that actually entails.

• What field other than your own do you find most interesting?

Hi!

If someone were interested in studying philosophy in their own time, how would you recommend they go about doing it?

I think I'd try to get a friend involved. One of the reasons philosophy is tricky is that the problems are often not very well defined. So it's hard to be sure whether your ideas are right. A friend can help by trying to find problems with your argument.

Another think you could do is take a MOOC or two, and go beyond the class by checking out the supplementary readings. Caspar Hare's Intro to Philosophy MOOC is really terrific!

I ask this because I'm an undergraduate student studying Mathematics & Computer Science,

Have you tried setting up an independent study with someone at your university's philosophy department? That might be a really helpful way of getting guidance!

What is research actually like?

Believe it or not, I once made an effort to write up a text explaining to non-specialists what I do. If you really want to know, you can check it out here:

http://web.mit.edu/arayo/www/Non-Philosophers.pdf

But the truth is that it's hard to get excited about the sorts of topics I work on without having some background. (That means that a philosopher's work can feel a little lonely sometimes.)

What field other than your own do you find most interesting?

I love math, and I love music. (I would have definitely chosen a career in music over a career in philosophy if I had any musical talent!)

Can you think of any examples of time travel stories that play by internally consistent rules and don't create paradoxes?

Hi Leyrue!

I think Twelve Monkeys and The Time Traveler's Wife are both consistent. (They're also excellent movies!)

How do philosophers grapple with infinity?

Hi gullu129!

The basic insight was actually do to a mathematician: Georg Cantor.

What Cantor discovered is that the best way of comparing the size of infinite sets is by using bijections. (More specifically, proposed that infinite set A is the same size of infinite set B if and only if there is a bijection from A to B.)

This way of thinking about size has proved to be extraordinarily fruitful. One can prove, for example, that there is an (infinite!) hierarchy of sizes of infinity, which has all sorts of interesting properties.

(You can learn about the basics by watching this video.)

What Cantor did is, in effect, to tame infinity. Before his big insight it seemed impossible to theorize about infinity in a rigorous way; after him, infinity is an active field of mathematical research.

Hello, professors! I am currently enrolled in the Paradox and Infinity course on edX. As a general layperson, I am excited to have my brain's ass get kicked, in a manner of speaking. Thank you for providing an opportunity for me and anyone else out there to expand their knowledge base. Professor Rayo, what do you think is the biggest hurdle to leap for someone enrolled through edX compared to being physically on campus for lectures and conversations?

Hi naturalbrianandrews!

I think the main obstacle is to do with the sorts of issues that came up in conversation with anintrovertedrobot and SnakeDevil: it's hard to do philosophy without having a community do discuss one's ideas with.

If you're still interested in philosophy after taking the MOOC, what I'd recommend is that you try to get in touch with the philosophy department of your local university. Most philosophy departments have a colloquium series, and they're typically very welcoming!

if Godel's Theorem establishes inherent limitations on most mathematical axiomatical systems, could it be possible that his own system is wrong?

Yes!

An important corollary of Gödel's Theorem is that no (interesting) mathematical system can prove it's own consistency (unless it's inconsistent -- in which case it can prove anything, including its own consistency).

So, in particular, Gödel couldn't have used the system in which he proved his famous theorem to show that that very system is consistent.

In practice, however, we can be confident that his system is consistent -- not because we have a meaningful proof, but because it is very well trodden mathematical terrain.

What books would you recommend relative to this course? And, do you have complete videos of lectures from the residential version of the class you teach at MIT? If so, can you give me a link?

Hi xiading!

You'll get access to full lectures of the residential version of the class if you sign up for the MOOC. (Which is free!) To find the lectures, go to the "Further Resources" section at the end of each Topic.

You'll also get plenty of reading suggestions in the MOOC, but my favorite introduction to the topic is Hofstadter's Gödel, Escher, Bach.