Course: Paradox and Infinity
School: MIT through edX (free; see below)
Instructors: Agustin Rayo
There is one kind of philosophy that I am especially interested in. And that’s, for lack of a better name, mathy philosophy. It’s very hard to say something more specific because really what unites the different problems that mathy philosophy is about is the fact that they interact with mathematics and, more generally, with the use of formal methods – so for example, the use of probabilities to model beliefs.
So say that this is mathy philosophy. So if one is to talk about mathy philosophy, there are basically two options. Option one is to do it properly…. We would spend all semester doing the groundwork. And some of it would be brutally dull. And then, finally at the end, there would be this jewel. And it would be wonderful. But it would only be for the brave.
So what I’ve decided to do is not do it properly.
So there are all these delicious fruit within mathy philosophy. And what I’m going to do is I’m going to go straight for the fruit.…In some ways, that will make the class more difficult because, if you have the details, if you have the proof, you can go and check it.
And the only thing between you and the fruit is work.
The short version: Great class. But if you don’t know what you’re getting into – if you think, “Oh, cool, time travel, science fiction!” – you might be disappointed by Week 2. Not to mention overwhelmed.
And what are you getting into? Like the man said: Mathy philosophy. Probabilities, omega sets, infinite series, Turing machines… yep. Mathy. I knew that going in. The course description and teaser video (which is pretty great, btw, worth watching just for fun) don’t quite make clear just how mathy it is. I’ve encountered many of these terms before, so I had some idea what was coming. Yet, as it turned out, I didn’t know the half of it. I had fun anyway. And I think I may have learned a few things.
The first week was the least mathy, with a look at time travel. I found it to be the least interesting of all the topics. Don’t get me wrong, I love me my Star Trek time loop episodes, but I’ve taken two philosophy courses that dealt with time travel and I still haven’t seen anything to equal Robert Heinlein’s “By His Bootstraps” in terms of dealing with the predetermination factor.
Then we spent some time on probability, one of my weakest areas (not that I have any mathematical strengths), which generates some interesting paradoxes. Monty Hall did not show up in this course, though his cousin Newcomb did. The connection to free will was tenuous, but I suspect the whole “time travel and free will” was marketing.
In the second three-week module, we ended up in set theory and abstract algebra, the “infinity” part of the course, which, it turns out, I love, with or without the melons (used to illustrate the conversion of the Cayley graph into the Banach-Tarski paradox). I’ve heard of Banach-Tarski before, but rather than just saying it exists, the objective here was to show how it works (Coincidentally, just as the class was concluding, VSauce put up a video explaining some of the topics in this unit). Much to my surprise, I kinda sorta get it. Heavy on the kinda sorta, since I’m still a little hazy on equivalence relations and cells. It was frustrating, difficult – and fun. It’s the area I most want to understand better.
The last three weeks were devoted to Turing machines, and while I got the basics of the beginning, I got lost in the middle and kind of gave up, so I never got to Godel’s Theorem. I was done. But overall, it was a course very much worth taking. And surprisingly, I “passed” – thanks to a very low bar, and an exam structure that allowed a basic understanding to suffice. I’d like to know more, and hope I can do some additional work on these topics. But given my capacity to screw up math, the pass was somewhere between a gift and a miracle. I’m not fooling myself, thinking I “know” anything – but it’s a start.
Worth special mention is the inclusion of several “Meet the Expert” videos – lectures, interviews, conversations, collaborations with a variety of professors with special interest in the topic at hand. These included physicist Alan Guth on time travel, epistemologist Susanna Rinard on a number of paradoxical probability examples, logician Graham Priest on dialetheism (my new word for the day, meaning statements can be true and false simultaneously – such as “This statement is false”), Steve Yablo on ω-sequences, and computer scientist Scott Aaronson on computational complexity. Some of these were advanced topics; others were very accessible.
One of the problems I had was that I needed more basic explanations than were included in the course materials. The lectures were from the actual MIT course Rayo teaches, and while it was fun to see all those MIT nerds (I use that term with great respect, btw) coming up with perfect explanations for why we can’t know whether Thompson’s lamp is on or off at midnight, there’s a lot missing from those video clips. And, sure, there was written material, but I have this problem reading math: by the time I figure out what all the symbols mean, I forget what I’m doing there in the first place. Fortunately, I was able to find more elementary materials. And, incredibly enough, I was able to get help through the edX forums.
I say “incredibly enough” because the edX forums are notorious for being less than user friendly, and it matters to me most in courses like this one where I often need a bit of advice, or just reassurance. Because “daily digest” notifications of replies are only sent every 24 hours (Coursera sends them nearly instantly), conversation requires a great deal of motivation on the part of participants; I’ve learned to check the forums for replies often during the day, but if my interlocutors don’t do the same, the lag time is extreme. Then there’s the opt-in setup for “follow” on replies, meaning no notifications on replies are ever received, and it’s very difficult to find a reply at a later time; if follows were simply changed to opt-out for replies, as they are for initial posts, it would help tremendously. But either the edX powers that be have decided this requires too much work/cost, or they aren’t interested in improving forum functionality. In any case, conversation, collaboration, and support is possible, but it takes people who are aware of system limitations and are willing to put in the extra steps to compensate.
But it worked here because I was lucky to encounter an old friend. I first met Purgatorio last summer in SVCalc – he was one of the inspirations for the Dante work I did this year, and I think it’s fitting that Purgatorio turned out to be my favorite canto (maybe I just like middles). Dear Purgy is rigorous about “mathematical truth” and has little patience for fakery or shortcuts. This sometimes causes a bit of friction for him. Fortunately, since I enter every math course with a white flag raised in surrender (I keep saying I can’t afford ego when it comes to math), we’ve struck up a friendship. He’s also very funny, sometimes unintentionally so (while highly educated and fluent, English is not his native language, so occasionally a colloquialism comes out a little sideways). He has also shown a great deal of patient willingness to walk me through concepts when I struggle, and though I don’t always get the whole point, I do always take away something of value. In this course, the combination of banter, and explanation of fine points of set theory, got me through what might otherwise have been a lost cause, and I’m very grateful.
Another aspect of this course worthy of note: the “human-graded” option. MOOCs depend on machine grading, of course – multiple choice tests, numerical answers. As restrictive as this is, there are some courses (and this is one of them) that manage to include some reasoning-intense questions in these formats. Some courses use essays graded via peer assessment by other students. The results, in terms of grades, are inconsistent, though I usually find the experience worthwhile in that I have to consolidate and organize my thoughts in order to write a coherent essay in the first place, and the papers I’m assigned to review often include some wonderful insights I hadn’t considered. This course, depending on multiple choice with a number of optional, non-graded questions sprinkled throughout, offered something else: a $300 human-graded option. That’s what it’s called, honest.
Would you like to have your work graded by humans? If you sign up for a verified certificate, you will be assigned problems that are graded by teaching assistants and given professional written feedback. This will bring your learning experience one level closer to that of residential students at MIT. And if you pass the class, you will receive an MITx certificate, in addition to edX’s Verified Certificate of Achievement.
This included some odd notations on the course forums; I’m not sure if human-graded students had their own forums or what, but some posts were specified as “Non-human graded cohort only” which seemed… a little creepy. I’m not sure if the optional questions were human-graded, which would make sense. I can see how this could have appeal to certain people in a position to take full advantage of it. I fully support MOOCs doing what they need to do to survive, though I sadly realize that eventually the “free” part of all this will end, and the door to all this education will slam shut in my face. That’s why I’m trying to cram it all in, while I can.
In spite of the difficulty level, I enjoyed this course. Rayo is an engaging lecturer, the material is interesting, and I can highly recommend it for those who are more prepared than I – that is, those for whom things like set theory, equivalence relations, and Turing machine processing are not completely new concepts, or who can easily acquire that material from rather cursory explanations. I would imagine math/CS students would love the opportunity to peek into an MIT classroom. As for the rest of us, I can still recommend it for those who are less prepared but are interested in the topics, have a high frustration tolerance and are willing to find more elementary material to supplement course materials.