# Course Review – Introduction to Mathematical Philosophy

I really struggled with the Introduction to Mathematical Philosophy MOOC, largely due to my eccentric learning challenges with regard to mathematics.

Brockhaus and Efron Encyclopedic Dictionary via Flickr

These go back to my original college days when I was able to master whatever math was needed to learn a science subject (such as quantum mechanics or thermodynamics) regardless of how many integral signs the professor threw at the front of an equation. But in calculus or linear algebra courses that taught math outside of concrete problems in chemistry or physics, my mind would blank and I had to work overtime just to stay in B territory.

Given that Coursera’s Introduction to Mathematical Philosophy promised to address philosophical issues using mathematical methods, I had hoped that taking this course would more resemble my experience in science than math classes. But despite the fact that the professors anchored each lecture in some philosophical topic (such as infinity, truth and decision theory), this was very much a math class vs. a philosophy class that happened to be talking about mathematical tools and concepts.

Such personal challenges aside (which I suspect didn’t impact students with strong math backgrounds), Mathematical Philosophy was one of the most intriguing courses I’ve taken all year, especially since it demonstrated how mathematical techniques not available to the ancients (and probably not understood by most modern philosophers) can solve some of history’s more challenging philosophical riddles while also creating (or at least exposing) new conundrums to puzzle over.

Take for example Zeno’s Paradox which says motion is impossible since every journey requires you to get halfway to your destination, but to get to this halfway point you must traverse half that distance and so on. The conclusion drawn from this thought experiment is that you can never arrive at any destination, or even move significantly far from where you are, since you are always stuck trying to get to some point that has an infinite number of halfway points between you and where you are trying to go.

Now one can always solve this problem pragmatically by simply moving from here to there, ignoring the philosophical implications along the way. But proving the original paradox wrong requires mathematics, specifically calculus. For the paradox implies that when you add up an infinite number of fractions that mark your various halfway journeys (adding one half and one quarter and one eighth, ad infinitum) you end up at infinity (implying that every journey must be infinitely long). But some simple calculus applied to this fractional addition problem shows that these fractions don’t total to infinity but rather converge to one. (In your face Zeno!)

In addition to either solving, or providing new mathematical models for considering classical philosophical problems, such as the Liar Paradox, Mathematical Philosophy also introduced new challenges to thinking and perception, my favorite being the Monty Hall Problem. This is the one that places you on the game show Let’s Make a Deal where the host, Monty Hall, asks you to pick one of three doors where a car is behind one door and goats behind the other two. If you pick door 1 and Monty opens door 3 to reveal a goat, you actually stand a much better chance of winning if you switch your pick to door 2 (a phenomenon provable using computer simulations, regardless of how counter-intuitive the assertion I just stated might seem).

But as fascinating as were each of the topics making up the course, the course as a whole had that semi-apologetic “we can only scratch the surface” tone that I’ve seen in many shorter MOOCs dealing with complex topics. In fact, during an Easter-egg bonus lecture the professors, Stephan Hartmann and Hannes Leitgeb from Ludwig-Maximilians-Universität München, responded to criticism that their lectures seemed dry and boring by claiming that the topics they covered could only really be understood through careful study, engagement with open-ended problems and further reading. Which is why they eschewed course elements like weekly graded homework and required reading assignments in exchange for problem sets and extensive reading recommendations embedded directly into the slides making up their lectures.

The problem was that integrating these type of elements directly into the lecture (rather than using Coursera features that allow you to link automated assessments, assigned readings and video lectures into a weekly package with associated deadlines) made everything beyond watching the lectures seem optional or (in the case of un-prioritized reading lists that went on for pages) overwhelming.

And to make things even more problematical, the technical method used to embed these problems and reading recommendations into lectures meant that those elements — so important to the professors — weren’t included in downloaded versions of those videos.

Fortunately, other Coursera classes demonstrate how Mathematical Philosophy can be re-build if the professors decide to give the course a second go (which I hope they do). [Editor’s note: A new run of the course is on the Course catalog for April 2014.] For example, Coursera’s logic course Think Again from Duke University did a very able job integrating regular homework problems and challenging quizzes into the course, giving students the means to put what they learn each week to work. And Stanford’s Einstein class asked students to solve complex, open-ended problems on a weekly basis (even if input of final answers – only provided days after the problem set was released – was via a multiple choice form).

I’d also like to note that the final surprise lecture I just mentioned, one in which the professors talked to one another about issues raised by their involvement with the course, revealed them to be anything but “dry and boring.” Which is why they might want to follow the lead of their European cousins at University of London whose English Common Law class featured brief weekly talks in which the professor addresses specific questions and concerns that had arisen in discussion groups in a more casual give-and take manner.

Even with the newness of MOOCs, there has been enough experimentation with different tools and techniques to allow professors to leverage work that’s come before rather than start from scratch or assume the medium can’t overcome perceived limitations. And learning how to take advantage of these tools should be child’s play to anyone who has grasped the infinity of infinities and lived to tell the tale.

Editor’s note: This guest post is from Jonathan Haber at Degree of Freedom, who is tracking his progress in trying to learn in just twelve months everything he would if enrolled in a four year liberal arts BA program and using only free resources. Along the way he is writing reviews of courses he completes, some of which he generously allows us to republish here. To get all of Jonathan’s MOOC reviews, and more, be sure to sign up for the weekly Degree of Freedom Newsletter.

Jonathan Haber (19 Posts)

Jonathan Haber is a Boston-based writer and educational specialist whose Degree of Freedom project is experimenting with whether it's possible to learn everything you would get from a four year liberal arts degree in just twelve months using only free educational resources. You can follow his progress at www.degreeoffreedom.org.