Tuesday, May 26, 2015

Next MOOC: Paradox and Infinity

UPDATE 6/8: The course is about to start. Sign up here if interested (it's free to audit, as I'm doing), and if you're taking it, feel free to be in touch.

This course looks promising: "Paradox and Infinity," at EdX, by MIT philosopher Agustín Rayo, and I'm planning to take it as the next step in my online math education. It looks like it will be lively and interesting, for instance based on this earlier clip of some of the material, and I've gotten some assurance from course staff that there will be opportunity to build math skills.


I'm also, when I have a chance, going through the self-paced Calculus One at Coursera, taught by mathematician Jim Fowler of Ohio State University. That's a subject I studied long ago and never really understood. Down the road, I expect to take courses in statistics, probability and more; as an economics major, I learned some statistics, but that was never a strong point of mine either. Given the general level of mathematical interest and knowledge among journalists, I'm hoping to be an outlier.

Saturday, May 23, 2015

X+Y → A Brilliant Young Mind

This movie looks promising. Released in the UK as "X+Y," it's coming out here, at an unspecified date, as "A Brilliant Young Mind." Here's a scene I like.

Monday, May 18, 2015

Rejuvenated brain considerations

Here's an interesting story: "Neurobiologists restore youthful vigor to adult brains." In mice, that is. Excerpt:
UC Irvine neurobiologist Sunil Gandhi and colleagues wanted to know whether the flexibility of the juvenile brain could be restored to the adult brain. Apparently, it can: They've successfully re-created a critical juvenile period in the brains of adult mice. In other words, the researchers have reactivated brain plasticity—the rapid and robust changes in neural pathways and synapses as a result of learning and experience.
And in doing so, they've cleared a trail for further study that may lead to new treatments for developmental brain disorders such as autism and schizophrenia. Results of their study appear online in Neuron.
Me: Without meaning to dismiss the various good things that may come from developments such as this becoming applicable to humans, I think they might also lead to some perverse incentives, eg: "Why should I study this now rather than leave it for my midlife brain rejuvenation?" In my own math studies that I've described recently, I've been surprised by how some concepts that would have or did baffle my 19 year old self (e.g. in real analysis) make more sense now that I actually am interested in them. A key challenge as neurobiological tinkering becomes more doable and mainstream will be resisting the temptation to use it as a substitute for motivation and diligence.

UPDATE 5/19: Recommended reading: Daniel Klein (worried) and David Henderson (more upbeat) on designer babies. I lean toward Klein's outlook. Via Walter Olson.

Friday, May 8, 2015

Leftists beyond orbit

I've seen hints over the years that someday anti-space exploration would become a driving force on the left, for example long ago when I reported on the anti-Cassini movement. Well, maybe the day has come. Here's Rand Simberg, writing in PJ Media: "Social Justice Warriors Make Their Claim on Space." Excerpt:
People are starting to take the notion of large-scale habitation of space seriously, and some of the Social Justice Warriors, fresh from their recent bloodying with GamerGate and the Hugos, seem to be switching their sights to a new target. A few weeks ago, Elon Musk, Bill Nye and Neil DeGrasse Tyson had a conversation about (among other things) the importance of becoming a multi-planet species (one of Musk’s driving concerns, and the reason he started his company SpaceX). 
Well, D. N. Lee, a biology blogger at Scientific American, found the discussion “beyond problematic” (one of the SJWs’ favorite words)...
Rand goes on to discuss a Guardian piece that focused on literal off-planet rape. Rand's conclusion:
There is a moral case to be made for settling space by humanity, warts and all, and we have to be prepared to make it.
Me: I agree, and count me in on the pro-space side. What I wonder, though, is how the political sides will line up over time. Will the liberal space enthusiasts at places like Scientific American defend exploration and (gasp) colonization against the hard left? Or will the hard left manage to intimidate a substantial portion of the political spectrum into at least falling silent amid attacks on "White Colonialism Interstellar Manifest Destiny Bullshit"? Interesting times.

Wednesday, May 6, 2015

Review: Birth of a Theorem

I almost did not bother to read most of Birth of a Theorem: A Mathematical Adventure, by Cédric Villani. After reading several chapters, it was clear that I wasn't going to understand the mathematics in this book, and via Twitter I came across some negative reviews that emphasized the book's inclusion of incomprehensible material. Moreover, in keeping with my recent hobby of studying math, I've been very much in the mode of wanting to actually do math, rather than just observe it in some vague way. But I plowed ahead with Villani's book, and I am certainly glad I did.

Birth of a Theorem gives a compelling and personal picture of what it is like to do math at an extremely high level; for example, to think you've solved a longstanding problem and then find that you haven't, or to wake up with a momentous realization that "You've got to bring over the second term from the other side, take the Fourier transform, and invert in L2"--and then write a note on a scrap of paper before rushing to get the kids dressed and onto the school bus.

Some pages of the book are laden with equations, and a note on translation at the end states: "No attempt has been made to expand upon, much less to explain, fine points of mathematical detail, many of which will be unfamiliar even to professional mathematicians. The technical material, though not actually irrelevant, is in any case inessential to the story Cédric Villani tells in this book."

I would have preferred it if, at some point, there had been a diagram with annotations summarizing, term by term, what a key equation means. As it was, though, I had some fun picking out the symbols I did understand--an ∈ here, an ∀ there, a sup somewhere (all of which I learned fairly recently), and I agree that this book tells a valuable story even while displaying so much unexplained math. The only pages that I didn't find interesting were ones devoted to a long listing of musicians and bands the author likes. (Some readers may like this part, however, especially if they were attracted to the book by Patti Smith's blurb on the back cover.)

So, Birth of a Theorem is recommended. It is a very different offering from Edward Frenkel's Love and Math: The Heart of Hidden Reality, which strives to make some very difficult math comprehensible to a lay audience. Still, I suspect that some people will pick up Villani's book and end up being drawn further into mathematics, as well.

Monday, May 4, 2015

Financial history lessons

My latest at Research magazine: I interview historians Richard Sylla and Robert E. Wright about their new book Genealogy of American Finance (Columbia Business School Publishing). Excerpt:

Does the subject of financial history get as much attention as it ought to from financial professionals? How about from the general public?

Wright: Financial professionals, policymakers, and the general public do not pay enough attention to financial historians when times are good. When times are bad, the stock of financial historians does increase but then it is too late to do much good. We were much in demand in 2008–9 as journalists, policymakers, investors, and voters tried to wrap their heads around the financial crisis but it would have been better for everyone if they had paid attention to us in 2002–7! Ken Snowden, for example, had shown that six previous mortgage securitization schemes had blown up between the Civil War and World War II. While his historical analysis did not conclusively “prove” that trouble loomed (the past can never be used to predict the future with certainty because the past rhymes rather than repeats) it should have set off more alarm bells, as it did for our colleague at NYU-Stern (where I taught from 2003–9), Nouriel Roubini, one of the few economists to make accurate predictions of the impending disaster.

Studying financial history, all forms of the past for that matter, can help to create good, old-fashioned judgment, the “soft” skills that help financiers like Henry Kaufman to discern the difference between junk mathematical models and the real deal.

Sylla: Most financial professionals pay too little attention to financial history, which ought to instruct them. In the wake of the recent crisis, a good number of them became more interested in financial history for the perspectives it shed on what had happened, and some even advocated more study of it. The CFA Institute has been studying ways of adding more financial history to its educational programs for finance professionals. But as the crisis fades in memory, finance professionals talk less and less about history's importance. Its cautionary lessons might interfere with taking the next big risk to make the next fast buck. One of the great lessons of financial history is that a lot of finance professionals over the decades and centuries never learn, and so they repeat the mistakes of the past. The general public ought to learn more financial history to protect themselves from short-sighted finance professionals!

Friday, May 1, 2015

Progress report on learning math online [updated and moved to top]

3/25/15: I've been studying math lately, as noted in the post below. I'm now in the 6th week of Prof. Keith Devlin's "Introduction to Mathematical Thinking," which runs either 8 or 10 weeks depending on whether you stop after the Basic Course or continue through the Extended Course. The latter is described in the course description as being particularly difficult:
Students who struggle with the Basic Course are likely to find the additional two weeks of the Extended Course extremely difficult, if not impossible. Note also that the final two weeks of the Extended Course are more intense than the Basic Course, being in part designed to give students a sense of the pace of a university-level course in pure mathematics. Moreover, in Week 10 there is a series of fairly tight deadlines you must meet, with 48 hour turnaround times. [Emphasis in the original.]
Me: We'll see how that goes, as I've been hoping to make it through the Extended Course and get the Statement of Accomplishment With Distinction that comes from passing it. I've found the course highly interesting and will have learned a great deal regardless of how far I get. The overall course emphasizes logic and proofs, and is designed to give a sense of how mathematicians think (in contradistinction to the emphasis on following instructions that characterizes much K-12 math). At times, I have struggled with the concepts, though my weekly test results have been generally decent (with one exception), with scores equivalent to about 87, 87, 41 (oops), 96 (comeback) and 86.

The course is not particularly oriented toward visual thinking, but I have found some visualization helps in grasping the concepts. Here is what I drew and wrote for a homework problem. [Added: SPOILER: Don't read the image first if you want to answer the question yourself.] The question was: "Prove or disprove the claim that there are integers m, n such that m2+mn+n2 is a perfect square." I drew and wrote the below, and posted it to a class discussion board (along with a question as to whether and to what extent "visual proofs" are acceptable):

A fellow student pointed out that my diagram doesn't match the algebraic expression m2+mn+n2, which is true. If I were to do it over, I'd leave one of the mn boxes out or cross-hatch it or something. My basic idea that the claim is verified by making n zero seems to have some merit.

Here is a report from someone who took the same course a couple of years ago. I agree with many of the sentiments expressed, including about finding the course very interesting and enjoyable, and also about this:
Assignments are not submitted for marking (but a helpful feedback video is made available in the next week in which Devlin explains how to answer a selection of the questions). In the first few weeks of the course Devlin puts a very prominent amount of emphasis on the need for all students to discuss the course with others in an informally established study group. In my case I chanced on and joined a Google Group called “Mathematical Thinking UK Discussion Group”. This initially had about 40 members. About seven were helpfully active in the first three weeks, but the study group has seemingly since ceased to function. So I am on my own: it feels a bit late in the day to try to find another study group, nor to attempt to breath life into this one.
Me: That mirrors my experience very closely, as I was part of a weekly Google Hangout study group, which started off very promising and then progressively wound down into nothingness. I gather the dissolution of such groups has a lot to do with people quitting the course (considerably more that, I suspect, than with people finding the course easy and deciding they don't need a study group). The statistics posted by the professor weekly show that, while a large number of people are enrolled (over 38,000, from all over the world), they vary in activity or lack thereof; and those who actually hand in the weekly test, or Problem Set, are a small and declining subgroup (under 1,500 at last count).

In any case, I've become a big fan of online courses, a remarkable and unprecedented resource (and one that for now at least is often free). Edward Frenkel, who did much to inspire my newly heightened interest in math, has an upcoming (and seemingly not too time-consuming) class, which I've signed up for as well. Whatever limits I encounter in my online education, it's clear to me that I've learned much already, and that this has been time well-spent.

The above was originally posted 3/25/15.

UPDATE 4/28/15: I've completed the course, including the extended course, and am now awaiting my Statement of Accomplishment With Distinction, which I believe I earned. Someone pathetically tried to disrupt the peer-review portion of the extended course by placing numerous bogus "zero" reviews, but this was detected and deleted. I am now on to "Introduction to Mathematical Philosophy," which looks to have many fascinating concepts but also be less time-consuming (a good thing, given my schedule, though it also means it will involve less hands-on learning, as this course, unlike the previous one, does not involve handing in weekly problem sets for grading).

UPDATE 5/1/15: My SOA With Distinction. I was one of 275 students to get one of these; another 991 earned an SOA for completion of the Basic Course.

A few additional thoughts before closing this post:

1. "The only way to really learn math is by doing it." This is something one hears now and then, and I can affirm it based on my experience in this course. If I had only listened to the professor's lectures or read his book Introduction to Mathematical Thinkingwithout actually trying to solve the (ungraded) assignments and (graded) problem sets myself, I would not have gotten nearly as much out of this course.
2. "You don't really understand something until you've taught it." Another bit of repeated wisdom that I find has validity. In evaluating other students' work and trying to explain things to them, I found I had to learn the material better than I would have just from handing in my own work.

3. My final grade was 98.5%, which is of course good but needs some context. The numerical grades are described by the professor as "akin to the points awarded in a video game: significant within the game, but only within the game." Under the scoring system, it is possible for students to get more than 100%, which then gets normalized to 100%. My grade was boosted by certain features of the scoring, such as that it includes only your highest score out of three "evaluation exercises" (evaluating proofs and seeing how close your score is to the professor's; I did great on the third one); in general, the grades are a marker of progress and persistence more than performance.

4. A substantial portion of this course is about language and communication; such as in converting between natural language and symbolic logic, and in assessing the clarity of proofs. When I signed up for "Introduction to Mathematical Thinking," I wondered if I might be going too far afield in spending valuable time on something unrelated to my financial journalism day job, Erie Canal book project and overall writing career. I'm pleased to find the course was plenty relevant to writing.

UPDATE: More here.