But the early universe was very hot, very dense, and gravitationally very different from the comfortable-to-us 1 g we experience today on the surface of the earth. Einstein has convincingly shown that spacetime is accordingly divorced from that human experience. Clocks, for example, are affected by gravity and satnav constellations have to take this into account. Did the first three minutes flow the same way three minutes flow in the here and now? I sent that question to Chris Impey’s online office hour and he kindly answered. It is a tantalizing response and one that will require substantial further study to fully appreciate – perhaps finally diving into the guts of GR. It makes me wonder even more intensely why we anthropomorphize those intervals the way we do.
I’ve got only a superficial understanding of the revolutionary incompleteness theorems that Gödel brought forth while a preternaturally gifted young man. It will be the work of a remaining lifetime to learn it in any detail but it stimulates periodic revisits. This podcast from the BBC’s In Our Time series smacked my gob when I came across it: A very well-prepared moderator and threedistinguishedscientists discuss its impact. It starts with a punch and soars from then on. Yes, it is true that the posh accents predispose to trust but beyond that it is 40 minutes of serious yet freewheeling fluffless conversation that compels the listener to pay attention in the now and do homework afterward. The presumption on the listener to be prepared, attentive, and engaged is exhilarating. I can’t imagine such an exchange prepared for broadcast in the modern United States even by the remaining cultural outposts like NPR.
The outsized impact of Teutonic peuples on the sciences requires the rest of the world to get acquainted with the nuances of German pronunciation what with the umlauts, diphthongs, capitalizations, and veryLongwordscomposedofötherwordswIthnospacesinbetwëën. Berkeley chemistry majors of my vintage needed to take two terms of German so that we could read the older literature before English more-or-less took over as lingua franca. Some of us actually enjoyed it, most cursed it. Speaking of franca, the French were/are no slouches when it comes to scientific impact but we were not required to take French as part of the curriculum.
Vacky Deutscher JensFehlau offers a basic pronunciation course in typical vacky, flammable, and completely not-safe-for-work idiom forgoing his usual chalkboard for a MySpace page come to life. He vents justifiable rage against those profaning the great Leonhard Euler as ‘Youler’ or even ‘Wheeler’. For an existence proof of the former, see UCL mathematician Hannah Fry deadpan it repeatedly in a discussion of map projections. What in the actual f, indeed? The definitive guide to speaking the insufficiently known Emmy Noether is a major service.
Let us now consider other Wheelers (not Youlers or Oilers) that have loomed in the physical sciences. Most famous is Princeton’s late John Archibald, coiner of terms (black hole, wormhole, it-from-bit) and who begat Feynman, Thorne, and other luminaries. There is also Reed’sNicholas Wheeler whose lecture notes have become the stuff of legend. Now the A. A. Knowlton Professor Emeritus of Physics, Wheeler created many courses over a six decade career developing first his own approach to a topic, writing about it, and then taking his students through it. It is a pedagogical road taken by a select few. He writes,
I learned early on in my undergraduate education that while it is instructive to read, and to attend to the words of informed speakers, I cannot gain the feeling that I “understand” a subject until I have done my best to write about it. So much of my time these past sixty years—even when seemingly involved with other things—has been spent pondering the outlines of what I would write when I returned to my desk, “composing the next sentence.”
Which means that I have been engaged more often in trying to write my way to understanding than from understanding
When thinking through a subject in preparation for a class I have no option but to write my way through the subject, and then to lecture from my own notes. I find it much more pleasant and productive to spend an afternoon and evening writing than arguing with the absent author of a published text.
Reed has placed the original handwritten notes in their Special Archives for consultation. Fortunately, Wheeler also took pains to meticulously typeset a large fraction of these notes in and put them on his website. It is heady stuff this idiosyncratic guide through the highways, byways, and backroads of mathematical physics. Wheeler’s writing is alternately informal then precise, rigid then fluid, purposeful and then digressive. Each polished chapter contains seeds and fruits from all the others just as pieces of a hologram recapitulate the whole. There are treasures enough for many lifetimes. We can only marvel at undergraduates who had both the fortune to experience this ‘drawing out and not a putting in,’ as well as the ability to absorb and understand so much material coming from so many directions. Click the image to go the site and rejoice in each folder and its branches.
Facility with calculus, specifically integration and differentiation, is mandatory for just about any technical discipline. Slog the first is setting up the problem, Slog the second is hacking through whatever differential equations and/or integrals present themselves. For those of us of a certain age, getting to calculus in high school was a badge of honor and being able to evaluate difficult integrals through clever substitutions and grit a point of pride.
Time has passed and the standards have gone up. Way way up. The kids these days are learning more, learning it earlier, and are scaling peaks we didn’t know existed. The mathematics subculture on the Internet is fueling this fire and a particular segment of Youtube is devoted to these calculations both for fun and for education. MIT has even held an “Integration Bee” for many years where students go head-to-head under time pressure.
Leading this pack is Jens Fehlau, an early twenty-something bro from Germany whose skills and presentation style make us glad we aren’t in a class where he’s ruining the curve. His Flammable (formerly Fappable) Maths channel has a strong following with early videos in German and more recent ones in excellent English with calming sounds of chalk on a chalkboard. Fehlau also reminds me of a college classmate of mine who did exactly that and is now an eminent professor of chemistry.
Here’s one of his playlists. Fair warning – the language can get salty at times.