Monday, 11 July 2016

Toomas Karmo (Part A): Practical Tips for Maths-Physics Studies

Quality assessment:

On the 5-point scale current in Estonia, and surely in nearby nations, and familiar to observers of the academic arrangements of the late, unlamented, Union of Soviet Socialist Republics (applying the easy and lax standards Kmo deploys in his grubby imaginary "Aleksandr Stepanovitsh Popovi nimeline sangarliku raadio instituut" (the "Alexandr Stepanovitch Popov Institute of Heroic Radio") and his grubby imaginary "Nikolai Ivanovitsh Lobatshevski nimeline sotsalitsliku matemaatika instituut" (the "Nicolai Ivanovich Lobachevsky Institute of Socialist Mathematics") - where, on the lax and easy grading philosophy of the twin Institutes, 1/5 is "epic fail", 2/5 is "failure not so disastrous as to be epic", 3'5 is "mediocre pass", 4.5 is "good", and 5/5 is "excellent"): 4/5. Justification: There was enough time to develop the relevant  points to  reasonable length.

Revision history: 

UTC=20160712T0001Z/version 1.0.0: Kmo uploaded base version (and planned to upload in the ensuing four-hour interval, without formal documentation in this revision history, nonsubstantive revisions, as versions 1.0.1, 1.0.2, 1.0.3, ...) . 

[CAUTION: A bug in the blogger software has in some past weeks shown a propensity to insert inappropriate whitespace at some late points in some of my posted essays. If a screen seems to end in empty space, keep scrolling down. The end of the posting is not reached until the usual blogger "Posted by Toomas (Tom) Karmo at" appears.]

0. Prefatory Remarks,
Regarding Purpose and Audience of this Essay

In pondering the impending destructive work of Fossil Fuel Depletion and Climate Change (these two factors are already exacerbating our general cultural decline, and eventually a Dark Age looms), I have for the moment finished my discussion of that engrossing question, "Is Science Doomed?" 

Admittedly, I could have done better in treating the question. 

I could, in particular, have added, in last week's "Part E", some detail regarding the Soviet and Oxford academic evaluation systems - only briefly cited in "Part E", as I managed to write it up. While Oxford assigned its commendable broad categories of "First Class Honours", "Upper Second", and so on only to entire degrees, I now suspect - contrary to an impression I conveyed last week - that the Soviet system was more fine-grained than Oxford's. I now suspect that Soviet tertiary institutions assigned their own  commendably broad "5, 4", and so on, not (or at any rate not exclusively) to entire degrees, but also to results in individual examinations - perhaps, say, to each of the individual papers in that scary suite of mathematical-physics examinations which I think to this day defines Moscow's "Landau Minimum".

And I should have added something about the question of "Appeals". What happens if, in the impending Dark Age, you take, say, the "Chicago Finals in General Mathematical Physics", under correct ten-day invigilation in your local warlord's little police station, and Chicago eventually awards you a Lower Second, and you think that you instead deserved at worst an Upper Second, more plausibly a First? Have you any recourse to a court of appeal, as people nowadays sometimes have when some judge hands down a verdict in some civil or criminal trial?


I write "have for the moment finished" because I can already at this point see what will later have to be done. 

(a) It will later be necessary to reproduce some remarks from astrophysicist Fred Hoyle, in his essay entitled "The Anatomy of Doom", in his 1965 book Encounter with the Future, regarding the role of libraries in our impending Dark Age. (Hoyle back then was as confident as I am now that a Dark Age impends, for reasons quite similar to mine.) 

(b) It will later be necessary to examine and comment on Martín López Corredoira's circa-2013 book The Twilight of the Scientific Age, brought to my notice in the last few days by a commenter at John Michael Greer's blog

(c) It will later be necessary to address two remarks from John Michael Greer, at this same, made by way of comments (i)  to me and (ii) to some other reader. 

(i) In a posting there timestamped by the blogger software as ""7/7/16, 11:20 PM", Mr Greer writes, by way of comment for me, /.../ it's /.../ invalid to assume /.../ that science can't run up against the law of diminishing returns. There is, after all, no prima facie reason why there must be a limitless number of things that can be understood using the particular set of human activities we call science. Logically speaking, both possibilities have to be held in abeyance pending the arrival of new data. 

(ii) In a posting there timestamped by that same software as "7/9/16, 11:11 PM" (this means 2016-07-09, not 2016-09-07; and the software's clock reference is either to Pacific-USA winter civil time or to Pacific-USA summer civil time), Mr Greer writes in a comment for some reader other than me (with reference to physicist Jeremy England):   We already know a lot about thermodynamics; we also know a lot about biological evolution; if we fuse the two into a single overarching theory, and that theory works, we take a huge jump closer to the point at which the only meaningful questions that can be asked within the overall paradigm of modern science are questions of detail. Now of course that paradigm will eventually be set aside -- but remember that the last change in grand paradigms was the transition between the logical method pioneered by the Greeks and the scientific method, and the next one will thus in all probability go to something that isn't science in any sense of the word we would recognize at all.


For the moment, I work on another task examining the practical nuts and bolts of study in maths and physics. 

My remarks here are envisaged as possibly useful to people in the humanities (such workers will have a legitimate curiosity regarding science pedagogy), but as more immediately and clearly useful to science students themselves (perhaps especially to those, like me, studying at a mere early-undergrad level). 

1. The Goal of Working Things out for Oneself

One goal dominates in maths-and-physics study. We must arrange our work in such a way that, so far as humanly possible, we work things out for ourselves. I think everyone who stays in maths and physics at all accepts this, eventually. We may indeed start with the vulgar goal of "getting through the course". In that ignoble endeavour, we simply work through the textbook author's proof of, say, some theorem, reading the published paragraph clause by clause, and agreeing with each successive elegant step in the published demonstration. Eventually, however, we see that this is insufficient. Books are only guides to what we have to do for ourselves.

One of the best minds in mathematical physics ever, Lev Davidovich Landau (1908-1968; Lubjanka prison only briefly, in 1939 April; 1962 Nobel in physics for mathematics-of-superfluidity) was famous for his refusal to pore over the literature. The Landau method, I gather, is to glance at a journal article just long enough to ascertain its results. One then has, I gather, to put the article aside, deriving its results independently. 

Perhaps all maths-physics workers have their own individual, private, Instant of Epiphany for this. My own is rather silly (well, I work at a rather silly level: 1996 Honours B.Sc., essentially as the University of Toronto equivalent of UK "Upper Second", with later-1990s enhancements in astrophysics). All the same, I do want to put it on record here. 

For some reason, quite possibly after taking that 1996 B.Sc., I decided that I might as well be thorough on the matter of coefficients in the various terms making up the sum which is the expansion, for arbitrary n in {1, 2, 3, ... }, of (x+y)-to-the-power-n. (An example: if n is 4, then the expansion is x^4 + 4 x^3 y + 6 x^2 y^2 + 4 x y^3 + y^4, with the respective coefficients in this particular case accordingly  1, 4, 6, 4, and 1. - A further example: if n is 5, then the expansion is the sum of 6 terms, the respective coefficients now being 1, 5, 10, 10, 5, and 1.) 

The coefficients happen to be related to the number of selections of pebbles from a bowl of n pebbles. 

For a bowl containing five pebbles, we have the following: 

  • one possible selection of 0 pebbles (trivially; so 1 is "5 choose 0") 
  • five possible selections of 1 pebble (rather trivially; so 5 is "5 choose 1")
  • ten possible selections of 2 pebbles (not quite so trivial: if the five pebbles are labelled a, b, c, d, and e, then the ten possible two-pebble selections are
    {a, b},      
    {a, c},
    {a, d},
    {a, e},    
    {b, c},
    {b, d},    
    {b, e},    
    {c, d},
    {c, e},
    {d, e}
    (so 10 is "5 choose 2")),
  • ten possible selections of 3 pebbles (trivial given the previous remark, on the case of 2 pebbles, but I nevertheless spell it out:
    {c, d, e},
    {b, d, e},
    {b, c, e},
    {b, c, d},
    {a, d, e},
    {a, c, e},
    {a, c, d},
    {a, b, e},
    {a, b, d},
    {a, b, c}
     (so 10 is also "5 choose 3")), 
  • five possible selections of 4 pebbles (namely - this is, to be sure, rather trivial -
    {b, c, d, e},
    {a, c, d, e},
    {a, b, d, e},
    {a, b, c, e},
    {a, b, c, d}
    (so 5 is "5 choose 4")),
  • one (trivially) possible selection of 5 pebbles, namely {a, b, c, d, e} (so 1 is "5 choose 5"). 

In general, for any n in {1, 2, 3, 4, ... }, and for any k in {1, 2, 3, .. , n, n+1}, the kth term in the (n+1)-term sum which is the expansion of (x+y)^n is "n choose k-minus-1". One might call this fact the "Combinatorial Fact Regarding Coefficients in the Binomial Expansion". 

I set myself the task of demonstrating the "Combinatorial Fact", and I guess I succeeded. At any rate, I have in later years taken this modest piece of work, and the surprising feeling of confidence which it generated, as a demonstration to myself that I can and must do things to some significant extent on my own. 

2. Means to the Goal:
(a) Suppressing Comparison of Oneself against Others

Perhaps the most glaring fault of the present system of maths-and-physics education - even worse than its failure, outside a few good places like Oxbridge, to separate the teaching personnel from the examining personnel - is its propensity to pit student against student, in a competition analogous to a race. Or, to vary the simile, it is an exercise analogous to the grading of farm produce. (In the country in which I am writing, we have, for such things as cucumbers or cherries or turnips, such categories as "Canada Fancy" and "Canada Extra Fancy" and "Canada Choice".) We may expect the fault to be resolutely remedied at the "Other Place" I imaginatively sketched last week, at the end of the "Is Science Doomed?" essay. 

Competition is often a little subtle. I picture it thus, recalling in part what I observed, or at any rate rather justifiably feared, at the University of Toronto: 

The late-1990s astrophysics class is not big, and the prof and Teaching Assistant are nice, and the students get on just fine with one another. Now we have some kind of big assignment, and the two specially good people in the small class are Toomas (surprise, surprise: but I, unlike everyone else, had plenty of time for study) and "PQR". Now what happens? We cannot really expect everyone to get top marks. If, per impossibile, there is a significant plurality of nearly-perfect grades, then the prof and T.A. who teach-and-also-examine may expect trouble with some Head of Department, or even with some Dean. 

Toomas gets an exceptionally high mark, something like a "95% A+", with Toomas being on this occasion deemed a particularly impressive cucumber or turnip. And now, as Prof. Tom Lehrer sings in his "Lobachevsky Song", in skilfully done fake-Soviet accent - , perhaps - "Ha-ha begins the fun." For prof and T.A. will be under some quiet pressure, duly internalized, but always reinforceable as required by Departmental Head or Dean, to put PQR a bit lower. It does not look good to have both PQR and Toomas getting 95%. 

And really, objectively speaking, some difference can be found to rank PQR and Toomas differently, given that we have not the sensibly broad classifications of Soviet "5, 4, 3, 2, 1", but instead have the absurdly fine-grained "100%, 99%, 98%, 97%, ...". 

The upshot of this little race is that Toomas has "come in first", and poor PQR has "come in second", and now PQR might not find it quite as easy as she or he otherwise might to get into, say (I make my example up) Princeton grad school, as opposed to, say (I make my example up), University of Melbourne grad school. 

If PQR does get into Princeton, or whatever, all the same a bit of damage has been done to his or her soul. She or he feels a bit lowered, a bit shopworn. Toomas, of course, suffers a spiritual damage equally real, but tending in the opposite direction from PQR's - namely, a bout of Swelled Head Syndrome. 

Well, I hope I am forgiven for retailing this, if anyone from my cohort - PQR, for example - ever happens to be reading. We were a nice class, as I say, and I think I did successfully resist my one significant bout of temptation-to-tease-PQR. (He or she had temporarily dyed her or his hair a vivid red. Mindful of vividly red 6563-ångström hydrogen-emission light in the Be "shell" class of stars, I perhaps nevertheless did manage to resist the temptation to chant, as in playground in Grade Two, "Nah-nah-nah-nahnah: PQR is in emission, throwing out a shell.") 


How do we resist the temptation to compare ourselves against others? To some extent, we wrestle with this demon forever. But a small meditation does help:

The year is 1920, or something, and one is Albert Einstein, or somebody. Oh, so scary, oy veh gewalt: one compares oneself against de Broglie and Madame Curie, and things feel perhaps okay, and then one compares oneself against Newton, or somebody, and one feels insecure. 

Is this line of thought reasonable, or is it silly? 

It additionally helps to note that no matter how far down one may be on the scientific totem pole, one can always find some contribution to make. I for my part, for instance, not only volunteer 40 or 60 hours' work annually updating the Royal Astronomical Society of Canada Observer's Handbook bright-stars table (for benefit of amateurs), but keep working on topics in vector calculus. The vector calculus should eventually be useful for writeups in physics-of-radio, ultimately to the benefit of the ham radio community. And less ambitiously, the vector calculus, or things related to it - geometry-of-determinants, for instance - should rather soon prove useful for small writeups on vector calculus itself. 

In the wake of Brexit, HM the Q helpfully drew general British public notice to the spirit of an old wartime poster - the one which, anticipating the possibility of a Wehrmacht invasion, bore the helpful words "Keep Calm and Carry On." The Web has over the last few years shown also the commercial availability of coffee  mugs, tee-shirts, and the like, inscribed with such helpful things as "Keep Calm and Knit On," and "Keep Calm and Learn Latin."  

To this small genre of literature, I would like to add "Keep Calm and Carry on Promoting  a Formalism for Multivariate Calculus in which the Leibniz 'd' and curly-d are Suppressed, with Due Recourse Made Instead to the Alonzo Church Lambda-Abstraction Notation in the Spirit of Sussman-Wisdom's Structure and Interpretation of Classical Mechanics." 

One thing that I could do in vector calculus is promote an Alonzo Church notation, with specific reference to Jacobean determinants and change-of-variable in integration. 

Here I would also write up a derivation of the formula for, say divergence (now nicely purged of the obfuscatory curly-d) in the setting of, say, polar coordinates, with due attention to the question, "But what does it MEAN?" ("Vot MEENZ?", as they say beside the samovar in my grubby imaginary "Nicolai Ivanovitch Lobachevsky Institute of Socialist Mathematics": the fatal flaw in Leibnizian notation for derivatives is that it sacrifices conceptual clarity to ease of symbolic manipulation, as traditionally desired in pencil-and-paper computations. )

 3. Means to the Goal:
(b) Securing Multi-Year Continuity of Effort

Working things out for oneself requires continuity of effort, over periods of months and years, and potentially even of decades.

So, for instance: This is the year that rotation matrices have to be studied. Later this year, or even next year, a good grasp of these matrices will make it possible to grasp, indeed to some extent to do on one's own, those particular textbook examples of tensors which involve the rotation of coordinate axes.

However working paper may happen to get shelf-filed and computer-indexed (this is a topic on which I will say more in my next section, on "Available Physical Supports"), it is at any rate essential that the filing system render working papers efficiently retrievable even some years after their creation.

In securing such multi-year continuity of effort, I find it helpful to keep several kinds of formal computer log. In a Debian GNU/Linux /usr/bin/vim text-editor timelog, I track time spent on various particular topics. Here, to illustrate, is my timelog for work on rotations and rotation matrices, started on 2016-04-08 at a time which I rather arbitrarily have deemed to be  UTC=121314Z:

20160415=02h05->0012h34__learned of diff between alibi (active)
                         and alias (passive) transformation, LAVS DEO  
                         (_this overcomes the seeming discrepancy
                           between my work and FraBea+Wikip) 
20160521=05h04->0050h18__started special sub-initiative, 
                         re rotn matrices-and-orthonormality-and-        Euler
                       __did rotn_matr_orth_Eule 01h31
                         __rotn_matr_otrh_Eule__aggr = 01h31
20160525=01h03->0051h21__having got correct Rx(theta), seemed to have
                         today ERRED, alas, in Ry(theta) 
20160526=00h21->0051h42__MEMO: "p. 11, recipe for finding
                         sphe coords, as opposed to mere multipliers
                         of sphe unit vectors, given rect coords" 
                       __MEMO: Durell & Robson, _Advanced Algebra_ 
20160531=02h42->0056h06__started reading closely the Wikip article
                         on Euler angles
20160601=03h17->0059h23__started also reading closely MacRae on rotns
20160602=03h24->0062h47__much work with clay-and-stick models, 
                         in connection with Wikip article

The cryptic "FraBea" is a reference to a clear, straightforward, not very challenging text on linear algebra, by Fraleigh and Beauregard. "MacRae" is an ancient, but seemingly promising, UK early-undergrad book, from the era of Neville Chamberlain and Vera Lynn. The first "MEMO" tells me where to find a rather troublesome point in my stack of papers. 

Perhaps most importantly for purposes of the present discussion, the time gap this summer,  spanned by the discrepant dates 2016-06-11 and 2016-07-08, shows how it can be necessary to drop something for a whole month (I had to attend to the Observer's Handbook 2017 bright-stars table) before returning. With my papers in rather good order, it turned out not to be too hard to pick up again from where I had left off. 

It does rather help, in securing continuity of effort, to keep a special log with reminders of tasks that are coming up. I make my log swiftly retrievable at the computer, with the command-line prompt taking some Debian GNU/Linux bash-shell "alias". Here is what I have been using for rotation of coordinate axes: 

((STACK QUANDO="20160612T011730Z"))
!_do handwrite for generality of extrinsic TaitBry scheme
  (_not in Wikip)
!_do handwrite for generality of intrinsic TaitBry scheme
  (_in Wikip)
!_do "early" handwrite for generality of extrinsic Euler scheme
  (_not in Wikip)
  (_I have already done generality of intrinsic Euler
    __under timestamp 20160611T0257Z)

(The cryptic "TaitBry" is a reminder that it will be necessay to look not only at Euler angles, but also, eventually, at Tait-Bryan angles.) 


It is humiliating that things take so long when done on one's own. But with timelogging, and with little computer-stored memoranda about such upcoming tasks as the investigation of Tait-Bryan, the pain of the humiliation is at least lessened, to the point of becoming bearable. 

It admittedly does additionally help to picture monks patiently binding books, or cultivating fields, without haste or impatience - or even to view such monks via Google Images or in YouTube vids. 

And it admittedly does help to remind oneself of two lines from T.S. Eliot's Four Quartets

The only wisdom we can hope to acquire
Is the wisdom of humility: humility is endless. 

[To be continued, and probably concluded, next week, in the upload of UTC= 20160719T0001/20160719T0401Z.] 

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