Monday, 5 December 2016

Toomas Karmo: Remarks for Mathematics Students and Teachers, Including Individuals Possibly Seeking Tutoring

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"): 5/5. Justification: Kmo, while slipping on his schedule, nevertheless  had time to do a decidedly complete and (within the framework of the version 3.0.1, 3.0.2, .. process) reasonably polished job.

Revision history:

220161206T2049Z/version 3.1.0: Kmo found a really bad mistake: he had omitted a premiss in his example of a valid non-syllogistic inference.  He also noted that aside from little errors of scant consequence, he had mangled Sit Arthur Conan Doyle, inadvertently turning Sherlock Holmes's "the game is afoot" to "the fame is afoot". - Having corrected all the mistakes he could at this stage find, he reserved the right to make further  minor, nonsubstantive, purely cosmetic, tweaks over the coming 48 hours, as here-undocumented versions 3.1.1, 3.1.2, 3.1.3, ... .

20161206T1601Z/version 3.0.0: Kmo finished converting his point-form outline into coherent prose. He reserved the right to make minor, nonsubstantive, purely cosmetic, tweaks over the coming 48 hours, as here-undocumented versions 3.0.1, 3.0.2, 3.0.3, ... . 

20161206T0310Z/version 2.0.0: Kmo overcame the inadequacies in his point-form outline, and was now at last ready to start converting it - behind schedule - into coherent prose. He was not sure if he could get the job done in the next two hours, or if he would have to resume the following Toronto morning.

20161206T0007Z/version 1.0.0: Kmo had time only to upload a (nearly complete) point-form outline. He hoped to convert this to coherent prose in a succession of uploads, finishing at some point in the next 4 or 5 hours.

[CAUTION: A bug in the blogger server-side software has in some past weeks shown a propensity to insert inappropriate whitespace at some 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. - The blogger software has also shown a propensity to generate HTML that is formatted in different ways on different client-side browsers, perhaps with some browsers not correctly reading in the entirety of the "Cascading Style Sheets" which on many Web servers control the browser placement of margins, sidebars, and the like. If you suspect "Cascading Style Sheets" problems in your particular browser, be patient: it is probable that while some content has been shoved into some odd place (for instance, down to the bottom of your browser, where it ought to appear in the right-hand margin), all the server content has been pushed down into your browser in some place or other. - Anyone inclined to help with trouble-shooting, or to offer other kinds of technical advice, is welcome to write me via]

0. Preliminary Remarks (on Tutorial Fees, and on Other Points)

Today, as on some previous occasions over the past months, I post to this blog in ways which I hope will prove mildly useful to mathematics students and teachers. I have, today as always, a wide constituency in mind: 

  • students of mathematics in the formal "K-12" school system, and in conventional colleges and universities, and their teachers at all these various levels
  • parents attempting homeschooling
  • individuals studying mathematics privately, whether as juveniles or as adults


It is necessary to get an awkward, dispiriting question out of the way first: what am I, as the author of this blog posting, liable to say if someone approaches me for tutoring and the question of fee comes up? 

An initial, mildly comforting, answer is that the question of fee need not come up. If I am approached by someone poor enough to be either homeless or verging on homelessness, or by someone who has not approached me before and needs just an hour's help, or by someone who can ask questions through e-mail or telephone in such a way that I do not in answering have to spend more than 20 minutes on any one day, then there is no point in my exacting a fee. 

What, on the other hand, if I am working with someone who does not meet any of the extenuating conditions just stated? In that case, I would, while resisting the temptation to be mercenary, nevertheless have to avoid going so cheap as to undercut other Greater Toronto Area (GTA) tutors. Those others need, no less than I do, to pay their rent, to buy their food, and to maintain their telephone and Internet connections. 

A good guide is provided by Saint Benedict of Nursia, in his early-Dark-Ages Rule for monastics: let the monastery charge for the things it sells to the wider community, even while charging a little under the usual rate. In my particular case, I would set a fee (were this some day to prove necessary) by first ascertaining the approximate median rate for mathematics tutoring of the particular level contemplated (is it at university-or-college level, or merely at K-12 school level?), and then subtracting 12 percent from the median. 

I would also have to charge for travel to and from my residence to any place beyond walking distance from my residence (i.e., more than 3 kilometres away), if the student did not wish to take tea or coffee in my own snug book-lined parlour, and if that travel could not be fitted into my normal pattern of movements. My normal pattern involves a weekly journey from Richmond Hill down to Toronto. My various Toronto errands are then normally confined to the 3-kilometre stretch bounded by Union Station on the south and the streets a couple of hundred metres beyond Bloor Street on the north. My normal stretch thus contains the University of Toronto St George campus, plus Ryerson University, plus the Toronto Reference Library. 


It is also appropriate at this preliminary stage to describe my own three tiny previous brushes with mathematics tutoring.

Here I protect privacy and conceal some details, by reserving the right to alter three (but only three) kinds of factual detail - the portion of the GTA involved, and the gender of my cited individuals, and the marital status of my cited individuals:

(1) In the GTA municipality of "Markham" is a student named "Mrs Physics", who has for many years had an interest in physics, and additionally in finite mathematics (including combinatorics). Mrs Physics has for years been ringing me up, on average perhaps once every five or ten weeks, with some minor question, that I can as a rule answer right away, in under 20 minutes - as it might be, the question how, in "The number of ways of taking exactly 3 numbered vehicles out of a car park containing exactly 17 vehicles, where we do not care which particular vehicle in the trio is selected first, and which second, and which third, is 17-factorial divided by the product of 14-factorial and 3-factorial," we justify the "product of 14-factorial and 3-factorial".

I cannot recall ever being depressed by the queries from Mrs Physics. They seem always to serve as useful reminders of the intrinsic interest of the topic at hand.

(2) Enrolled at the University of Toronto a few years ago was "Mr Astrophysics". Upon learning of my connection with the David Dunlap Observatory (perhaps including my repeated public lecturing there a while ago, to tourists), he begged me to take him on as a tutorial pupil in the initial University of Toronto undergraduate sophomore-level "Astrophysics Specialist" theoretical course. This could have been exciting, especially since Mr Astrophysics was so insistent on becoming my student. But for reasons not known to me, Mr Astrophysics changed his enrollment early in the semester, forsaking for the time being the Astrophysics Specialist programme, and thereby to our mutual regret revoking his intention to work with me.

(3) Somewhere in GTA is "Mr Troubled". Mr Troubled is a young adult at one of the "Community Colleges", a rung below GTA's universities. Mr Troubled has a sister, "Mrs Serene", I think a little older than he, and enjoying a combination of good health and stable (if perhaps tedious) employment. Mrs Serene makes to me the following points: (a) Mr Troubled is studying something very elementary indeed in his college (not calculus at all, but something remedying high-school deficiencies), with the ultimate ambition of proceeding to a certain branch of medical or psychological or veterinary science. (From what I can gather, his ambition seems realistic.) (b) Mr Troubled has (as is common in GTA) experienced some angst in recent years. (c) Mr Troubled and Mrs Serene need a book recommendation for the pre-calculus curriculum, with Mrs Serene now proposing to tutor Mr Troubled. - Well, good luck with that, say I, in my hard-nosed way: sooner or later, Mrs Serene and Mr Troubled are liable to be darkening my doorway, jointly seeking my help, despite anything I can now say to deflect them. If they darken it in more than the trivial ways I laid out above - I mentioned, above, the first free hour, and the subsequent 20-minutes-per-day maximum - then I will have to do what much or little I can to help them, and I will have to charge accordingly, for any services successfully rendered.

For the time being, I have managed to sidetrack this particular two-person problem by making an appropriate book recommendation to Mrs Serene - Leonard Irvin Holder's pre-calculus textbook entitled Primer for Calculus. I used the book heavily almost 30 years ago, in its fourth edition, and now there is even a sixth edition. Further, I find the book (as I would expect, given my own positive experience with it) favourably reviewed toward the bottom of the Web page

1. Mathematics in the Alternative Education System

Here in the GTA, the mainstream education system comprises the public-sector and private-sector "K-12" schools, plus the colleges and universities.

High on our rather modest local educational pyramid sits the rather dismal University of Toronto - perpetually vying with McGill University in Montréal for the honour of being accounted Canada's top university, but in world overall-excellence rankings emerging perhaps twenty-fifth. Having intimate knowledge both of 1970s Oxford and of the 1990s University of Toronto, I would say that the former stands to the latter as champagne does to vin du pays, or as a concert pianist stands to a nightclub jazz pianist - by which I do not at all mean to insinuate either that vin du pays is toxic or that the typical nightclub jazz pianist plays with two fingers.

Surprisingly, I have seen one recent survey ranking our rather dismal Local as thirteenth in the world, albeit in research excellence as distinct from overall excellence.

Some day, I must begin learning about a research-and-teaching physics establishment of clear international pre-eminence, just outside GTA, the Perimeter Institute (

Additionally, in some sense within the University of Toronto administratively, and right on the University's downtown campus, is a pure-mathematics establishment of clear international pre-eminence, the Fields Institute ( However, I do not know if the Fields does significant teaching, as distinct from research.

Outside the mainstream educational system there used to be, and perhaps at present is not, and perhaps is destined some day once again to be, Toronto's "Underground University". The "Underground" in past years undertook some minor teaching in electronics, even while devoting much of its effort to minor arid-seeming seminars in social critique. Perhaps this little informal initiative, or some like-spirited successor to it, might some day harbour pure mathematics?

Further, Toronto's Hacklab ( has offered minor workshops and classes, in a kind of engineering-education outreach.

Then there is the interesting, if worrisome, example of a post-secondary establishment robustly and commendably countercultural, Our Lady Seat of Wisdom Academy (; - admittedly, far from GTA, in the Ottawa-region hinterland which is the Madawaska Valley.

When I write here "worrisome", I have two points in mind. (a) The Academy seems to take calculus no farther than univariate - in other words, up to the old Soviet equivalent of Grade Nine or Grade Ten. (b) The Academy seems from its Web publicity to confine the teaching of logic (a proper preliminary or adjunct to maths, in Britain perhaps sometimes available in high school) to traditional, pre-19th-century, syllogistic. If it is indeed so confining itself, then the Academy omits much logic that is mathematically crucial - for instance, the validity of the non-syllogistic, because unavoidably relational, inference "No S-ish A is R'd by any B; some Bs are As; all Bs R themselves; therefore at least one A is not an S-ish A" (e.g.: "No pulp novelist is admired by any critics; some critics are novelists; all critics admire themselves; therefore at least one novelist is not a pulp novelist").

Our Lady Seat of Wisdom, with also its small, struggling, countercultural equivalents outside Ontario, is bound to take on an increasing importance later this century, should it succeed in surviving. If it survives, its importance will rise once our globalized economy, with its elephantine mega-universities (universities, moreover, dangerously close to the business world)  disintegrates a few decades from now, under the twin stresses of resource depletion and climate change.

With such now-obscure post-secondary institutions as Our Lady Seat of Wisdom, we may expect to see thriving also, in our coming troubled decades, a countercultural primary- and secondary-education element already far from obscure - namely, homeschooling.

Given this present and this unhappy future, mathematics tutors are needed already and will in future continue to be needed.

A glance at University of Toronto notice boards suggests the present availability of tutors from a university tradition reaching back at least into Victorian Britain. In those remote times, Cambridge mathematics was centred on the final suite of exams, a gruelling "Tripos". I presume the same was the case also at Oxford and at the then-junior British universities, notably the University of London. Students would prepare for their final-examination day or (more likely) days not only with the help of their individual Cambridge colleges (and I presume their intramural counterparts at the University of London, and the like) but also with the help of private, extramural, "coaches".

I have the impression that those "coaches", their rather derogatory appellation notwithstanding, could on occasion be mathematicians of eminence. Sherlock Holmes's nemesis, the mathematician-astronomer Professor Moriarty, for instance, is said in "The Final Problem" not only to have attracted general European notice with his treatise on the "Binomial Theorem" (author Sir Arthur Conan Doyle was here fantasizing) and to have held a chair at one of the then-junior British universities, but additionally to have coached. The best coaches would (such is my impression) vie with each other for outstanding results, each striving to propel as many of his own pupils as possible onto the published "First Class Honours" examination rankings.

(And I also add, timidly, that in my own small circle of acquaintances is N1, who knows N2, who knows N3, ... , who knows Nk, for some positive integer k, and that Nk got tutored by the most eminent living mathematician, Dr Grigori Yakovlevich Perelman in Sankt Peterburg. As USA mathematician Prof. Tom Lehrer puts it in his immortal Lobachevsky Song: "I have a friend in Minsk,/ Who has a friend in Pinsk,/ Whose friend in Omsk/ Has friend in Tomsk/ With friend in Akmolisnk..." - Readers who have not already heard the Song are herewith urged to stop wasting their time on me and go to some convenient YouTube upload - for instance, to the YouTube upload 2007-11-01, by YouTube user "nanioushka", under title "LOBACHEVSKY - Tom Lehrer", with the undeservedly low total of 117,103 views as of UTC=20161206T162504Z. In my corner of the Internet, nanioushka's URL is

We may piously hope that the British Victorian extramural tutorial tradition gains further ground in the Ontario of the troubled future - perhaps with a shrinking, impoverished University of Toronto concentrating less on teaching than on rigorous examining, and with a little army of private coaches taking up the slack as university budgets shrink and shrink.

We may also piously hope that when, in our social decline,  homeschooling gathers even more momentum than it has now, parents will see the wisdom of handing at least some subjects off to private coaches, rather than struggling to teach them around the kitchen table - French to people from Montréal (or wherever), with a duly clean pronunciation; piano, at any rate in its higher reaches, to people with some conservatory background; and mathematics, at any rate in its higher reaches, to people of a suitable mathematics specialization.

(I might add here, again in timid parentheses, that in a healthy homeschooling culture there would also be a market for the private physics lab, and that this idea of outsourced physics makes a little more vivid my idea of outsourced maths. For a while, experiments can be done at home, by parents who are themselves not too deeply trained. A point is reached, however, at which time is best booked by the homeschooling parents in a lab for some appropriate weekly fee, and at which Junior is best set to work with real vernier calipers, with a real mercury manometer, with a  real interfereometer immobilized on a real stone bench, and so on.)

2. Practicalities of Mathematics Study: Timelogs, Libraries, Filing

I have written it before on this blog, and I write it again today, and I cannot write it too often: (a) to learn anything, we have to work; (b) to be sure we are working, we do best to keep timelogs. 

Here is my timelog for overall maths studies (on the Debian GNU/Linux box which I assembled from components early in 2013, populating its drive from the latest in a  procession of previous machines): 

20071123=00h38z+00h00c->00000145h15z+000796h02c__least sqrs

This extract shows that on 2007-11-23, I was working on least-squares approximations as a means of fitting a curve or line to a set of data points, and that on 2007-11-23 I put in just 0 hours, 38 minutes of "z", or "snoring", work (no calculations; just thinking or reading), and a humiliating 0 hours, 0 minutes of "c", or "calculuations", thereby taking my cumulative total, since this particular record-keeping began (that was back in June of the year 2000), to 145 hours, 15 minutes of "snoring" and 796 hours, 2 minutes of "calculations". This extract additionally shows that by 2016-12-02, my cumulative totals for snoring and calculations stood, respectively, at 1078 hours, 59 minutes and 1856 hours, 29 minutes. 

Alongside my overall maths timelog I find timelogs for a clutch of little projects - notably conic sections, from 2005 August; statistics and probability, from 2006 March; complex variables,  from 2009 April; sequences and series, from 2009 April; elementary Euclidean geometry, from 2009 May; vector calculus for spacecurve kinematics, from 2011 January (this had reached an unsusually high total of 635 hours, 16 minuts by 2013-05-14); theory of measurement and analysis of  laboratory uncertainties, from 2012 May; real analysis in several separate projects (one of them from 2013 May); a self-confessed "nonrigorous" initiative on Jacobi-Green-Stokes-Gauss in vector calculus, from 2014 September; linear algebra, with a view to determinants and tensors, from 2015 January; rigorous Euclidean geometry, from 2016 July (this is what stimulated me to blog on this present server this summer, expounding the virtues of geometer Moise); and topology rudiments, my current interest, from 2016 September.

Here is an illustrative excerpt from the topology log:

20160907=00h54->0000h54__did skim in Munkres, basically chap01
                         __bought Munkres 160.00 CAD used UofT LAVS DEO
20161130=04h11->0114h39__Munkres; did first 00h20 of work on secn17exerc
20161201=03h32->0118h11__Munkres; continued with secn17exerc

My current aim is to do 200 hours of topology or less, and then to switch, in the inevitable flagging of enthusiasm that sets in after the first 100 or 150 hours, to some new topic - this northern-hemisphere winter very likely to spherical geometry, with the help of my now-ordered, and soon-to-arrive, Budapest "Lénárt sphere" ( 


I have not previously blogged on the need for mathematics students to know the libraries in their city. 

The above-mentioned Mrs Serene and Mr Troubled can no doubt find various copies of Leonard Holder, or of other books as good or better, in Toronto library stacks. (I gather from Mrs Physics that Holder aside, one possibility might conceivably be Stanley I. Grossman's  Precalculus with Applications.) 

It turns out, a little surprisingly, that the University of Toronto Gerstein Science Centre on King's College Circle is not a mere subset of the Department of Mathematics library on adjacent St George Street, but actually has some materials absent from the departmental library. 

The various University of Toronto constituent-college libraries (these are proper university institutions, as distinct from mere Community Colleges), such as my own St Michael's, might or might not prove helpful. St Michael's College, at any rate, seems to be overwhelmingly humanities-oriented. 

The Faculty of Engineering library, on or (depending upon how you look at it) just off King's College Circle, might sometimes help. I do seem to recall it being helpful for a topic outside pure mathematics, namely computer architecture - and perhaps also for something in geometry. 

Additionally, some mileage might be had from the Toronto Reference Library. I have been struck to find on those shelves a rigorous, earnest treatment either of elementary Euclidean geometry or of elementary analytical geometry. Further, it is at least suggestive that the Toronto Reference Library has - this is admittedly outside the pure-maths domain - a multi-volume set of Einstein's collected works, and at another point in its capacious stacks a treatment of computer architecture (at the conceptually helpful  level of registers, adder, RAM address bus, and the like). 

Finally, I cannot forget my experience with a collection that one might at first be tempted to write off, the minor-seeming Richmond Hill Public Library. Glancing at a whim over its sparse mathematics holdings, I found Munkres on topology. I was gripped at once, with Munkres emerging from just a half-hour or one-hour perusal as an author of the highest calibre - as a veritable Moise, or a veritable Spivak, of point-set topology. Soon afterwards I learned, from glancing at Web reviews, that Munkres is indeed held in the highest professional regard by essentially everyone. So, just a fortnight later, I bought a copy for myself, with confidence, the painfully high used-copy price notwithstanding. 

The stacks of all the above-mentioned libraries are open to the general public, with reading tables handy, and with the various predictable restrictions on borrowing. Within the University of Toronto system, for instance, the borrower is normally a University of Toronto student or University of Toronto employee or University of Toronto alumnus. I do think, without being quite sure, that the general public can also buy a one-year borrower's card, for some tens of dollars. And I remember that some years ago, a specially low rate applied if one wanted to borrow from a constituent-college library, without seeking entitlement to borrow from every normally-lending library in the big University of Toronto collection of libraries. 

For the greater part of one's trawling through libraries, it is helpful to know a bit about the Library of Congress ("LC") mathematics classification headings. Municipal libraries, such as the Toronto Reference Library and the Richmond Hill Public Library, tend to use the comparatively weak Dewey Decimal System. However, more serious North American libraries, among them most or all of the many dozen University of Toronto normally-lending libraries, favour LC. 

Within the overall domain of knowledge, the LC heading Q is used for science. Under this, as subdivisions, are QA, QB, QC, and QD for mathematics, astronomy, physics, and chemistry,  respectively - with QA therefore the heading relevant to this present blog posting. We find, for instance, the second (the current, year-2000) edition of Munkres's Topology assigned the Library of Congress within-publication call number QA611.M82 2000), This same call number, or something not too unlike it (perhaps QA611.M82 2000X?) may be expected to appear on the spine of any copy of this book in any LC-based library, including even peers of the small (LC-based) library at Our Lady Seat of Wisdom Academy.

Within the broad QA classification, QA611 is, as is to be expected from my Munkres example, the, or at any rate a, topology classification heading. 

Some day I will have to track down, and perhaps even post here, the full set of QA subheadings. At the moment, I can give only a partial picture, reflecting my own library work so far as I have taken it:

  • QA39 is used for (at least some) precalculus, at the level of the above-mentioned Holder and Grossman. The above-mentioned Mr Troubled and Mrs Serene would consequently find it useful to visit one of the University of Toronto libraries - the Gerstein Science Centre is especially convenient, being open into the late evening - and look over the (half-dozen? dozen? three dozen?) books shelved there under QA39. They would be best advised to confer also a few minutes' inspection on the immediately adjacent shelving, for prudence. 
  • QA73.73 is used for (at least some) programming languages.
  • QA76.76 is used for (at least some) computer markup formalisms, such as SGML and its child-or-ward-or-protégé HTML. I believe this classification is for some (not quite logical?) reason used also for at least some aspects of operating systems, including Linux. 
  • QA184 is used for (at least some) linear algebra.
  • QA276 is used for (at least some) statistics-and-probability.
  • QA300, QA303, and QA311 are (at least some of) the classifications used for various topics in calculus - with the low-level, frosh-or-sophomore, nonrigorous stuff finding its way at least in some instances to QA303, and rigorous real analysis in at least some instances to QA300, and rigorous complex analysis in at least some instances to QA331. (A little while ago, I muttered to myself on the seeming illogicality of bracketing nonrigorous books in QA303 between two bins of rigour, QA300 and QA311. But the arrangement does, on closer reflection, make some sense: let us first have some bins, QA300 and QA303 among them, for calculus done entirely on the reals, and let us reserve some later bin(s), such as QA311, for the elaboration off the real line into the richer structure of the Argand plane, which embeds the reals as a special case.) 
  • QA403 is used for (at least some) Fourier methods.
  • QA433 is used for (at least some) tensors.
  • QA453 and QA455 are used for (at least some) Euclidean geometry. Some day I will have to try to find out what the difference is supposed to be. 
  • QA611 houses, as I keep remarking the point of tedium, among other topologists Munkres. 
  • QA612 is used for "differential toplogy", which is supposed to include Spivak's universally dreaded Calculus on Manifolds. (That particular bin is so used even though that particular book might equally be regarded as rigorous real analysis. Perhaps the classifiers have here taken the view that Spivak goes beyond Green-Stokes-Gauss, as in real analysis, to a real-spaces discussion, in a late chapter, of manifolds generally - including those specially hideous objects - the Möbius strip, for instance, and as a surface in four-dimensional space the Klein bottle - which are the non-orientable manifolds. This would be reason enough for putting Spivak into a "differential topology" bin, as opposed to the less imposing "analysis" bin.) 
It can be seen from this partial picture that classification is difficult, combining elements of science and black art. Even the specialists at the actual Library of Congress in Washington, DC, might conceivably, on some occasions, end up making arbitrary or questionable decisions. I well recall a professional librarian at the University of Toronto Department of Astronomy and Astrophysics telling me that in her experience, Washington's LC designation for a newly acquired astrophysics book sometimes needs to be altered before she can admit the book to her catalogue and shelving.

But partial knowledge, and that of a system perhaps not everywhere fully logical, is better than none. 

I think of the library as a dark, brooding place, from which secrets are to be unearthed, as one pulls fifteen books from the stack and inspects each in turn, in quest of some Really Helpful Thing. Perhaps one will find some rather clear and undemanding exposition of 3-by-3 matrices for rotation of three-dimensional rectangular coordinates, with due reference either to Euler angles or to Tait-Bryan angles. Perhaps, again, one will find, upon riffling through those fifteen books cunningly lugged from shelves to reading desk - one of the rules is to take everything from the 1930s seriously, as liable to be deep - some clever little page with a drawing of springs in a three-dimensional array, explaining why classical mechanics needs tensors rather than mere vectors. 

The maths shelving is, in general, akin to a Scotland Yard records room - a point on which I will touch again at the very end of today's posting. 


With at least some grip on the LC headings, one can bring order into the filing of one's polished work.

I have already, in some previous blog posting on Moise's rigorous Euclidean-geometry treatise, remarked on the importance of organizing papers. But I will repeat the central point today - namely, that it is advisable not only to create rough work but to convert the rough work into polished writeups. It is sometimes the case that something which looks okay at the rough-work stage proves incomplete - perhaps as omitting discussion of some obscure degenerate case - at the polished-writeup stage. (In working with topological spaces on a set X, we do have to be able to answer the question, "What if X is the empty set?") Still worse, there is the possibility that something which looks okay on the rough papers will prove fallacious once a polished writeup is attempted. Has one, perhaps, confused some "all" with an "only", some "if...then" with an "only if...then"? Has one, perhaps, forgotten that in general, in topology, "Set A is closed" does not entail "Set A is non-open," but only "The complement of set A is open"?

I find that as keeping timelogs raises morale, so also is morale raised by making the polished notes clear (with even the handwriting tidy), and by then filing the polished notes in a duly polished filing system.

Surely everyone will file polished notes in a system that is formally, from an abstract data-structures standpoint, a tree: in my LC-driven case, all studies in science under the LC "Q", with studies in physics under "QC", but studies in mathematics under "QA"; and then, for those polished notes which happen to relate not only to topology, as QA611, but specifically to the year-2000 edition of Munkres, to folders on whose tab is "QA611.M83.2000", or something close to that (some slight vagaries are possible as one tries to match one's folder tab to the exact usage of one's local university librarians); followed by other indications localizing the papers still further, to particular chapter, and ultimately either to general-workthrough-of-chapter or to author-assigned problems-from-chapter.

Tracking all this on my Debian GNU/Linux box, I find that I have created on the computer  a directory






I then find that within this directory I have recorded, by creating suitably named 0-byte files, the existence on my actual physical workroom shelves of cardboard filing folders (so far) "LBNN____comprehension__chap02..." (for general workthrough of the first half of chapter 2), "LBON____comprehension__chap02..." (for general workthrough of the second half of chapter 2), "QBNN____exercises__chap02..." (for my problem-set answers from the first half of chapter 2), and "QBON____exercises__chap02..." (for my problem-set answers from the second half of chapter 2).

Since all my paper and cardboard is tracked on the Debian GNU/Linux box, and since I have already made corresponding pencil notes in the margin of my copy of Munkres, I can be pretty sure of being able to locate a necessary cardboard folder, and within it the requisite pieces of paper, even some years from now.

At the level of paper, I note that, for instance, I have created a clean-pencil sheet headed in blue ink as "~ /*stud* /Q_* /QA_* /QA00000611_* /QA00000611.M82.2000X* /AANN* /OBNN*" ("private studies/ science/ maths/ topology/ Munkres topology book in its year-2000 edition/ Munres-year-2000-topology-book study initiative computer-logged as 'AANN____studium_of_20160907T160000Z'/ cardboard folder computer-logged as 'OBON__exercses__chap02_secn17ff'"), and with an upper-right-hand-corner blue-ink annotation


That corner annotation indicates that I am on question 5 in the problem set from chapter 2, section 17, as worked in an attempt that started at the UTC time 20161201T2257Z. Additionally, the corner annotation indicates that this is sheet 8 in that particular - rather tediously bulky - sheaf of sheets. The bulk is due to the mildly troublesome nature of the problem. One of Munkres's special virtues is his asking open-ended questions, compelling the student to think like a research mathematician. So Munkres asks only, in an ever-so-innocent way, that the student explore under what "conditions" (he does not write "necessary", and he does not write "sufficient"), in an order topology, the closure of the interval (a,b) succeeds in being (not merely contained-as-subset-within, but actually identical with) the interval [a,b]. I grimly took it that I should through my exploration be so thorough as to supply nontrivial necessary conditions for "actually identical with" and nontrivial sufficient conditions for "actually identical with". Happily, it eventually turned out that I could get a single tidy condition both necessary and sufficient: whether or not the universe of the order topology has a min, and whether or not it has a max, and if so, then whether or not a is that min and whether or not b is that max, in any case the following conjunctive condition is both necessary and sufficient: a has no immediate successor and b has no immediate predecessor.

At the level of filing-folder cardboard, I note that this sheet lives in a folder on whose tab I have written, in my signal blue ink, in the spirit of adequately exact Scotland Yard filing, the following (matching the Debian GNU/Linux log):




3. Four Meditations for Moral Uplift

Discouragement is only a mild occupational hazard in astronomy (a subject which generates its own perpetual excitement, to a still greater extent than the admittedly already exciting subjects of history, languages, and Law do). The hazard looms larger in physics. In mathematics the hazard looms at its largest, mirroring the status of mathematics as the most exacting of disciplines.

I finish today's blog posting by writing out four moral-uplift meditations, for the student's possible encouragement. Here I have very much in mind Mrs Physics, Mrs Serene, and Mr Troubled, all of whom are liable to be reading today's posting at some point, as I sooner or later draw it to their attention.

The first meditation (unlike the other three) is one which I have already to some extent covered elsewhere on this blog, in some previous month:

One is so terribly, tragically, inadequate. One is somewhere high up in the Royal Society, with the letters "F.R.S." after one's name, and with a knighthood in prospect. But oh, how terribly, terribly stupid one is, in comparison with Prof. Albert Einstein.

Or, again: One is so terribly, terribly inadequate. One is way down in the slums of Toronto, sleeping a little east of Jarvis, begging for dollar coins on Bay Street as the black-suited lawyers stride elegantly between commuter train and office tower (or, perhaps, trying to sell trinkets to the lawyers: my nemesis Mr David Bronskill, who defends the David Dunlap Observatory and Park property developer, strides past on his way to Goodmans LLP (, declining the proffered trinket, and yet ever-so-generously chucking a tuna sandwich into the pathetic baseball cap: gee whiz, it would be helpful if Mr Bronskill's team happened to be skipping my blog this week). One is barely able to multiply two three-digit numbers by hand. And oh how terribly, terribly stupid one is, in comparison with the people who can manipulate fractions.

This meditation can be developed first in one comic direction and then in another comic direction, as one reminds oneself first in one way and then in some contrary way that it is pointless to compare one's work against the work of others (however much our so-competitive University of Toronto may encourage those empty comparisons).

As reinforcement for this first meditation, the student can usefully take one of John Mighton's writeups on teaching - perhaps his book The Myth of Ability. John Mighton was one of my classmates in 1991-1992 MAT257 at the University of Toronto. The course was taught by a highly skilled professor, with a highly skilled Polish teaching assistant, from that sinister barrel of torments, from that warm spider consommé, from that steaming bowl of cream-of-toad, from - words all but fail me - that gruesome rite de passage for Norh America's pubescent ambitious scientific acne-suffering teens which is the Spivak Calculus on Manifolds.

I remember Dr Mighton (as he now is) for his optimism and good cheer, amid the torments. And so I am today not at all surprised to find that Dr Mighton gets a biographical note in you-know-what: In (if I recall correctly) The Myth of Ability, Dr Mighton makes exactly the right points - contrary to what our perniciously competitive educational system would have us fear, there is no "barrier of ability" confining mathematics to some narrow elite; anyone willing to put in the effort, and to rise above discouragement, will progress.

This concludes my first meditation.

Here is my second:  We are in a workshop. Before us sit chisels, planes, a lathe, a light hammer, a carpenter's hand drill, a jeweller's hand drill, an awl, and assorted other tools, as suitable for close work. Here we might fashion birch into a nice mitre-jointed box, suitable for holding telescope eyepieces, or for holding geological specimens, or for holding some leather-bound mediaeval manuscript. Our mathematics desk is itself quiet and orderly, like this workshop.

Here is the third: It is the long-anticipated Day in Court (as, perhaps, right now, in the early December of 2016, with the Supreme Court at Westminster hearing arguments on Article Fifty and Brexit). Wigged barristers rustle their papers. All is orderly, all is calm.

The point of this meditation is that mathematics - like Canada's, Estonia's, and Britain's Supreme Courts - is a place for the unhurried, meticulous application of logic.

Here is the fourth: "Come, Watson, come. The game is afoot" - and indeed the brown night fog swirls silently up gas-lit Baker Street. As we stride softly forth to (say) Paddington, with our dark-lantern, magnifier, revolvers, and as a final precaution a few pairs of Bracelets, we know our criminal quarry to be even now approaching the web we have so diligently spun for the benefit of Scotland Yard and Queen Vicky.

The point of this meditation is that the defeat of ignorance is akin to the triumph of justice. Both pursuits presuppose a pitiless and implacable attention to detail. And both pursuits offer, on occasion, amid all the hard slog, a certain sober thrill of the chase.

[This concludes this blog posting.] 

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