Tuesday, 20 December 2016

Toomas Karmo: DDO&P: Open Letter to Advertising Standards Canada, with Municipality and Others

Revision history:

20161220T11702Z/version 1.0.0: Kmo published on this blog, having a few minutes earlier sent the same material in e-mail to Advertising Standards Canada (info@adstandards.com), with cc to himself and to clerks@richmondhill.ca, MPozzebon@metrusdev.com, info@observatoryhill.ca, dbronskill@goodmans.ca, officemayor@richmondhill.ca, vito.spatafora@richmondhill.ca, brenda.hogg@richmondhill.ca, greg@gregberos.com, tom.muench@richmondhill.ca, castro.liu@richmondhill.ca, david.west@richmondhill.ca, karen.cilevitz@richmondhill.ca, godwin.chan@richmondhill.ca, attorneygeneral@ontario.ca, tmcmeekin.mpp.co@liberal.ola.org, bmauro.mpp.co@liberal.ola.org, cballard.mpp.co@liberal.ola.org, premier@ontario.ca, rmoridi.mpp.co@liberal.ola.org, kmcgarry.mpp.co@liberal.ola.org, commissioner@eco.on.ca, david@donnellylaw.ca, anne@donnellylaw.ca, KZarzour@yrmg.com, newsroom@yrmg.com, ptyson@skyandtelescope.com, news@nowtoronto.com, editorial@torontolife.com, Newsroom@globeandmail.ca, city@thestar.ca. 



1. Background: Potentially Misleading Advertising by Subdivision Developer


The Ontario property developer DG Group (http://www.dggroup.ca/; but trading before 2015 April as Metrus Development) in 2008 created a subsidiary, Corsica Development Inc., for the purpose of building a subdivision on part of the former David Dunlap Observatory and Park lands in Richmond Hill, Ontario. Their subdivision was to be called "Observatory Hill". The envisaged subdivision is promoted, although not explicitly under the names of Corsica or DG Group, at http://observatoryhill.ca/ and http://myobservatoryhill.ca/


110 acres of the Observatory Hill project are now confirmed as destination green space in Richmond Hill and will be supported by a $54 million budget. 

In its first phase, the new plan will focus on general maintenance, but subsequent phases will introduce renovations to the David Dunlap Observatory, its beautiful administration building, plus the addition of exciting and interactive spaces like a skating trail, visitor centre, planetarium, tennis courts, and family-friendly picnic areas. /.../ 

I believe the published promotional material to be in the following respects inaccurate or misleading: 

  • The name "Observatory Hill" is attached to an envisaged 110-acre (45-hectare) municipal remnant park, which in fact is differently named, and is separate from (although adjacent to) the envisaged 32-hectare subdivision. 
  • The wording does not make it clear that the "$54 million" designated for the remnant park is public (i.e., taxpayer) money, rather than money put up by the developer. 
  • The wording presents a "planetarium" as a decided-upon feature of the remnant park, whereas the Town of Richmond Hill has indicated in its park-planning documentation only that it will be studying the possible eventual feasibility of building a planetarium. 

2. Possible Corrections


Something like the following would serve the interests of accuracy:


110 acres (45 hectares) of the remnant municipal park, conserved from the pre-2008 188- or 189-acre (77-hectare) David Dunlap Observatory and Park, are now confirmed as destination green space in Richmond Hill, and will be supported by a $54 million budgetary allocation from the Town of Richmond Hill. This conserved remnant park is adjacent to the projected 79-acre (32-hectare) Observatory Hill subdivision. 

In its first phase, the new Town of Richmond Hill remnant-park plan will focus on general maintenance. But subsequent phases will introduce renovations to the David Dunlap Observatory,  its beautiful administration building, plus the addition of exciting and interactive spaces. Current Town park planning envisages a skating trail, a visitor centre, tennis courts, and family-friendly picnic areas, and also envisages a feasibility study for a possible planetarium. 


3. Request for Action from Advertising Standards Canada

I request that Advertising Standards Canada acknowledge this communication from me and indicate what step(s) it is now able to take, working with the Corsica legal team. 

I am proposing to use my blog, http://toomaskarmo.blogspot.ca/, as a vehicle for keeping the public informed of my progress with the various authorities in this case, the most notable of these presently being Advertising Standards Canada. 


4. Request for Action from Town of Richmond Hill

I request that clerks@richmondhill.ca (a) acknowledge this communication from me, (b) confirm for me that the office of Clerks is forwarding my communication to the appropriate officer(s) within the Legal department of the Town, and (c) confirm for me that there will by 2017-03-31 be some followup communication from the Town to me regarding the appraised legality or illegality of Corsica's herein-quoted advertising.

APPENDIX: Disclosure of Personal Interest



(A) I was employed at the David Dunlap Observatory on a pair of concurrent part-time contracts, as telescope operator and as research assistant, from 2006 November to the end of 2008 June or the beginning of 2008 July.

(B) I largely financed, as a now-impoverished private donor, the subdivision-opposing work of the Richmond Hill Naturalists (http://www.rhnaturalists.ca) at the Ontario Municipal Board in 2012, at Divisional Court in 2013, and at the Ontario Municipal Board in 2014. 

(B) In the spring of 2014, I was threatened with legal action by municipal politician Karen Cilevitz - since the autumn of 2014, Town Councillor Cilevitz. Ms Cilevitz has since 2011 been a public supporter of the projected Corsica subdivision.  She and I settled out of court. Our casework is documented at http://www.karen-vs-toomas-blog.ca and http://www.karen-vs-toomas-legaldocs.ca/.

(D) I have blogged on the various aspects of the David Dunlap Observatory and Park heritage-conservation case at http://toomaskarmo.blogspot.ca:


  • 2016-11-28 or 2016-11-29, under heading "DDO&P Sewage-Works Stormwater Facility: Queries for Province and Town, and Suggestions for Residents"
  • 2016-11-25, under heading "DDO&P: Letter to Ontario's Ministry of Municipal Affairs, on OMB Reform" 
  • 2016-11-07 or 2016-11-08, under heading "USA Election, and Government Generally (DDO File as an Indicator)"
  • 2016-11-07 or 2016-11-08, under heading "Theology-of-Civic and DDO (Oak Ridges Moraine Aquifer Cap"
  • 2016-09-26 or 2016-09-27, under heading " DDO&P: Submission to Town Council"
  • 2016-09-19 or 2016-09-18, under heading "DDO&P: (A) Submission to C.O.W.; (B) RASC-TC communication"
  • 2016-09-12 or 2016-09-13, under heading "Minor DDO&P News: (1) Dr R.Chou Now Heads RASC-TC; (2) DND Memo Now Updated"
  • 2016-08-29 or 2016-08-30, under heading "Open Letter re DDO&P DND (Department of National Defence) Implications" 
My posting of 2016-08-29 or 2016-08-30 has a list of my earlier DDO&P-relevant blog postings. 





[End of communication.]

Monday, 19 December 2016

M.P.Montague: "Twenty Minutes of Reality" (1917; with minor 2016 comments from Toomas Karmo)



[This essay first appeared in 1916 or 1917 or so, anonymously, in  The Atlantic Monthly.  Margaret Prescott Montague (1878-1955; https://en.wikipedia.org/wiki/Margaret_Prescott_Montague) then publicly acknowledged herself to be the author, republishing her essay together with some letters from Atlantic Monthly readers as a book entitled Twenty Minutes of Reality: An Experience: With Some Illuminating Letters Concerning It  (New York:  E.P.Dutton & Company, 1917). Her book may nowadays be downloaded in various formats, including PDF, from https://archive.org/details/twentyminutesre02montgoog. The book appears from that website to be nowadays free of copyright encumbrances. - In reproducing Margaret Montague's essay here, I (Kmo) have altered spelling, and in some very minor respects punctuation, to achieve conformity with 2016-era British book-publishing conventions. I otherwise keep the essay intact (the italicized emphases are the author's), checking my own blogger typesetting against a PDF of the book. - I append a few minor comments of my own at the end, in a bolded italic typeface.]  

Revision history:


  • 20161220T0246Z/version 3.0.0: Kmo finished converting his outline(s) to 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, ... . 
  • 20161220T0132Z/version 2.0.0: Kmo added a coarsegrained outline for his envisaged minor comments on Margaret Montague. He hoped to convert this first into a finegrained outline, then into coherent prose, by 20161220T0301Z, in a series of successive uploads.  
  • 20161220T0001Z/version 1.0.0: Kmo uploaded base version (having had time to check his work carefully against a Web-published PDF of the M.P.Montague's book). 


As a child I was afraid of world without end, of life everlasting.  The thought of it used to clutch me at times with a crushing sense of the inevitable, and make me long to run away. But where could one run?  If never-ending life were true, then I was already caught fast in it, and it would never end. Perhaps it had never had a beginning. Life everlasting, eternity, forever and ever: these are tremendous words for even a grown person to face; and for a child - if he grasp their significance at all - they may be hardly short of appalling. The picture that Heaven presented to my mind was of myself, a desperate little atom, dancing in a streak of light around and around and around forever and ever. I do not know what could have suggested such an idea; I only know that I could not think of myself caught there in eternity like a chip in a whirlpool, or say "round again, and round again, and round again" for more than a minute, without hypnotizing myself into a state of sheer terror. Of course, as I grew older I threw off this truly awful conception; yet shorn of its crudeness and looked at with grown-up eyes, there were moments when, much as I believed in, and desired, eternal life, that old feeling of "round again, and round again" would swoop back upon me with all its unutterable weariness, and no state of bliss that I could imagine seemed to me proof forever against boredom. Nevertheless, I still had faith to believe that eternity and enjoyment of life could in some way be squared, though I did not see how it was to be done. I am glad that I had, for I came at last to a time when faith was justified by sight, and it is of that time that I wish to write here. 

If this paper ever chances to be printed, it will be read, I think, by two sets of persons. There will be those who will wonder if I speak of something that is really there, or who will be quite sure that I do not - that I either imagined or made up the whole thing, or else that it was entirely due to the physical condition of convalescence. Others there will be who will believe that I am speaking of the truth that is there, because they, too, have seen it. These last will think that it was not because I was returning to health that I imagined all life as beautiful, but that with the cleared vision that sometimes attends convalescence I "saw into reality", and felt the ecstasy which is always there, but which we are enabled to perceive only on very rare and fleeting occasions.

It is these last for whom I wish to write. If this clearing of the vision is an occasional occurrence of convalescence, then what I saw is of far more value than it would be had my experience been unique.

I do not really know how long the insight lasted. I have said, at a rough guess, twenty minutes. It may have been a little shorter time, it may have been a little longer. But at best it was very transitory.

It happened to me about two years ago, on the day when my bed was first pushed out of doors to the open gallery of the hospital. I was recovering from a surgical operation. I had undergone a certain amount of physical pain, and had suffered for a short time the most acute mental depression which it has ever been my misfortune to encounter. I suppose that this depression was due to physical causes, but at the time it seemed to me that somewhere down there under the anaesthetic, in the black abyss of unconsciousness, I had discovered a terrible secret, and the secret was that there was no God; or if there was one, He was indifferent to all human suffering.

Though I had hardly reëstablished my normal state of faith, still the first acuteness of that depression had faded, and only a scar of fear was left when, several days later, my bed was first wheeled out to the porch.  There other patients took their airing and received their visitors; busy interns and nurses came and went, and one could get a glimpse of the sky, with bare grey branches against it, and of the ground, with here and there a patch of melting snow.

It was an ordinary cloudy March day. I am glad to think that it was.  I am glad to remember that there was nothing extraordinary about the weather, nor any unusualness of setting - no flush of spring or beauty of scenery - to induce what I saw. It was, on the contrary, almost a dingy day. The branches were bare and colourless, and the occasional half-melted piles of snow were a forlorn gray rather than white. Colourless little city sparrows flew and chirped in the trees, while human beings, in no way remarkable, passed along the porch.

There was, however, a wind blowing, and if any outside thing intensified the experience, it was the blowing of that wind. In every other respect it was an ordinary commonplace day. Yet here, in this everyday setting, and entirely unexpectedly (for I had never dreamed of such a thing), my eyes were opened, and for the first time in all my life I caught a glimpse of the ecstatic beauty of reality.

I cannot now recall whether the revelation came suddenly or gradually; I only remember finding myself in the very midst of those wonderful moments, beholding life for the first time in all its young intoxication of loveliness, in its unspeakable joy, beauty, and importance. I cannot say exactly what the mysterious change was. I saw no new thing, but I saw all the usual things in a miraculous new light - in what I believe is their true light. I saw for the first time how wildly beautiful and joyous, beyond any words of mine to describe, is the whole of life.  Every human being moving across that porch, every sparrow that flew, every branch tossing in the wind, was caught in and was a part of the whole mad ecstasy of loveliness, of joy, of importance, of intoxication of life.

It was not that for a few keyed-up moments I imagined all existence as beautiful, but that my inner vision was cleared to the truth so that I saw the actual loveliness which is always there, but which we so rarely perceive; and I knew that every man, woman, bird, and tree, every living thing before me, was extravagantly beautiful, and extravagantly important. And, as I beheld, my heart melted out of me in a rapture of love and delight. A nurse was walking past; the wind caught a strand of her hair and blew it out in a momentary gleam of sunshine, and never in my life before had I seen how beautiful beyond all belief is a woman's hair.  Nor had I ever guessed how marvellous it is for a human being to walk. As for the interns in their white suits, I had never realized before the whiteness of white linen; but much more than that, I had never so much as dreamed of the mad beauty of young manhood. A little sparrow chirped and flew to a nearby branch, and I honestly believe that only "the morning stars singing together, and the sons of God shouting for joy" can in the least express the ecstasy of a bird's flight. I cannot express it, but I have seen it.

Once out of all the gray days of my life I have looked into the heart of reality; I have witnessed the truth; I have seen life as it really is - ravishingly, ecstatically, madly beautiful, and filled to overflowing with a wild joy, and a value unspeakable. For those glorified moments I was in love with every living thing before me - the trees in the wind, the little birds flying, the nurses, the interns, the people who came and went. There was nothing that was alive that was not a miracle. Just to be alive was in itself a miracle. My very soul flowed out of me in a great joy.  

No one can be as happy as I was and not have it show in some way. A stranger passing paused by my bed and said, "What are you lying here all alone looking so happy about?" I made some inadequate response as to the pleasure of being out-of-doors and of getting well. How could I explain all the beauty that I was seeing? How could I say that the grey curtain of unreality had swirled away and that I was seeing into the heart of life? It was not an experience for words. It was an emotion, a rapture of the heart.

Besides all the joy and beauty and that curious sense of importance, there was a wonderful feeling of rhythm as well, only it was somehow just beyond the grasp of my mind. I heard no music, yet there was an exquisite sense of time, as though all life went by to a vast, unseen melody.  Everything that moved wove out a little thread of rhythm in this tremendous whole. When a bird flew, it did so because somewhere a note had been struck for it to fly on; or else its flying struck the note; or else again the great Will that is Melody willed that it should fly. When people walked, somewhere they beat out a bit of rhythm that was in harmony with the whole great theme.

Then, the extraordinary importance of everything! Every living creature was intensely alive and intensely beautiful, but it was as well of a marvellous value. Whether this value was in itself or a part of the whole, I could not see; but it seemed as though before my very eyes I actually beheld the truth of Christ's saying that not even a sparrow falls to the ground without the knowledge of the Father in Heaven. Yet what the importance was, I did not grasp.  If my heart could have seen just a little further I should have understood. Even now the tips of my thoughts are forever on the verge of grasping it, forever just missing it. I have a curious half-feeling that somewhere, deep inside of myself, I know very well what this importance is, and have always known; but I cannot get it from the depth of myself into my mind, and thence into words. But whatever it is, the importance seemed to be nearer to beauty and joy than to an anxious morality. I had a feeling that it was in some way different from the importance I had usually attached to life.

It was perhaps as though that great value in every living thing was not so much here and now in ourselves as somewhere else. There is a great significance in every created thing, but the significance is beyond our present grasp. I do not know what it is; I only know that it is there, and that all life is far more valuable than we ever dream of its being. Perhaps the following quotation from Milton may be what I was conscious of:-

         What if earth 

Be but the shadow of Heaven, and things therein 

Each to each other like, more than on earth is thought. 

What if here we are only symbols of ourselves, and our real being is somewhere else, - perhaps in the heart of God? Certainly that unspeakable importance had to do with our relationship to the great Whole ; but what the relationship was, I could not tell.  Was it a relationship of love toward us, or only the delight in creation?  But it is hardly likely that a glimpse of a cold Creator could have filled me with such an extravagant joy, or so melted the heart within me. For those fleeting, lovely moments I did indeed, and in truth, love my neighbour as myself. Nay, more: of myself I was hardly conscious, while with my neighbour in every form, from wind-tossed branches and little sparrows flying, up to human beings, I was madly in love. Is it likely that I could have experienced such love if there were not some such emotion at the heart of Reality? If I did not actually see it, it was not that it was not there, but that I did not see quite far enough.

Perhaps this was because I was still somewhat in the grip of that black doubt which I had experienced, and of which I have spoken. I think it was owing to this doubt also that afterwards I had a certain feeling of distrust. I was afraid that all that beauty might be an uncaring joy. As if, though we were indeed intensely important in some unguessed way to the great Reality, our own small individual sorrows were perhaps not of much moment. I am not sure that I actually had this feeling, as it is very difficult, after the lapse of almost two years, to recapture in memory all the emotions of so fleeting and so unusual an experience. If I did, however, I comfort myself, as I have said, with the thought of the intense joy that I experienced. The vision of an uncaring Reality would hardly have melted me to such happiness. That the Creator is a loving Creator I believe with all my heart; but this is belief, not sight. What I saw that day was an unspeakable joy and loveliness, and a value to all life beyond anything that we have knowledge of; while in myself I knew a wilder happiness than I have ever before or since experienced.

Moreover, though there was nothing exactly religious in what I saw, the accounts given by people who have passed through religious conversion or illumination come nearer to describing my emotions than anything else that I have come across.

These testimonies I read almost a year after my hospital episode. I came upon them by chance, and was astonished to find that they were describing very much what I had passed through. I think if I had had nothing to match them in my own experience I should almost certainly have felt sure that these people, because of the emotional excitement within themselves, imagined all the beauties that they described. Now I believe that they are describing what is actually there. Nor are poets making up - as the average mind believes, and as I think I always believed - the extravagant beauty of which they sing. They are telling us of the truth that is there, and which they are occasionally enabled to see.

Here are some of the testimonies offered by people who have experienced illumination in one form or another.

"Natural objects were glorified," one person affirms. "My spiritual vision was so clarified that I saw beauty in every natural object in the universe." Another says, "When I went into the field to work, the glory of God appeared in all his visible creation. I well remember we reaped oats, and how every straw and beard of the oats seaned, as it were, arrayed in a kind of rainbow glory, or to glow, if I may so express it, in the glory of God." The father of Rabindranath Tagore thus describes his illumination: "I felt a serenity and joy which I had never experienced before ... the joy I felt ... that day overflowed my soul.... I could not sleep that night. The reason of my sleeplessness was the ecstasy of soul; as if moonlight had spread itself over my mind for the whole of that night." And when Tagore speaks of his own illumination he says, "It was morning; I was watching the sunrise in Free School Street. A veil was suddenly drawn and everything I saw became luminous. The whole scene was one perfect music; one marvellous rhythm."  (Note his sense of rhythm, of which I was also conscious.) "The houses in the street, the children playing, all seemed part of one luminous whole - inexpressibly glorified." (Perhaps the significance of that tremendous importance which I felt, but failed to grasp, was that we are all parts of a wonderful whole.) "I was full of gladness, full of love for every tiniest thing."

And this was what - in a smaller degree - I, too, saw for those fleeting moments out there upon the hospital porch. Mine was, I think, a sort of accidental clearing of the vision by the rebirth of returning health. I believe that a good many people have experienced the same thing during convalescence. Perhaps this is the way in which we should all view life if we were born into it grown up. As it is, when we first arrive we are so engaged in the tremendous business of cutting teeth, saying words, and taking steps, that we have no time for, and little consciousness of, outside wonders; and by the time we have the leisure for admiration life has lost for us its first freshness. Convalescence is a sort of grown-up rebirth, enabling us to see life with a fresh eye.

Doubtless almost any intense emotion may open our "inward eye" to the beauty of reality. Falling in love appears to do it for some people. The beauties of nature or the exhilaration of artistic creation does it for others.  Probably any high experience may momentarily stretch our souls up on tiptoe, so that we catch a glimpse of that marvellous beauty which is always there, but which we are not often tall enough to perceive.

Emerson says, "We are immersed in beauty, but our eyes have no clear vision." I believe that religious conversion more often clears the eyes to this beauty of truth than any other experience; and it is possible that had I not still been somewhat under that black cloud of doubt, I should have seen further than I did. Yet what I did see was very good indeed.

The following quotation from Canon Inge may not be entirely out of place in this connection : "Incidentally I may say that the peculiar happiness which accompanies every glimpse of insight into truth and reality, whether in the scientific, aesthetic, or emotional sphere, seems to me to have a greater apologetic value than has been generally recognized. It is the clearest possible indication that the truth is for us the good, and forms the ground of a reasonable faith that all things, if we could see them as they are, would be found to work together for good to those who love God."

In what I saw there was nothing seemingly of an ethical nature. There were no new rules of conduct revealed by those twenty minutes.  Indeed, it seemed as though beauty and joy were more at the heart of Reality than an over-anxious morality. It was a little as though (to transpose the quotation),

I had slept and dreamed that life was duty,

But waked to find that life was beauty. 

Perhaps at such times of illumination there is no need to worry over sin, for one is so transported by the beauty of humanity, and so poured out in love toward every human being, that sin becomes almost impossible.

Perhaps duty may merely point the way. When one arrives at one's destination it would be absurd to go back and reconsult the guidepost. Blindness of heart may be the real sin, and if we could only purify our hearts to behold the beauty that is all about us, sin would vanish away. When Christ says, "Seek ye the Kingdom of God; and all these things shall be added unto you," He may mean by "all these things" spiritual virtues even more than things temporal, such as what we shall eat, and wherewithal we shall be clothed. It may be that He stood forever conscious of a transcendent beauty, and joy, and love, and that what grieved Him most was mankind's inability to behold what was there before their very eyes.

Perhaps, too, this may be the great difference between the saints and the Puritans. Both are agreed that goodness is the means to the end, but the saints have passed on to the end and entered into the realization, and are happy. (One of the most endearing attributes of saints of a certain type was - or rather is, for one refuses to believe that saints are all of the past - their childlike gaiety, which can proceed only from a happy and trustful heart.) The Puritan, on the other hand, has stuck fast in the means - is still worrying over the guideposts, and is distrustful and over-anxious.

It is like walking and dancing.  One could never dance unless he had first learned to walk, or continue to dance unless walking were always possible; yet if one is too intent upon the fact of walking, dancing becomes impossible. The Puritan walks in a worried morality; the saint dances in the vision of God's love; and doubtless both are right dear in the sight of the Lord, but the saint is the happiest.

Father Tyrrell says, "For Jesus the moral is not the highest life, but its condition."

Some may object that I preach a dangerous doctrine; others, that I am trying to whip a mad moment of Pagan beauty into line with Christian thought. Possibly I am; yet I am trying not to do the one or the other.  I am merely wondering, and endeavouring to get at the truth of something that I saw.

And all the beauty is forever there before us, forever piping to us, and we are forever failing to dance. We could not help but dance if we could see things as they really are. Then we should kiss both hands to Fate and fling our bodies, hearts, minds, and souls into life with a glorious abandonment, an extravagant, delighted loyalty, knowing that our wildest enthusiasm cannot more than brush the hem of the real beauty and joy and wonder that is always there.  

This is how, for me, all fear of eternity has been wiped away. I have had a little taste of bliss, and if Heaven can offer this, no eternity will be too long to enjoy the miracle of existence. But that was not the greatest thing that those twenty minutes revealed, and that did most to end all dread of life everlasting.  The great thing was the realization that weariness, and boredom, and questions as to the use of it all, belong entirely to unreality. When once we wake to Reality - whether we do so here or have to wait for the next life for it, - we shall never be bored, for in Reality there is no such thing.

Chesterton has pointed out the power for endless enjoyment of the same thing which most children possess, and suggested that this is a Godlike capacity; that perhaps to God His creation always presents itself with a freshness of delight; that perhaps the rising of the sun this morning was for Him the same ecstatic event that it was upon the first day of its creation. I think it was the truth of this suggestion that I perceived in those twenty minutes of cleared vision, and realized that in the youth of eternity we shall recapture that God-like and child-like attribute which the old age and unreality of Time have temporarily snatched from us.

No; I shall have no more fear of eternity. And even if there were no other life, this life here and now, if we could but open our dull eyes to see it in its truth, is lovely enough to require no far-off Heaven for its justification. Heaven, in all its springtide of beauty, is here and now, before our very eyes, surging up to our very feet, lapping against our hearts; but we, alas, know not how to let it in!

Once again, when I was almost recovered, I had another fleeting visitation of this extreme beauty. A friend came into my room dressed for the opera. I had seen her thus a great number of times before, but for a moment I saw her clothed in all that wild beauty of Reality, and, as before, my heart melted with joy at the sight. But this second occasion was even more transitory than the first, and since then I have had no return.  Tagore's illumination, he says, lasted for seven or eight days and Jacob Boehme knew a "Sabbath calm" of the soul that lasted for seven days, during which he was, as it were, inwardly surrounded by a divine light.  "The triumph that was then in my soul," he says, "I can neither tell nor describe; I can only liken it to a resurrection from the dead."

And this miraculous time was with him for a whole week, while I have only tasted it for those few short minutes! But he was a saint, and had really ascended to the holy hill of the Lord through clean hands and a pure heart, while I was swept there momentarily, and, as it were, by accident, through the rebirth of returning health. But when the inspired ones testify to a great joy and a great beauty I, too, can cry, "Yes, I have seen it also! Yes, O Beauty, O Reality, O Mad Joy! I, too, have seen you face to face!" And though I have never again touched the fullness of that ecstatic vision, I know all created things to be of a beauty and value unspeakable, and I shall not fail to pay homage to all the loveliness with which existence overflows.  Nor shall I fear to accord to all of life's experiences, whether sad or gay, as high, as extravagant, and as undismayed a tribute of enthusiasm as I am capable of.

Perhaps some day I shall meet it face to face again. Again the grey veil of unreality will be swirled aside; once more I shall see into Reality. Sometimes still, when the wind is blowing through trees or flowers, I have an eerie sense that I am almost in touch with it. The veil was very thin in my garden one day last summer. The wind was blowing there, and I knew that all that beauty and wild young ecstasy at the heart of life was rioting with it through the tossing larkspurs and rose-pink Canterbury bells, and bowing with the foxgloves; only I just could not see it. But it is there - it is always there - and some day I shall meet it again.  The vision will clear, the inner eye open, and again all that mad joy will be upon me. Some day - not yet perhaps - but some day!

****

The sequence of events is interesting, in suggesting a careful observer. First comes Montague's experience of eternity. Then comes her realization, from her reading "almost a year later", that others have had experiences resembling hers. Finally, about two years on, comes her publication - first anonymously, in an eminent literary periodical, and then in her duly acknowledged book. 

The reference to "cleared vision" recalls a common experience in mathematics study. When we return to some problem after an hour's break, the mind moves with a sudden easiness, rendering possible, and even pleasant, what had previously seemed out of reach.

The reference to "rhythm" recalls the timeless, and yet rhythmic, aesthetic of mathematics, as in the intricacies of the circular functions (including Fourier series), or the intricacies of transfinite ordinals, or the intricacies of topology. One does, however, think also of the actual intricacies of the physical world, in which mathematics becomes so-to-speak incarnate - as when, for instance, a violin string is observed to vibrate, or the Sun's outermost opaque layer to pulsate, in multiple, arithmetically intricate, modes. We have a similar experience of incarnation when we find electrons to be standing waves in Schrödinger's formalism, akin to the waves on a violin string. 

Specially helpful are Montague's remarks on the importance of physical things (she refers, for instance, to human hair, to white linen, and to sparrows): "I know very well what this importance is, and have always known; but I cannot get it from the depth of myself into my mind, and thence into words."

My guess is that there is an importance to things which is somehow simple, and is in a sense already known to us, even though we cannot yet raise it out of our depths to the level of our discursively reasoning minds. Our difficulty calls to mind a dictum from Plotinus: "As speech is the echo of the thought in the Soul, so thought in the Soul is an echo from elsewhere: that is to say, as the uttered thought is an image of the soul-thought, so the soul-thought images a thought above itself /.../" 

In general, I like to think of eternity on the following analogy: You die, and two seconds after you die you see Marilyn Monroe walking arm in arm with Pope John XXIII, and Pope John says in bad English, "It's a nice-a place-a we gotta here," and Ms Monroe (chewing on her bubble gum) says, "We was worried about you, Honey." My point here is that there is some kind of surprise - some kind of unexpected thing - in store, which when encountered will seem at first shocking, but on closer consideration will prove exceedingly natural, logical, and simple. 

I do respectfully dissent from Margaret Montague's tentatively offered suggestion that "here we are only symbols of ourselves".  I think of the familiar physical world as fully real and as acquiring its theological worth from the fact of its reality. Admittedly, I have no argument to offer in defence of this dissenting opinion, beyond the argument that only the real can possess, in Margaret Montague's arresting language, "importance". 

Margaret Montague's passage regarding the garden is particularly fine: "The veil was very thin in my garden one day last summer. The wind was blowing there /.../"

Have a very blessed Christmas, everyone! 

[This is the end of the present blog posting. Although there might be something in the next few days from my desk on the perennial problem of David Dunlap Observatory and Park conservation, I think that I shall omit to do any blogging in the normally scheduled four-hour window of UTC=20161227T0001Z/20161227T0401Z, and that I shall return to the normal blogging to the normal blogging routine a fortnight from now - that is to say, in the four-hour window of UTC=20170103T0001Z/20170103T0004Z.] 


Toomas Karmo: Christmas preparations nearly complete

A screenshot from one of my four Debian GNU/Linux desktops. Anticlockwise, from top left: operations clocks (as always, in green for the local civil time, and in red for UTC); a closeup view of my big mountain of Christmas presents, awaiting distribution later this week; a more remote view of the same, showing the minimalist, Paris-or-Manhattan, nominal Christmas tree, or rather Christmas shrub, which until Epiphany decorates my snug book-lined parlour; a shot of flamed plum pudding, from http://www.goodfoodrevolution.com/young-blood-sommelier-nathan-nate-morrell/; an excerpt from Dickens's description of plum pudding in the Cratchit family, in a "glass teleteype", or Debian GNU/Linux /usr/bin/xterm window; the Cratchits as realized in a stage play, from http://www.lasplash.com/publish/Entertainment/cat_index_chicago_performances/the-christmas-carol-at-goodman-theatre-review.php; the David Dunlap Observatory 1.88-metre telescope dome, from back when life was normal, in the sense that astrophysics research was proceeding at the spectrograph (from some winter in perhaps the 1980s or 1990s). As with all my top-of-blog-entry images, an enlarged view can in most or all graphic Web browsers be had by left-clicking. 



Revision history:
  • 20161220T0122Z/version 2.0.0: Kmo added caption under image, plus a few paragraphs of text also under the caption. Kmo reserved the right to make minor, nonsubstantive, purely cosmetic, tweaks, as here-undocumented versions 2.0.1, 2.0.2, 2.0.3, ... . 
  • 20161220T1902Zapprox/version 1.0.0: Kmo uploaded image without significant accompanying text. 

The winter scene from the David Dunlap Observatory, just beside the operations clocks in this image, recalls our last DDO staff Christmas party, held in a Richmond Hill Hungarian restaurant in the December of 2007. For our bittersweet occasion, I composed a variant on Irving Berlin, and the twenty or thirty of us around that long table sang it. 

I will have to reproduce the words now from memory. 

The phrase "guiding star" is a reference less to the Magi than to an aspect of operations in the 1.88-metre telescope dome: it is necessary to "guide" on a star of interest, continually tweaking the motor controls in the warmroom, to ensure that starlight at all times proceeds correctly through the (in our operations, generally 303-micron-wide) spectrograph slit: 

I'm dreaming of the stars this Christmas

Just like the ones we've come to know, 

Where the treetops glisten, and children listen

To hear wildlife in our snow. 

I'm dreaming of the stars this Christmas

With every Christmas card I write: 

May your days be merry and bright, 

And may you find your guiding star tonight.  

****

What can possibly be diuvulged about Christmas preparations? This year I have the good fortune to be shopping for a rather large crowd - essentially (leaving aside a couple of people in peripheral positions) four adults, plus four children aged 5-or-so, 8-or-so, 9, and 11.

Of the four children, I deem at least two to be "Corresponding Members" of a "Section" in that  figment of my overheated imagination which is the Nikolai Ivanovitch Lobachevsky Institute of Socialist Mathematics. As good luck would have it, these two children not only live under dark skies but also have have ready access to their late great-grandfather's 10-centimetre refractor. 

Much support is needed. I have accordingly obtained for our "Section" a second-hand copy of the Dickinson-Dyer Backyard Astronomer's Guide, adding to this  my spare copy of the 2017 Royal Astronomical Society of Canada Observer's Handbook. (Being a contributing author, I get sent a copy over and above the copy that I receive in the normal course of events qua rank-and-file RASC member.) 

This leaves, however, the problem of detailed star maps, left pretty much unaddressed both by Dickinson-Dyer and by the Handbook. We need not only to learn the constellations, but within the constellations to do as much as we can to learn off by heart the Bayer designations of stars (memorizing, for instance, the sequence "zeta-epsilon-delta" for the stars in Orion's belt, appearing in the northern hemisphere from left to right, and "theta-beta-alpha-gamma-delta-epsilon-iota" for the stars in Corona Borealis: it is appropriate to mumble to oneself, on inspecting these two respective things, "zed" and "th-BAG-dei"). 

Perhaps the best tool for learning Bayer designations is the sequence of naked-eye sky charts (with constellations duly Bayer-labelled, and with unusually and exceptionally intelligent connecting lines added, to jog the visual memory in just the right way) which appeared in Sky and Telescope up to perhaps around 1995. These charts were the work of planetarium lecturer George Lovi. As good luck would have it, my book-lined parlour has among its treasures a couple of document cases with old numbers of Sky and Telescope.

My plan is accordingly to complete Christmas preparations by taking appropriate photocopies and putting them into a special bundle for the Lobachevsky Institute. The slip of calligraphy-grade paper visible on top of my Sky and Telescope number in my photos is already inscribed, in anticipation of tomorrow's work at the copy shop, in part as follows: In support of astronomical operations at the Nikolai Ivanovitch Lobachevsky Institute of Socialist Mathematics, Ottawa Valley Section, with particular reference to the concrete operational requirements of Agents E and R

Perhaps it will later be possible to chronicle on this blog the various kinds of progress being made by the Ottawa Valley Section. 

[This is the end of the blog posting.] 




Monday, 12 December 2016

Toomas Karmo: Resisting One's Depression: The Example of Wartime Britain

A screenshot from one of my four Debian GNU/Linux desktops. Anticlockwise from upper right: my own operations clocks, disciplined about once in 24 hours under Network Time Protocol by /usr/sbin/ntpdate, interrogating pool.ntp.org (my local software-clock drift is sometimes mere tens of milliseconds over 24 hours, sometimes a few hundred milliseconds; as always, I use green for displaying my local civil time, red for Universal Coordinated Time); the 1940 operations board at RAF Uxbridge, showing the status lamps to which Churchill refers in his here-quoted memoir (what is shown is surely historically accurate, and yet does not fit perfectly with the details that Churchill supplies: the image is possibly from  http://www.airshowspresent.com/raf-uxbridge-battle-of-britain-bunker.html); the 1940 plotting table at RAF Uxbridge, also mentioned by Churchill (and possibly likewise from http://www.airshowspresent.com/raf-uxbridge-battle-of-britain-bunker.html); prop and engine of a museum Spitfire (I am sorry not to know the provenance of the image I have herewith lifted off the Web); a museum radar transmitter, such as was used in 1940 (the image is possibly from http://www.broadlandmemories.co.uk/blog/2012/05/neatishead-air-defence-radar-museum/); the wartime BBC "Bush House" control room, from which I imagine were coordinated broadcasts of Vera Lynn and the like (I believe the image is from 1943, and that it comes from http://www.bbc.com/news/magazine-27340358); and a Debian GNU/Linux xterm, showing a judiciously selected excerpt from my private flat-ASCII study notes. - As with my other images on this blog, an enlargement can be viewed by left-clicking, I believe in most or all mainstream Web browsers.


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: Kmo, while slipping on his schedule, nevertheless  had time to do a complete  and (within the framework of the version 4.1.1, 4.1.2, 4.1.3, ... process) reasonably polished job.





Revision history:



  • 20161216t0156Z/version 4.1.0: Kmo added to his discussion of Finland a short reference to the problems of Karelia and Viipuri. He also slightly clarified his assessment of Julius Caesar (making it now clear that it is as a prose stylist, not necessarily as a politician, that Caesar at all times attracts universal admiration.)  - He reserved the right to make minor, cosmetic, nonsubstantive tweaks, over the coming 48 hours, as here-undocumented versions 4.1.1, 4.1.2, 4.1.3, ... . 
  • 20161214T1743Z/version 4.0.0: Kmo adjusted his praise of Churchill (he had previously generated a misleading impression of unqualified enthusiasm), and added a reference to the Battle appraisal of circa-year-2000 analyst Bungay. He reserved the right to make minor, cosmetic, nonsubstantive tweaks, over the coming 48 hours, as here-undocumented versions 4.0.1, 4.0.2, 4.0.3, ... . 
  • 20161213T1714Z/version 3.2.0: Kmo added bibliographic details, as best he could, for the materials incorporated in his image, and repaired a very bad typo ("do not know" mistyped as "do now know", with reference to member "ABC" of the Reich forces), and deleted a questionable reference to a firestorm in Nagasaki (it seems that combustion in Nagasaki was less catastrophic than in Hiroshima), and added a Wikipedia link for Lord Haw-Haw. He also made various minor, cosmetic, nonsubstantive tweaks. He reserved the right to make further such tweaks over the coming 48 hours, as here-undocumented versions 3.2.1, 3.2.2, 3.2.3, ... .  
  • 20161213T0508Z/version 3.1.0: Kmo added an image. He reserved the right to make minor, cosmetic, nonsubstantive tweaks, over the coming 48 hours, as here-undocumented versions 3.1.1, 3.1.2, 3.1.3, ... . 
  • 20161213T0450Z/version 3.0.0: Kmo finished converting his point-form outline into coherent prose. He now prepared for uploading an image, and for checking punctuation (and other such cosmetic, non-substantive, points) in his prose. 
  • 20161213T0253Z/version 2.0.0: Kmo uploaded a now fully polished point-form outline. 
  • 20161213T0001Z/version 1.0.0: Kmo had time only to upload a sketchy  point-form outline. He hoped to convert this into first a more polished outlines, and then into coherent prose,  in a succession of uploads, finishing at some point in the next 4 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 Toomas.Karmo@gmail.com.]



1. Background for the Battle of Britain



When I am depressed (as  has been the case over the last few days), I am liable to think of our probable bleak future, decades and centuries hence. As I imagine it, World War 3 never does come. What cannot, however, be dodged (I have pointed this out in previous writing on this blog) is a steep social decline - possibly a Dark Age - as resource depletion and climate change erode our economic foundations. 

****

At such times of recurrent depression, I am liable not only to look forward in gloom, but also to look in gloom in the other direction. I look back to the experiences of my parents, aunts, and uncles, when the 1939-1945 Hitler war and its 1946-1991 aftermath dealt blow upon blow on half of Europe, in a situation which seemed permanent until some little distance into the surprising General Secretaryship of one of humanity's benefactors, M.S.Gorbachev. 

Start north of the Article Circle, at the northern extremity of the Gulf of Bothnia. Trace a (somewhat undulating) curve southward, down the Gulf into the more open Baltic waters, passing between the Estonian islands of Saaremaa and Hiiumaa - they are to lie to the east of this line - and the Swedish island of Gotland, to the west of the line. After Gotland, bear a little more to the west, hitting the mainland coast just east of Jutland. Then head south across the Continent, setting a  course for the Adriatic. 

On the one side of this (somewhat undulating) line is that part of Europe which made an essentially full recovery in the generation following 1945. On the other is that portion in which the spiritual, political, emotional, and material recovery, from 1945 right up to the Gorbachev era, was only partial. 

Three qualifications are, admittedly, necessary. 

Greece and Austria, despite lying so dangerously far to the east, escaped incorporation into the Soviet sphere. Perhaps, for all I know, they achieved their respective escapes rather narrowly, through some combination of statesmanship and blind luck. Additionally, and thirdly, Finland, despite lying so dangerously far to the east, was Sovietized only in its postwar foreign policy, on the domestic front retaining its laws (including its civil liberties) and its parliamentary institutions (except that the culturally significant Karelia district, and with it the economically significant town of Viipuri, got swallowed by the USSR). In Finland's complex case, then, neither of the two labels "partial recovery" and "full recovery" seems entirely apt. 

Estonia, like so many of the larger nations, should be considered to have made, from 1945 through 1991, a mere partial recovery - indeed, in my own bleakly realpolitisch assessment, to have made a recovery so far from full as to considered in and of itself, without comparison against the still harsher war years, disastrous. 

Within my own immediate circle of Estonian friends and Estonian family, I ponder "ABC". 

"ABC" had in 1939 tried in vain to join the British forces, being temporarily in Switzerland rather than back home in Estonia, and consequently applying through the British diplomatic authorities in Switzerland. Alas, the Chamberlain government seems to have not at that point been set up to receive military volunteers from that particular corner of Mitteleuropa. ABC had later, upon returning to Estonia, thought it right to resist the Red Army by fighting in the Reich forces. (I for my part do not know quite what to make of the wisdom or folly of ABC's ultimate choice.) 

VE Day found ABC in good standing with the Danish Resistance, but with little additional cause for celebration. He marked, at any rate, the change in European affairs by flushing his Reich medals down a Copenhagen toilet. 

I ponder also the lady whose letter, reporting to friends or family in (I think) Estonia her deportation to Siberia, is reproduced in someone's Estonian-language memoirs. I recall just the general gist of her Estonian text. And of course tonight I have to put everything into English: Our trip was wonderful. But my saucepan broke. I also buried my baby in the railway ditch. It is terrific to travel. The weather has been great. 

What can be made of so strange a letter? Perhaps we have here a hard-headed woman who has to get the news back to Estonia, and so deliberately adopts the tone most likely to facilitate the safe passage of her sheet of paper past NKVD censorship desks in the labyrinthine Soviet postal system. Perhaps, on the other hand, her letter is just what it purports to be - namely, the letter of a person who, under pressure of cold and malnutrition, and of fear, and above all of grief for her child's death during those interminable days or weeks in her crowded, slowly advancing cattle car, has gone insane. 

****

As a counterweight to my depressed reflections, I ponder Britain's successful wartime resistance. In the war as Britain experienced it, things went dim without going black. Although there were lots of bombs, there were no firestorms after the pattern of Dresden and Hamburg in the European theatre, or of Tokyo (let alone of Hiroshima) in the Pacific. Food was generally available, to a point at which the British 1945 national health statistics were said to improve on the corresponding statistics from 1939.

The tenor of those dark-and-yet-not-black British times is captured by modern comedians Armstrong and Miller, in a YouTube upload of 2010-12-08 by YouTube user "discodenys", under title "The Armstrong and Miller Show - Hitler sketch". In my particular corner of the Internet, the URL for the Armstrong-and-Miller material is https://www.youtube.com/watch?v=MetBQSkDUoA. But a different URL is liable to be needed in at least some parts of the world, remote from my screen here in Canada.

I do have two small notes of caution on Armstrong-and-Miller. (a) Their comedy routine is ribald, to the point of being only marginally suitable for adolescent and pre-adolescent viewers. (b) Their routine, although running on YouTube for 4 minutes, 16 seconds, nevertheless needs cutting off at 3:43 or 3:45 or so, before it slides (as comedy routines unfortunately tend to) into a pointless, cheap, and degrading vulgarity.

Armstrong and Miller present music of a sort. Strictly speaking, they suggest, with just a few bars and a few words, a handful of songs, and they sing in historical accuracy the complete first stanza of one song from the actual war - this one being, I presume, the most raunchy to emerge in actual semi-polite British wartime society.

We may supplement Armstrong and Miller with the immortal, and still living, and never ribald, Vera Lynn - above all, with her "We'll Meet Again."

We might add also a few other selections, here resisting the temptation to wallow in the Lale Andersen or Marlene Dietrich (or Suzy Solidor, or 1990s-Milva) "Lili Marlene". One other that comes especially to mind is a signature number from Gracie Fields, "Wish Me Luck (As You Wave Me Goodbye)".

One has also a special fondness for the rather plentiful YouTube uploads of the BBC wartime "Music While You Work", with their perky suggestion of Glen Miller and the Big Band.

In the domain of British history as a record of political and military events, we may look past the dark and shameful - past Bomber Harris, notably; and past the appeasement of Stalin at the Yalta Conference, under the nose of a Prime Minister who had in his previous political incarnation denounced Chamberlain's appeasement of Hitler - and ponder Bletchley Park.

Or we can, with equal benefit, ponder a thing on which I specially direct my blogosphere searchlight tonight, the Battle of Britain.


2. The Battle of Britain in its Wider Strategic and Diplomatic Context


It is sometimes suggested that Britain faced invasion in the 1940 northern-hemisphere autumn, through the Reich's projected "Operation Sealion". The suggestion is implausible.

Crossing the Channel is unlike crossing the Rhine. In particular, the (quite wide) Channel would have received a (quite competent) defence in 1940 from a (quite desperate) Royal Navy, no matter how crushing the Luftwaffe's eventual air supremacy might have been. So while we cannot call "Sealion" an impossibility, we cannot regard it as much better than a gesture. The envisaged operation was less a bold, Napoleonic gamble than a public-relations gambit, conceived by a propagandistic politician. The actual stakes were not military but political, with "Sealion" part of the Reichsführer's ramped-up rhetoric.

This is the same Reichsführer as once got photographed in a thoroughly comic sequence of twenty or so separate poses, gesticulating with hands held low and hands held high, now with face stern and now with face less stern, practicing and practicing, in the privacy of some camera-equipped rehearsal room. That particular aspect of the Reich reminds me of what I have been told by a friend in Australia, not in politics - told accurately, I do hope - regarding Baroness Margaret Thatcher. She is supposed to have been recorded, practicing with her diction coach: "Enough IS enough." - "No, no, Mrs Thatcher, that is too petulant." "Enough is enough." - "Well, that's better; could you try putting more stress on your first word?" - "ENOUGH is enough." - "Ah yes, that should do it."

Had Hermann Göring won the Battle of Britain for his sordid Reichsführer, Prime Minister Churchill's position in Cabinet would have been rendered intolerable. We see from postwar Britain, in the examples of Sir Anthony Eden, Sir Harold MacMillan, and Baroness Thatcher - and indeed we see already from the war, in the May-of-1940 example of Neville Chamberlain - how ready the Conservative Party inner ranks are to dump a Prime Minister when a change appears in a given political emergency to serve the national or Party interest.

So let us, as an exercise in speculation, posit that Göring presses home his September advantage, destroying aerodrome upon aerodrome, in his here-posited operation sagaciously resisting the temptation to divert his aeroplanes into the (tactically futile) bombing of London civilians.

In real life, the RAF was in September feeling itself strained. In my alternative scenario, the Luftwaffe, by resolutely continuing the tactics which had proven advantageous in August, gains control of British skies by October or November.

There is now no need for the Reichsführer to mount the risky "Sealion". Mere diplomatic bluff, of the kind one today comes to associate with that specialist in Chechen, Georgian, Crimean, Donbass, and Syrian affairs,  Mr V.V.Putin, should now suffice. Let there now be a few carefully tempered, ostensibly moderate, speeches in the Reichstag, from the Führer and others. Let these be followed up, over the coming hours and days, with a handful of "commentaries" or "analyses" from Lord Haw-Haw (https://en.wikipedia.org/wiki/Lord_Haw-Haw), to his wide British radio audience - an audience surely including not just common folk occasionally wary of the BBC, but high civil and military leaders. Let the general Reich message, first-drafted by the Führer, then fine-tuned by the Führer's PR architect Dr P.J.Goebbels, and finally disseminated by the more subordinate Haw-Haws and Leni Riefenstahls, be this: We are ever so anxious, from considerations as much humanitarian as military, to negotiate a peace; unlike the Soviets, we are no barbarians, but instead constitute cultured continental Europe's last, best hope in the face of the advancing Communism; Germany and Britain are not even traditional rivals. 

In the face of such a message, I suggest, Cabinet would have proved restive. According to 1999 analyst Jon Lukacs in Five Days in London: May 1940 (and I believe contrary to Churchill's own late-1940s writings), Cabinet had already been wavering in May. How much worse would the Prime Minister's political situation have become now, in October or November, with the RAF shattered!

Here is one possible, even probable, eventual outcome:
  • Lord Halifax, or someone of his mind, takes over as Prime Minister, with the extremist Churchill now permanently discredited. (He continues, as I imagine it, in Cabinet, while now safely displaced from the absolute centre of power. He has become just one personality, among many potentially competing personalities, around the conference table at Number Ten.) 
  • A peace is concluded which leaves Britain unoccupied, and with that millstone-around-the-neck which is its overseas empire cunningly left intact. The peace is nevertheless realized by everyone in Moscow, Berlin, Paris, London, and (crucially) Washington to be a thinly veiled capitulation. 
  • The Roosevelt government sees no alternative to taking the advice of its London head-of-diplomatic-mission, Ambassador Joseph P. Kennedy, thereby writing off Britain and the Continent. 
  • Eventually, somehow, in 1941 or 1943 or 1945 or thereabouts, some kind of war erupts between the Greater Reich and the USSR (perhaps with the Greater Reich invading eastward, but perhaps - cf the bleak 21st-century analyst V.B.Rezun, publishing under the nom-de-plume "Viktor Suvorov" - with the USSR invading westward). 
  • However this Reich-USSR conflict ends, Continental Europe, all the way from the Urals to the Atlantic, sooner or later settles into a period of war-free totalitarian rule (whether fully Nazi, or fully Soviet, or in some Nazi-Soviet partition in the spirit of the 1939-08-23 Molotov-Ribbentrop Pact), lasting for at least some decades. Confronted with this peaceful Continental reality, Britain becomes a progressively less relevant - a progressively more quaint - bystander. 
In my speculative scenario, eccentric, endearing little Britain, with its old-fashioned rhetoric of "freedom" and "values", proves in the long run tolerable enough to Hitler, Stalin, and their various eventual Nazi or Soviet successors. Then the 1914 prophecy of Sir Edward Grey, "The lamps are going out all over Europe, and we shall not see them lit again in our own life-time" (already, in the actual world, regarded as rather accurate) takes on the firm, bleak, ring of hitting-the-bullseye prophecy, as we know it, with reference to the fall of Judah, from Isaiah and Jeremiah.  

****

To study the Battle, I have in my poverty of both time and money resorted to a well-written, agreeably inexpensive, eminently modern (year-2000) Penguin paperback, running to a mere 170 or 180 pages, Richard Avery's  The Battle. I have also found YouTube useful, especially as supplying period ciné footage of Filter Room and aerodrome activities. Further, YouTube makes available a compelling 1942 film, "The First of the Few", on the life and work of Spitfire designer Reginald Joseph Mitchell, CBE (1895-1937). (One can perhaps take the upload of YouTube user "Billy Gellings", from 2016-08-12, under the title "The First Of The Few 1942 Full Movies". In my particular corner of the Internet, the upload is available through the URL https://www.youtube.com/watch?v=DP3Cd6g2Q9Y.)

I have the impression, from casual investigations in a bookstore and on the Internet, that a history of the Battle by Stephen Bungay, under some such title as The Most Dangerous Enemy, from around the year 2000, is now widely held to be authoritative. This author is said in reviews to stress the strength of the RAF and the inherent weaknesses of the Luftwaffe, and to correct successfully the romantic (in the sense of untruthful, unrealistic) old 1950s-through-1990s impressions of the Battle, as perhaps very particularly served up on television. The romantic old histories represented the Battle as a heroic defence mounted by an inherently weak Britain. Assuming the coldly realistic Bungay to be right - I have not yet had time to do more than glance at him - I add my own small comment: if the RAF was indeed inherently strong in 1940, then the Battle was fought from the late 1930s onward, being in its key opening phases a work of engineers (notably Mitchell, and on the radar side Sir Robert Alexander Watson-Watt) and administrators (notably the 1st Baron Dowding). The aerial operations of 1940 become, on this conceptualization, a small and specially noisy part of a bigger, predominantly quiet, whole.

If future general World War 2 histories adopt this broader perspective, they will do well to draw parallels between, on the one side, the silent meditative labours of Mitchell, Watson-Watt, and Lord Dowding, and on the other hand the silent meditative labours of Prof. Turing, and his colleagues, at Bletchley Park.

3. Sir Winston Churchill's Account of Operations on Sunday 15 September


And then there is a passage from Churchill's memoirs (from Book Two, or Alone - marketed, however, as a part of a volume entitled Their Finest Hour - under the chapter title "The Battle of Britain"). This passage is so good as to be part of permanent world literature. It rises easily to at least the heights of, say, Caesar documenting Vercingetorix. Much though I personally mistrust the bellicose Churchill (he called the saintly, and correctly pacifist, Mohandas Gandhi "this half-naked fakir";  and in the postwar House of Commons he referred to the very solid Aneurin Bevan as a "squalid nuisance", and to the very solid Clement Attlee as a "sheep in sheep's clothing"), I have in my reluctant admiration to concede that his passage is liable to find itself reproduced in school anthologies even thirty generations from now. That will be a time when sulky schoolchildren struggle to decode their archaic, lapidary "Standard English" texts, with  lexicon and grammar-book close at hand, under teachers rather similar to the kindly Latin master Chipping in "Good-Bye, Mr Chips". (Since our future is bleak, these schoolchildren are perhaps to be pictured as construing not from printed books but from tattered, dog-eared manuscripts, in impoverished monastery schools.) It will be similar to what we have now, but with "sheep in sheep's clothing", or "I may be drunk, Miss, but in the morning I will be sober and you will still be ugly," or other such bits of lapidary Churchilliana, in place of such present-day schoolroom philological puzzles as Gallia est omnis divisa in partes tres, quarum unam incolunt Belgae /.../ . 

I do have to take care not to overstate my case regarding "permanent world literature", as indeed I overstated it in all versions of this blog posting up to and including 3.2.0, 3.2.1, 3.2.2, ... . The idea of inclusion in "permanent world literature" must not be taken for some supreme compliment, such as would be merited by Wordsworth's "Intimations of Immortality". In putting Churchill into the bin of the Permanent, I proceed, rather, pragmatically.

Anthologies of English letters in the very remote future will of course have to include Wordsworth's "Intimations", and again Hamlet's soliloquy, and again two or three or four or five things from the professionally theological English writers (from Dame Julian of Norwich, for instance, or from John Henry Cardinal Newman). Anthologists, however, will have to have something also from the world of practical affairs, illustrating the prose of an actual leader - something, in particular, to capture and hold the attention of Young Persons of a domineering and valorous and rugby-loving spirit, aged between thirteen and eighteen. These are, in any society and historical period, juveniles temperamentally averse to the poetic and philosophical, and for good or ill intended by their families to become leaders in their nation's legal and political life.

In classics-oriented British schools, the purpose has been served, for generation upon generation, by Caesar's Commentarii de Bello Gallico. Caesar, although everywhere and at all times admired as a prose stylist, is not considered the acme of Latinity. I gather that if you really want to progress in Latin farther than I have, to the point of writing your own serious Latin essays and speeches, then you are best advised to imitate not Caesar's sparse wooden-bench practicality, but instead to use as a model the more rounded, the so-to-speak more upholstered, Cicero.

As for Latin now, so also, then, for the "Standard English" in the Year of Grace 3016 or so: Churchill's destiny is not to occupy the apex of the conserved English literary pantheon, as Wordsworth's "Immortality" or the Middle English world-as-hazelnut passage from Dame Julian eventually might, but all the same to fill a classroom niche.

Before I reproduce Churchill's passage, I caution the reader to consider with special care the paragraph beginning "The young officer, to whom this seemed a matter of routine". In that particular paragraph Churchill is echoing, whether consciously or at some eerie Freudian level of pre-consciousness, his account of an operation in the Battle of France that summer, from elsewhere in his memoir. Having asked Gamelin, or Weygand, or someone, "La masse de réserve?" ("The reserves?"), he had been given the despairing answer, "Aucune" ("There aren't any").

Here, then, is what Churchill writes:

It was one of the decisive battles of the war, and, like the Battle of Waterloo, it was on a Sunday. I was at Chequers. I had already on several occasions visited the headquarters of Number 11 Fighter Group in order to witness the conduct of an air battle, when not much had happened. However, the weather on this day seemed suitable to the enemy, and accordingly I drove over to Uxbridge and arrived at the Group Headquarters. Number 11 group comprised no fewer than twenty-five squadrons covering the whole of Essex, Kent, Sussex, and Hampshire, and all the approaches across them to London. Air Vice-Marshal Park had for six months commanded this group, on which our fate largely depended. From the beginning of Dunkirk, all the daylight actions in the South of England had already been conducted by him, and all his arrangements and apparatus had been brought to the highest perfection. My wife and I were taken down to the bomb-proof Operations room, fifty feet below ground. All the ascendancy of the Hurricanes and Spitfires would have been fruitless but for this system of underground control centres and telephone cables, which had been devised and built before the war by the Air Ministry under Dowding's advice and impulse. Lasting credit is due to all concerned. In the South of England there were at this time Number 11 group H.Q. and six subordinate fighter station centres. All these were /.../ under heavy stress. The Supreme Command was exercised from the Fighter Headquarters at Stanmore, but the actual handling of the direction of the squadrons was wisely left to Number 11 Group, which controlled the units through its fighter stations located in each county.

The Group Operations Room was like a small theatre, about sixty feet across, and with two storeys. We took our seats in the dress circle. Below us was the large-scale map-table, around which perhaps twenty highly trained young men and women, with their telephone assistants, were assembled. Opposite to us, covering the entire wall, where the theatre curtain would be, was a gigantic blackboard divided into six columns with electric bulbs, for the six fighter stations, each of their squadrons having a sub-column of its own, and also divided by lateral lines. Thus, the lowest row of bulbs showed as they were lighted the squadrons which were "Standing By" at two minutes' notice, the next row those "At Readiness," five minutes, then "At Available," twenty minutes, then those which had taken off, the next row those which had reported having seen the enemy, the next - with red lights - those which were in action, and the top row those which were returning home. On the left-hand side, in a kind of glass stage-box, were the four or five officers whose duty it was to weigh and measure the information received from our Observer Corps, which at this time numbered upwards of fifty thousand men, women, and youths. Radar was still in its infancy, but it gave warning of raids approaching our coast, and the observers, with field-glasses and portable telephones, were our main source of information about raiders flying overland. Thousands of messages were therefore received during an action. Several roomfuls of experienced people in other parts of the underground headquarters sifted through them with great rapidity, and transmitted the results from minute to minute directly to the plotters seated around the table on the floor and to the officer supervising from the glass stage-box.

On the right hand was another glass stage-box containing Army officers who reported the action of our anti-aircraft batteries, of which at this time in the Command there were two hundred. At night it was of vital importance to stop these batteries firing over certain areas in which our fighters would be closing with the enemy. I was not unacquainted with the general outlines of this system, having had it explained to me a year before the war by Dowding when I visited him at Stanmore. It had been shaped and refined in constant action, and all was now fused together into a most elaborate instrument of war, the like of which existed nowhere in the world.

"I don't know," said Park, as we went down, "whether anything will happen today. At present all is quiet." However, after a quarter of an hour the raid-plotters began to move about. An attack of "40 plus" was reported to be coming from the German stations in the Dieppe area. The bulbs along the bottom of the wall display panel began to glow as various squadrons came to "Stand By." Then in quick succession "20 plus," "40 plus" signals were received, and in another ten minutes it was evident that a serious battle impended. On both sides the air began to fill.

One after another the signals came in, "40 plus," "60 plus"; there was even an "80 plus." On the floor table below us the movement of all the waves of attack was marked by pushing discs forward from minute to minute along different lines of approach, while on the blackboard facing us the rising lights showed our fighter squadrons getting into the air, until there were only four or five left "At Readiness." These air battles, on which so much depended, lasted little more than an hour from the first encounter. The enemy had ample strength to send out news waves of attack, and our squadrons, having gone all out to gain the upper air, would have to refuel after seventy or eighty minutes, or land to rearm after a five-minute engagement. If at this moment of refuelling or rearming, the enemy were able to arrive with fresh unchallenged squadrons, some of our fighters could be destroyed on the ground. It was, therefore, one of our principal objects to direct our squadrons so as not to have too many on the ground refuelling or rearming simultaneously during daylight.

Presently the red bulbs showed that the majority of our squadrons were engaged. A subdued hum arose from the floor, where the busy plotters pushed their discs to and fro in accordance with the swiftly changing situation. Air Vice-Marshal Park gave general directions for the disposition of the fighter force, which were translated into detailed orders to each fighter station by a youngish officer in the centre of the dress circle, at whose side I sat. Some years after I asked his name. He was Lord Willoughby de Broke. (I met him next in 1947, when the Jockey Club, of which he was a member, invited me to see the Derby. He was surprised that I remembered the occasion.) He now gave the orders for the individual squadrons to ascend and patrol as the result of the final information which appeared on the map-table. The Air Marshal himself walked up and down behind, watching with vigilant eye every move in the game, supervising his junior executive hand, and only occasionally intervening with some decisive order, usually to reinforce a threatened area. In a little while all our squadrons were fighting, and some had already begun to return for fuel. All were in the air. The lower line of bulbs was out. There was not one squadron left in reserve. At this moment Park spoke to Dowding at Stanmore, asking for three squadrons from Number 12 Group to be put at his disposal in case of another major attack while his squadrons were rearming and refuelling. This was done. They were specifically needed to cover London and our fighter aerodromes, because Number 11 Group had already short their bolt.

The young officer, to whom this seemed a matter of routine, continued to give his orders, in accordance with the general directions of this Group Commander, in a calm, low monotone, and the three reinforcing squadrons were soon absorbed. I became conscious of the anxiety of the Commander, who now stood still beneath his subordinate's chair. Hitherto I had watched in silence. I now asked, "What other reserves have we?" "There are none," said Air Vice-Marshal Park. In an account which he wrote about it afterwards, he said that at this I "looked grave". Well I might. What losses should we not suffer if our refuelling planes were caught on the ground by further raids of "40 plus" or "50 plus"! The odds were great; our margins small; the stakes infinite.

Another five minutes passed, and most of our squadrons had now descended to refuel. In many cases our resources could not give them overhead protection. Then it appeared that the enemy were going home. The shifting of the discs on the table showed a continuous eastward movement of German bombers and fighters. No new attack appeared. In another ten minutes the action was ended. We climbed again the stairways which led to the surface, and almost as we emerged the "All Clear" sounded.


[This is the end of tonight's blog posting.] 



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 Toomas.Karmo@gmail.com.]



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 https://www.amazon.ca/Primer-Calculus-Leonard-Irvin-Holder/dp/0534177484.


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 (http://perimeterinstitute.ca/).

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 (http://www.fields.utoronto.ca/). 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 (https://hacklab.to/) 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 (https://en.wikipedia.org/wiki/Our_Lady_Seat_of_Wisdom_Academy; https://www.seatofwisdom.ca/) - 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 https://www.youtube.com/watch?v=RNC-aj76zI4.)

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
20071124=00h33z+00h00c->00000145h48z+000796h02c
20071126=00h24z+00h00c->00000146h12z+000796h02c
20071127=01h33z+00h00c->00000147h45z+000796h02c__conics
20071201=00h20z+00h00c->00000148h05z+000796h02c__conics
(((SNIP))
20161130=03h51z+00h20c->000001078h59z+001851h35c
20161201=00h00z+03h32c->000001078h59z+001855h07c
20161202=00h00z+01h22c->000001078h59z+001856h29c

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
20160908=00h41->0001h34__Munkres
20160909=03h35->0005h09__ditto
20160910=02h26->0007h35__ditto
20160914=01h17->0008h52__ditto
20160915=04h26->0013h18__ditto
20160916=03h14->0016h32__ditto
20160917=03h03->0016h35__ditto
20160920=02h06->0018h41__ditto
                         __bought Munkres 160.00 CAD used UofT LAVS DEO
20160922=02h11->0020h52__Munkres
((SNIP))
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" (https://en.wikipedia.org/wiki/L%C3%A9n%C3%A1rt_spherehttp://lenartsphere.com/). 

****

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

/home/karmo/AANN____workworld/

  GNNN____studies_private/Q_______science/

      /QA____maths/QA00000611____topology/

          /QA00000611.M82.2000X____munkres__topology__2nd_edn/

               AANN____studium_of_20160907T160000Z

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

#2.17.5::08
20161201T2257Z

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):

~/*stud*/Q_*/QA00000611_*/QA00000611.M82.2000X*munkres*topol*2nd*/

    AANN*studium*20160907*/

        OBON*exercises*chap02*secns17ff*



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 (http://www.goodmans.ca/), 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: https://en.wikipedia.org/wiki/John_Mighton. 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.]