Toomas Karmo

Monday, 5 September 2016

Toomas Karmo (Part C): In Praise of Moise's "Elementary Geometry from an Advanced Standpoint"

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 had time to do a reasonably complete and (within the framework of the version 1.0.1, 1.0.2, .. process) reasonably polished job.

Revision history:

UTC=20161003T1529Z/version 1.2.0: Kmo made a tiny adjustment: a little more elegant than "positive square root 49" is "nonnegative square root of 49" (since in the reals, what we mean by the surd sign is not "the one and only positive square root of", but "the one and only nonnegative square root of": working on the real-number line, as opposed to the more generous Argand complex-number plane, it makes sense to prefix the surd sign even to "0). Kmo added some remarks on his translation of Aristotle. - Kmo reserved the right to make further minor, cosmetic, nonsubstantive tweaks over the coming 48 hours, as here-undocumented versions 1.2.1, 1.2.2, 1.2.3, ... .
UTC=20160906T0030Z/version 1.1.0: Kmo finished everything off, adding during this operation brief necessary references to elliptical geometry, hyperbolic geometry, and Eudoxus. - Kmo reserved the right to make further minor, cosmetic, nonsubstantive tweaks over the coming 48 hours, as here-undocumented versions 1.1.1, 1.1.2, 1.1.3, ... .
UTC=20160906T0002Z/version 1.0.0: Kmo uploaded a base version, rather hastily, and without quite finishing the last few paragraphsd. He hoped to finish at some point in the next two hours.

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

[I continue now with my Section 1, "Introductory Comments, on the Inadequacy of Traditional Geometry Textbooks". For the convenience of readers I start by repeating the last three sentences of my 2016-08-22 or 2016-08-23 posting, in which I took that section just partway to completion.]

I have the time and patience to cite one further example of Godfrey-and-Siddons's logical failings, concerning their already-cited notion of "equality". I have quoted them as asserting that figures that can be made to coincide are "equal in all respects". Well, as they say around the samovar in my imaginary Nikolai Ivanovitch Lobachevsky Institute of Socialist Mathematics (across that imaginary potholed Siberian street from the "Alexandr Stepanovitch Popov Institute of Heroic Radio"): "Vot MEENZ 'equal'?" Can two distinct triangles be "equal in all respects"? Can two distinct angles, for instance the base angles of some isosceles triangle, be "equal"?

On the clearest way of using language, one doe not write the term "equal" at all, but merely writes "identical". Two distinct congruent triangles should not be asserted to have corresponding pairs of "equal sides", but rather to have corresponding pairs of sides whose lengths are identical. Instead of writing that in an isosceles triangle (say, for concreteness, the internal angles at B and C) are equal, one should write that the "degree measure of B is identical with the degree measure of C" (being, as it were, the real number 22.5), or again that the "radian measure of B is identical with the radian measure of C" (being, as it were, the real number pi/8). Similarly, one should write not that the line segments AB and AC in this isosceles triangle are "equal", but that they have "lengths which are identical", or "have the same length".

This point is made vivid by the pre-eminent pioneer of mathematical logic, Gottlob Frege (1848-1925): mathematics, he rightly insists, operates with the same notion of "identical" or "same" as the Greek astronomers would have used, in announcing their discovery that the Evening Star (the "Hesperus" of their unscientific ancestors) is identical with the Morning Star (the "Phosphorus" of their unscientific ancestors).

Admittedly, I will some day have to blog on the possible need for a refinement involving "relative identity", following British Catholic analytical philosopher-logician Peter T. Geach (1916-2013). On Geach's analysis of entityhood, x can be the same A as, and yet a different B from, y. Here "A" and "B" are what the analytical philosophers, notably including Prof. Geach, call "sortals".

I worked out some of the possible formalism when speaking at the Cambridge Moral Sciences Club in 1982 or 1983. I suggested on that evening, contrary to Prof. Geach, that the very notion of an equivalence relation is sortal-relative: it can, I suggested, be true that (1) for every A x and for every A y and for every A z, (1.1) x is the same A as x and (1.2) if x is the same A as y then y is the same A as x and (1.3) if x is the same A as y and y is the same A as z then x is the same A as z and yet false that (2) for every B x and for every B y and for every B z, (2.1) x is the same A as x and (2.2) if x is the same A as y then y is the same A as x and (2.3) if x is the same A as y and y is the same A as z then x is the same A as z.

It is possible that this relativization-of-identity idea, whether in Prof. Geach's own articulation or in my variant on it, has ramifications for analytical metaphysics. Ramifications could, for instance, arise when we try to render precise some ideas of Aristotle and Aquinas, on "being". (I probably did not consider these ramifications when briefly speaking in 1980s Cambridge.) Geach's ideas, or some variant on them, might in particular prove useful in rendering precise the sense in which different parcels of matter can at different times be "the same human being as" the vigorously metabolizing Socrates, with Socrates himself (as Aristotle would put it) "a form IN matter", and indeed (so Aristotle ought to put it) a "form migrating THROUGH matter".

I would myself here compare the vigorously metabolizing Socrates to a knot slipping along a rope, and would stress the need in a formal analytical metaphysics for explaining how being knotted differs from being a knot. ("There is a scientific discipline,": writes Aristotle at Metaphysics I.1 as an odd-but-deep early scientist, "which scrutinizes being insofar as it is being, and scrutinizes the various properties entailed by it insofar as it is being" - I translate myself here, not in the admittedly very fine literal-English manner of my one-time mentor Prof. J.Ackrill (1921-2007), but instead in a manner intended to amplify the conceptual resonances or conceptual overtones of "episteme tis" and "theorei" and "hyparchonta".)

It is perhaps advisable to add here that once I explained some of these knot-in-rope ideas to Prof. Geach himself, in some private chat or other, he said, slowly and ponderously, "A human being ... is very ... unlike ... a knot." But what did Prof. P.T.Geach understand? I dunno about you, Gentle Reader. I for my part, at any rate, am only too willing to correct, to rebuke, to adjust, to needle, and in general to make fun of our so-serious professors.

For the purposes of mathematics in its familiar forms, these Geachean refinements can fortunately be neglected, and the austere simplicities of Frege allowed to stand. In familiar mathematics - in calculus, in geometry as in Moise and Hilbert and Birkhoff, in set theory as in Zermelo-Frankel or their expositors (Rotman-and-Kneebone, Halmos) - it is enough, in Prof. Geach's framework, to introduce just one "sortal", namely "classical-maths entity". We can then write, blandly enough, "the positive square root of 49 is the same classical-maths entity as the sum of 4 and 3," and "The empty set is a different classical-maths entity from the set whose sole member is the empty set," and so on.

2. An Overview of Moise's Excellence

I have by now flipped through much of Moise's final (1990, third) edition. Further, I have started from the beginning, and have read carefully, and have corrected such few inelegances as I can in my scientific poverty find, and have done perhaps two thirds (or more) of all the problems in the many problem sets up to page 68 - logging over the period 2016-08-11/2016-09-02 a total of around 80 deskwork hours. The Third Edition of this treatise on two-space (on "two-dimensional geometry"), O Gentle Reader, in fact ends in a blaze of glory with the exhibition of a field Euclidean-and-yet-not-Archimidean, on p. 495.

On the strength of my admittedly so far slender acquaintance, I make bold to pick out four specific virtues in Moise's treatment.

(I) Moise discusses non-Euclidean geometry in an appropriately general setting.

He not only introduces Riemann's elliptical geometry and Lobachevsky's hyperbolic geometry, but also introduces Bolyai's idea of "absolute" geometry. "Absolute" (sometimes also called "neutral") geometry commits itself to denying Riemann, and yet stays neutral between Euclid and Lobachevsky. It stays neutral in that while affirming that (A) given a line and a point off the line, there exists at least one parallel to the given line through the given point, it neither affirms nor denies that (B) given a line and a point off the line, there exists at most one parallel to the given line through the given point.

I believe one of the exciting things about geometry, as the professionals might practice it within the august walls of Toronto's Fields Institute (or whatever) is that much can already be proved within the seemingly barren "absolute" framework.

Moise has even educated me on a topic pertinent to the history of Catholicism, by pointing out the non-Euclidean contribution of Father Giovanni Girolamo Saccheri (1667-1773), in the Jesuit order: Fr Saccheri accurately explored the dramatic consequences of denying "(B)" above, even while inaccurately regarding his deduced dramatic consequences as a reductio ad absurdum of the denial.

(II) Moise discusses the formalization of Euclid in both of its contemporary variants.

As is generally rather notorious, Euclid's lapses in rigour are due to a failure to axiomatize some foundational concepts, notably the concept, applied to the three distinct collinear points A, B, C, "B lies between A and C." - This was a fact I did manage to pick up in my misspent youth at Dalhousie University in Nova Scotia, before unwisely forsaking any real efforts in exact science in favour of my Two Largely Lost Decades of logic-and-philosophy. As many know - this emerged when I poked around the Dalhousie maths library, in a few end-of-semester days during my just-cited jeunesse dorée - Euclid got rigorously formalized around 1895, at Göttingen, by David Hilbert (1862-1943).

But only this summer, in 2016, did I gather from poking around University of Toronto libraries that Euclid also got rigorously formalized, in a different and simpler way, down in interwar Hah-vud, by George David Birkhoff (1884-1944). The Birkhoff idea is to assume the properties of the real numbers, as an ordered field, and to postulate that every line can be mapped surjectively to the reals, and then to introduce in terms of such mappings an appropriate definition for betweenness. (I think Hilbert, by contrast, takes betweenness as a primitive notion, for which he then has to set up a small handful of separate postulates.)

Amazingly, Moise discusses both strategies - at first developing everything on Birkhoff's lines, but then explaining how Hilbert could be used instead.

In general, I would suggest that this is the one fully right way to do axiomatizations. Axiomatization, I would suggest, is not a goal in its own right. The work is not complete until (I would suggest) we can see that a topic can be axiomatized both in this way and in that way, with the same ultimate total corpus of propositions emerging, in both cases as the union of some small set of postulates and some set of theorems.

(III) Moise discusses the geometrical relevance of real-number-line completeness properties.

Amazingly, Moise goes a long way without assuming the full strength of real-number-line completeness. (He thus works for a long time without, e.g., assuming that every set of reals bounded from above has a least upper bound.) His initial, and for a long time sole pertinent, assumption is only that every positive number has a positive square root. Eventually, this minimalist approach lets him prove one or more of the celebrated impossibility results, which I used to think in my despairing way would be the exclusive province of graduate students and profs (as opposed to in-essence-undergrads, like me): he proves that no mere straightedge-and-compass construction in Euclidean two-space is guaranteed to trisect an arbitrary angle.

Somewhere in all this is subtle, insightful stuff about the ever-so-complete "real line" and some weakening thereof, the subtly gappy "surd line".

Equally amazing is Moise's ability to draw connections between Dedekind, as a father of modern analysis, and Eudoxus. Until stumbling across Moise, I had written the Greeks off, as bright-but-wacky people without an inkling of Dedekind completeness. Moise, however, argues that Eudoxus's theory of proportion (cunningly designed to accommodate even the proportions between things not commensurate - certainly between the diagonal and the side of a square, and for all I presently know perhaps even, at a deeper level of subtlety, between the circumference and the radius of a circle) anticipated the modern Dedekind-cuts construction of the reals.

(IV) Moise presupposes almost nothing.

To be more precise, here is his programme of work (as explained in his "Preface"; it is to be borne in mind that the book title is Elementary Geometry from an Advanced Standpoint):

The title of this book is the best brief description of its content and purposes that the author was able to think of. These purposes are rather different form those of most books on "higher geometry" or the foundations of geometry. The difference that is involved here is somewhat like the difference between the two types of advanced calculus courses now commonly taught. Some courses in advanced calculus teach material which has not appeared in the preceding courses at all. There are others which might more accurately be called courses in elementary calculus from an advanced standpoint: their purpose is to clean up behind the elementary courses, furnishing valid definitions and valid proofs for concepts and theorems which were already known, at least in some sense and in some form. [I, Kmo, might as well add here that suitable textbooks for such cleaning-up calculus courses would be the celebrated introductions-to-calculus by Michael Spivak and Tom Apostol.] One of the purposes of the present book is to reexamine geometry in the same spirit.

If we grant that elementary geometry deserves to be thoroughly understood, then it is plain that such a job needs to be done; and no such job is done in any college course now widely taught. The usual senior-level courses in higher geometry proceed on the very doubtful assumption that the foundations are well understood. And courses in the foundations (when they are taught at all) are usually based on such delicate postulate sets, and move so slowly, that they cover little of the substance of the theory. The upshot of this is that mathematics students commonly leave college with an understanding of elementary geometry which is not much better than the understanding that they acquired in high school.

The purpose of this book is to elucidate, as thoroughly as possible, both this elementary material and its surrounding folklore. My own experience, in teaching the course to good classes, indicates that it is not safe to presuppose an exact knowledge of anything. Moreover, the style and the language of traditional geometry courses are rather incongruous with the style and the language of the rest of mathematics today. This means that ideas which are, essentially, well understood may need to be reformulated before we proceed. /.../ In some cases, the reasons for reformulation are more compelling. For example, the theory of geometry inequalities is used in the chapter on hyperbolic geometry; and this would hardly be reasonable if the student had not seen these theorems proved without the use of the Euclidean parallel postulate.

For these reasons, the book begins at the beginning. Some of the chapters are quite easy, for a strong class, and can simply be assigned as outside reading. Others /... / are more difficult. These differences are due to the nature of the material: it is not always possible to get from one place to another by walking along a path of constant slope.

/.../

The book is virtually self-contained. The necessary fragments of the theory of equations and the theory of numbers are presented in Chapters 28 and 29, at the end. At many points, ideas from algebra and analysis are needed in the discussion of the geometry. These ideas are explained in full, on the ground that it is easier to skip explanations of things that are known than to find convenient and readable references. The only exception to this is in Chapter 22, where it seemed safe to assume that epsilon-delta limits are understood.

/.../

This is exactly what is appropriate. Although some of what Moise deploys scientifically impoverished people like me do already know (he does not in my particular case have to belabour, for instance, the concepts of injection and surjection), nothing can be taken for granted across his entire heterogeneous, multicultural, many-decade, numerous readership except for the very thing he himself does take for granted, namely epsilon-delta.

I hope I will be forgiven here for retailing my all-time-favourite maths story, the epsilon-delta story from my "Utopia 2184" essay at http://www.metascientia.com/PNNN____lit/BXPF____utopia__chap01.html:

Mathematics. Professor Michael Edelstein, born 1917 March 21. Departed this life 2003 January 27, of natural causes. Arrived Jerusalem 1937, thus escaping Holocaust. From 1964 onward, a founder of the research programme at Dalhousie University in Nova Scotia.

A small seminar room in my first undergraduate year, in the early months of 1971. Back then, I must explain, my school was Dalhousie University in Halifax, not the University of Toronto. ('Oh,' said the Secretary of St John's College, Oxford, a few years later, when I had succeeded in temporarily escaping Canada. She was eyeballing my file. 'Oh, you're from Dal-HOO-sie.' - 'We call it 'Dal-HOW-sie, actually.' - 'Oh. I'm sure you do.' - But I digress.)

What do we mean, asked Professor Edelstein's chain of reasoning - we had to write our own textbook, from the lectures and Professor Edelstein's own spirit-duplicator crib notes - what do we mean when we claim that the limit of, say, the ratio of x to sine x (the sine here being of course evaluated in the manner of the severer mathematicians, in radian measure), as x approaches zero, is one? Mathematics stands outside time, so the language of 'approach' can at best be metaphor. But it is easy: we mean merely that for any positive epsilon, no matter how small, there exists some positive delta such that if the distance of x from zero is a nonzero number less than delta, then the discrepancy between x/sin x and 1 is less than that excruciatingly tiny preassigned epsilon. For any epsilon, no matter how small. After a mere hour's reflection, it makes sense.

And then Professor Edelstein, pleading for comprehension, in the heavy accents of Estonia or Germany or his native Poland or the old 1930s Mt Scopus campus of Hebrew U. in Jerusalem: 'Fullerton, Guptill, Karmo. You geeev me ENNNNNY epsilon ... and I can proe-DEUCE a delta.'

Those last words in roughly the tone of the Handel aria about knowing that my Redeemer liveth.

Then worse: 'If you do not understand ziss defini-shun, you stay outSIDE muss-em-mah-tiks.'

Dear Professor Edelstein, undoubtedly one of my three best teachers in any branch of science or humanities, in any department on any campus, but all the same adept at escalating the terror.

[To be continued, and perhaps even concluded, as "Part D", perhaps in the upload normally scheduled for the UTC interval 20160913T0001Z/20160913T0401Z.]

Toomas Karmo: Paramonastic Scientists and the Theology of Disability

I took this photo late in August of 2016. The Salvation Army shelter on Toronto's McCaul Street was always a sad place, with an outward air of decay, and with unhappy-looking residents standing outside on the sidewalk, perhaps seeking a bit of fresh air and a brief smoke. But now, with the unhappy-looking residents gone, it is infinitely worse. - Poverty, as I remark in today's essay, assumes diverse forms.

Quality assessment:

Revision history:

UTC=20160906T0113Z/version 1.1.0: Kmo added a photo, illustrating the material side of Toronto poverty. He retained the right to make tiny, cosmetic, nunsubstantive tweaks, as here-undocumented version 1.1.1, 1.1.2, 1.1.3, ..., over the coming 48 hours.
UTC=20160906T0001Z/version 1.0.0: Kmo uploaded a base version, rather hastily. - Kmo retained the right to make tiny, cosmetic, nunsubstantive tweaks, as here-undocumented version 1.0.1, 1.0.2, 1.0.3, ..., over the coming 48 hours.

For perhaps 10 or 15 years, I have had a dream or vision that refuses to go away. Here it is:

Around the world emerges a little grouping of Catholics and friends-of-Catholics, held together by bonds of personal support. The bonds notably include bonds of mutual prayer.

In its informality, the grouping resembles the Catholic Worker movement that sprang up in 1930s New York around philosopher Peter Maurin and social activist Dorothy Day, and which has since spread to several countries.

Most members in the grouping are in the laity, in many instances forswearing academic ambition, and in all instances committed to the joint ideals of Poverty and Science.

For want of a better name, let us for the moment simply call it "Movement XYZ".

Movement XYZ has no monastic garb. Perhaps it does not even have a motherhouse, a corporate library, a bank account, or other significant material assets.

Informal "Movement XYZ" groups, however, are to be found around many of the current nuclei of exact-sciences research - around the University of Toronto, for instance (perhaps in some way linked with the University of Toronto Catholic chaplaincy, the "Newman Centre"), or around a few of the world's makerspaces, such as the Toronto "Hacklab" or the Kitchener-Waterloo "Kwartzlab". Some may even be found around such specialized, eminent, exact-science bodies as the Fields Institute (Toronto's senior centre for mathematics research) and the Perimeter Institute (a Fields equivalent for theoretical physics, in Kitchener-Waterloo - and therefore an hour or two away from the Fields Institute, by commuter train).

Persons of any faith, or of none, are welcome. But Catholics in good standing tend to predominate.

Movement XYZ draws what support it can from formal structures of spiritual direction (i.e., of spiritual counselling) within the Church. One might imagine, for instance, a cordial relationship developing with the Jesuit-run Vatican Observatory Research Group, active these days both at Castel Gandolfo and in Arizona. Additionally, I would imagine some kind of cordial links forming, eventually and somehow, at Città del Vaticano itself, with at least some tolerant, kindly members of the Pontifical Academy of Sciences.

A key feature of this Movement, as of Catholic Worker, would be the youthfulness of many of its more active members.

As the membership would tend to self-select in the direction of Catholics in good standing (despite the lack of formal membership prerequisites), so also would it tend to self-select in the direction of the exact sciences. People in the Movement would, in other words, tend to come from pure and applied maths, from theoretical computer science, from theoretical and experimental physics and chemistry and (to a perhaps lesser degree?) from various branches of engineering. With the sofware engineers (the kind of people who propose improvements to the Linux and GNU Hurd kernels) and the theoretical computer scientists (the kind of people who study "Cook's Theorem" and "NP Completeness"), there might also be at least a few people pioneering in whatever disciplines might now be emerging at the interface of biology and formal computer science. (These, as I in my ignorance imagine it, are the pioneers examining questions like "the possible syntactical role of junk DNA", or at any rate "the bitcount or bytecount of the human genome".)

The Movement would not feature, at any rate in any very central way, humanists who study history-and-philosophy-of-science. Always the emphasis, as I dream of it, would be on the scientifically engaged doers, rather than on the humanistically engaged commenters.

****

I imagine poverty in scientific ability or scientific attainments to be no disqualification. In this respect, the Movement would be unlike those groupings of the eminent which are the national Academies - Britain's Royal Society, for instance, or indeed the Pontifical Academy of Sciences at Città del Vaticano.

Generally speaking, poverty is a topic of theological interest.

Obvious is material poverty. In Toronto alone, this has generated a cohort of possibly five, seven or ten thousand homeless individuals or near-homeless individuals.

Anyone wanting a reminder of the cohort is welcome to walk south from the University of Toronto campus, along the west side of McCaul. Where the Salvation Army used to house dozens of indigent men, in at least some kind of semi-private semi-dignity, there is as of recent weeks or months just a sign, announcing that the "Hope Shelter Operations are now closed."

Then we have the emotional poverty of the financially well-off, such as I cited in my posting of 2016-08-08 or 2016-08-09, under the heading "Muzzo-family/Toomas Conciliation Project (DDO&P and Convict)". Readers should note from that posting, and study, a frightening YouTube clip, in which a person of marked financial affluence boasts of her repeated visits to the Playboy Mansion, and of other cognate (to any sane mind, negative) things in her life.

We have a kind of cultural poverty, as when some eminent mathematician or physicist proves uninformed on Dante or Bach. Although I have not myself encountered such cases of philistinism, I am willing to bet on their existence. Indeed we get more than a hint of the various dark possibilities upon reading the Leavis-Snow 1950s "Two Cultures" debate, with physicist-novelist C.P.Snow himself looking a bit shaky. (Prof. Leavis, as a literary critic of eminence, is pitiless in ridiculing Snow's condescending attitude to Ibsen.)

And we have scientific poverty.

I know scientific poverty when I tremble over my C-minus in 1990s third-year quantum mechanics, and I think also in 1990s fourth-year nonlinear physics. I know scientific poverty when I tremble over my failure to have retained anything at all from the theory of complex-numbers integration, beyond the standard joke:

QN: What is the contour integral of western Europe?

ANSW: Zero. Because all the Poles are in eastern Europe.

We do well at all times and in all social contexts to remember our Church's "Preferential Option for the Poor". All the more urgent, and all the more joyous, is this injunction now, in the wake of Mother Theresa's 2016-09-04 canonization.

****

I am reminded of the joy possible in poverty not only by this past weekend's ceremonial, but also by some unexpected, almost exactly contemporaneous, e-mail correspondence from a visually impaired (perhaps even fully blind) Catholic lady in Australia.

My correspondent writes: Within Catholic Christianity, at its best, people with disability are viewed as having full agency and personhood, possessing an equal share in Christ's plan of salvation.

She goes on to remark the following: /.../ a monastic tradition has arisen in which the gifts and charism of persons with a disability particular to said disability are celebrated within orders such as the Benedictine Sisters of Christ Crucified, the Blind Presentationists (a beautiful order for women with little or no sight that sadly is confined to Italy and parts of South America), plus the Little Sisters Disciples of the Lamb (an order for women with Down's Syndrome). For the first two of these, I have not yet managed to find Web materials. For the third, however, Web documentation is quickly retrieved, by googling on the six-word string little sisters disciples of the Lamb.

****

And a further reminder of the joy possible in poverty (this more particularly addresses my own personal condition) comes from http://en.radiovaticana.va/news/2014/11/22/pope_break_down_isolation,_stigma_of_autism/1112005. Here Pope Francis is quoted as saying:

Everyone should be committed to promoting acceptance, encounter and solidarity through concrete support and by encouraging renewed hope. In this way we can contribute to breaking down the isolation and, in many cases, the stigma burdening people with autism spectrum disorders, and just as often their families.

This must not be an anonymous or impersonal accompaniment, but one of listening to the profound needs that arise from the depths of a pathology which, all too often, struggles to be properly diagnosed and accepted without shame or withdrawing into solitude.

****

So what is it that the dreamed-of "XYZ Grouping" actually does?

I dare not write too much.

There will, of course, be bonds of prayer.

And there will be forms of service, of the most varied sort, both through scientific-engineering publication and through hands-on work. Two examples, from opposite ends of the theory-down-into-applications continuum, must suffice for today.

(1) I wish myself to write, eventually, a set of "White Papers" in radio physics, for the particular benefit of the ham-radio communities in countries including Estonia and Canada. In the "White Papers" would be duly careful discussion of things I have in many cases not yet mastered, and yet am working on in a spirit of cheerful scientific poverty - most importantly, the application of tensors, and eventually of Special Relativity, to the mathematics of electromagnetic waves. (It will be appropriate, in particular, to have a White Paper revealing that the difference between an electrostatic field and a magnetic field is not intrinsic, but frame-relative: a magnetic field is an electrostatic field in something other than the rest frame of the field source.)

(2) It would be good to see someone, in a makerspace spirit, develop a robust food-refrigeration solution, appropriate for manufacturers in a cooperative, worker-owned economy, perhaps in a country (Sudan? Kenya? Peru?) that cannot attain, or else that has resolutely forsworn, western affluence. Engineers might develop - I have also pointed this out today in a comment at http://thearchdruidreport.blogspot.com - a Peltier-cooler ice wand. This device would have no moving parts beyond one microswitch, and could create a fist-sized or melon-sized ball of ice when immersed in a tub of water. The waste heat from the hot side of the thermocouple would be conducted some suitable distance away from the water by a heat pipe, such as is used with fans in upmarket computer-CPU cooling systems.

The microswitch would be useful for turning current on and off. Let it, for instance, switch current off to the Peltier thermocouple whenever the ice ball exceeds a certain size, i.e., whenever the buoyancy of the wand-and-ice-assemblage exceeds a certain limit.

The ice wand might perhaps be developed into a more robust small-scale refrigeration solution than the silly "bar fridges" that are now sold at WalMart and the like, under such names as "Danby". These silly "bar fridges" last for only two or five years, have lots of moving parts and lots of electronics, and are said by repair specialists to be not worth repairing once they break.

A single-person household could keep most perishables cold and dry in glass jars in a single chilled-water tub, while keeping such water-tolerant things as beets and carrots and eggs right in the chilled water.

For a family, a couple of tubs, and a couple of wands, would be appropriate.

Monday, 29 August 2016

Toomas Karmo: Remark in Lieu of Forthcoming Part C of Essay-in-praise-of-Moise

Perpetually at my desk is the obituary, from Toronto's Globe and Mail of 2003-02-07, of Prof. Max Edelstein (written by his daughter, the British Columbia mathematician Leah Keshet). Prof. Keshet and I have been in correspondence in recent years, and I am inexpressibly grateful to her for the gift of two books from her distinguished father's personal library. - Prof. Edelstein, as my teacher in a special analysis course at Dalhousie University in the 1971 spring semester, conceived the perfect method of teaching. There was no exam. There was no mid-term. There were not even any problem sets, in the normal sense. The sole requirement was that we, the students in his tiny cohort, would write our own textbook, proceeding from hints and questions and challanges handed out by Prof. Edelstein in blue spirit-duplicator coursenotes. My own textbook, which I have kept carefully and have now also sent in photocopy to Dalhousie University, starts with the definition of a field, soon proceeding to a run of theorems. It is just like the first chapter of the wonderful Moise. But I do notice a tiny blemish, common to Prof. Edelstein and the immortal Moise. We really should not say, in postulating the additive identity, 'There exists a unique 0 such that for every element a in the field, 0 + a = a" (and likewise for the mulltiplicative identity). It suffices to say "There exists at least one element 0 such that for every element a in the field, 0 + a = a" (and similarly for the multiplicative identity). Uniqueness is then something we prove, from the field postulates (both in the case of the additive identity and in the case of the multiplicative identity). - The shiny ceramic sphere, with its blue, green, and red triangles, persuades me that in spherical geometry, in contrast with Euclidean plane geometry, isosceles triangles with a common apex can fail to be similar: the equal base angles of my red isosceles triangle, in particular, on the far side of the sphere, are obtuse, in startling contrast with the equal-and-acute base angles of my blue isosceles triangle, and in startling contrast with the equal-and-acute base angles of my green isosceles triangle.

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 had time to do a reasonably complete and (within the framework of the version 1.0.1, 1.0.2, .. process) reasonably polished job.

Revision history:

UTC=20230526T0813Z/version 2.1.0: Kmo rectified an embarrassing error. He had written "eight-element" where "sixteen-element" was needed.

UTC=20160830T0233Z/version 2.0.0: Kmo managed to get rid of the point-form outline and to a proper upload. He reserved the right to do tiny cosmetic, nonsubstantive, tweaks over the coming 48 hours, as here-undocumented versions 2.0.1, 2.0.2, 2.0.3, ... .

UTC=20160830T0003Z/version 1.0.0: Kmo lacked time to be thorough, and so uploaded mere point-form outline. He resolved to get this (admittedly short) blog posting correctly polished by 20160830T0401Z .

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

Here is a mere public-service announcement.

I had no time this week for writing up "Part C" of my essay praising Moise, but I hope to write it up next week (with upload in the normal four-hour interval, in this case the four-hour interval UTC=20160906T0001Z/20160906T0401Z).

In the mean time, readers may wish to follow me in proving an easy general result pertinent to Moise's first chapter, on fields. (The result emerges quickly from thinking about Moise, although Moise does not himself include it in the chapter problem sets.) Suppose F is any field - finite or infinite. F might, for instance, be the tiny two-element field which Prof. Edelstein drew to the notice of his 1971-spring-semester special Dalhousie University analysis class. Or F might be the rational numbers with standard addition and multiplication. Or F might be the real numbers with standard addition and multiplication. No matter what F is, we can construct a further field, whose elements are pairs (a,b) of F-elements, as follows: addition is defined in terms of F-addition, as (a, b) schplus (c,d) = (a plus c, b plus d), with (0,0) as an (and, we can quickly prove, as the unique) additive identity element; multiplication is defined in terms of F-multiplication, and the additive inverse "minus" of F, as (a, b) schtimes (c, d) = ((a times c) minus (b times d), (a times d) plus (b times c)), and with (1,0) as the identity element. (The task is to prove that the set of such pairs, with schplus and schtimes, is itself a field.) - So given the reals, we can construct the complex numbers as pairs of reals; and given Prof. Edelstein's tiny two-element field, we can construct a four-element field as pairs of Edelstein-tinies; and given the just-constructed four-element field, we can construct a sixteen-element field; and so on.

Toomas Karmo (=VA3KMZ): Moral-Uplift Images Pertinent to Radio and Department of National Defence

Imitating my practice on earlier occasions, I once again show a screenshot of one of my (at all times foufold) Debian GNU/Linux desktops.

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Here, as in previous screenshots, I show in the upper right-hand corner a pair of operations clocks - green for Toronto civil time, and red for UTC - generally attaining a precision tighter than plusminus 200 milliseconds. These clocks are disciplined over Network Time Protocol essentially once in 24 hours, on essentially all occasions against that peer of tock.utoronto.ca and chime.utoronto.ca which is tick.utoronto.ca.

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Today I have caused two Debian GNU/Linux /usr/bin/xterm windows to show excerpts from some flat-ASCII private study notes on radio.

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The two depicted rigs are from "back Home", i.e., from Estonia. They saw civic service in the most serious European crisis since the 1939/1945 war, in 1991 August. (European history has thus just this month, in 2016 August, passed a quarter-century milestone.)

In the 1991 crisis, Estonia, Latvia, and Lithuania regained their interwar independence. The Ukraine of 1991 August attained, or moved to attaining, the independence which many of its brave people had sought in vain during the post-1917 Bolshevik-Menshevik civil war. And similar developments, of a generally promising character, occurred elsewhere that anxious August, in the now-imploding USSR.

At the onset of the crisis, a putsch had temporarily removed M.S.Gorbachev (1931-) from Kremlin power - leaving, however, B.N.Yeltsin (1931-2007) able to play a constructive role, in defiance of the absurd would-be junta.

The Red Army, prima facie putsch-supporting, had occupied terrain at the base of the Tallinn Television Tower. Pro-independence Estonian personnel in the tower were asserting that if this facility - crucial to Estonian public broadcasting, and to Estonian peacetime communications abroad - came under Red Army attack, they would respond by activating their building's halon fire-suppression system. They would thereby in one stroke be both ending their own lives and demonstrating to world opinion the firmness of Estonian political resolve.

Had the widely feared attack occurred, events could (I now in hindsight suggest) have escalated out of control, ultimately to the grief not of Estonia alone but of many nations.

It was therefore essential for authorities in Estonia's principal nucleus of government, Tallinn's Toompea citadel, to monitor the radio traffic of Red Army units within Estonia; to monitor public broadcasting services in Russia and the West; and to maintain their own external communications, inbound and outbound. There was additionally some mission-critical jamming of Red Army signals.

The rig on the left (so explains http://raadiosoda.blogspot.com) is a model of an HF ("high frequency", "shortwave") receiver used widely in the USSR army and navy, and available also to USSR hams, and kept in production from 1948 right up to 1981: Tema väga oluliseks omaduseks oli äärmiselt suur töökindlus. Seade ei läinud praktiliselt mitte kunagi täielikult rikke ja mõne raadiolambi halb töötamine sai uue vastu väljavahetamisega kiiresti kõrvaldatud. ("Its exceptionally relevant feature was its extreme robustness. The set would practically never get so out of order as to be beyond repair. A deficiency in some valve would be quickly remedied, by swapping the old valve out for a new one.")

This set was used at Toompea for monitoring public broadcasts, both within the USSR and outside it; for receiving ham-operator situational reports; and for monitoring the movements of a Red Army Pskov (I suspect, a paratroop) division.

The homebrew transceiver on the right (so explains http://raadiosoda.blogspot.com) was used by a Tartu ham, the either-then-or-eventual ES5MC, in support of the Tallinn-Vilnius intergovernmental hotline. My lookup today, in the Toronto afternoon of 2016-08-29, at http://www.qrz.com indicates that ES5MC (Mr Arvo Pihl) is still licensed under that callsign. Mr Pihl is indeed shown in my http://www.qrz.com database lookup to have an e-mail adress on the Estonian equivalent of Canada's ham-radio rac.ca server, namely erau.ee.

At the lower right is the most photogenic, though not necessarily the most operational, part of the the Toompea facilities, the 1360-1370 "Pikk Hermann" ("Tall Hermann") tower.

An English writeup correspoding to the just-cited http://raadiosoda.blogspot.com may be found at http://estonianelectronicwarfare.blogspot.com.

Further, an English writeup of Pikk Hermann, placing it into its wider architectural and governmental context, can be found on the Estonian parliamentary Web server, at http://www.riigikogu.ee/en/visit-us/toompea-castle/tall-hermann-toompea-towers/.

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In the centre image is a small part of a many-hectare thing I have seen on dozens of occasions from the VIA train window, namely the antenna farm on the Tantramar Marshes outside Sackville, New Brunswick.

It is a standing joke in radio engineering that an antenna farm is optimally sited on a mountaintop salt marsh. Tantramar has the desirable radio-reflective salty (and so electrically conductive) salt water, although it is, alas, at no considerable elevation.

The main user of the farm was Radio Canada International (RCI). The RCI shortwave service was discontinued, in favour of an Internet service with a reduced number of languages, in 2012. But according to https://en.wikipedia.org/wiki/CKCX, this same antenna farm was also used under time-share arrangements by Radio Japan, China Radio International, the Voice of Vietnam, the BBC World Service, Deutsche Welle, and Radio Korea. The same source indicates that the farm was demolished in 2014.

Kipling writes, in a poem that attracted the admiration even of T.S. Eliot,

Far-called, our navies melt away;
On dune and headland sinks the fire.

My last train-window glimpse of the imposing setup was in the summer of Mum's death, in 2011.

Toomas Karmo (=VA3KMZ): Open Letter re DDO&P DND (Department of National Defence) Implications

This map, ultimately from materials published by the Town of Richmond Hill in a hyperlink tree rooted at http://www.richmondhill.ca/subpage.asp?pageid=DDO_park, documents the 72-hectare David Dunlap Observatory and Park (DDO&P) "Trapezoid" and the 5-hectare DDO&P "Panhandle". The "Panhandle" (in white) is to become part of the envisaged approximately 45-hectare rump park. The remaining approximately 40 hectares of the rump park is that part of the Trapezoid which is here coloured green. Also shown is the envisaged, but to many of us entirely unacceptable, approximately 32-hectare subdivision (with its lane, its 14 streets, its stormwater sump, and its 520-plus homes). The northern edge of the Trapezoid, along Hillsview Drive, is around 1.2 km long. The Trapezoid is bounded on the east by Bayview Avenue and on the west by the CNR "Bala Subdivision" line. This is a line with some DND relevance, but now most prominent in the public mind as the GO commuter line, running on about 30 km of not-quite-straight railbed to Toronto Union Station. (As the crow flies, the distance is more like 20 km.) Yonge Street lies a little west of the railway, about 1 km to the west of the Panhandle.

20160912T1810Z/version 2.1.0: Kmo added points on Toronto-Ottawa landline and Toronto-Ottawa satellite links, and on the suitability of the DDO Administration Building roof for solar-panel arrays, and on the ability of the USSR to run a large radio installation without trimming back too much woodland. He reserved the right to upload minor, cosmetic and nonsubstantive, tweaks over the coming 48 hours, as here-undocumented versions 2.1.1, 2.1.2, 2.1.3, ... .
20160830T0122Z/version 2.0.0: Kmo finished uploading a reasonably polished version, and accordingly deleted his point-outline material. He reserved the right to upload minor, cosmetic and nonsubstantive, tweaks over the coming 48 hours, as here-undocumented versions 2.0.1, 2.0.2, 2.0.3, ... .
20160830T0001Z/version 1.0.0: Kmo uploaded base version, unfortunately in part merely in point-outline form. He resolved to polish things adequately by 20160830T0401Z.

Some of my readers, particularly at the Department of National Defence (DND), will be unfamiliar with the David Dunlap Observatory and Park (DDO&P) land-conservation file.

I have blogged on this file extensively in the last month and a half, as follows:

on 2016-08-22 or 2016-08-23, under the title "Open Letter re DDO&P Breach-of-Aquifer Question (Town et al)"
on 2016-08-15 or 2016-08-16, under the title "Failure of 2016-08-14 Archdiocese-Outreach Effort (Pertinent to DDO&P Conservation)"
on 2016-08-15 or 2016-08-16, under the title "Failure of 2016-08-14 Muzzo-Outreach Effort (Pertinent to DDO&P Conservation)"
on 2016-08-15 or 2016-08-16, under the title "RASC-TC Leadership Change, and Further RASC-TC Statement on DDO&P"
on 2016-08-08 or 2016-08-09, under the title "Muzzo-family/Toomas Conciliation Project (DDO&P and Convict)"
on 2016-08-08 or 2016-08-09, under the title "DDO&P Karen-and-Toomas Reconciliation Project"
on 2016-08-08 or 2016-08-09, under the title "Commentary on Useful RASC-TC Editorial on DDO&P Conservation Case"
on 2016-08-01 or 2016-08-02, under the title "Open Letter re DDO&P to Town Council, Addressing the 2016-07-29 Concerns of Councillor Karen Cilevitz " (but fortunately the comments by Councillor Cilevitz, arguably inconsistent with the Municipal Code of Conduct in its call for Councillors to be courteous toward the public, have since been removed - I presume by her - from her Facebook page)
on 2016-07-25 or 2016-07-25, under the title "Open Letter re DDO&P to Muzzo and DeGasperis Families, and Others"
on 2016-07-22, under the title "CIVIC EMERGENCY: RASC Pulls out of DDO 2016 September"

The posting of 2016-07-22 incorporates, under the section heading "1. Background to the 2016-07-22 RASC Crisis", a full set of descriptive references to my earlier DDO&P-pertinent http://toomaskarmo.blogspot.com blog postings. I do here apologize for not having had the time to set up a proper indexing formalism for http://toomaskarmo.blogspot.com. If time were to permit, I would have to create a hyperlink index of all my principal blog topics, imitating the model of Sister Laurel O'Neil at http://notesfromstillsong.blogspot.com. Sister O'Neil's right-hand column, some distance down her page, is headed "Labels". Here are given appropriate subject references, running from such alphabetically early headings as "Abba John Colobos" and "Abuses of Canon 603" all the way down to such alphabetically late headings as "white martyrdom" and "wordliness".

1. Update:
Ongoing Oak Ridges Moraine Aquifer Cap Investigation

Before today embarking on my main DDO&P topic, I should give a short update on my topic from 2016-08-22 or 2016-08-23, the "Open Letter re DDO&P Breach-of-Aquifer Question (Town et al)".

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As of UTC=20160829T2131Z, I had no information from the Town regarding its state of knowledge on the physical integrity of the aquifer cap. The Town had, however, assured me at the end of the previous week that it was working on a reply, and that it would contact me once it had consulted an appropriate member of its Staff. This official was due back from holiday by 2016-08-29. I for my part thanked the Town for their constructive communication, remarking that I had directed a parallel enquiry to Ontario's Ministry of Natural Resources and Forestry, noting to the Town that the Ministry undertakes to answer queries in 15 days, and stressing to the Town that accuracy in our aquifer-cap question is preferable to speed.

2. Fresh Considerations: DDO&P and the Department of National Defence

2.1: Background Regarding Greater-Toronto-Area Resilience

From http://www.yorkregion.com/news-story/1434067-video-aurora-house-holds-secret/ (the Aurora Banner, 2012-03-23, under headline "Aurora house holds secret") one can see the alarmingly modest scale of Cold War efforts for the civil defence of Toronto. In an era in which the United Kingdom had rudimentary UKWMO ("Warning and Monitoring Organization") bunkers across the country, in at least many instances at some such tight spacing as 20 kilometres, it rather appears that Toronto civil defence was to be coordinated from a single Aurora bunker, at 220 Old Yonge Street. The bunker possessed a (surely inadequate) hundred or so phone lines, plus a map room and assorted minor auxiliaries.

The 220 Old Yonge property has at some point subsequent to 2012 been put up for sale. Whether a buyer has been found I do not know.

My experience in Melbourne civil defence, as an early-1980s volunteer in the State Emergency Service (under the Government of Victoria) suggests a similarly modest scale of disaster preparation in Australia.

The Toronto and Melbourne examples are to be contrasted not only with Cold-War-era UKWMO, but still more tellingly with Switzerland. Perhaps it was in part because Switzerland was serious about civil defence in the two troubled decades following World War I that it managed to avoid Austria's sad 1938 fate.

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Disaster scenarios are, admittedly, different now from what planners faced in the eras of Anschluss and Cold War. The immediately plausible threats are no longer of a self-evidently extreme character. if we leave the most dramatic of the various possibilities aside today (therefore today staying silent about, in particular, a possible repeat of the Carrington Event, as discussed at https://en.wikipedia.org/wiki/Solar_storm_of_1859 and http://science.nasa.gov/science-news/science-at-nasa/2009/21jan_severespaceweather/), then two special vulnerabilities suggest themselves:

there could be a repeat of the 2003-08-14 blackout, for instance in winter, under the stress of some region-wide ice storm paralleling the storms of 1998 January and 2013 December
there could be an Internet degradation, for instance from military or terrorist hostilities occurring far outside Canada

DND will have performed some risk appraisals in connection with at least the second of these two scenarios. Although I cannot for my part claim to be at all well read, I nevertheless do offer the following scenario points here (subject to correction, either by DND itself or by competent others):

The worldwide Domain Name Service (DNS) root name servers - one, no doubt physically distributed over multiple machines, governs URL-to-and-from-IP-address translation for the entire *.com domain; another does this job for the entire *.ca domain; a third does this job for the entire *.uk domain; and so on - are a point of Internet vulnerability. They indeed have been asserted (whether accurately or inaccurately, I do not know) to have already, in some recent year, come under attack.
The DNS root-level servers aside, DNS is vulnerable from a conceivable subversion of the Network Time Protocol. The problem here is that routers are, for compelling engineering reasons, barred from updating their routing tables in the event of chronometry inconsistencies.
Toronto has in past years (I was told this by an appropriate professional around 2000 or 2005) had essentially one single Internet chokepoint - i.e., one single building, the loss of which would isolate Toronto from the world's Internet backbones.
If this single chokepoint has since that time been replaced with multiple chokepoints, the number of such present chokepoints at any rate cannot, for compelling reasons of engineering and economics, be large.

The two vulnerabilities are not guaranteed to be independent.

On the one hand, a grid failure has Internet implications - notably when the backup systems for routers fade away, with their diesel generator fuel running out or their battery banks running down.

And conversely, an Internet degradation may in turn affect the "SCADA" control systems in power stations, triggering a collapse of the grid. - I was given an assurance to the contrary by a professor some years ago at the Royal Canadian Institute, in a Sunday-afternoon lecture series. But I am now reluctant to believe him. If the Internet and the grid are decoupled, as the professor cheerfully asserted when I approached him at post-lecture mix-and-mingle time, then how (I now ask) has it been possible for USA SCADA (or similar) power-plant systems to be remotely hacked?

2.2: General National-Resilience Implications of Radio

Governments planning for continuity of operations need VHF/UHF radio at all levels. This much they have probably had since soon after the Hitler war. Indeed I believe that some thought has even been given to VHF/UHF disaster planning at the humbler, mere university level: I think a friend of mine has to some degree raised the question of VHF/UHF links for coordinating, in a situation where Internet and grid (and of course landline and cellular telephone) fail, the disaster-recovery efforts of the University of Toronto downtown, Mississauga, and Scarborough campuses.

Something else, however, is needed too. And this is a thing easily overlooked.

Disaster planning requires attention to radio links beyond the VHF/UHF horizon, i.e., to distances in excess of (nominally) 50 kilometres. How, in a disaster degrading the Internet, can Toronto-based municipal, provincial, and federal authorities be guaranteed links with over-the-horizon Kingston and Windsor? And how, above all, can they be guaranteed links with Ottawa?

It would be foolish to rely on chains of VHF/UHF repeater stations, with communications handled through intricate sequences of relay points. The only robust solutions over Ontario distances, now even as before the Hitler war, lie in the HF ("high-frequency", "shortwave") regime - or still lower, in the MF ("medium-frequency", including and flanking the AM broadcast band) regime; or lower again, in the exotic marine-relevant LF ("low-frequency") regime.

Admittedly, there is likely to be a government emergency land-line link, whether in fiber or in copper, between Toronto and Ottawa, separate from the commercial telecommunications infrastructure. But here we run into the same problem of a repeater-station chain as besets long-haul VHF/UHF.

The Victorians ran telegraph copper all the way across the Atlantic seabed without repeaters. In compensation, however, they had to endure an impossibly low bandwidth, permitting them on the order of 10 just words per minute. At faster speeds, the dots and dashes in their Morse blurred together. What the modern copper-engineering tradeoff is between high bandwidth and paucity of repeaters I do not know. For fiber optics, we have of course bandwidth adequate for government needs - data, voice, and more - but at the high price of a repeater every 100 kilometres or so.

Additionally, landline is vulnerable to sabotage, and to seismic events.

One hopes that government has, apart from dedicated Toronto-Ottawa landline, a dedicated Toronto-Ottawa satellite link. However, satellite, too, has its vulnerabilities, as in military and solar-storm scenarios. For satellites in lowish, rather than in geosynchronous, orbits, there is also the emerging "space junk" vulnerability: https://en.wikipedia.org/wiki/Kessler_syndrome.

For HF and below, the principal vulnerabilities are jamming (but this is nowadays addressed, if it has to be addressed, through burst transmission technology), and Sun-driven ionospheric disturbance. As one goes to lower and lower frequencies, and therefore to longer and longer wavelengths, reliability of communications in the face of ionospheric disturbance increases.

HF is conventionally defined as extending from 3 MHz to 30 MHz. The less distance-capable VHF runs from 30 to 300 MHz, with 300 MHz marking the lower boundary of UHF.

Perhaps governments can at present operate with only a minor HF capability. In future decades, however, as Canada's security situation deteriorates under the twin impacts of climate change and fossil-fuels shortfalls, more will be needed. The principal operational HF-or-lower requirement in the troubled 2020s, or at any rate in the troubled 2030s, 2040s, 2050s, and beyond, will be the high-gain antenna farm, with appropriate directionality. Such antenna farms, with their feedlines and transmitters and receivers, are appropriately located at the edges of Canadian cities. From the suburban HF-or-lower rigs, VHF/UHF links can be run down into the actual urban nuclei of government - in Toronto's case into the Premier's offices beside the University of Toronto downtown campus; into downtown police headquarters; and into the existing Toronto-based Emergency Operations Centre, with its existing radio desks. The result would be that communications could be established between a Toronto nucleus of government and some corresponding nucleus in, say, Ottawa with just three hops: from the Premier's office (e.g.) on VHF/UHF, out to the HF-or-lower setup, with its antenna farm, in the Toronto suburbs; then out on, say, an HF wavelength of around 40 metres or 80 metres to a corresponding HF antenna farm in the Ottawa suburbs; and then on down, with VHF/UHF, to some appropriate situation room housing the Federal cabinet.

Amateurs using HF get by with antenna farms of a rather modest scale. At the University of Toronto Hart House Amateur Radio Club, around 2005 or 2010, I worked two-way Morse with the USA, and on a few happy occasions also with Europe, on that glorified copper clothesline which is the dipole. For government, however, more is needed, on a physical scale no longer modest. The 40-mere dipole, appropriate for telegraphing Europe on just over 7.0 MHz, is physically about 20 metres long, and so could fit (in its for-us-optimal north-south orientation, for strong east-west response) onto the Hart House roof.

In government, however, reliability is paramount. To get through at the appropriate times of day even with a balky ionosphere, officials need something like the log-periodic, the "curtain", or the rhombic design.

For concreteness, let us take the rhombic. At http://www.w8ji.com/rhombic_antennas.htm we find the suggestion that for the 40-metre amateur band (a good choice for Toronto-to-Ukraine in the evening, and for Toronto-to-Ottawa in the morning), "414 feet" is an appropriate overall length. In SI units, to two significant figures, this becomes 130 metres.

But government will surely also want to transmit on a wavelength close to the 80-metre (circa 3.5-MHz) ham band. That would for its part require an antenna farm about 260 metres across at its widest point, i.e., along the major diagonal of the rhombus.

There might additionally be some need for robust government transmissions on a wavelength close to the amateur 160-metre band (in other words, around 1.8 MHz). In that daunting case, the major diagonal of the four-mast rhombus would have to stretch to a length of around 520 metres. (Does one perhaps at this point add intermediate, insulated, masts, to keep the long runs of horizontal wire from snapping under their own weight? Or does one abandon rhombic for curtain - or for some beaming configuration, optimally over watery ground, of vertical-antenna masts?)

2.3: DDO&P Implications of National-Resilience Radio

These HF (or, if we want also something close to the 160-metre ham band, even MF) arrangements are perhaps not too difficult in Ottawa. One Ottawa possibility is an old federal facility, the Experimental Farm. And I know from direct inspection that there is plenty of open farming country around the notorious old Cold War "Diefenbunker", municipally within Ottawa boundaries, and in fact in rural Carp, near the blink-and-you-have-missed-it hamlet of Kinburn.

In Toronto, the task becomes harder. An open site is needed, 20 or so kilometres outside the immediate prospective downtown blast-and-infestation zone (ensuring that the HF capability stays up even if downtown administration is destroyed), and yet not blocked from downtown Toronto by any significant hills.

In our problematic built-up environment, DDO&P proves perhaps the best site available.

Admittedly, some thought will have to be given also to the University of Toronto Mississauga and Scarborough campuses, with their open spaces, and to the various facilities - there is one even in outlying King City - of Environment Canada.

With DDO&P, at any rate, we have a wide terrain, allowing an abundant spread of multiple rhombics for multiple frequencies and multiple beam directions, and space to spare for such possible further requisites as a heliport.

Conveniently also, DDO&P is bounded by a passenger railway and a principal traffic artery (many-lane Bayview Avenue), and is just an easy walk away from Yonge Street.

Additionally, DDO&P offers something perhaps not available elsewhere, namely a heavy masonry structure with free lab space. In the DDO Administration Building basement are three big rooms which over the period 1935/2008 respectively served as a woodworking shop, as an optics shop, and as a metal-machining shop. On the upper floor of the Administration Building is an erstwhile photographic darkroom, whose multiple large sinks could be repurposed for life science - notably in a time of pandemic, or of other regional biohazard. Adjacent to the erstwhile darkroom are two large rooms, each capable of accommodating lab benches for 5 or 10 workers.

The most important asset from a DND-radio perspective, however, is the electronics lab, on this same upstairs level. Around the lab runs an abundance of shelving or cabinet space, appropriate for a busy electronics-repair facility. In the centre of the room is a long two-sided bench, topped in slate for soldering, and with space for 6 - or even for 10 or 12 - individual soldering stations.

There remains space in this big room for some to-me-indeterminate number of desks. Admittedly, desk space is available also in a set of conventional offices on the same (upper) floor, and in a similar set of conventional offices on the ground floor.

The Administration Building roof, apart from three telescope domes, is flat, and communicates with lower floors by staircase rather than by utility ladder. These architectural features would benefit personnel installing or maintaining photovoltaic arrays.

A planning scenario now suggests itself. I offer it as a possible point of departure for DND analysts working in concert with Parks Canada, with the Province, and with the Town of Richmond Hill:

DND embarks on discussions with the would-be DDO developer, DG Group (formerly Metrus) subsidiary Corsica. DND points out that an acquisition would relieve the developer of its growing public-relations embarrassment, while at the same time serving the national interest over coming decades.
DND acquires the prospective 32-hectare subdivision land on fair terms from the developer, conceivably in a land-swap context.
DND does as little as possible with the land in the short term, mindful of the imperative to minimize peacetime defence expenditure. In the short term, DND welcomes citizen groupings to undertake the slow, hard work of reforestation.
DND puts into place legal machinery ensuring that in an emerging national crisis, it can with the full cooperation of the Town of Richmond Hill erect any necessary antenna farm on its 32-hectare terrain, and if necessary even put radio installations onto the 45-hectare Town-of-Richmond-Hill rump park, and can erect any needed outbuildings on its 32-hectare terrain, and can requisition any needed lab and solar-panel space in and on the DDO Administration Building. (In general, the erection of a rhombic requires some trimming in whatever forest may in future be present, and yet - so I conjecture, subject to correction - is unlikely to require a wholesale clear-cut. My discussions with an ex-Soviet specialist, " Mr/Miss/Mrs DEF", this summer are instructive. Mr/Miss/Mrs DEF had seen what was perhaps the principal long-distance HF jamming facility of the USSR, used rather unsuccessfully by the USSR authorities to try to block reception of VOA, BBC, Radio Free Europe, and the like by the USSR citizens. Although this malign transmitter installation radiated far more power than any Canadian government could want for its Toronto-Ottawa link, the scrub forest - the ground was swampy, and so that Siberian scrub included birch - was left rather largely intact.) DND and the Town are in making these contingency plans guided by the precedent set in the Hitler war, in which DND as a DDO on-the-grounds guest carried out research into magnetism (in some such field as the degaussing of naval vessels, as a precaution against magnetic mines) - housing its then personnel, in fact, in what was later to become a DDO building, the extant 1960s-onwards "Radio Shack".

Monday, 22 August 2016

Toomas Karmo (Part B): In Praise of Moise's "Elementary Geometry from an Advanced Standpoint"

[In anticipation of follow-on writing, I have now added the necessary "Part A" to last week's title, and the requisite "Part B" to this week's. Additionally, I have now added to "Part A" the necessary section heading, "1. Introductory Comments, on the Inadequacy of Traditional Geometry Textbooks". - I now continue in this "Section 1", for the reader's convenience repeating my last two sentences.]

One could not agree more. But now what, by way of "clear and distinct abstract reasoning", do we actually encounter in traditional presentations of Euclidean geometry, such as in that disappointing 1919 Godfrey-and-Siddons?

Early on, in fact as Godfrey-and-Siddons "Theorem 12", there is the familiar "pons asinorum" proposition regarding triangles that are either isosceles-and-not-equilateral or equilateral: If two sides of a triangle are equal, the angles opposite to these sides are equal. It is given that ABC is a triangle which has the length AB the same as the length AC. The proof is now via the familiar "S.A.S.", in Godrey-and-Siddons's exposition "Theorem 10": If two triangles have two sides of the one equal to two sides of the other, each to each, and also the angles contained by those sides equal, the triangles are congruent. To make S.A.S. applicable, the unhappy pair of authors consider a line bisecting the Theorem 12 triangle at A. Here they say, "Let it cut BC at D."

But (Moise rightly presses this question) how do we know the existence of such a point D? The postulates offered in Godfrey and Siddons do not suffice for an existence proof.

A similar problem, involving an unsubstantiated existence claim, occurs with the Godfrey-and-Siddons discussion of an actual method for bisecting an angle (I quote verbatim, except that for maximum clarity I reletter): "From PQ, PR cut off equal lengths PX, PY"; "With centres X and Y and any convenient radius describe equal circles intersecting at Z." If the existence of equal-length segments PX, PY is charitably granted (this itself requires some discussion), there nevertheless remains the leap to the assertion that point Z exists. How, in general, do we know that given any points X and Y in Euclidean 2-space, there is some distance r such that the circles at X and Y, of radius r, have at least one point Z in common?

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Turning now to the Godfrey-and-Siddons proof of the underlying "Theorem 10", the "side-angle-side" congruence principle used in proving their Theorem 12, we find a further gap, of an interestingly different kind. Their proof begins, "Apply triangle ABC to triangle DEF so that A falls on D, and AB falls along DE." Their notion of "applying" conceals subtleties. We in fact need to note (as I think, on my cursory revisit to their book this week, Godfrey and Siddons fail to note) two different situations in Euclidean 2-space.

(a) Triangles ABC and DEF may possibly be such that through some combination, in some order, of zero or more within-the-Euclidean-plane translations of ABC and zero or more within-the-Euclidean-plane rotations of ABC, the vertices and angles of ABC and DEF may be brought into coincidence - as at least one of the conceivable correspondences ABC-DEF, or ABC-DFE, or ABC-EDF, or ABC-EFD, or ABC-FDE, or ABC-FED. In this case, let ABC and DEF be called "within-Euclidean-plane superposable".

Since the just-proffered definition says "zero or more", not "one or more", it is trivially true that for every triangle ABC, ABC and ABC are within-Euclidean-plane superposable. This small fact will soon prove helpful, as a lead-in to Pappus's improvement on Euclid's "pons asinorum" page.

(b) Triangles ABC and DEF may possibly be such that through some combination, in some order, of zero or more within-the-Euclidean-plane translations of ABC and zero or more within-the-Euclidean-plane rotations of ABC and zero or more within-the-Euclidean-plane reflections of ABC through some line, the vertices and angles of ABC and DEF may be brought into coincidence - as at least one of the conceivable correspondences ABC-DEF, or ABC-DFE, or ABC-EDF, or ABC-EFD, or ABC-FDE, or ABC-FED. In this case, let ABC and DEF be called "congruent".

The concept of congruence proffered here matches the traditional plane-geometry textbook concept of congruence, but (this is my main point) is looser than the concept of triangles that are within-Euclidean-plane-superposable.

If triangles ABC and DEF are congruent, and yet are not within-Euclidean-plane-superposable, then ABC can admittedly be brought into coincidence with DEF by taking ABC out of the Euclidean plane, and flipping it over in Euclidean 3-space, and returning it to the Euclidean plane, and doing some translations and/or rotations within the plane.

The distinction is slurred over by Godfrey and Siddons, who unfortunately write of figures (for instance, triangles) that "when applied to one another can be made to coincide (i.e. fit exactly)": such figures, they add, "must be equal in all respects", and "figures which are equal in all respects are said to be congruent". This language is unfortunate. "Can be made to" is not spelled out. The meaning must be "can be made to by a process of translation and rotation in Euclidean 3-space, not necessarily just in Euclidean 2-space". But how is the hapless reader supposed to divine that meaning?

When we leave two-dimensional Euclidean geometry for Euclidean solid geometry, the problem ramifies, becoming an illustration of the topic of "handedness", or (after an ancient Greek root) "chirality" - important in electromagnetics, and if we are not careful leading us into the terrifying world of hyperspace.

For consider two modest solids. One is a Roman "R", with its various straight and curved strokes of some nonzero width, and with the whole letter of some nonzero thickness. The other is a Cyrillic "ya" (the backwards R, the last letter of the Russian alphabet), of the same volume, with matching segment lengths and matching arc radius and matching angles, and so on - being in fact a three-dimensional mirror image of the given R.

The R and the ya are in a straightforward sold-geometry sense - analogous to what we have already developed for Euclidean plane geometry - not only congruent, but even superposable. (As in the two-dimensional case superposition is to be achieved by rotations and translations in 2-space, so in this present, Euclidean solid-geometry, case superposition is to be achieved by rotations and translations in 3-space.)

Imagine, now, the R and the ya to be decorated, each with one small hemispherical dimple in its front surface, where the oblique stroke meets the bowl. If the tiny dimples are scooped out in such a way that the R and the ya remain mirror images, then the two solids are in a straightforward sense congruent-and-yet-not-superposable.

This is like the difference in electromagnetics between a right-handed coil and a left-handed coil. A mirror-image pair of such coils, subject to identical conditions of time-varying magnetic flux, will by Faraday's law of induction develop equal-but-opposite voltages across their terminals.

It seems that similar examples can be found in stereochemistry. The picture of amino-acid "enantiomers" at the beginning of https://en.wikipedia.org/wiki/Chiral is suggestive.

The language of Godfrey and Siddons notwithstanding, there is no sense in which the dimpled R and the dimpled ya "can be made to coincide" - unless, indeed, we leave Euclidean 3-space for the terrifying, unvisualizable, world of Euclidean 4-space.

A more satisfying approach to the "Theorem 12" of Godfrey and Siddons is one pioneered a couple of decades after Godfrey and Siddons, by Harvard authority George David Birkhoff (1884-1944; he is considered one of the major American mathematicians of his era). Birkhoff was writing for a school-mathematics community, as a collaborator with educator Ralph Beatley, in a book bearing the title Basic Geometry. Their third and final edition was printed, if not earlier, then at any rate in 1959, at the publishing house Chelsea. A sign of excellence is that this same edition was reissued in 2000 by the American Mathematical Society, under ISBN 978-0-8218-2101-5.

Introduce (says Birkhoff-with-Beatley; I paraphrase from memory) S.A.S. as a postulate, in the form "Let triangles ABC and DEF be congruent in the sense that AB and DE are of equal lengths, and AC and DF are of equal lengths, and the internal angles at A and D are of equal measure; then the sides BC and EF are of equal lengths, and the sides CA and FD are of equal lengths, and the internal angles at B and E are of equal measure, and the internal angles at C and F are of equal measure."

We can now rapidly prove "Theorem 12", without recourse to the problematic claim that a bisector of the apex angle intersects the triangle base. If sides AB and AC are of equal length, triangle ABC satisfies the S.A.S. postulate, in that ABC is congruent with itself (in the protasis, we innocently take D to be A, cunningly take E to be C, cunningly take F to be B); with S.A.S. assumed, and with the S.A.S. protasis found to be satisfied, each part of the apodosis can now be asserted. In particular, the internal angle at B is now guaranteed to have the same measure as the internal angle at C.

Moise has not only the virtue of rigour, but the virtue of classical erudition. For in pointing this terse proof out, and remarking that Euclid's proof runs by contrast to the length of a whole printed page, he addes that the terse proof was known in antiquity to Pappus.

Moise has also a third virtue, namely the ability to be witty at the expense of computer scientists. He points out that the simple proof weas inadvertently rediscovered in a comp-sci lab:

Not many years ago - or so the story goes - an electronic computing machine was programmed to look for proofs of elementary geometric theorems. When the pons asinorum theorem was fed into the machine, it promptly printed Pappus' proof on the tape. This is said to have been a surprise to the people who had coded the problem; Pappus' proof was new to them. /.../ if you want a machine to get the idea that the triangles /... / in the SAS postulate are supposed to be different, you have to say so explicitly. It didn't occur to anybody to do this, and so the machine proceeded, in its simple-minded way, to produce the most elegant proof.

What I have just just quoted is from pp. 105 and 106 of Moise's final, third, edition, published in 1990. The first edition, from an earlier era in computing, has the same anecdote, but with some additional disparaging remark on "vacuum tubes".

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[To be continued next week, I hope, as "Part C". - To anticipate a little, "equal" has to be unpacked in the manner of mathematical logician Frege, with "x equals y" having just one correctly clear meaning: "x is the same object as y". I guess I will be citing here Frege's dictum that the only acceptable mathematical sense for "equals" is the one in which the heavenly body Hesperus "=", i.e., is the same planet as, the heavenly body Phosphorus: the thing you occasionally see in the evening sky, and are wont to call Hesperus, is the same entity, being the planet Venus, as the thing you occasionally see in the morning sky, and are wont to call Phosphorus.]

Monday, 5 September 2016

Toomas Karmo (Part C): In Praise of Moise's "Elementary Geometry from an Advanced Standpoint"

2. An Overview of Moise's Excellence

Toomas Karmo: Paramonastic Scientists and the Theology of Disability

Monday, 29 August 2016

Toomas Karmo: Remark in Lieu of Forthcoming Part C of Essay-in-praise-of-Moise

Toomas Karmo (=VA3KMZ): Moral-Uplift Images Pertinent to Radio and Department of National Defence

Toomas Karmo (=VA3KMZ): Open Letter re DDO&P DND (Department of National Defence) Implications

1. Update: Ongoing Oak Ridges Moraine Aquifer Cap Investigation

2. Fresh Considerations: DDO&P and the Department of National Defence

2.1: Background Regarding Greater-Toronto-Area Resilience

2.2: General National-Resilience Implications of Radio

2.3: DDO&P Implications of National-Resilience Radio

Monday, 22 August 2016

Toomas Karmo (Part B): In Praise of Moise's "Elementary Geometry from an Advanced Standpoint"

1. Update:
Ongoing Oak Ridges Moraine Aquifer Cap Investigation