__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"): 3/5. Justification: I had to make this installment of my essay shorter than would have been desirable, since time was running out.**

__Revision history:__

**UTC=20160823T0002Z/version 1.2.0: Kmo improved title, adding a necessary "Part A", and added a necesssary section heading.****UTC=20160816T1355Z/version 1.1.0: Kmo corrected a really bad mistake. He had unthinkingly asserted that every isosceles-but-not-equilateral triangle has exactly two dominant angles. (If the equal angles, at the base, are each less than 60 degrees, then the isosceles triangle has exactly one dominant angle, at its apex.) - He retained the right to make nonsubstantial tweaks, as here-undocumented versions 1.1.1, 1.1.2, 1.1.3, ...., over the coming 48 hours.****UTC=20160816T0004Z/version 1.0.0: Kmo uploaded base version. He retained the right to make nonsubstantive tweaks, as here-undoumented versions 1.0.1, 1.0.2, 1.0.3, ... , over the coming 48 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.]**

## 1. Introductory Comments, on the Inadequacy of Traditional Geometry Textbooks

There is a line somewhere in liturgy about "all who bring hope to this world".
English-reading devotees of undergraduate mathematics know that among those who bring hope is the still-living Prof. Michael Spivak (1940 - ), in his rigorous presentation of univariate calculus.

Many will additionally know at least a little of Spivak's contemporary, Tom Mike Apostol (born 1923, departed this life 2016-05-08), who expounds univariate and multivariate calculus in a bold two-volume work. It takes a radical defiance of classroom convention to develop integral calculus (the study of single-variable Accumulations) before univariate differential calculus (the study of single-variable Rates). But Prof. Apostol did, I suspect in his process somehow - this is a bit I have not yet glanced at - underscoring the inevitability of the "Fundamental Theorem", which ties those two branches of calculus together (asserting the Accumulation, from x=a to x=b, of rate-of-change-in-F, to be none other than F(b) - F(a)).

How many, on the other hand, know of Edwin E. Moise (1918-1998), as a correctly rigorous expositor of geometry?

****

Geometry has long been in a bad state.

A little earlier this summer, aware that one of the leading twentieth-century mathematicians was University of Toronto geometer Harold Scott MacDonald Coxeter (1907-2003), I cracked open a library copy of some appropriate book. This was perhaps Coxeter's 1961

*Introduction to Geometry*.*Ai-ai-ai-ai-ai*, and*oy veh*. Here was an "Introduction" in the sense in which some Nobel laureate - some Claude Cohen-Tannoudji - might write an "Introduction to Quantum Mechanics".
And yet I had persistently in my mind the uneasy, almost heretical, thought that it might be geometric truths, not algebraic truths, that are fundamental. I was therefore determined to keep working, even in the teeth of discouragement.

****

We come up here against a mystery, a

*mysterium tremendum ac fascinans*. Everyone connected, however weakly, with the world of science must have at least a few nightmarish thoughts. I wonder, admittedly as a non-biologist, whether we might not some day have to reject wide swathes of the Theory of Evolution, even while continuing to profess the veracity of the fossil record.
Here, for instance, is a beehive in August. The young worker bees are inside, building their comb and tending to their queen's larvae. On the landing board arrives an older scouting worker bee, approaching the end of her so-brief summer life. In addition to her minuscule load of nectar or pollen, she bears Information, of a character potentially vital to the winter survival of her "colony" (in the specialized language of apiculture), or in more mundane language her city.

Making her way inside, into the warm darkness, she starts her dance.

I believe the dance was deciphered by Karl Ritter von Frisch (1886-1982; Nobel Prize in Physiology or Medicine, 1973, with Tinbergen and Lorenz): the nature of her discovered nectar-and-pollen source, and its direction and distance, get duly communicated to her attentive colleagues. After a short while, ten or twenty or more of those colleagues are briefed, almost in a tiny simulacrum of the RAF. Having understood, they fly forth on their own massive foraging, to the correctly indicated destination, perhaps even two or three or four or (in extreme cases) five or more kilometres away.

That collective foraging - perhaps it will be on a hundred-metre long stand of goldenrod along some sunny highway edge, just coming into its late-August bloom - will turn the tiny initial donation of pollen or nectar into a winter resource big enough to underpin the survival of the entire 60,000- or 30,000-citizen metropolis.

How, I uneasily ask, could Darwinian selection explain such a remarkable attainment, involving as it does even the use of symbols?

****

And a little more to the present point, I have, in my admitted capacity as the Village Idiot in all things scientific, intermittent nightmarish thoughts on Mathematics.

Could it be that progress in physics is currently being blocked, somewhere, by a dearth of mathematical machinery sufficiently close to underlying physical realities - even as the physics of the Greeks was frustrated by their lack of a formalism for describing those centrally important Rates which are accelerations? Should we, perhaps, be using the Interval Arithmetic of https://en.wikipedia.org/wiki/Interval_arithmetic rather than the familiar infinite-precision real number line to handle laboratory measurements? Might it, perhaps, be even some kind of subtle error or oversimplification to formulate quantum mechanics - beset as it is with problems of philosophical interpretation - in terms of that familiar, tidy, Argand complex-number plane?

Might there even (this is one of the ultimate nightmares) be some sense in which the mathematics of physics has to start from geometry, with our so slick, so facile contemporary algebra becoming a mere ancillary?

****

So, as I say, I was wrestling with the "mysterium tremendum ac fascinans", finding Coxeter too advanced to permit safe flying.

I had, to be sure, logged a total of 203 hours, 3 minutes over the period 2009-05-05/2009-08-25 from working the

*Elementary Geometry: Practical and Theoretical*of G.Godfrey and A.W.Siddons (3rd edition; Cambridge University Press, 1919). But this was a vexing experience, which in the end felt only moderately better than time-wasting busy-work.
Geometry, it is said, trains the intellect in rigour. Well, do allow me some comment.

I try to keep an occasional eye on two small institutions which might mark a way forward for Catholic tertiary education, as our cultural decline morphs over the next few generations into a Dark Age. I have already cited the two in my blog posting from the first week in 2016 July, headed "Part E" of "Is Science Doomed?", under section heading "Practicalities of Triage: Education in a Catholic Setting".

(1) At Our Lady Seat of Wisdom Academy in rural Ontario, mathematics, notably including euclidean geometry, is celebrated in the following terms, in a discussion somewhere under the "Academics" tab at https://www.seatofwisdom.ca:

*Mathematics, primarily via Calculus, Statistics and Euclidean Geometry, leads the mind to clear and distinct abstract reasoning, prescinding from matter.*
(2) From the other institution, the rather rural Wyoming Catholic College, comes an almost identical sentiment, on a Web page - http://www.wyomingcatholiccollege.com/academics/Curriculum/mathscience/index.aspx - decorated with a proof either of the Pythagorean Theorem ("If a triangle is a right triangle, the area of a square erected on the side opposite a dominant angle equals the sum of the areas of the squares erected on the other two sides") or of its converse ("Only if a triangle is a right triangle does the area of a square erected on the side opposite a dominant angle equal the sum of the areas of the squares erected on the other two sides"; by "dominant angle", I mean here "Any angle of the given triangle than which there is none greater in that triangle" - so that every triangle which is isosceles-and-not-equilateral has either exactly two dominant angles or exactly one dominant angle, and every equilateral triangle has three dominant angles, and every triangle outside the two just-cited special cases has exactly one dominant angle). Wyoming Catholic College writes as follows, with reference to "Arithmetic and Geometry":

*/.../ its highly logical structure and freedom from the imprecision of material being makes it ideal for elementary training in reasoning, where the beauty of mathematics both inspires wonder and makes evident the human mind's thirst for understanding.*
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?

**[To be continued next week, in the upload scheduled for the four-hour UTC interval 20160823T0001Z/20160823T0401Z. This entire essay will have to run over something like two or three or four installments, therefore probably finishing either late in August or at some point in September.]**

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