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

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.] 

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