Polished papers from my study notes on Moise, showing (as I discuss today) a suitably flexible logical formalism for the laying out of proofs. |

__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 has time to make the necessary points to adequate length.**

__Revision history:__**20161011T1502Z/version 2.1.0: Kmo improved a couple of tiny mathematical points (noting, for instance, that surface integrals can be defined even on a non-orientable manifold, and that it is rather in the more demanding case of the flux integral that orientability becomes mandatory; and he also improved his off-the-cuff remark on Dilemma and inclusive-versus-exclusive-disjunction). He reserved the right to upload further cosmetic, non-substantive, revisions over the coming 48 hours, as here-undocumented versions 2.1.1, 2.1.2, 2.1.3, ... .****20161011T0418Z/version 2.0.0: Kmo finished uploading a version with all necessary points written out, and with the outline-form lines removed. He reserved the right to upload further cosmetic, non-substantive, revisions over the coming 48 hours, as here-undocumented versions 2.0.1, 2.0.2, 2.0.3, ... .****20161011T0001Z/version 1.0.0: Kmo uploaded only part of the base version, putting the rest down merely in outline form. He hoped to fill everything in over the coming four 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.]__How much formal-logic machinery should be exhibited in our proofs, as we work on geometry in the rigorous spirit of Moise, deducing theorems from postulates?__

**How Moise readers might be encouraged to achieve a Golden Mean in logical formalism:**
I suspect it is common for school pupils to write out geometry proofs in columns, with steps on the left accompanied by justifications on the right. But I suspect professional mathematicians do not write this way in the primary, research-disseminating, journals. Such a two-column style is certainly not prominent among the good maths authors on my (admittedly modest, and admittedly merely English-language) shelves - these good authors being in my (modest) case analyst Spivak, analyst Apostol, geometer Moise-as-borrowed-from-library, analyst-and-toplogist Munkres (I will come back to him at the very end of today's blog posting), and a few others. (Sorry, folks. I am way too far down on the scientific totem pole to be able to put onto my shelves the Real Heavies, such as Russian Grigorii Mikhailovich Fichtenholz (1888-1959; more formally Григорий Михайлович Фихтенгольц; I commented on him here, in a blog upload of 2016-07-04 or 2016-07-05) or German David Hilbert (1862-1943).) I find a different formalism helpful in my polished private notes.

But before explaining this, let me develop a little more the concept of "polishing".

"Polished" private notes are to be distinguished from "rough" private notes.

It proves helpful when reading to close the book, to try proving the author's theorem or lemma or off-the-cuff remark on one's own, and then to compare one's own work against the author's proof. (Often, as it turns out, the author will be more concise, and on various occasions also the author will show some special, superior, insight.)

In such work, one wants first to write things out informally and hastily, on scrap paper, drawing whatever quick diagrams might be useful (erasable coloured pencils can make things vivid), and to use colloquial language - as it might perhaps be, "The Red Thingie and the Blue Thingie can be proved congruent if I can first establish that the Green Angles are of identical measure".

One then wants to transfer the scrap-paper solution to polished-work papers, ensuring in the process of transfer that all the colloquial language gets replaced with language at the Spivak-Apostol-Moise-Munkres level of professionalism, and that the gamut of possible cases gets covered adequately by one's polished-work diagrams. It may, for instance, prove necessary to replace one single, impressionistic, diagram from the rough-work stage with two separate diagrams, explicitly separating the cases in which a given triangle-of-interest has its largest angle

Disillusionment is possible in this process of transfer. What looked rigorous on scrap paper might prove, at the time of polished writing, confused or gappy. (For instance - a common kind of gap - it might turn out that one's rough work was basically acceptable, and yet omitted to discuss one or two pathological, "extremely special and rare-in-practice", cases.)

For prudence, I file all polished-work paper, even putting onto each sheet an annotation of Universal Coordinated Time (UTC) year, month, day, hour, and minute, and in quite a few specially weighty cases making a further Debian GNU/Linux log of the completed polished work.

With the polished proof finally brought to a finished stage (perhaps scrap paper will have to be laid out more than once), and duly compared against the now-opened textbook (if things have gone badly, my censorious red pen might now be making ugly adverse comments on my polished proof), the scrap paper can be discarded.

I have sought a formalism which is less confining than the two-column straightjacket, and which makes it possible (as the two-column formalsm does not) to see quickly how even an intricate proof is structured.

I present my formalism by taking one example.

On going through my folder Debian GNU/Linux-indexed as

But before explaining this, let me develop a little more the concept of "polishing".

"Polished" private notes are to be distinguished from "rough" private notes.

It proves helpful when reading to close the book, to try proving the author's theorem or lemma or off-the-cuff remark on one's own, and then to compare one's own work against the author's proof. (Often, as it turns out, the author will be more concise, and on various occasions also the author will show some special, superior, insight.)

In such work, one wants first to write things out informally and hastily, on scrap paper, drawing whatever quick diagrams might be useful (erasable coloured pencils can make things vivid), and to use colloquial language - as it might perhaps be, "The Red Thingie and the Blue Thingie can be proved congruent if I can first establish that the Green Angles are of identical measure".

One then wants to transfer the scrap-paper solution to polished-work papers, ensuring in the process of transfer that all the colloquial language gets replaced with language at the Spivak-Apostol-Moise-Munkres level of professionalism, and that the gamut of possible cases gets covered adequately by one's polished-work diagrams. It may, for instance, prove necessary to replace one single, impressionistic, diagram from the rough-work stage with two separate diagrams, explicitly separating the cases in which a given triangle-of-interest has its largest angle

*(a)*less than or equal to a right angle, and*(b)*greater than a right angle. Ot it might prove necessary to distinguish the case of*(a)*a convex*n*-gon from*(b)*one which is not convex, and instead insultingly sports one or more internal angles each of degree measure greater than 180. And in work involving trigonometry, one might want four diagrams, adequately covering a situation in which the directed angle-of-interest, while having as its initial ray the non-negative*x*axis, might have its terminal ray in any one of the four quadrants.Disillusionment is possible in this process of transfer. What looked rigorous on scrap paper might prove, at the time of polished writing, confused or gappy. (For instance - a common kind of gap - it might turn out that one's rough work was basically acceptable, and yet omitted to discuss one or two pathological, "extremely special and rare-in-practice", cases.)

For prudence, I file all polished-work paper, even putting onto each sheet an annotation of Universal Coordinated Time (UTC) year, month, day, hour, and minute, and in quite a few specially weighty cases making a further Debian GNU/Linux log of the completed polished work.

With the polished proof finally brought to a finished stage (perhaps scrap paper will have to be laid out more than once), and duly compared against the now-opened textbook (if things have gone badly, my censorious red pen might now be making ugly adverse comments on my polished proof), the scrap paper can be discarded.

I have sought a formalism which is less confining than the two-column straightjacket, and which makes it possible (as the two-column formalsm does not) to see quickly how even an intricate proof is structured.

I present my formalism by taking one example.

On going through my folder Debian GNU/Linux-indexed as

**/home/karmo/AANN____workworld/**

**GNNN____studies_private/**

**Q_______science/QA____maths/**

**QA00000455____geometry_elementary_euclidean__also/**

**QA00000455.M58.1990____moise_elementa**

**ry_geometry_from_an_advanced_standpt__edn3/**

**AANN____studium_of_20160811T214337Z/**

**LACN____comprehension_chap03__distance_and_congruence____HARDCOPY**
I find a polished-notes comment relating to Moise p. 65, and spanning four clean-writeup sheets (actually "p. 065@0.2", or "remark in textbook 20% of the way down p. 65"). I show two sheets from the four in a screenshot at the top of this blog posting.

We see in the upper-right-hand corner of each of the two sheets an identifier relating the paper to the book ("p. 065@0.2" means, to reiterate, "this relates to p. 65 of the book, a remark 20% of the way down the page"), with a timestamp "20160903T1907Z", indicating that I embarked on this particular writeup at 19:07 UTC, i.e., at 15:07 EDT, on 2016-09-03.

I have blocked out irrelevant, and potentially distracting, material in my two exhibited sheets, by piling on some writing or drawing tools - very notably three things mission-critical in rough-paper thinking, an erasable blue pencil and an erasable green pencil and an erasable red pencil.

In what remains visible, it can be seen that I am as a step in my wider argument seeking to establish, from postulates or previously proved therems, that for every point

Various helpful steps in the work I label, with little circled indicators: we see here "ASS P [is-a-member-of] [segment AB]", "102", "103, "104", and "201". It is perhaps just barely necessary to reassure the reader that "201" does not mean that there are 200 previous cited lines! I attach labels in any way convenient, often using "1", "2", and "3", and then "11", "12", and then "101" and "102" and the like, and then perhaps "201", with also many wordy labels, such as "ASS foobar" or "ASSU foobar" for an "assumption foobar".

It will be seen that rather early in this chain of reasoning, I introduce an assumption, that

To some extent, the problem may stem from an insufficient acquaintance with formal logic. This is one failing that can be alleviated rather easily.

In the best of all possible worlds, students will have an explicit formal-logic course or two, either before they start pure mathematics or running concurrently with their earliest pure-mathematics courses.

Departments of Philosophy are in some respects dreary. The cruel old joke says it all:

But such Departments do have an honourable tradition, honed as of 2016 through four or five, or on some campuses even as many as eight, decades, in teaching formal logic to a mathematically adequate standard. Drawing on best international Departments-of-Philosophy practice, one can confidently enough propose a syllabus for a course or a two-course sequence:

We see in the upper-right-hand corner of each of the two sheets an identifier relating the paper to the book ("p. 065@0.2" means, to reiterate, "this relates to p. 65 of the book, a remark 20% of the way down the page"), with a timestamp "20160903T1907Z", indicating that I embarked on this particular writeup at 19:07 UTC, i.e., at 15:07 EDT, on 2016-09-03.

I have blocked out irrelevant, and potentially distracting, material in my two exhibited sheets, by piling on some writing or drawing tools - very notably three things mission-critical in rough-paper thinking, an erasable blue pencil and an erasable green pencil and an erasable red pencil.

In what remains visible, it can be seen that I am as a step in my wider argument seeking to establish, from postulates or previously proved therems, that for every point

*P*which is a member of line segment*AB*,*P*is a member of the ray from*A*through*B*.Various helpful steps in the work I label, with little circled indicators: we see here "ASS P [is-a-member-of] [segment AB]", "102", "103, "104", and "201". It is perhaps just barely necessary to reassure the reader that "201" does not mean that there are 200 previous cited lines! I attach labels in any way convenient, often using "1", "2", and "3", and then "11", "12", and then "101" and "102" and the like, and then perhaps "201", with also many wordy labels, such as "ASS foobar" or "ASSU foobar" for an "assumption foobar".

It will be seen that rather early in this chain of reasoning, I introduce an assumption, that

*P*both is a member of the line containing*A*and*B*and is between*A*and*B*, and cite a Moise theorem (3.4.B-2, which I think of as the "exactly-one-sits-in-the-middle theorem" - this has to do with points and betweenness), and deduce from the assumption that*P*is a member of the ray from*A*through*B*, and then (by drawing a box) discharge the assumption, inferring a conditional: "If*P*both is a member of the line containing*A*and*B*and is between*A*and*B*, then*P*is a member of the ray from*A*through*B*." This conditional I label "102". By further reasoning, inside a box, I on the next sheet of paper deduce the conditional labelled "103", and then (I write into my box merely a note, that previous reasoning is to be imitated) the conditional labelled "104". It is now time to pull things together (toward the bottom of the second sheet), drawing an appropriate inference from 101, 102, 103, and 104 jointly. And so on.**In classes, teachers are liable to find their students hampered by lack of practice in writing polished mathematics. The students will complain that they know what they want to argue, and yet cannot lay it all out in a correctly disciplined way.**__How Moise might be supplemented with a small formal-logic workbook:__To some extent, the problem may stem from an insufficient acquaintance with formal logic. This is one failing that can be alleviated rather easily.

In the best of all possible worlds, students will have an explicit formal-logic course or two, either before they start pure mathematics or running concurrently with their earliest pure-mathematics courses.

Departments of Philosophy are in some respects dreary. The cruel old joke says it all:

*For a University Dean, a physicist is frightfully expensive. You need the liquid helium, the X-ray machine with its lead shielding, the cobalt-60 and other sources, the lathes and glass-blowers, the supercomputing cluster, the cleanroom, and other costly things.*

*A pure mathematician, on the other hand, is a Dean's delight. In this case you need only supply a desk, a desklamp, paper, pencils, and a wastebasket.*

*A philosopher is from a Dean's standpoint better still. In this case, you need not supply a wastebasket.*But such Departments do have an honourable tradition, honed as of 2016 through four or five, or on some campuses even as many as eight, decades, in teaching formal logic to a mathematically adequate standard. Drawing on best international Departments-of-Philosophy practice, one can confidently enough propose a syllabus for a course or a two-course sequence:

- Let two-valued classical propositional logic, and on top of it first-order quantification theory, be developed principally and foremost in the framework of "Natural Deduction". (This is a natural tightening-up of the expository style illustrated in today's two-sheet screenshot. When I was teaching in the 1980s, John Lemmon's
*Beginning Logic*was found good for the job.) But let there also be some explicit reference to Gentzen's style of writing out "natural deduction", and let there additionally be some explicit development of "semantic tableaux". (I used to find Wilfrid Hodges' 1977 paperback,*Logic - An Introduction to Elementary Logic*fine for tableaux.) - In at least one of these settings, let there be, perhaps via lecturer coursenotes, a development of canonical-model completeness proofs for first-order quantification. - Let a lecture or two or three or so be devoted to the difficult old 1920s way of doing this same stuff, as the "axiomatics" prominent in the Polish school. (That is the school of Łukasiewicz
*et al.*, from well before the Hitler war: at this point some Poles were busily pioneering the decryption of Enigma, and I like to think - I have not researched this yet - that the Polish cryptanalysts will have had at least some degree of connection with the just-cited school. - Arthur Prior is an Anglo-Saxon expositor of prewar Polish logic, sensitive to the fine points in their axiomatics. - Let there be some pretty careful further development, to a length of a whole semester, of Gödel, Turing, and related incompleteness topics (with a full development of Turing machines: students should write and run their own machines, on a computer simulation, to keep everything vivid). Let there be some concomitant philosophy-of-mathematics discussion, on the "inevitable limitations of postulate systems".
- Let one further semester be devoted to modal logic, both in calculi of necessity (with special emphasis placed on Kripke model theory, and on system K, the "Brouwersche" system, S4, and S5) and obligation (as when we note that even the empty set of premisses has to entail "It is obligatory that if it is obligatory that p then p," i.e., "It is obligatory that whatever is obligatory be actual").
- Let this further semester make at least some mention of quantified modal logic and quantified tense logic. (This is a field in which I myself did some unpublished 1980s work, explaining how by assigning scopes to free variables one can block the undesirable modal-logic inference from "Possibly some entity S-eth" to "Some entity is such that possibly it S-eth", and correspondingly can block the undesirable tense-logic inference from "In future some entity S-eth" to "Some entity is such that in future it S-eth." I did canonical-model completeness proofs for quite a few systems.)

In the absence of such a course, much good could still, I conjecture, be done by having the students go through a simple workbook. This would be a workbook embodying the educational philosophy of astrophysicist Prof. Fred Hoyle, as I quoted him in my blog upload of 2016-06-15 or 2016-06-16.

Hoyle, it may be recalled, recommended having students discover things for themselves, typically through sequences of rather simple confidence-building exercises. So in this workbook, there will be lots of confidence-building exercises, intended to convey the general spirit and method of formal logic without setting up detailed (Lemmon, Hodge, Gentzen, ... ) formalizations.

Hoyle, it may be recalled, recommended having students discover things for themselves, typically through sequences of rather simple confidence-building exercises. So in this workbook, there will be lots of confidence-building exercises, intended to convey the general spirit and method of formal logic without setting up detailed (Lemmon, Hodge, Gentzen, ... ) formalizations.

The workbook would, for example ask students to explain, in any appropriately clear informal terms, which of the following inferences are valid, and which invalid:

*If the precipitate is brown, then the solution contained iron ions; the solution did not contain iron ions; therefore the precipitate is not brown;**Only if the precipitate is brown did the solution contain iron ions; the solution did not contain iron ions; therefore the precipitate is not brown;**The precipitate is brown if the solution contained iron ions; the solution did not contain iron ions; therefore the precipitate is not brown;**The precipitate is brown only if the solution contained iron ions; the solution did not contain iron ions; therefore the precipitate is not brown.*

There would have to be some question or questions probing DeMorgan with inclusive-or and DeMorgan with exclusive-or (for which of these two kinds of "or" is "not both p and q" equivalent to "either not p or not q"?), and in the context of the two kinds of "or" addressing also Dilemma ("Either the particle is a muon or it is a pion; if the particle is a muon, then it decays under Schema A; if the particle is a pion, then it decays under Schema B; therefore the particle decays either under Schema A or under Schema B").

After a few further such topics, notably Reduction ad Absurdum, in classical two-valued propositional logic, it will be time, in this modest workbook, for quantifiers.

The workbook should have some exercises developing an appreciation for the difference between "for all...there exists" and "there exists...for all" ("Every critic admires some novelist or other"; "There is a novelist who is admired by all critics"). Russell's 1905 Theory of Descriptions, with its analysis of "The one and only entity which is an S is a P" (a topic I discussed in my posting of 2016-10-03 or 2016-10-04) can get a couple of exercises, involving writing out binary quantifiers with explicit bound variables (formalized relative pronouns), as I did in that posting.

A couple of pages can be devoted to Aristotle's Square of Opposition, underscoring the need to expand this into an Octagon of Opposition: "Every entity which is an S is a P"; "No entity which is an S is a P", "Only those entities which are S are P" (i.e., "No entities but those which are S are P"); "All entities but those which are S are P", along with their negations. Here the student can be alerted to the difference between Euler diagrams - often misdescribed as Venn diagrams - and Venn diagrams in the strict sense, and can be asked to draw Venn-diagrams-in-the-strict-sense for all four of the just-cited propositions, along with their four respective negations. Some exploration should be made of logical relations among the various propositions in the Octagon - a topic on which I wrote in my 1980s article in the

What to do in such an overview, or "propadeutic" - or indeed in a full-blown logic course or pair of courses - about Heyting, Intuitionism, Brouwer, and the scary vistas of A.S.Essenin-Volpin (more formally Александр Сергеевич Есенин-Вольпи; 1924-2016) in philosophy of mathematics and Prof. Sir Michael Dummett (1925-2011) in general logic-driven philosophy I know not. But Catholic schools will have to note that Prof. Sir Michael Dummett is acknowledged a pre-eminent authority in the ranks of modern Catholic thinkers. I presume Deans in such schools will some day have to go even where I today fear to tread, as they decide what teaching to arrange.

##
POSTSCRIPT: Moise's later-life contribution

As the eminent George David Birkhoff (originator of the geometry-axiomatization approach chiefly stressed by Moise) brought out, with a collaborator, a school textbook, so also did Moise, at the very end of his long life - well after producing the university textbook which is here mainly under discussion.

I have already noted on this blog that in Prof. Birkhoff's case, the book was

In Moise's case, we find

Here, then, is an elementary treatment under Moise's imprimatur which we may suspect avoids the infelicities of the 1919 Godfrey-and-Siddons - even while not being able to probe as deeply as a university textbook, such as Moise's

##
POST-POSTSCRIPT: But as for me,

I have to end by confessing to forsaking university-standard geometry, at least for the time being. We must work at the most fundamental level possible, seeking to grasp concepts as deeply as our time and our nerves may allow. Alas, it seems to me that the most important thing for me to study now is not, after all, geometry!

At a deep level, we must ask, "What is a space, in the most general sense?" Here one prescinds even from questions of angles and distances, so central both in Euclidean geometry and in its non-Euclidean alternatives.

I used to wince inwardly at the mention of "topological spaces", thinking them the exclusive preserve of the scientifically eminent. But it does seem that it is topological spaces that are spaces in the most general sense. Moreover, it does now seem that topology is at the foundation of rigorous multivariate real analysis - central to astrophysics, and to the physics of radio (to cite two fields in which I want to make progress). Many of my longstanding calculus insecurities, I now feel, stem from my ignorance in the rudiments of topology.

I take as an instance the very book that has caused me unparalleled grief over the last twenty-odd years, Spivak's

So away, for the time being, with earnest, plodding Geometry, and All Hail, for the time being, Topology.

One might well say: Hail, Horror! -

As I end this six-part essay I sally forth, then, clutching my expensive-and-yet-second-hand Munkres

After a few further such topics, notably Reduction ad Absurdum, in classical two-valued propositional logic, it will be time, in this modest workbook, for quantifiers.

The workbook should have some exercises developing an appreciation for the difference between "for all...there exists" and "there exists...for all" ("Every critic admires some novelist or other"; "There is a novelist who is admired by all critics"). Russell's 1905 Theory of Descriptions, with its analysis of "The one and only entity which is an S is a P" (a topic I discussed in my posting of 2016-10-03 or 2016-10-04) can get a couple of exercises, involving writing out binary quantifiers with explicit bound variables (formalized relative pronouns), as I did in that posting.

A couple of pages can be devoted to Aristotle's Square of Opposition, underscoring the need to expand this into an Octagon of Opposition: "Every entity which is an S is a P"; "No entity which is an S is a P", "Only those entities which are S are P" (i.e., "No entities but those which are S are P"); "All entities but those which are S are P", along with their negations. Here the student can be alerted to the difference between Euler diagrams - often misdescribed as Venn diagrams - and Venn diagrams in the strict sense, and can be asked to draw Venn-diagrams-in-the-strict-sense for all four of the just-cited propositions, along with their four respective negations. Some exploration should be made of logical relations among the various propositions in the Octagon - a topic on which I wrote in my 1980s article in the

*Australian Logic Teachers Journal*. (Sorry, folks, but that is the title of the journal:*Teachers*, where correctness makes it very nearly mandatory, alas, to have*Teachers'*.)What to do in such an overview, or "propadeutic" - or indeed in a full-blown logic course or pair of courses - about Heyting, Intuitionism, Brouwer, and the scary vistas of A.S.Essenin-Volpin (more formally Александр Сергеевич Есенин-Вольпи; 1924-2016) in philosophy of mathematics and Prof. Sir Michael Dummett (1925-2011) in general logic-driven philosophy I know not. But Catholic schools will have to note that Prof. Sir Michael Dummett is acknowledged a pre-eminent authority in the ranks of modern Catholic thinkers. I presume Deans in such schools will some day have to go even where I today fear to tread, as they decide what teaching to arrange.

##
POSTSCRIPT: Moise's later-life contribution

to secondary (as opposed to tertiary) education

As the eminent George David Birkhoff (originator of the geometry-axiomatization approach chiefly stressed by Moise) brought out, with a collaborator, a school textbook, so also did Moise, at the very end of his long life - well after producing the university textbook which is here mainly under discussion.

I have already noted on this blog that in Prof. Birkhoff's case, the book was

*Basic Geometry*(in a collaboration featuring as second author Ralph Beatley; as I have remarked, there is a "third edition" at Chelsea Publishing in 1959 and a year-2000 reissuing by the American Mathematical Society).In Moise's case, we find

*Geometry*, by Moise in a 1991 collaboration with second author Floyd L. Downs, Jr., published at Addison-Wesley.Here, then, is an elementary treatment under Moise's imprimatur which we may suspect avoids the infelicities of the 1919 Godfrey-and-Siddons - even while not being able to probe as deeply as a university textbook, such as Moise's

*Elementary Geometry from an Advanced Standpoint*, can and must.##
POST-POSTSCRIPT: But as for me,

"Fresh woods, and pastures new"

I have to end by confessing to forsaking university-standard geometry, at least for the time being. We must work at the most fundamental level possible, seeking to grasp concepts as deeply as our time and our nerves may allow. Alas, it seems to me that the most important thing for me to study now is not, after all, geometry!

At a deep level, we must ask, "What is a space, in the most general sense?" Here one prescinds even from questions of angles and distances, so central both in Euclidean geometry and in its non-Euclidean alternatives.

I used to wince inwardly at the mention of "topological spaces", thinking them the exclusive preserve of the scientifically eminent. But it does seem that it is topological spaces that are spaces in the most general sense. Moreover, it does now seem that topology is at the foundation of rigorous multivariate real analysis - central to astrophysics, and to the physics of radio (to cite two fields in which I want to make progress). Many of my longstanding calculus insecurities, I now feel, stem from my ignorance in the rudiments of topology.

I take as an instance the very book that has caused me unparalleled grief over the last twenty-odd years, Spivak's

*Calculus on Manifolds*. (This is his on-all-campuses-dreaded thin second-year book, as opposed to his on-all-campuses-applauded thick first-year, merely univariate, book.) The very title says much. For what is a "manifold"? The surface of a sphere in Euclidean three-space surely counts as a manifold; so, surely, does the surface of a torus in Euclidean three-space. I also kinda-sorta think that the (two-dimensional, locally plane-like) surface of a Klein bottle in Euclidean four-dimensional space is a manifold (even though this surface is so ill-bred, I kinda-sorta gather, as not to be an "orientable" manifold). In general, we have first to set up the concept of a manifold rigorously - specially smooth, specially differentiation-welcoming mappings called "homeomorphisms", or "diffeomorphisms", or something, have something to do with this - and then to construct "surface integrals", "flux integrals", and the like, on objects embedded in manifolds. (Or at least - at this early point in my studies I wave my hands wildly, like any USA presidential candidate - in the case of the flux integrals building on objects embedded in manifolds which are so well-bred as to be "orientable".)So away, for the time being, with earnest, plodding Geometry, and All Hail, for the time being, Topology.

One might well say: Hail, Horror! -

*O Ave, horror desolatioque! o vae!*But I am resolute. For while it is something of a discovery to stumble across Moise as a geometry authority, a mere glance at the Web or at the University of Toronto bookstore shelving reveals a safe, universally applauded, topology authority - the still-living James Raymond Munkres (1930-), in his second edition. Perhaps never have I seen a technical book praised so highly by its various Amazon reviewers.As I end this six-part essay I sally forth, then, clutching my expensive-and-yet-second-hand Munkres

*Topology*(in the second edition), while quoting Milton:*And now the Sun had stretch'd out all the hills,*

*And now was dropt into the Western bay;**At last he rose, and twitch'd his Mantle blue:**Tomorrow to fresh Woods, and Pastures new.*

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