__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: Kmo did not have time to make as many points as would be desirable: a lot will have to be done in upcoming Parts E and, as it now appears, F. He did manage (within the framework of the version 1.0.1, 1.0.2, .. process) to do a reasonably polished job.**

__Revision history:__

**20160913T1501Z/version 1.2.0: Kmo added a small remark on Moise's proof that in the ordered field which is the reals, the multiplicative identity is strictly greater than the additive identity. Kmo reserved the right to upload minor (i.e., cosmetic, as opposed to substantive) tweaks over the coming 48 hours, as here-undocumented version 1.2.1, 1.2.2, 1.2.3, ... .****20160913T0100Z/version 1.1.0: Kmo supplied the missing material, as five paragraphs under the heading "Moise's discussion of unusual fields". He reserved the right to upload minor (i.e., cosmetic, as opposed to substantive) tweaks over the coming 48 hours, as here-undocumented versions 1.1.1, 1.1.2, 1.1.3, ... .****20160913T0003Z/version 1.0.0: Kmo uploaded base version, while under time pressure leaving some material out(with a marker flagging his omission). He reserved the right to upload minor (i.e., cosmetic, as opposed to substantive) tweaks over the coming 48 hours, as here-undocumented versions 1.0.1, 1.0.2, 1.0.3, ... .**

**[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.]**

In general, we find in Moise a careful mathematical formalism, consistent with the ideals of advanced-undergraduate Pure Mathematics, generally undergirded with a further virtue - a professional logician's sense, or at any rate something suitably close to that sense, of logic.

In what remains of this essay, I will discuss his excellence in formalism and logic in detail, mixing into my general approbation also some mild criticisms.

## 3. Points of Detail from Moise

### 3.1 Formalism, Outside the Strict Domain of Logic

**I noted under the photo accompanying my blog posting of 2016-08-29 or 2016-08-30 that my Dalhousie University introduction-to-analysis, in 1971, seems to have laid down as clauses in the definition of a field "There is a unique element which serves as an additive identity, and there is a unique element which serves as a multiplicative identity." Moise does likewise, additionally laying it down that every element has a unique additive inverse, and that every element which is not an additive identity has a unique multiplicative inverse.**

__Uniqueness of identities (and related point regarding inverses):__
But it suffices to lay down merely "There is at least one element which serves as an additive identity, and there is at least one element which serves as a multiplicative identity, and every element has at least one additive inverse, and every element which is not an additive identity has at least one multiplicative inverse." Given the other field axioms (associativity, commutativity, distribution of multiplication over addition, and the demand that no multiplicative inverse be also an additive inverse) one can then deduce uniqueness of additive identity, uniqueness of multiplicative identity, the existence for every field element

*a*of some unique additive inverse*a'*, and the existence for every field element*b*which is not an additive identity of some unique multiplicative inverse*b'*.**In introducing the concept of a function, Moise is correctly at pains to distinguish image from range. On the reals, the squaring function has as image the nonnegative reals. After going through various preliminaries, Moise makes the correct remark (the emphases on the words "defined" and "is" are his), "/.../we have explained the conditions under which a function is**

__Functions: "a range" versus "the range":__*defined*, but we have not said what kind of object a function

*is*." Next, he fills that gap in. For Moise, a function is a set of ordered pairs. He defines "a function with domain

*A*and range

*B*" as a collection

*f*of ordered pairs

*(a,b)*such that (I) for each pair, the first element is in

*A*, and such that (II) for each

*a*in

*A*,

*a*is the initial term of exactly one pair

*(a,b)*in

*f*, and (III) such that the second term of every pair in the collection is in

*B*.

Here, however, it is for clarity necessary to add what Moise neglects to add, namely that (to take an example) the collection of pairs

*(a,b)*of reals such that

*b*is the same entity as

*a times a*(this is a function with, I reiterate, the set of nonnegative reals as its image) is a function with infinitely many different ranges. One of these ranges is the just-cited image set. Another of its ranges is the set of reals (as one indeed might be liable to aver, upon recalling how people everywhere are liable to write, "The squaring function is a function mapping the reals into the reals"). A third of its ranges is the set which is the union of the nonnegative reals and the three-element set {-1, -17, -99}; and so on. In general, on Moise's definition of what a function is, we must not write of "the range" of a function

*f*, but of "a range" of a function

*f*- with, indeed, the image of

*f*being a range of

*f*, but additionally with any superset of the image of

*f*being a range of

*f*.

I do gather in my general Mathematics-Department ignorance that on some accounts of function, a function is not a bare set of ordered pairs meeting Moise's conditions, but is a Moise set of ordered pairs along with a range specification. On this competing terminology, we define two different functions when we write on the one hand "

*g*has as domain the reals, and is such that for every*x*in the reals*g(x)*is*x times x*, and*g*has as range the nonnegative reals" and write on the other hand "*h*has as domain the reals, and is such that for every*x*in the reals*h(x)*is*x times x*, and*h*has as range the reals."**Moise correctly devotes a short Chapter One section to "The Language and Notation of Sets" before proceeding in his Chapter Two to introduce the idea of a set of points, and to lay down the first five of his postulates. These initial postulates (he uses at this early stage in his book no notion of distance between points, and no notion of betweenness) pertain to a structure in which**

__Degenerate spaces: point P versus singleton {P}:__*S*is a set of points, and "curly-

*L*" (the set of lines) is a collection of subsets of

*S*, and "curly-

*P*" (the set of planes) is a collection of subsets of

*S*: "All lines and planes are sets of points"; "Given any two different points, there is exactly one line containing them"; "Given any three different noncollinear points, there is exactly one plane containing them"; "If two points lie in a plane, then the line containing them lies in the plane"; and "If two planes intersect, then their intersection is a line." Now, however, (on p. 45) comes something which is not quite true: "If you check back carefully, you will see that [these five postulates] are satisfied by the 'geometry' in which there is exactly one point

*P*in

*S*, and this point

*P*is both a line and a plane." What Moise should have written instead, with due regard for his earlier section on "The Language and Notation of Sets", is, rather, "If you check back carefully, you will see that [these five postulates] are satisfied by the 'geometry' in which there is exactly one point

*P*in

*S*, and the singleton set {

*P*} is both a line and a plane."

****

I end this discussion of formalism-outside-the-strict-domain-of-logic on some positive notes.

**Evidence of care on Moise's part is his treatment of the tiny field, comprising just an additive identity and a multiplicative identity. (This is inevitably a two-element algebraic structure, not a one-element algebraic structure, since a clause in the standard definition of a field - Moise briefly highlights its importance - is that the additive identity and the multiplicative identity are distinct. - Or rather, being very pure indeed, with a correct regard for the point I made above, under the heading "uniqueness of identities", we ought to postulate simply "No additive identity is also a multiplicative identity" (i.e., "However big or small may be the set of additive identities, and no matter how big or small may be the set of multiplicative identities, at any rate the intersection of these sets is empty").) Moise carefully gives the usual definitions for addition and multiplication in the two-element structure, and then simply asks his students, in Chapter One, section 1.2 problem set, as problem 19, "Which of the field postulates would hold true?" (A less skilled teacher would, by contrast, somehow hint at the answer, for instance by writing "Prove that all the field postulates hold true.")**

__Moise's discussion of unusual fields:__
Since Moise is in general careful with fields, and is in general so kind to his students, I felt emboldened to go a little beyond him, as I explained in my blog posting of 2016-08-29 or 2016-08-30:

*S**uppose 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 an eight-element field; and so on.*

Having established this result for an arbitrary field

*F*, I for the first time in my life feel rather cheerful about complex numbers, i.e., about the Argand plane. Argand-Schmargand, I now say, in a suitably dismissive tone:*any*field can with a mildly intricate definition of multiplication be used to generate a new, more elaborate, field, as the the complex numbers with their mildly intricate definition of multiplication are generated from the field of reals.
It is rather important to be able to say, dismissively, "Argand-Schmargand." For what is the altenative? For years - nay, for decades - I had an inappropriate feeling of superstitious awe regarding the Argand plane, thinking to myself that it conceals some kind of deep, and ultimately debatable, philosophical presupposition, somehow and somewhere. It of course does not at all help that people have historically spoken of "imaginary" numbers, and have loudly insisted that "Imaginary numbers, with Argand-plane sums of imaginary numbers and real numbers, are just as legitimate as real numbers." One becomes uneasy precisely because the insistence is delivered loudly - being delivered, in fact, in the earnest declamatory tones of a government intent on deceiving its electorate, or in the earnest declamatory tones of a phone-company marketing rep.

I will also be forever grateful to Moise for distinguishing carefully between the concept of a field with an ordering relation and the more stringent concept of an ordered field. He fortunately asks in the problem sets for a proof that an ordering relation can be imposed on the complex numbers (Chapter One, problem set 1.5, problem 5), while also asking (same problem set, next problem; he gives a hint) for a proof that the field of complex numbers cannot be given an ordering relation that turns it into an ordered field.

And it is good to compare Moise's first edition with his final (third) edition, in their respective treatments of the additive and the multiplicative identity in an ordered field. While the first edition is silent, the third edition helpfully proves as a theorem that in the ordered field which is the reals - in fact, we may add, his theorem holds for an arbitrary ordered field - the multiplicative identity is strictly greater than the additive identity. Moise's comment on this shows the kindly regard he has for potential difficulties amid the less sophisticated ranks within his big cohort of readers (the emphasis on "not" is his):

And it is good to compare Moise's first edition with his final (third) edition, in their respective treatments of the additive and the multiplicative identity in an ordered field. While the first edition is silent, the third edition helpfully proves as a theorem that in the ordered field which is the reals - in fact, we may add, his theorem holds for an arbitrary ordered field - the multiplicative identity is strictly greater than the additive identity. Moise's comment on this shows the kindly regard he has for potential difficulties amid the less sophisticated ranks within his big cohort of readers (the emphasis on "not" is his):

*The question here is*__not__*whether the real number 1 is greater than the real number 0; everybody knows that it is. The question is whether the statement 1 > 0 follows from the postulates and definitions we have written down so far. If this statement does not follow, then we need another postulate.***Moise, I presume like everyone who introduces the abstract-algebra concept of a field and avers that the reals are a field, postulates the associativity of addition and multiplication in terms of triples: for all**

__Moise's careful discussion of "The Generalized Associative Law":__*a*,

*b*,

*c*in the field,

*(a+b)+c*is postulated to be the same field element as

*a+(b+c)*; for all

*p*,

*q*,

*r*in the field,

*(p times q) times r*is postulated to be the same field element as

*p times (q times r)*(with similar in-terms-of-pairs postulates for commutativity of addition and commutativity of multiplication:

*t + u*is postulated to be none other than

*u + t*, and

*v times w*to be none other than

*w times v*, for all field elements

*t, u, v, w*).

One might well imagine a careless author leaving associativity there, taking it it as somehow "evident that", or as something "left to the reader to prove", that for any natural number

*n*,

*n*-fold sums and

*n*-fold products are freely associative.

Moise, on the other hand, fills this in with care, taking fully four pages, with a formal inductive proof, and in the course of his work answering that very necessary question - I imagine it said at the samovar in my imaginary, grubby, Nikolai Ivanovitch Lobachevsky Institute of Socialist Mathematics, on the scarier side of the Urals - "Vot MEENZ 'freely associative'?". He starts his four-page discussion as follows (the emphasis on the word "pairs" is his):

*In practice /.../, as soon as you get past Chapter 1 of anybody's book, you are writing n-fold sums*

*a1 + a2 + ... + an,*

*and n-fold products*

*a1 a2 ... an,*

*for n > 3. We insert and delete parentheses in these sums and products, at will. All this is fine, but it has not been connected up, so far, with the operations that are supposed to be given for*

*pairs**of numbers (a,b) and with the associate laws for triplets (a, b, c). It would be a pity if mathematics appeared to be split down the middle, with the postulates and definitions on one side, and the mathematical content on the other. let us therefore bridge the gap between our postulates and the things that we intend to do.*

**[To be continued, and perhaps concluded, soon (perhaps next week), as "Part E".]**

## No comments:

## Post a Comment

All comments are moderated. For comment-moderation rules, see initial posting on this blog (2016-04-14).