2008-03-08 11:44MATH 880 PROSEMINAR JT SMITHOUTLINE 19 SPRING 20081. Assignmenta. Have we neglected anything for the in-class outline?b. Are there further questions about the social organization of mathematics?c. Continue reading Gillman 1987 and prepare to discuss it.2. Gillman, chapter 3. We began this discussion, led by Mr. Kifer. Comments:a. Sometimes, if you put an otherwise unnecessary symbolic name in the state-ment of a theorem, you can avoid a whole sentence in the proof. For example,considerTheorem. A differentiable function is continuous.Proof. Let f be a differentiable function. Then... .The first sentence of the proof can be omitted if you just rephrase the state-ment of the theorem: a differentiable function f is continuous. It’s a trade-off. Either wording has an advantage and a disadvantage.b. Gillman doesn’t stress enough the problems with English articles a, an, andthe. He himself recommended changing A to every in the statement of theprevious theorem, lest some reader should interpret A as meaning some, whichit sometimes does. But Every doesn’t sound right to me in the modifiedversion. The problems are particularly bad for writers whose native languagesdo not use articles. I recommend that all such writers ask a native Englishwriter to check just that aspect of their drafts.c. We’ll continue this discussion.d. In particular, I think I’ve found the idea of a proof from Zorn’s lemma thatany two bases of the same vector space are equinumerous. In class, we’ll turnthat into a full proof. 3. Where to put history in the in-class outline. We considered inserting history intothe outline: topics to cover and where to do so.a. Although the introduction was initially suggested, it was noted that this mightlead to overloading it, and in any case requires describing history of a subjectbefore the subject itself, which is probably not feasible.b. A history section near the end was suggested. This is feasible: I did that inSmith 2002. Disadvantage: it looks like the history is an afterthought.Moreover, assuring readers that the paper would treat history might requiresome awkward wording in the earlier text.c. Mixing history in with the earlier content was considered as well. The disad-vantage is that the logical order of subjects might be very different from theirhistorical order, and thus it might be difficult to maintain an interesting storyline. Also, historical excursions during discussions of content might bedistracting.4. What history?a. Students suggested that of the basic theorems, and I added basic definitions.Page 2 MATH 880 SPRING 2008 OUTLINE 192008-03-08 11:44b. Students also would be interested in the motivation for that work.c. And its applications—they agreed that motivation and application ought tobe tied.d. We discussed the history of the approach to the subject. That might involvemuch motivation but little application.e. Notice: no one suggested covering lives of the “saints”. In fact, in linearalgebra it’s hard to suggest individuals for such canonization.f. What history to include is certainly dependent on what is easily available.Here some comments from my experience. i. It’s relatively easy to gain information about the history of the currentapproach to the subject. The original 1931 edition of Van der Waerden[1937] 1953 was about the first major text on “modern” abstract algebra,but its approach to linear algebra was rather different from ours. Myown instructor in the equivalent of Math 335 followed it. But for thesecond course, the equivalent of Math 325 and 725, he chose Birkhoff& Mac Lane 1941, which was the first English-language text organizedlike today’s courses. The books are available and their historical back-ground relatively easy to find.ii. Van der Waerden’s framework stems from a later (I think) edition ofthat book, and was massaged in a number of other papers to allowvarious applications in mathematics itself. These sources are largelyaccessible to you, but unless guided there you might not find them.iii. The notion of a vector space is rather recent: it stems from Peano’swork around 1890. I don’t want to dig out the reference now, but Iremember showing and translating it in Math 300 last year. I think themotivation there was to develop an approach to vector calculus, whichwas in its infancy then.iv. I don’t know the background of theorems such as the rank-nullitytheorems. The writer of this paper might be able to dig that out.g. I’ll work a few placeholders into the outline to represent where this historicalinformation might go if it should prove feasible to include it with the maincontent.h. But I’ll put a comment at the top about those placeholders’ being tentative,that if no coherent story line should arise to tie them in, they might better beput together in a history section near the end.5. Here’s the current state of the outline.a. Introductioni. This paper presents the main features of the theory of linear depend-ence.(1) This theory includes several of the major concepts and theoremsof linear algebra,(2) commonly introduced in first courses in linear algeba.ii. But its context is considerably more general than that,(1) to permit a much broader scope of application.MATH 880 SPRING 2008 OUTLINE 19 Page 32008-03-08 11:44iii. The theory is presented in the context of a vector space V over a divi-sion ring K of scalars.(1) Footnote: an alternative term for division ring is skew field.(2) Postulates for a division ring.(3) Examples: real numbers, complex numbers, rational numbers.(Others later.) (4) Postulates for a vector space V over a division ring K, whoseelements are called scalars.(5) Examples: real and complex n-space. (Others later.)(6) Definition of a subspace of V.(7) Examples: {0}, V, lines and planes given by parametric equationsin any n-space, hyperplanes and hyperlines given by linear equa-tions in any n-space.iv. The presentation follows that of Van der Waerden [1937] 1953.(1) In this framework, (a) dependence of a vector on a set of vectors is defined first,(b) and few basic theorems (¿ › A BETTER TERM ?) are proved.(c) All subsequent definitions and theorems are based solely onthese.(2) This permits all these definitions and theorems to be carried intactinto the development of any theory that includes an analogousconcept of dependence that satisfies those few basic theorems.(3) Moreover, by eliminating