2008-03-04 10:04MATH 880 PROSEMINAR JT SMITHOUTLINE 17 (Tentative) SPRING 20081. Assignmenta. Have we neglected anything for the in-class outline?b. Are there further questions about the social organization of mathematics?c. Begin reading Gillman 1987. I posted a portion of it temporarily on the coursewebsite. Ms. Morgan and Mr. Kifer have agreed to lead our discussion of thefirst sections of this manual.2. A question arose in class about use of the passive voice. I just noticed a goodexample the account of a Math 880 discussion last year. It concerned an editingtechnique: surgery on individual sentences or pairs of sentences to shorten themwithout changing the semantics. Applied to something written with less care, thiscan often cut the length significantly. This sort of editing is often extremelybeneficial in reducing clutter in abstracts and presentations, and in increasingeffectiveness in general writing. The example was from an abstract in our elevatorlobby intended to attract an audience to a talk by a candidate for our then-openchairmanship. (The candidate was not successful.)a. First sentence: “Recent advances in computational modeling and imagingtechnology are making it possible that computational models can be builtbased on patient-specific image data and accurate predictions can be made forrealistic clinical applications.” (33 words, 205 characters)b. Edited version: “Recent technological advances let us base computationalmodels on patient-specific image data to make accurate predictions for clinicalapplications.” (19 words, 130 characters)c. This is a 42% or 37% reduction, depending on what you count.d. Repeated phrases and use of passive voice in a passage are indicators that thissort of editing can help. Also suggestive are indirections such as “are makingit possible that ... can be built,” which I generally describe uncharitably as waysto avoid responsibility for what one says. Those eight words here are replacedby three: “let us build ....”e. Passive voice is overused as a way of avoiding writer’s responsibility. It’s amark of a bureaucrat-in-training.3. I digressed to explain how the ability to do linear algebra without requiringcommutativity of the scalars benefits the study of foundations of geometry.a. A common strategy is to develop a synthetic axiom system (about points, lines,planes, etc.) sufficiently to introduce the scalars as points on a given line,define their addition and multiplication, prove the major properties of thoseoperations, assign coordinates to points, and show that lines and planes aregraphs of linear equations of some sort. For the latter proofs, it’s nice to usetools of linear algebra. If those were available only for commutative scalars,we would have to prove commutativity before dealing with linear equations.Although commutativity of scalars follows easily from the famous Pappus-Pascal theorem about collinearity of the intersections of opposite edge linesPage 2 MATH 880 SPRING 2008 OUTLINE 172008-03-04 10:04of a hexagon inscribed in a degenerate conic, that theorem is very hard toprove from synthetic axioms. That’s why it doesn’t appear in school geometrytexts. If we build the linear algebra tools without assuming commutativity,we can show that most of analytic geometry is independent of the apparatusnecessary to prove the Pappus-Pascal theorem.4. In-class outline. We discussed inclusion of examples in the outline.a. Some organization is required, since some examples depend on previous ones,and some could be included in more than one place.b. I’ll put some very brief placeholders for examples into the outline.c. The current state of the outline is included at the end of this document.5. Materials on mathematical writing. Last year, students and I reviewed some suchguides, made some comments, and decided to discuss part of one of them in detail,together in class. Here I’ll mention those we considered. I’ve posted Gillman 1987online temporarily, so that you can read ahead and be prepared to start a discussionof it during the next meeting.a. Chicago 1993 is the bible. Both Wiley and Birkhäuser simply told me toadhere to its rules and suggestions. You should become acquainted with it.b. Finkel et al. 2000 has some interesting stuff, intended for applied mathematicsPhD students. Some attention to abstracts. It refers to the research paperStewart 2000 and its abstract, Trefethen 2000.c. Fowler 1958 is a wonderful source of tips on writing English. It’s divided intoseparate articles on topics such as “than”. For that reason, you’ll find itperplexing: the question you’re asking is hard to find. Solution: use it forbedtime reading. It’s often hilarious, and that makes its tips memorable.d. Gillman 1987 seems hard to obtain, but is concise, memorable, and appearedthe most useful. The author, a noted researcher in functional analysis, waseditor of the American Mathematical Monthly. We’ll discuss about 2/3 of thisin class. It’s on my website temporarily.e. Higham 1993 is more comprehensive than Gillman 1987, but did not seem togenerate as much interest.f. Krantz 2005 is mostly unsuited for class discussion, but its chapter 5 does havesome very interesting stuff on a wide variety of topics. Krantz will give acolloquium talk here this spring.g. Steenrod et al. 1973 includes an article on writing by Halmos that is memora-ble, but not really close to my taste. h. Strunk 2000 is a modern edition of a classic style guide often referred to asStrunk and White, that served as a model for many later style guides. Isuppose I’ve read and used it in the past, but don’t seem to have a copy now.6. 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,MATH 880 SPRING 2008 OUTLINE 17 Page 32008-03-04 10:04(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. iii. 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