A Handbook of Mathematical Discourse Charles Wells Case Western Reserve University Charles Wells Professor Emeritus of Mathematics Case Western Reserve University Affiliate Scholar, Oberlin College Drawings by Peter Wells Website for the Handbook: http://www.cwru.edu/artsci/math/wells/pub/abouthbk.html Copyright c 2003 by Charles Wells Contents Preface v Introduction Alphabetized Entries Bibliography 281 Index 292 Preface Overview This Handbook is a report on mathematical discourse Mathematical discourse as the phrase is used here refers to what mathematicians and mathematics students say and write • to communicate mathematical reasoning, • to describe their own behavior when doing mathematics, and • to describe their attitudes towards various aspects of mathematics The emphasis is on the discourse encountered in post-calculus mathematics courses taken by math majors and first year math graduate students in the USA Mathematical discourse is discussed further in the Introduction The Handbook describes common usage in mathematical discourse The usage is determined by citations, that is, quotations from the literature, the method used by all reputable dictionaries The descriptions of the problems students have are drawn from the mathematics education literature and the author’s own observations This book is a hybrid, partly a personal testament and partly documentation of research On the one hand, it is the personal report of a long-time teacher (not a researcher in mathematics education) who has been especially concerned with the difficulties that mathematics students have passing from calculus to more advanced courses On the other hand, it is based on objective research data, the citations The Handbook is also incomplete It does not cover all the words, phrases and constructions in the mathematical register, and many entries need more citations After working on the book off and on for six years, I decided essentially to stop and publish it as you see it (after lots of tidying up) One person could not hope to write a complete dictionary of mathematical discourse in much less than a lifetime The Handbook is nevertheless a substantial probe into a very large subject The citations accumulated for this book could be the basis for a much more elaborate and professional effort by a team of mathematicians, math educators and lexicographers who together could produce a v definitive dictionary of mathematical discourse Such an effort would provide a basis for discovering the ways in which students and non-mathematicians misunderstand what mathematicians write and say Those misunderstandings are a major (but certainly not the only) reason why so many educated and intelligent people find mathematics difficult and even perverse Intended audience The Handbook is intended for • Teachers of college-level mathematics, particularly abstract mathematics at the post-calculus level, to provide some insight into some of the difficulties their students have with mathematical language • Graduate students and upper-level undergraduates who may find clarification of some of the difficulties they are having as they learn higher-level mathematics • Researchers in mathematics education, who may find observations in this text that point to possibilities for research in their field The Handbook assumes the mathematical knowledge of a first year graduate student in mathematics I would encourage students with less background to read it, but occasionally they will find references to mathematical topics they not know about The Handbook website contains some links that may help in finding out about such topics Citations Entries are supported when possible by citations, that is, quotations from textbooks and articles about mathematics This is in accordance with standard dictionary practice [Landau, 1989], pages 151ff As in the case of most dictionaries, the citations are not included in the printed version, but reference codes are given so that they can be found online at the Handbook website I found more than half the citations on JSTOR, a server on the web that provides on-line access to many mathematical journals I obtained access to JSTOR via the server at Case Western Reserve University vi Acknowledgments I am grateful for help from many sources: • Case Western Reserve University, which granted the sabbatical leave during which I prepared the first version of the book, and which has continued to provide me with electronic and library services, especially JSTOR, in my retirement • Oberlin College, which has made me an affiliate scholar; I have made extensive use of the library privileges this status gave me • The many interesting discussions on the RUME mailing list and the mathedu mailing list The website of this book provides a link to those lists • Helpful information and corrections from or discussions with the following people Some of these are from letters posted on the lists just mentioned Marcia Barr, Anne Brown, Gerard Buskes, Laurinda Brown, Christine Browning, Iben M Christiansen, Geddes Cureton, Tommy Dreyfus, Susanna Epp, Jeffrey Farmer, Susan Gerhart, Cathy Kessel, Leslie Lamport, Dara Sandow, Eric Schedler, Annie Selden, Leon Sterling, Lou Talman, Gary Tee, Owen Thomas, Jerry Uhl, Peter Wells, Guo Qiang Zhang, and especially Atish Bagchi and Michael Barr • Many of my friends, colleagues and students who have (often unwittingly) served as informants or guinea pigs vii Introduction Note: If a word or phrase is in this typeface then a marginal index on the same page gives the page where more information about the word or phrase can be found A word in boldface indicates that the word is being introduced or defined here In this introduction, several phrases are used that are described in more detail in the alphabetized entries In particular, be warned that the definitions in the Handbook are dictionary-style definitions, not mathematical definitions, and that some familiar words are used with technical meanings from logic, rhetoric or linguistics Mathematical discourse Mathematical discourse, as used in this book, is the written and spoken language used by mathematicians and students of mathematics for communicating about mathematics This is “communication” in a broad sense, including not only communication of definitions and proofs but also communication about approaches to problem solving, typical errors, and attitudes and behaviors connected with doing mathematics Mathematical discourse has three components • The mathematical register When communicating mathematical reasoning and facts, mathematicians speak and write in a special register of the language (only American English is considered here) suitable for communicating mathematical arguments In this book it is called the mathematical register The mathematical register uses special technical words, as well as ordinary words, phrases and grammatical constructions with special meanings that may be different from their meaning in ordinary English It is typically mixed with expressions from the symbolic language (below) dictionary definition 70 mathematical definition 66 mathematical register 157 register 216 conceptual 43 intuition 161 mathematical register 157 standard interpretation 233 symbolic language 243 • The symbolic language of mathematics This is arguably not a form of English, but an independent special-purpose language It consists of the symbolic expressions and statements used in calculation and d sin x = cos x presentation of results For example, the statement dx is a part of the symbolic language, whereas “The derivative of the sine function is the cosine function” is not part of it • Mathematicians’ informal jargon This consists of expressions such as “conceptual proof ” and “intuitive” These communicate something about the process of doing mathematics, but not themselves communicate mathematics The mathematical register and the symbolic language are discussed in their own entries in the alphabetical section of the book Informal jargon is discussed further in this introduction Point of view This Handbook is grounded in the following beliefs The standard interpretation There is a standard interpretation of the mathematical register, including the symbolic language, in the sense that at least most of the time most mathematicians would agree on the meaning of most statements made in the register Students have various other interpretations of particular constructions used in the mathematical register • One of their tasks as students is to learn how to extract the standard interpretation from what is said and written • One of the tasks of instructors is to teach them how to that Value of naming behavior and attitudes In contrast to computer people, mathematicians rarely make up words and phrases that describe our attitudes, behavior and mistakes Computer programmers’ informal jargon has many names for both productive and unproductive Citations 408 (125, 264) Underwood, R S (1954), ‘Extended analytic geometry as applied to simultaneous equations’ American Mathematical Monthly, volume 61, pages 525–542 [p 525 Lines 17–20.] The example also illustrates the point that it is not always easy to decide whether two equations in more than two unknowns are consistent, quite aside from the matter of producing in that case a real solution 408 Citations 409 (34, 139) van Lint, J H and R M Wilson (1992), A Course in Combinatorics Cambridge University Press [p 35 Lines 8–4 from bottom.] We shall show that a larger matching exists (We mean larger in cardinality; we may not be able to find a complete matching containing these particular m edges.) 409 Citations 410 (15, 47, 198, 261) Van Douwen, E K., D J Lutzer, and T C Przymusi´ nski (1977), ‘Some extensions of the Tietze-Urysohn Theorem’ American Mathematical Monthly, volume 84, pages 435 [p 435 Theorem A.] If A is a closed subspace of the normal space X then there is a function η : C ∗ (A) → C ∗ (A) such that for every f ∈ C ∗ (A), η(F ) extends F and has the same bounds as F 410 Citations 411 (168) Vaught, R L (1973), ‘Some aspects of the theory of models’ American Mathematical Monthly, volume 80, pages 3–37 [p Lines 6–10.] For example, each of the properties of being a group, an Abelian group, or a torsion-free Abelian group os expressible in the so-called elementary language (or first-order predicate calculus) Thus, instead of saying that the group G is Abelian, we can sat it is a model of the elementary sentence ∀x∀y(x ◦ y = y ◦ x) Such properties are also called elementary 411 Citations 412 (56, 101, 210) Vaught, R L (1973), ‘Some aspects of the theory of models’ American Mathematical Monthly, volume 80, pages 3–37 [p 11 Lines 19–22.] the existence of such an A is of basic importance in the remarkable work of Găodel and Cohen on the consistency and independence of the continuum hypothesis and other basic propositions in set theory 412 Citations 413 (57, 276) Vaught, R L (1973), ‘Some aspects of the theory of models’ American Mathematical Monthly, volume 80, pages 3–37 [p 25 Lines 12–11, 6–2 from bottom.] T is called ω-complete if whenever T ∀v0 φ (v0 ) It is ∀v0 (θn (v0 ) → φ (v0 )) for all n, then T worthwhile noting that in contrapositive form the definition of ω-complete reads: (3) T is ω-complete if and only if for any 1-formula φ, if T + ∃v0 φ (v0 ) has a model, then for some n, T∃ v0 (θn (v0 ) ∧ φ (v0 )) has a model 413 Citations 414 (32, 171) Verner, J H (1991), ‘Some RungeKutta formula pairs’ SIAM Journal on Numerical Analysis, volume 28, pages 496–511 [p 501 Lines 1–2 under formula (21 ).] This may be written as bi aij j=4 i · ajk cqk k cq+1 j − q+1 =0 by invoking (15) to imply that the first bracket is zero for j = 2, Since the second bracket is zero for ≤ j ≤ by (17 ), and 414 Citations 415 (32) Wallach, N R (1993), ‘Invariant differential operators on a reductive Lie algebra and Weyl group representations’ Journal of the American Mathematical Society, volume 6, pages 779–816 [p 786 Lines 8–7 from bottom.] If f, g ∈ P(V0 × V0∗ ) then let {f, g} (the Poisson bracket of f and g) be as in Appendix 415 Citations 416 (28, 229) Waterhouse, W C (1994), ‘A counterexample for Germain’ American Mathematical Monthly, volume 101, pages 140–150 [p 141 Lines 9–6 from bottom.] Suppose n is an odd prime In modern terms, Gauss has shown that the field generated over the rationals by the n-th √ roots of unity contains ±n; here we must take the plus sign when n is of the form 4k + and the minus sign when n is of the form 4k + 416 Citations 417 (250) Weiss, G (1970), ‘Complex methods in harmonic analysis’ American Mathematical Monthly, volume 77, pages 465–474 [p 465 Lines 1–10.] First, we plan to show how properties of analytic functions of a complex variable can be used to obtain several results of classical harmonic analysis (that is, the theory of Fourier series and integrals of one real variable) This will be done in Section Second, in Section we shall indicate how some of these applications of the theory of functions can be extended to Fourier analysis of functions of several variables Some of these applications of the theory of functions seem very startling since the results obtained appear to involve only the theory of functions of a real variable or the theory of measure 417 Citations 418 (147) Whittlesey, E F (1960), ‘Finite surfaces a study of finite 2-complexes’ Mathematics Magazine, volume 34, pages 11–22 [p 15 Lines 1–7.] To determine the global structure of a complex we need to have first a knowledge of the local structure, of behaviour in a neighborhood of a point; with this understanding of local structure, we can solve the problem of recognition of a 2-complex by putting the pieces together, as it were In the case of graphs, the problem of local structure is completely resolved by a knowledge of the degree of a vertex The matter is less simple for surfaces 418 Citations 419 (250) Wilson, E B (1918), The mathematics of aăeerodynamics American Mathematical Monthly, volume 25, pages 292–297 [p 295 Lines 22–25.] Now by the method of “conformal representation” of the theory of functions the pressure exerted on a plane wing of various shapes, by the motion of the air, may in some cases be calculated, and the center of pressure may also be found 419 Citations 420 (31, 131, 135) Wilf, H S (1985), ‘Some examples of combinatorial averaging’ American Mathematical Monthly, volume 92, pages 250–261 [p 253 Lines 14–16.] Therefore, if n > we have       n n   (n − 1)! + (n − 2)! Q(n) =  n!    i,j=1  i=1  i=j What sort of creature now inhabits the curly braces? 420 Citations 421 (51, 130, 251) Williams, H P (1986), ‘Fourier’s method of linear programming and its dual’ American Mathematical Monthly, volume 93, pages 681–695 [p 682 Lines 9–5 from bottom.] Constraint C0 is really a way of saying we wish to maximise z where z ≤ −4x1 + 5x2 + 3x By maximising z we will “drive” it up to the maximum value of the objective function It would clearly be possible to treat C0 as an equation but for simplicity of exposition we are treating all constraints as ≤ inequalities 421 Citations 422 (116) Wilf, H S (1989), ‘The editor’s corner: The white screen problem’ American Mathematical Monthly, volume 96, pages 704–707 [p 704 Lines 11–8 from bottom.] To translate the question into more precise mathematical language, we consider a grid of M N lattice points G = {(i, j) | ≤ i ≤ M − 1; ≤ j ≤ N − 1} and we regard them as being the vertices of a graph 422 ... of the abstract version of C that were not originally thought of as being part of C Example The concept of “group” is historically an abstraction of the concept of the set of all symmetries of. .. the side of inclusivity Although the entries are of different types, they are all in one list with lots of cross references This mixed-bag sort of list is suited to the purpose of the Handbook, ... perceived similarity between some part of one and some part of the other Analogy, like metaphor, is a form of conceptual blend Mathematics often arises out of analogy: Problems are solved by analogy