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A Handbook of Mathematical Discourse

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A Handbook of Mathematical Discourse A Handbook of Mathematical Discourse Version 0.9 Charles Wells April 25, 2002 Charles Wells Professor Emeritus of Mathematics, Case Western Reserve University Affiliate Scholar, Oberlin College Address: 105 South Cedar Street Oberlin, OH 44074, USA Email: [email protected] Website: http://www.cwru.edu/artsci/math/wells/home.html Copyright c 2002 by Charles Wells Contents List of Words and Phrases Entries Preface Bibliography About this Handbook Citations Overview Point of view Index Coverage Left To Do Descriptive and Prescriptive Citations Citations needed Presentation Information needed Superscripted cross references TeX Problems Acknowledgments Things To Do contents wordlist index List of Words and Phrases abstract algebra arbitrary brace compartmentaliza- abstraction argument by bracket tion abuse of notation analogy but componentwise accumulation of argument calculate compositional attributes argument call compute action arity cardinality concept image ad-hoc article cases conceptual blend polymorphism assertion case conceptual affirming the assume category concept consequent assumption character conditional aha at most check assertion algebra attitudes circumflex conjunction algorithm addiction back formation citation connective algorithm bad at math classical category consciousness- alias bare delimiter closed under raising all barred arrow code example all notation codomain consequent always bar cognitive consider analogy behaviors dissonance constructivism and be collapsing contain angle bracket binary operation collective plural context-dependent anonymous bind college algebra context notation black box college contrapositive another planet boldface mathematics convention antecedent both college converse error any bound identifier colon equals converse any bound variable combination coordinatewise APOS bound comma copy 4 contents wordlist index coreference relation equipped find corollary dependent variable equivalence relation first order logic counterexample notation equivalent first order covert curriculum descriptive equivalent fix crisp descriptivist establish notation following are curly brackets determinate eternal equivalent dead metaphor dictionary eureka follow defined in definition evaluate formal analogy defined on discourse analysis every formal language defined over discourse example formal logic definiendum disjoint existential bigamy formal proofs definiens disjunction existential formal defining equation disjunctive instantiation formula defining expression definition existential formula defining phrase display quantifier forward reference definite article distinct expansive fraktur definite description distributive plural generalization free identifier definite divide explicit assertion free variable definition by cases divisor expression free definition domain extensional function of n definition dummy variable external variables degenerate each representations functional degenerate easy example extrapolate function degree elementary factorial fundamentalist delimiter element factor fuzzy delimiting example empty set fallacy generalization from delineated encapsulation false symmetry examples denote endowed family resemblance generalization denying the enthymeme category given hypothesis entification family give dependency equations field global identifier 5 contents wordlist index global parameters in your own words lemmata object globally include lemma mathematical gothic indefinite article let register graph indefinite lg mathematical grasshopper description light bulb mistake structure Greek alphabet inert linear algebra mathematical grounding infinite literalist vernacular metaphors infix notation ln mean grouping injective local identifier member guessing input local mental hanging theorem insight logarithm representation hat instantiate logician’s semantics metaphor hidden curriculum integer look ahead mnemonic high school algebra integral lowercase model hold intensional Luddism modern algebra hypothesis interpretation malrule monster identifier inventory examples mapping motivating example identify inverse error map multiple meanings identity function isomorphic marking must identity italics matchfix notation myths if and only if I mathedu mailing namely iff JSTOR list name if jump the fence mathematical narrative style illustration just in case definition natural number image just mathematical necessary imaginary unit juxtaposition discourse negation implication labeled style mathematical never imply lambda notation education notation in general larger mathematical logic not in other words learned name mathematical mind now in particular left Polish notation mathematical number theory 6 contents wordlist index N Platonism provided that rule object plug into put R object plural quantifier sanity check obtain pointy brackets Q satisfy one-to-one Polish notation radial category say only if polymorphic radial concept schema onto pons asinorum range scientific register open sentence positive ratchet effect scope open sentence postcondition real number scope open sentence postcondition real self-monitoring operation postfix notation recall semantic operator power reconstructive contamination or equivalently precedence generalization semantics order of quantifiers precondition reductionist sentence order predicate redundant set comprehension orthogonal predicate register setbuilder notation or prefix notation reification set osmosis theory prescriptive relation set outfix notation prescriptivist relocator should output prime representation show overloaded notation process respectively snow parameter program result some parametric pronunciation reverse Polish space polymorphism proof by notation specific parenthesis contradiction revise mathematical parenthetic proof by instruction rewrite using object assertion proof definitions specification partial function property right Polish split definition pathological proper notation square bracket pattern recognition proposition root standard permutation prototype round parentheses interpretation 7 contents wordlist index statement symbolitis trivial example variable clash status symbol trivial variable structural notation synecdoche turf mathematical structure syntax-driven type labeling object student-professor syntax type variable problem synthetic typical variate identifier subexpression technical underlying set variate subscript term under verify substitute text unique Vulcanism substitution TFAE universal Vulcanize such that that is generalization walking blindfolded sufficient the following are universal well-defined superscript equivalent instantiation when suppose then universal quantifier where suppression of theorem university without loss of parameters there is unnecessarily weak generality surjective the assertion witness symbol thus unwind yes it’s weird manipulation tilde up to you don’t know symbolic assertion transformer uppercase shriek symbolic expression translation problem vacuous implication zero symbolic language trigonometric value Z symbolic logic functions vanish 8 contents wordlist index Preface You may click on words and phrases in red for more information. In this version if a phrase in red is split across two lines you must click on the part on the first line. I hope to repair this for the final version. About this Handbook Overview This Handbook is a report on mathematical discourse. Mathematical discourse includes the special dialect of English mathematicians use to communicate mathematical reasoning and the vocabulary that describes the behavior of mathematicians and students when doing mathematics as well as their attitudes towards various aspects of mathematics. The book is a report on language usage in mathematics and on the difficulties students have with the language; it is not a text on how to write mathematics. The usage is determined by citations, 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. The Handbook is intended for teachers of college-level mathematics, to provide some insight into some of the difficulties their students have with mathematical language, and for graduate students and upper-level undergraduates who may find clarification of some of the difficulties they are having as they learn higher-level mathematics. The earliest dictionaries of the English language listed only “difficult” words (see [Landau, 1989] , pages 38–43). Dictionaries such as Dr. Johnson’s that attempted completeness came later. This Handbook is more like the earlier dictionaries, with a focus on usages that cause problems for those who are just beginning to learn how to do abstract mathematics. It is not like a proper dictionary in another way as well: I include words and phrases describing behavior while doing mathematics and attitudes towards various aspects of mathematics that I think should be more widely used. 9 contents wordlist index Someday, I hope, there will be a complete dictionary based on extensive scientific observation of written and spoken mathematical English, created by a collaborative team of mathematicians, linguists and lexicographers. This Handbook points the way to such an endeavor. However, its primary reason for being is to provide information about the language to instructors and students that will make it easier for them to explain, learn and use mathematics. In the remainder of the Preface, I discuss some special aspects of book in more detail. Several phrases are used that are described in more
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