Why mathematics needs engineering Raymond Boute, INTEC, Ghent University [email protected] 6 1 0 2 n Abstract a Engineering needs mathematics, but the converse is also increasingly evident. J Indeed,mathematicsisstillrecoveringfromthedrawbacksofseveral“reforms”. 5 Encouraging is the revived interest in proofs indicated by various recent intro- ] duction to proof-type textbooks. Yet, many of these texts defeat their own M purpose by self-conflicting definitions. Most affected are fundamental concepts G suchasrelationsandfunctions, despite flawlessaccounts50yearsago. We take the viewpoint that definitions and theorems are tools for capturing, analyzing . h and understanding mathematical concepts and hence, like any tools, require t a diligent engineering. This is illustrated for relations and functions, their alge- m braicpropertiesandtheirrelationtocategorytheory,withtheHalmos principle [ for definitions and the Arnold principle for axiomatizationas designguidelines. 1 Keywords: algebra, analysis, calculus, category theory, codomain, definition, v design, domain, engineering, function, logic, mathematics, relation, soundness 9 8 9 0 1. Introduction: Mathematics and Engineering 0 . Mathematics has been intertwined with engineering since antiquity [12, 52]. 1 0 Kline notes that “More than anything else mathematics is a method” [32]. 6 Arguably, the primary purpose of this method is effective reasoning. This view 1 best explains what Wigner calls the unreasonable effectiveness of mathemat- : v ics [64], in particular its practical usefulness far beyond the originally intended i application areas. From this perspective, the dichotomy between Platonism X and formalism dissolves: mathematical objects do exist, albeit in an abstract r a universe. Formalism, definitions and theorems are the tools to study them. Tools, being artifacts, deserve careful design, borrowing criteria and guide- linesfromengineering. SomeofthesealsobeendiscussedbyJos´eOliveira[43]in anothercontext. Here wefocus onusingengineeringprinciplesinmathematics. Foremost is enhancing the effectiveness in reasoning. Symbolic notation properly designed and used yields extra guidance via the shape of the expres- sions. It should function like well-meshed gears in a Swiss precision clockwork. Aptness and economy in capturing the abstract objects of interest ensures conceptual malleability, generality and practical usefulness. Human factors are influentialhere,anditisoftenoverlookedthatthisisahighlyindividualmatter of temperament and background. Even so, everyone benefits from clear con- Preprint submitted toElsevier January 7, 2016 ceptualization and reasoning. For instance, separation of concerns avoids the common misconceptions caused by intellectual noise and conceptual tangling. In classical mathematics, methods and notations were often thought-out carefully. In algebra, for instance, symbolic notation started with Diophantus and evolved into its current form via Vi`ete and Descartes [6, 12], rarely violat- ing good design practices, thus making symbolic calculation today’s norm. In comparison,notationsfrom“modernmathematics”asusedineverydaypractice are substandard, hampering symbolic reasoningand thus making it unpopular. Thecauseofthisstagnationislargelyhistorical. Whenintroducingso-called “modernmathematics”,forgettingitsrootscausedseriouseducationalmistakes, denounced in rather strong terms by Arnold [3]. In a severe overreaction, the viewofmathematicsasamethodwassacrificedinfavorofmathematicsasabag oftricksandattemptingtoelicitmotivationbyso-called“real-life”examplesno more realistic than the farmer-sells-potatoes-type problems in grade school — and in PISA tests! The well-provenstructure definition-examples-theorems was frowned upon, and mathematical exposition had to become a “narrative”. Asaresult,classicslikeRudin’sPrinciples ofMathematicalAnalysis[51]are, as Krantz observes, “often no longer suitable, or appear to be inaccessible, to the present crop of students”[34]. Here the blamedoesnotfallonthe students. Narratives lack the punctuation provided by headings like “Definition” and “Theorem”,whichhelpnovicestodistinguishbetween,say,statementsthatcan be deduced from earlier ones and statements introducing new elements. If definition-examples-theorems expositions often deserve criticism, it is not for the usual reasons (take your pick), but because definitions are usually pre- sented as “given”, or as arbitrary points of departure for a game of logic. In fact, definitions are the result of design decisions. They also determine the flavor of the theorems (and proofs) derived from them. Hence it is crucial for understanding that these decisions are explained and justified. In mathematics texts, this is all too rarely done. One of the few exceptions is Halmos’s Naive Set Theory [26] which, if only for this reason (yet also for other reasons!), should be required reading for all beginning students — and many mathematicians as well. Halmos not only explains the design decisions andtheirshortcomingsformostconventions,butalsodoesnotshrinkbackfrom calling some poor practices “unacceptable but generally accepted”. Quine [47] evendesignateslesseroffensesas“glaringperversity”,whichseemsanaptchar- acterization of mathematicians acting against better judgment. Indeed, perceived“generalacceptance”isoftentakenasa licenceto perpet- uate junk conventions. Users ofinept designs typicallydefend them by feigning confusion at proper alternatives, calling them “nonstandard” even if they have been around for a long time and are routinely used by plenty of other authors. If an engineer is sloppy, his design may fail, even catastrophically. Math- ematicians often condone sloppiness, even if it sets bad examples and abuses confidence. Discerning students will be dissatisfied by the discrepancy between thereputationofmathematicsasbeingpreciseandactualpractice. Othersmay even get confused if insecure teachers insist on “doing things as in the book”. Yet, the engineering literature is not blameless either. Years ago Lee and 2 Varaiya [39] corrected many inept mathematical practices in signal processing. Playing down suchissues as “just a matter of notation” is misleading. Poor notationpreventstheshapeofexpressionsfromgivingguidanceinreasoning. It also reflects poor understanding, according to Boileau’s aphorism “Ce que l’on con¸coit bien s’´enonce clairement – Et les mots pour le dire arrivent ais´ement”. If authors misunderstand their own definitions, what about their students? “Ifitain’tbroke,don’tfixit”isanotherengineeringmaxim. Yet,asweshall see, even basic concepts that worked fine 50 years ago somehow got “broke”. This paper addresses the issue in the title by presenting a design view on variousconceptsfromtheliterature,butitisnotsomelinear,completeproposal. Often references include page numbers to make them truly useful for the reader. Forbrevity,co-authorsareomittedwhenmentioningnamesinthe text. 2. Case study A – Relations: two logically equivalent definitions 2.1. Simple and safe formulations The simplest “modern” definition of a relation is typical in older texts such as Bourbaki[9,p.71],Suppes[60,p.57],Tarski[61,p.3],butonlyinafewcurrent books, such as Jech [31, p. 10], Scheinerman [53, p. 73] and Zakon [65, p. 8]. Definition 1 (Relation). A relation is a set of ordered pairs. Equivalently, in symbols [9, 60]: Risrel≡∀z.z ∈R⇒∃x.∃y.z =(x,y) . Takingset andordered pair colloquially,andwith‘nonmathematical’examples, the word statement of Definition 1 is even accessible at grade school level. In this paper, when saying just “pair”, we always mean “orderedpair”. Somenotationaldesignissuesarisehere. First,anorderedpairiscommonly written (x,y). Some authors use hx,yi, a waste of symbols. In fact, one can even write x,y and reserve parentheses for emphasis or disambiguation, which also covers n-tuples like (x,y,z) and trees like ((x,y),z). Identifying (x,y,z) with ((x,y),z) as in Bourbaki [9, p. 70] is clearly a bad design decision. Second, the literature diverges about writing (x,y) or (y,x) and xRy or yRx. Quine[47,p.24]offersmanygoodreasonsforfollowingPeanoandG¨odel in using the natural order from spoken language, writing “a is the father of b” asaFb,and“aissmallerthanb”asa<b. Similarreasonswouldfavorwriting, for instance, “velocity versus time” as (v,t). However, mathematicians used to writing the “independent variable” first might feel disoriented—unlike novices! Tradition can be reconciled with reason by writing (x,y) ∈ R iff yRx. For human engineering reasons,such conventions should be stated conspicuously. Intermezzo: Quine, the angry notational engineer. In an uncharacteristic dia- tribeofnearlytwopages[47,p.24–26],Quinedeploresthe“sorrybusiness”and “glaring perversity” of ill-designed notations. He concludes (a) “I have given much space to a logically trivial point of convention because in practice it is so vexatious.”; (b) “[Whoever] switched a seemingly minor point of usage out of willfulness or carelessness cannot have suspected what a burden he created.”. 3 These remarks reflect typical engineering concerns. Indeed, (a) reminds us that avoidingflaws during the design phase is easy comparedwith repair after- wards (one line versus lengthy arguments), and (b) emphasizes the importance oftaking intoaccountthe interestsofthe users,namely,the future generations. Auxiliary notions. Mostauthorsusing Definition 1 addDefinitions 2, 3and4. Definition 2 (Domain, range). The domain of a relation R is the set of first members of the pairs in R. The range of a relation R is the set of second members of the pairs in R. Notation: theliteraturementionsvariousself-explanatorynotationssuchasDR or Dom(R) for the domain of R and RR or Ran(R) for the range of R. Definition 3 (Relation from X to Y). A relationfromX toYis a relation whose domain is a subset of X and whose range is a subset of Y. Equivalently, in symbols: Risrel(X,Y)≡Risrel∧DR⊆X ∧RR⊆Y . Notation: Writing R:X↔→Y or R:X ↔Y introducesa relationfromX to Y. Aside As in Meyer [42, p. 23], X↔→Y denotes the set of relations from X to Y. Also, the symbol : clearly distinguishes bindings from statements. A binding i : S, read “i in S”, introduces an identifier i for an object in a set S, whereas i ∈ S, read “i is in S” (or similar) is a statement in which i is used. Proper symbolism passes the prose test: transliterating formulas into words must yield sentenceswithcorrectgrammar. TheRHSofS ⊆T ≡∀x:S.x∈T isread“for all x in S, x is in T” or “every x in S is in T”. Lamport [36, p. 289] notes that the common forms {x∈S|p} and {e|x∈T}, where p is a boolean expression and e is any expression, are ambiguous if p is x ∈ T and e is x ∈ S. Writing {x:S|p} and {e|x:T} yields {x:S|x∈T}=S∩T and {x∈S|x:T}⊆B. Finally,S ini:S iscalledatypeandexpressesarangefori,notanattribute ofi. HencetheX andY figuringinR:X↔→Y areattributesofthetypeX↔→Y, not of R. This concludes the aside. The following formulation is independent of the (x,y) versus (y,x) issue. Definition 4 (Composition, converse). The composition S◦R of relations S and R is the relation such that z(S◦R)x≡∃y.zSy∧yRx. The converse R⌣ of a relation R is the relation defined by xR⌣y ≡yRx. Compositioniscalledrelative productbySuppes[60,p.63]andTarski[61,p.3], and resultant by Quine [47, p. 22]. Suppes writes S/R, the others S|R. 2.2. Entering murky waters: misconceptions, unsoundness and poor judgement Allbooksinourintroduction toproofsample[8,p.172],[14,p.101],[15,p.93], [19, p. 155], [20, p. 86], [23, p. 51], [24, p. 267], [27, p. 192], [49, p. 176], [55, p. 135], [62, p. 171] except [53, p. 73] combine Definitions 1 and 3 as follows. Definition 5 (Relation from X to Y). ArelationfromX toY is asubset of X ×Y. Equivalently, in symbols: Risrel′ (X,Y)≡R⊆X×Y . 4 Thisforcesdefiningrelationbackwards,asarelationfromX toY forsomeX,Y. Also,usingCartesianproductsatthisstagemaybeaneducationalburden[54]. A moreseriousproblemis thatdisregardingseparation of concerns carriesa heavypriceinunderstanding,apparentlyevenforthe authors. Mostfailtorec- ognize that Definitions 3 and 5 are equivalent: Risrel(X,Y)≡Risrel′ (X,Y). Indeed, manytextbooksstronglysuggestthatDefinition5somehow“glues” X and Y to the relation, and that one cannot even define just relation without adding “from X and Y”. Some texts remainvague here, but the litmus test for one’s understanding of a mathematical concept is the view on equality, in this case: when is a relation R from X to Y equal to a relation S from U to V? The answer was evident 50 years ago: with relations defined as sets, R=S iff both containthe same elements, regardlessofX, Y, U, V. Defining equality anew often causes unsoundness, typically by stating that R = S also requires X =U and Y =V [49, p. 179]. Indeed, if R⊆X×Y, X 6=∅ and Y ⊂V, then R⊆X ×V and Y 6=V, so R=R would require Y =V, a contradiction. Manytexts[49,p.180],[55,p.141],[62,p.236],[63,p.94]defineS◦Rfora relationR from X to Y and a relationS from U to V only for the case U =Y. Since R ⊆ X ×Y ⊆ X ×(Y ∪U) and S ⊆ U ×V ⊆ (Y ∪U)×V, this is not restrictive, unless one accepts the aforementioned unsound view on equality. Aside As a restrictivevariantof a relation from X to Y, Bourbaki[9, p. 72] defines a correspondence from X to Y as a triple (R,X,Y) where R⊆X ×Y, and restricts composition of (R,X,Y) and (S,U,V) to the case U =Y. 3. Case study B – Functions Poor design decisions for relations reappear for functions with a vengeance. This is especially unfortunate since, as Herstein puts it, Without exaggeration this [namely, a mapping or function] is probably the single most important and universal notion that runs through all of mathematics. [29]. The“modern”definitionoffunctionwasissue-free50yearsago,andstillisin analysis/calculusbooks[1,5,17,33,34,38,50,51,58,59,65],butunsoundness appears since 2005 in transition to proof texts [8, 13, 14, 15, 19, 20, 27, 49, 55]. In passing, we mention some harmful myths that can be read between the linesintextbooksbutsurfaceexplicitlyinoralandwrittenconversations. Myth #0 (a meta-myth actually)holds that divergencesin definitions just reflect dif- ferentneedsinvariousdisciplines[54]. However,oursamplescomefromalgebra, analysis/calculus, discrete math, logic, set theory, and reveal that nearly all of themusethesamefunctionconcept,differingonlyinthecaredevotedtodesign and formulation. Myth #0 is harmful in trying to divert closer scrutiny. Recurrentpointsofinterestsare: (i)definingfunction,(ii)functionequality, (iii)functionfromX toY,(iv)onto-ness,(v)functioncompositionandinverse. 3.1. Once again: simple and safe formulations (i) A relation R is called functional [9, 42] iff no two pairs in R have the same first member. Hence the following phrasings are equivalent; the choice depends onwhether relations areskipped, asinanalysis/calculustexts,ordefinedfirst. 5 Definition 6 (Function). (A) Apostol [1, p. 53]: A function f is a set of ordered pairs (x,y) no two of which have the same first member. (B) Bourbaki [9, p. 77]: A function is a functional relation. Definition6isalsofoundinDasgupta[16,p.10],Flett[17,p.4],Jech[31,p.11], Mendelson [41, p. 6], Scheinerman [53, p. 167], Suppes [60, p. 86], Tarski [61, p. 3], Zakon [65, p. 10]. Functionality justifies writing y = f(x) iff (x,y) ∈ f. As in [40, p. 1], one may write fx instead of f(x) when no ambiguity results. Authors using Definition 6 introduce the domain and range as in Definition 2. (ii) This results in the following theorem, quoted from Apostol [1, p. 54]. Theorem 1 (Equality). Functions f and g are equaliff (a) f and g have the same domain, and (b) f(x)=g(x) for every x in the domain of f. (iii) Authors starting from Definition 6, including Apostol [1, p. 578], Das- gupta [16, p. 10], Flett [17, p. 5], Jech [31, p. 11], Scheinerman [53, p. 169] and Zakon [65, p. 10], use the following common notions for classifying functions. Definition 7 (Function from X to Y). A function from X (in)to Y is a function with domain X and range included in Y. Notation: writing f :X →Y introduces a function f from X to Y. This is the ISO standard [30, p. 15], where a function is defined in broader terms, mentioned later. Divergent views on f :X →Y are nonstandard. It is convenient reading X → Y as the set of functions from X to Y and X→/ Y asthesetoffunctionalrelationsfromX toY [42,pp.25–26]. Suchtypes serve as partial specifications for functions. Tighter types are defined later. (iv) Next, we consider onto-ness as defined by authors using Definitions 6 and 7, for instance, Flett [17, p. 5], Jech [31, p. 11], Mendelson [41, p. 6], Scheinerman [53, p. 172], Tarski [61, p. 3] and Zakon [65, p. 11]. Definition 8 (Onto Y). For any set Y, a function is said to be onto Y, or surjective on Y, iff its range is Y. Note that onto is a preposition, and appears as such in Definition 8, which is also used by many authors (listed later) who do not start from Definition 6. The dual notion of “f is onto Y” (Rf =Y) is “f is total on X (Df =X). The dual notion of “f is into Y” (Rf ⊆ Y) is “f is partial on X” (Df ⊆ X). A function f : X→/ Y is often called a partial function, but is {(0,1),(2,3)} a partial function? Grammatically correct is: a function from part of X to Y. (v) For composition, we mention two equivalent formulations. Formulation (A) skips relations, as in Apostol [1, p. 140], Flett [17, p. 11], Mendelson [41, p. 7] and many others, listed later. Formulation (B) is based on relations. Definition 9 (Function composition g◦f). (for any functions g and f) (A) g ◦ f is the function whose domain consists of all x in Df that satisfy f(x)∈Dg and whose value (g◦f)(x) for arbitrary x in that domain is g(f(x)). (B) g ◦f follows Definition 4 assuming natural order as in Quine [47, p. 24]; otherwise f and g must be swapped, e.g., g ◦f = f/g in Suppes [60, p. 87], g◦f =f|ginTarski[61,p.3]. Proofobligation: showingthatg◦f isfunctional. 6 3.2. Wearing the ice thin: convoluted formulations (i–iii) Definitions 6 and 7 are sometimes crammed together, starting as early as Halmos [26, p. 30] and Herstein [29, p. 10], and more often in current texts including Krantz [34, p. 20], Velleman [62, p. 226] and others, listed later. Definition 10 (Function from X to Y). Let X and Y be sets. (a) A function f from X to Y, written f :X →Y, is a relation f ⊆X×Y satisfying the property that for each x in X the relation f contains exactly one ordered pair of the form (x,y). (b) The set X is called the domain of f. As a warning against uncritical copying, we reproduced the widespread but unacceptable phrasing, which suggests that the function is written f : X → Y (in fact, the function is written just f), and that f ⊆ X ×Y is a relation (in fact, it is a statement about the relation f). Proper phrasings are evident. All authors using Definition 10 overlook that part (b) requires proving that X is fullydeterminedbyf asdefinedin(a). This iseasy;the resultisX =Df. Moreimportantly,usersofDefinition10failtorealizeits logicalequivalence toDefinition 6(proof: exercise),including the standardmeaningoff :X →Y. Still, the different formulation has a huge impact on clarity. By disregarding separation of concerns, Definition 10 has given rise to Myth #1, which holds that one cannot define function by itself, but only function from X to Y. The common pitfalls are exposed by the litmus test: equality. Surprisingly few present-day texts using Definition 10 mention Theorem 1, found only in Daepp [14, p. 152], Gerstein [20, p. 113] and Smith et al. [55, p. 189]. Instead, many define equality anew, all too often unsoundly, as demonstrated later. (iv)“Classical”authorsusingDefinition10(orsimilar),includingBartle[5, p.13],Halmos[26, p.31],Herstein[29,p.12],Kolmogorov[33,p.5],saythatf is onto Y iff Rf =Y, as in Definition 8, using “onto”as a preposition. Some of the few “modern” users of Definition 10 who write “onto Y” are Gerstein [20, p. 118] and Smith [55, pp. xvii, 205], but their formulation lacks generality. (v) Most“classical”authors usingDefinition 10(or similar),including Bar- tle[26,p.40]andHalmos[26,p.40],defineg◦f forarbitraryfunctionsf andg, asinDefinition9. Onlyafewclassicaltexts[29,p.13][50,p.9]restrictcoverage of g◦f for functions f :X →Y and g :U →V to the special case U =Y. Most“modern”textssuccumbtothisrestriction,includingBloch[8,p.146], Roberts [49, p. 226],Scheinerman[53, p. 183],Smith [55, p. 197],Velleman [62, p.231]. An intermediateformrequiringRf ⊆Dg appearsinDaepp(2003)[14, p. 175], Daepp (2011) [15, p. 167], Jech [31, p. 11] and Krantz [34, p. 22]. The general form appears in Larson [38, p. 25] and Stewart [58, p. 40] which, not surpringly, are calculus texts, since restricted composition is impractical. 3.3. Falling through the ice: common yet unsound additions to Definition 10 (i–iii)Definition 10 is the sound part of definitions in transition to proof texts, but Bloch [8, p. 131], Chartrand [13, p. 216], Daepp [14, p. 147][15, p. 143], Garnier [19, p. 224] Gersting [20, p. 383], Hammack [27, p. 195], Roberts [49, p. 220], Smith [55, p. 185], Gilbert [22, p.13] and Wallis [63, p. 106], add 7 Definition 11 (Codomain). Definition 10(c) Y is called the codomain of f. However, just like Definition 10(b), adding 10(c) entails a proof obligation. Recognizingthisclearlyrevealsalogicalcontradiction. Indeed,thedefiniendum is codomain of f,the definiens is Y, butY is notuniquely determinedbyf. As for relations, letting f ⊆X ×Y ⊂X×Y′ reveals a contradiction. Still, some authors uphold Myth #2: writing f : X → Y makes Y an attribute of f by specifying f ⊆X ×Y. Yet, all this says about Y is Rf ⊆Y. Myth#3maintainsthatDefinition10containsambiguitiesallowingmultiple views, making logical contradictions just a matter of interpretation. Yet, in Definition 10(a), the definiendum and the definiens are clear (except as written in [8, p. 131]), using the unambiguous concepts subset and Cartesian product. Also, insofar as Definition 11 makes the perceptive reader wonder if the authorsreallymean“codomainof f”,further contextindicatesthey mostlydo. Again equality is most revealing. Apart from three exceptions mentioned, all transition to proof texts using Definition 11 overlook Theorem 1 and define equality anew. Roberts [49, p. 223] avoids conflict with Definition 10 by using thestatementofTheorem1. Others,e.g.,Bloch[8,p.136],Garnier[19,p.224], Hammack [27, p. 198] extend this statement with codf = codg, indirectly contradicting Definition 10. Interestingly, Smith [55, p. 189] explicitly states that function equality does not require equal codomains! (iv) Unsoundness also results from improper use of “onto” as an adjective, as inBloch[8, p.155],Daepp [14, p.163][15, p. 157],Gries [24, 282],Hammack [27, p. 199], Krantz [34, p. 22], Roberts [49, p. 231] and Velleman [62, p. 236]. Definition 12 (Onto). Afunctionf :X →Y isonto(surjective)iffRf =Y. WithDefinition12,thesamefunctioncanbebothontoandnotontodepending on whether or not the set Y appearing in f :X →Y happens to be Rf. (v)AlltextsaddingDefinition11requireforg◦f thatDg =Cf andforthe inverse f− that Rf =Cf. This is impracticalfor applications, e.g., in calculus. 3.4. Design considerations, variant concepts, and evaluation Many textbooks and countless blogs use the term codomain, typically in the unsoundwaydescribed. EventhoughmostauthorsusingDefinition10/11seem tosensetheproblemswithsqueezingincodomains,theysomehowfeelobligedto try. Ofcourse,theyneedn’t! Clearly,aproperaccountforcodomain is overdue. Inviewoftheearlieranalysis,thetermcodomainisbest(i)simplydiscarded, or (ii) used as the symmetric counterpart of domain, thus far called range, or (iii)recognizedasanattributeofatypelikeX →Y orX→/ Y,notofafunction. A quite different approachis defining a variantof the function conceptsuch that the set Y in f :X →Y is truly part of f, safely called the codomain of f. For instance, Bourbaki initially defines a function [9, p. 76] as a triple (F,A,B) where F is a functional relation with domain A and range included inB. Suchatripleissubsequentlycalledanapplication fromAintoB [9,p.76], which avoids confusion with using function for a functional relation [9, p. 77]. The term codomain is not mentioned in [9] — so let’s not blame Bourbaki! 8 The engineering question is: what purpose might such a variant serve? For the sake of generality, this issue is best disassociated from the set of pairs view, whose predominance in the discussion thus far just reflects random literature samples from diverse areas of mathematics. Many authors, including Lang [37, p. 38], Lee [39, p. 48], Royden [50, p. 8] and Spivey [56, p. 29], note thatasetofpairsisreallyarepresentation ofafunction,calleditsgraph. Hence let’s broaden the representational Definition 6 to a conceptual one, inspired by the Goursat/Courant style, but generalized to arbitrary domains and properly distinguishing f from f(x). For instance, the ISO standard[30, p. 15] just says that a function f assigns to each x in its domain a unique value f(x). Here “assigns” can be made precise by an assertion of the form A(x,f(x)), called relation by Bourbaki [9, p. 47], but not to be confused with a set of pairs. Equalityispivotalformathematicalobjects. Distinguishingfunctionsonthe basis of possible assignments outside their domain would be an useless compli- cation, hence these values are best declared irrelevant for a function. (A) The minimalist, “no frills” design reflecting this view is the following. Definition 13 (The function concept: minimalist design). i. A function f is an object fully specified by (a) a set Df, called thedomain of f, and (b) for each x in Df a unique value, written f(x) or fx. ii. The stipulation “fully specified” means that f = g if (a) Df = Dg and (b) f(x)=g(x) for all x in Df. (Note: “only if” by Leibniz’s principle [24]) iii. A function f from X to Y is a function such that (a) Df = X and (b)f(x)∈Y forallxinX. Suchafunctionisintroducedbywritingf :X →Y. Definition 13.iii simply follows the ISO standard: Df =X and Rf ⊆Y, where Rf ={f(x)|x:Df}. Types of the form X →Y are partial specifications. An illustration is defining sqrt : R → R with (sqrtx)2 = x. Composition is ≥0 ≥0 unrestricted: D(g◦f)={x:Df|f(x)∈Dg} and (g◦f)x=g(fx) as usual. Composition also supports specification by proxy: specifying g : Y → Z via f : X _ Y (_ indicating onto) and h : X → Z by g(fx) = hx. Well- definedness (functionality) of g requires h(x)=h(x′) whenever f(x)=f(x′). An example is defining the inverse: let g := f−, Y := Rf, Z := X and h:=id . Well-definednessoff−amountstof being1-1. Inthegeneralscheme, X iff is1-1,thengiswell-definedandg◦f =hisequivalenttog =h◦f−. Pattern matching is an instance: compare g(cons(a,x))=h(a,x) and gs=h(cons−s). Thesetofpairsviewwillremainausefulanalogy,e.g.,indefiningf− asthe object represented by the inverse relation, which is functional iff f is 1-1. (B) A typical non-minimalist design variant of Definition 13 would add: i.(c) a set Cf, called the codomain of f; ii.(c) Cf =Cg; iii.(c) Cf =Y. Logically,compositionand inversescould still be defined without restriction (exercise). However,authors using codomains do restrictg◦f by Dg =Cf and define inverses for 1-1 functions only if the latter are “onto” [their codomain]. So the issueboils downto: whatare the costsandthe merits ofcodomains? Shuard [54] published perhaps the only paper evaluating the function-with- codomainvariant. Hersingle(!) argumentinfavorisusingontoasanadjective. Yet, the ability to say that “f is onto Y but not onto Z” is more selective. 9 Shuard’s argument against is more solid: simplicity. She states: “Flett’s definition wins hands down as simplicity in analysis is concerned”. Her next statement, “In algebra, however, it is more convenient to start by mentioning the codomain of a function” (resembling Myth #0), is left unsubstantiated. Excellent algebra texts such as Herstein [29] do fine with the standard variant. InShuard’spaperandallothersourcesconsulted,suggestionsthatattaching acodomainmightbeconvenientturnsouttobefallacious,typicallyoverlooking that the standard view regarding f : X → Y already implies Rf ⊆ Y. This view covers all sensible purposes of f : X → Y, “mentioning” Y included, and without making Y a function attribute. Only for topology further study is neededtodeterminewhetherviewingY asanattributeofafunctionf :X →Y is just some tradition based on similar oversights or has genuine advantages. Still, what’s the harm in a function-with-codomain, beside complexity? From a conceptual and practical viewpoint, burdening a function with a codomain affects all other definitions, complicates equality and impoverishes the function algebra for inverses, composition, merge, override and so on [42]. Shuard [54, p. 10] notes that the only analysis book that she knows to use codomains [57] gets into trouble by defining the inverse of a 1-1 function f to havedomainRf asusualandignoringthatcodomainusersrequiresurjectivity. She adds that “the distinction between f : A → R and f : A → f(A) is so 1 tedious that it is clearly better forgotten at this stage”. One might say: “clearly better avoided from the start”, matching Smith’s view on equality [55, p. 189]. Aside: programming versus mathematics Types and signatures of “func- tions” in programming [19, Section 3.8] and some proof assistants typically are unique attributes by design. They are easier to implement than general sym- bolic computation of, say, D(g ◦f), but remain rather crude approximations of types as partial specifications following the ISO standard. Indeed, the gen- erality provided free of charge by the minimalist/standard view (illustrated for g◦f and f−) is common fare in mainstream mathematics, by which we mean: the mathematics routinely applied by the large majority of users, ranging from mathematicians active in analysis/calculus, linear algebra, discrete math etc. to engineers active in signals and systems. Such users would be justified in dis- missing as impractical any definition that infringes on these “acquired rights”. Similar considerations in the context of specification languages are found in [35]. In the specification language TLA+ [36, p. 48], the notation [X → Y] has the meaning of X →Y as defined by the ISO standard. Notsurprisingly,nearlyallcalculus/analysistextsavoidcodomainsandsim- ply proceed from Definition 6 or 13 or equivalent, the most complete picture being presented by Flett [17, pp. 4–6]. In this manner, calculus/analysis books succeed in giving a proper account in about one page, without being too terse, and often before page 10. Functions are too important to postpone their intro- duction beyond page 100, only to get them bogged down in unsoundness. Conclusion For mainstream mathematics, attaching a codomain onto a function has no verifiedmerits but increasescomplexity and reducesgenerality. Hence any definition that accepts such drawbacks entails a heavy obligation of justifying the design, even if it concerns only some niche area. 10