Sumterms, Summands, Sumtuples, and Sums and the Meta-arithmetic of Summation
aa r X i v : . [ m a t h . HO ] S e p Sumterms, Summands, Sumtuples, and Sums andthe Meta-arithmetic of Summation
Jan A. BergstraInformatics Institute, University of Amsterdam ∗ September 18, 2020
Abstract
Sumterms are introduced as syntactic entities, and sumtuples are in-troduced as semantic entities. Equipped with these concepts a new de-scription is obtained of the notion of a sum as (the name for) a role whichcan be played by a number. Sumterm splitting operators are introducedand it is argued that without further precautions the presence of theseoperators gives rise to instance of the so-called sum splitting paradox. Asurvey of solutions to the sum splitting paradox is given. ∗ email: [email protected], [email protected] . ontents N and of Z . . . . . . . . . . . . . 23 References 30
I will use the phrase meta-arithmetic to refer to reflections about arithmetic,without the connotation, known from meta-mathematics that such reflectionsare necessarily supported by or even consisting of mathematical work, or mainlyconsist of formal logic.With elementary arithmetic I will understand the activity of developing, con-firming and disconfirming, proving and disproving, open and closed identitiesover an arithmetical signature. I will restrict attention to the case of character-istic zero, and in fact I will understand arithmetic as school arithmetic ratherthan as its academic (somehow advanced) counterpart.The central question of this paper is: “what is a sum?” Unlike the relatedquestion “what is a fraction” the educational literature leaves this questionremarkably untouched. I propose to understand sum a a role name, rather thanas a noun. in arithmetic, the role of a sum is played by a number.
To the best of my knowledge there is no convincing definition of sums as acollection of entities. In this respect sums differ for instance from squares whichmay, but need not, be understood as a collection of numbers. Sum is a rolename, as in “ a is the sum of b and c ”, rather than a noun referring to an entityof a certain type. Moreover, in this case b has the role left summand , and c hasthe role right summand . Further a has as well the role function value , and both a and b have the respective summand roles and in addition to those role(s) thestill weaker role of (being an) argument .Objects, concrete as well as abstract, may have different roles at the sametime, and the same object may have different roles in different contexts. Unlikephysical objects one and the same number can appear in different contexts atthe same time. For instance in 3 = 2 + 1 and 5 = 2 + 3, two signs which coexistin time, the sub-sign 3 has different roles.Given a < b , a has the role of the smaller entity in an ordered pair. It ispointless to say that a is small (or smaller). Indeed a is element of an ordereddomain, which, however, is not the same as its role in a < b . Sum, summand,left summand, and right summand are roles which are plausible for numbers.Who thinks of sum and summand as roles, thinks of numbers as entities capable3f fulfilling such roles. The view that sum is a role instead of a predicate, or acollection denoted by a predicate, therefore comes with some form of realism onnumbers, because as entities these may fulfil one or more roles. I will assume that say 75 is a proper noun which stands for a sufficiently specificentity. Proper nouns are also called proper names. Alternatively, and lessplausible, one may contemplate the notion that 75 is a common noun (alsocalled generic noun) so that it stands for all entities which somehow qualifyas the number 75. Suppose it is agreed that x is a variable which ranges overnatural numbers (whatever these are, even if they don’t exist after all), onemay holds that, within a context aware of said agreement, x is a common nounbut not a proper noun. Now consider 75 + x . This sign may be considered todenote a parametric proper noun: a noun which comes about after obtaininga denotation for a parameter x from an environment. In fact x itself may alsobe considered a parametric proper noun. While “natural number” is a commonnoun ranging over the natural numbers, x names a not yet known, but onceknown also specific number. In logic the context providing an actual number towhich x refers is called a valuation.Sum is not a proper noun (which must refer to a specific entity, also calleda generic noun), as it does not refer to any specific entity. But sum is not acommon noun (ranging over some class of entities, alternatively called a genericnoun) either. I will qualify sum as a parametric proper noun: in the context oftwo summands sum is a proper noun with the summands as parameters. Thesummands act as parameters to be taken from a context which allow “sum” torefer to a specific value.Roles which are played by unique entities once key paramaters from a contexthave been determined may be considered parametric proper nouns.Nouns may also be qualified in terms of concreteness, where concrete nounsrange over collections of physical entities, and where abstract nouns range overnon-physical entities. So “natural number” may be considered an abstract com-mon noun, and sum may be considered an abstract parametric proper noun.The sign 25 is a physical entity, which may be uniquely determined in a givencontext, say on a piece of paper. Then “look at the 25 in the middle of thepage” contains 25 in the quality of a singular concrete proper noun. In “the25’s in this paper should be rendered boldface” 25 is embedded in a a pluralnoun. As it turns out the typing of grammatical units as signs (i.e. entities),nouns, proper nouns, common nouns, abstract nouns, concrete nouns etc. isnon-trivial and may give rise to polymorphism.A distinction between collective nouns and singular nouns may be made,besides and independently of, a distinction between plural nouns and singularnouns. Then “the arguments of function f ( − , − , − )” and “the values of function g ( − , − , − )” are collective nouns. Plural nouns are nouns which occur in pluralform, for instance the occurrence of times in “3 is three times as large as 1”.Plural noun is a linguistic notion without manifest logical content.4 summarize the above considerations regarding sums with the followingclaim. Claim 1.1. (Scepticism on sums) There is no convincing and generally accepteddefinition of the concept of a sum in elementary arithmetic. The notion of asum is too ambiguous for it to be used as a common noun or as a noun.
A sumtuple is an ordered triple ( a, b ; c ) such that c is the sum of a and b .So a sumtuple, in this case also called a sumtriple, contains a sum, and insomewhat less accurate language it has a sum. The sum of a sumtuple is simplyits third component. Once the terminology of sumtuples has been coined onemay imagine a distinction between valid sumtuples and invalid sumtuples. Fora valid sumtuple ( a, b ; c ) it is the case that a + b = c , and for an invalid sumtuplethat is not the case. In some cases a more detailed notation for sumtuples maybe useful, e.g. +( a, b ; c ) or plus ( a, b ; c ).The pair ( a, b ) of both summands of a sumtuple is important as well and Iwill refer to such a pair as a sumterm or as a plusterm. Below we will only usesumterm. Different notations for sumterms are useful, e.g. +( a, b ) or simply a + b . It is sumterms rather than sums which are equipped with two summands.Sumterms have a sum as well, not in the sense of containing a component whichhas the role of a sum, but in the sense of allowing a unique completion witha sum (a number with role sum) to a valid sumtuple. As it tuns out “havinga sum” is ambiguous (or rather polymorphic), it means something different forsumterms and for sumtuples.Conceiving a function as a graph its tuples contain result values, whereasupon viewing a function as a prescription there is no notion of containment of aresult. Having a value is a different matter for different conceptions of function,and the mentioned divergence of interpretation for having a sum correlates withdifferent ways of understanding addition. My objective in this paper is to sharpen the picture of sums which was sketchedin the preceding Paragraphs. Various questions immediately arise: (i) whatis a number, in other words which entities play the roles of sum and of therespective summand, (ii) what is the relation between syntax and semantics,(iii) is any of the two, numbers or expressions for numbers, more fundamentalthan the other, (iv) why not do without expressions (sumterms), (v) why notdo without sumtuples, (vi) what is the official academic perspective on thesematters, if any, (vii) what happens if one insists that a sum is a number withtwo components, a left summand and a right summand, (viii) what is the statusof sums of more than two numbers (are these roles just as well, or do nestedexpressions, or abbreviations thereof, enter the picture at this stage)?5
Sumterms a kind of arithmetical quantities
Sumterm is a noun ranging over expressions with addition as the leading functionsymbol. Using the notion of a sumterm requires acceptance of a distinctionbetween syntax and semantics. I will refer to an expression or term as anarithmetical quantity (abbreviated to AQ below).
Definition 2.1.
A sumterm is an AQ with addition (usually written + ) asits leading function symbol. With this definition comes some level of abstraction: 2 + 3 , plus (2 , , +(2 , Definition 2.2.
For a sumterm p + q the AQ p is its left (or first) summandand the AQ q is its right (or second) summand. Sumterms have a sum, which is another word for value in the special case ofsumterms. As was already stated above, unlike sumtuples, however, sumtermsdo not contain a sum.One may think of natural numbers as presenting straightforward context inwhich to contrast syntax and semantics. Remarkably, however, maintaining aconvincing distinction between syntax for naturals and semantics of naturals ischallenging.I assume that it is plausible to see some form of syntax at work for an agentwho makes sense of the sign 2 + (5 + (( −
3) + 0)). It is less obvious, however,what may count as semantics in relation to such kind of signs. The notionof an AQ comes with a syntactic bias just as the notion of a number comeswith a semantic bias. Both for numbers and for AQs, however, providing anunambiguous definitions seems to be impossible.AQs are abstract entities, not tangible signs. AQs may serve as interpre-tations of signs, and more specifically of arithmetical signs, amenable to beingconceived of as abstract entities. Conversely signs may represent AQs or partsthereof in a human readable manner. A sign may denote an AQ, but the AQdoes not inherit attributes like, size, place, time, color, authorship, etc. formthat sign. I will assume that meta-arithmetic takes for granted the existenceof AQs which can be imagined as abstract entities, in advance of any detailedunderstanding of arithmetic.Semantics enters the picture via the assumption that for AQs p and q , theidentity (equation) p = q expresses the notion that p and q have the samemeaning (or equivalently: denote the same entity). Semantics concerns ques-tions about the nature of meanings for various classes of AQs? Semantics hastwo aspects: ontology: what are semantic entities, and denotation: which se-mantic entity is assigned to a given AQ. For AQ p the latter entity is denotedwith J p K , which is itself a notation from meta-arithmetic, unlikely to show upin arithmetic proper. 6 efinition 2.3. A poly-infix sumterm is an AQ of the form t + . . . + t n fornatural n > . Poly-infix sums of naturals require naturals to be known for indices. It ispossible to introduce poly-infix operations for each arity one at a time, so thatin some stage only a limited number of such operations have been defined andno appeal is made to an infinite totality of natural numbers.
Definition 2.4.
Let k, n be decimal natural numbers (see 3.1 below) with k ≤ n .Then(i) length of the poly-infix sumterm t + . . . + t n is n , and(ii) the k -th summand (also denoted smnd k ( p ) ) of the poly-infix sumterm p ≡ t + . . . + t n , is t k . For n = 2, a poly-infix sumterm is just a sumterm, with smnd ( p ) as its leftsummand, and smnd ( p ) as its right summand. The sums of poly-infix sumtermsof different arities are related as follows: t + . . . t n − + t n + t n +1 = t + . . . + t n − + ( t n + t n +1 ).Adopting the presence of poly-infix sumterms can be contrasted with adopt-ing the view that say t + . . . + t abbreviates (( t + t ) + t ) + t n . By usingpoly-infix sumterms the use of inductively defined syntax can be delayed or evenavoided.The introduction of poly-infix sumterms suggests the introduction of abbre-viations as follows: e.g. t − t − t + t abbreviates the sumterm t + ( − t ) +( − t ) + t .Poly-infix sumterms arise in the teaching of arithmetic if for instance AQslike 3 + 7 + 9 appear without either a description of a convention on how toview the expression as an abbreviation of another expression which involvesadditional brackets, or alternatively an explanation on when brackets may beleft out, detailing the role of assiciativity for that matter. Definition 2.5.
The sum of AQs p and q is the value of the sumterm ( p ) + ( q ) . For instance: the sum of p ≡ q ≡ p ) + (0) = ( p ) + 0 = p , all numbers are such values, andfor that reason Definition 2.5 does not lead to a non-trivial notion of sum (asa proper noun). This situation may be contrasted with squares which can beusefully defined, among integers, as the members of the range of the squaringoperation.Given an equation p = Q with Q a sumterm or a poly-infix sumterm it maybe said that p is written as a sum. Thus “written as a sum” applies to AQs andabbreviates “written as a sumterm or written as a poly-infix sumterm”. I will discuss various ways in which one may imagine natural numbers as ab-stract entities. Remarkably the seemingly elementary question “what is a nat-ural number” has not received a definitive answer in mathematics, in logic, or7n philosophy. The conventional understanding of natural numbers as finite or-dinals in ZF set theory, thus following the classic proposals of J. von Neumann,seems not to provide an intuitive understanding of natural numbers which isboth convincing and is usable at an elementary level at the same time.In 3 below I will survey different options for the ontology of natural numbersand in addition I will motivate the choice just presented: in terms of ontology,natural numbers are AQs and as a semantics producing operation finding themeaning of an AQ with type natural number is a projection.Rather than leaving the ontology of natural numbers open, I will choose asmy preferred semantics of natural numbers a collection of arithmetical quan-tities, the so-called decimal naturals (see Paragraph 3.1 below). The semanticfunction λt. J t K is a projection (for each AQ t it is the case that JJ t KK = J t K ). Forexample: JJ
17 + ( − KK = J K = 16. One may speak of an arithmetical quantity t without having a precise definitionof the relevant syntax at hand. So 1 + 2 can be labeled an AQ even if (that isat a stage where) there is no answer to the questions like: (i) is (1) + 2 an AQ,(ii) is ((1)) + 2 an AQ , (iii) is (1 + 2) an AQ, and (iv) is +(2 ,
3) + 4 an AQ?Below it will be assumed that these questions are given an affirmative answereach, but that need not be the case.On AQ there is an equality = AQ , abstracting from redundant bracketing andspacing, including new lines and new pages. For instance 0 = AQ (0) = AQ ((0))and 1 + 2 = AQ AQ (1 + 2) = AQ ((1) + 2) while 1 + 2 = AQ = AQ (1 + 2) + 5. Substitution on AQs binds stronger than additionand it will introduce additional brackets as follows: [ t/x ] P [ X ] = AQ P [( t )]. Forinstance [1 + 2 /X ](3 + X ) = AQ (3 + (1 + 2)) = AQ /X ]3 + X = AQ (3 + X ) = AQ X . The let construct, however, works differently,by not introducing any additional brackets: let x = AQ in (3 + x ) = AQ (3 + 1 + 2) = AQ t/x ] A = AQ let x = AQ ( t ) in A .In general, for closed AQs X and sumterms Y , X = AQ Y = ⇒ X = Y butnot the other way around. Working with AQs amounts to the adoption of some form of syntax. However,adopting syntax without meaning is problematic for arithmetic. There must bea balance between syntactic considerations and semantic considerations. If X is considered an expression, or an AQ, the question arises what kind of entityis denoted by X . I will assume the following principle. Principle 2.1. (Balance principle for syntax and semantics.) Adopting AQsor any other form of syntactic entities, brings with it a complementary need toadopt semantic considerations.
Claim 2.1. (Skewed plurality claim for syntax and semantics.) Once a contrastbetween syntax and semantics is adopted for elementary arithmetic it is plausibleto acknowledge that (i) there is a plurality for both, while (ii) the divergence ofoptions and opinions regarding semantics of numbers exceeds the divergence ofviews regarding notations for numbers.
The skewed plurality phenomenon indicates that uniformity and consensusis skewed in the direction of syntax, form, and notation, with high uniformityand consensus on such matters in comparison to consensus on meaning. I willprovide some arguments for Claim 2.1 below.
I will assume that an individual P upon being taught arithmetic acquires whatI will call competence of elementary arithmetic (CoEA). Such competence isprobably best measured as some partial ordering. Instead I will distinguish threelevels of CoEA. Being easily able to answer standard questions at elementaryschool level amounts to basic CoEA, whereas the competence level of a teacheris denoted as professional CoEA and the level of a professional mathematicianis labeled as academic CoEA. Ontology of arithmetic may be based on various preconceptions, to mention: • Natural numbers can be considered ordinals as well as cardinals, and bothviews suggest different semantic options. • Natural numbers, if there are supposed to exist, are interpretations ofsuitable AQs and therefore the ontology of natural numbers may also bederived from syntax if syntax is supposed to exist in advance of arithmeti-cal semantic considerations. • Naturals are integers. In mathematics the informal use of language sug-gests that the natural numbers are subset of the integers. Although theassertion “3 is an integer?” is plausibly confirmed, strictly speaking mostconventional definitions of naturals and integers do not support this as-sertion. • It is plausible to view the collection of natural numbers as a set, with as aconsequence upon adopting ZF set theory that particular natural numbers9re sets as well. However, once taking ZF as the technical foundation, anysubsequent choice for a set N of naturals is quite arbitrary, and labeling aparticular set as a natural number (on the grounds of membership of N )can only be based on an arbitrary convention. • One may hold that natural numbers can and should be defined withouta simultaneous introduction of the core operations of addition and mul-tiplication, the introduction of which takes place at a later stage. Andone may alternatively hold the converse, claiming that without operationsthere is no point in viewing any objects as numbers, i.e. that naturalnumbers must come with an arithmetical datatype. • Contemplating the natural numbers as a collection is an option. Collectionis a form of aggregate which is more general than set and class. The naturalnumbers may als be chosen as urelements for a ZF style set theory.
A well-known construction due to J. V. von Neumann ([21]) determines thenatural numbers (i.e. the elements of the set N which itself is an ordinal aswell) as the finite elements of the class of ordinals.Now following the observations of Paul Benacerraf in [3] numbers of sort nat are not intrinsically unique abstract mathematical entities, at least not in ZFstyle set theory and under the assumption that N is a set, because, in spite ofthe technical efficiency of the well-known encoding as finite ordinals, there is nounique canonical representation of ordinal numbers in ZF.The von Neumann encoding of ordinals provides finite ordinals as a veryplausible structure for naturals but that design is not unique, and there is nocompelling reason to adopt that particular design so that there is no compellingreason to view the finite ordinals thus defined as the true natural numbers. Iwill refer to naturals thus defined as the ordinal natural numbers, notation N ord ,with the understanding that “ordinal” will be omitted on most occasions. For cardinal numbers one may distinguish concrete cardinal numbers, that isordinals which are cardinals at the same time, and abstract cardinals, i.e. theclass of all sets with the same number of elements. Thinking in terms of concretecardinals leads to the same natural numbers as thinking in terms of ordinals.Considering abstract ordinals that changes.Working in ZF set theory the sets with cardinality 17 constitute a properclass say C . Let ω denote the set of finite ordinals in ZF, and let ≡ denoteequicardinality on sets. Then C = { x | x ≡ } with 17 the 17th ordinalin ω . I will write [ A, B, C, . . . ] for a collection the elements of which are classesand may be proper classes. Now N card = [ C , C , C , ... ] is a collection10f classes which itself is not a class, because its elements are proper classes. Iwill refer to a collection of classes as a meta-class; meta-class is a special caseof collection.The meta-class N card cannot be found in ZF set theory but it exists invariations thereof, such as Ackermann’s theory of sets and classes (see e.g. [19]for a discussion of these matters). One may write N card = [ X ∈ Class | ∃ n ∈ N ord . ( X = { x ∈ Set | x ≡ n } )]where X ranges over all classes and x ranges over all sets. Forgetting that thedefinition of N card uses N ord one may take the extension of N card as the morefundamental notion which is available for conceptualising naturals. Howeverobvious the idea of N card may be, when contemplating finite cardinals fromfirst principles, there is not even a standard notation for it as the use of [ .. ] ismerely an ad hoc proposal for this occasion, and it lies outside the comfort zoneof ZF based academic mathematics. Nevertheless N card constitutes a credibleoption for an ontology of natural numbers. A classical idea due to Peano is to have a successor function S in mind and toidentify the sequence of naturals as the sequence: 0 , S (0) , S ( S (0)) , S ( S ( S (0))) , . . . .Now 17 is a notation for a natural number rather than a number itself.As a conceptual idea the main source of lack of uniqueness of number thusconceived comes from the choice of a name for the successor function. This pricemust be paid if one prefers not to consider naturals as a collection which is given(and thereby unique) as a a primitive set in advance. The Peano representationis technically very attractive, and has found many applications in theoreticalcomputer science. This approach and can be used to provide the semanticfoundation of elementary arithmetic without taking all of ZF on board. Onefinds e.g. J K = S ( S ( S (0))). One may view “3” as belonging to syntax and“ S ( S ( S (0)))” as a semantic entity. I will refer to these natural numbers asPeano’s natural numbers and I will refer to Peano’s natural numbers as N peano . An interpretation of N which is workable for basic CoEA as well as for profes-sional CoEA identifies the natural numbers simply as non-empty digit sequences.So it is assumed that N d ⊆ AQ, N d ⊆ AQ, and N d ⊆ AQ. A naive set theoryis adopted which does not give rise to the idea that digit sequences should beencoded in a setting of lower level primitives, and therefore does not come witha sense of arbitrariness of a specific encoding of these. With this interpretationsemantics of natural numbers becomes a projection of AQs.The elements of Z d may be termed decimal natural numbers in order not toclaim that a fully general treatment of natural numbers has been obtained andsuggesting instead that no such fully general treatment exists.11hus a single structure and domain for the interpretation of the sort nat ischosen. Insisting on ZF foundations the pluralistic nature of this approach isundeniable because encoding digit sequences in ZF can be doe in many ways,none of which can claim a foundational significance in excess of other encodings.I will adopt (this particular form of) projection semantics for naturals andintegers as a preferred option, and I consider this choice to be justified by a com-bination of its intuitive appeal with the absence of any other known option forthe semantics of natural (and integers) which is conceptually more convincing. Definition 3.1. (Decimal natural numbers). N + d is the collection of non-emptysequences of decimal digits starting with a non-zero digit as the first symbol. N d = { } ∪ N + d . N + d is understood as the collection of positive decimal natu-ral numbers, and N d is correspondingly understood as the collection of decimalnaturals. Naive set theory allows not to be bothered by the details of encoding se-quences of digits in ZF set theory. Such details unavoidably come with manyalternatives thereby refuting uniqueness of the concepts in a principled sense.The fact that at some stage a confrontation with classical paradoxes necessitatesmore caution is then taken for granted as these paradoxes are unlikely to showup in school arithmetic.An argument in favour of definition 3.1 is that it is as close as possible tothe idea that “17 is a natural number”. Using naive set theory one may claimthat all occurrences of the sign 17 share the underlying non-material meaning of17-hood which is codified by Definition 3.1. From the point of view of logic orof computer science it is a disadvantage of decimal natural numbers that thereare infinitely many constants. From a conceptual perspective this disadvantagemay considered an advantage instead because it is simpler. By taking only afinite collection of constants into account one obtains useful simplified and finiteapproximations of arithmetic.
Given the naturals, say N ord , N card , N peano , and N dec different constructionsfor integers can be applied used. I will mention some options without any claimof completeness. Below α ranges over { ord, card, peano, d } .1. Consider the integers as equivalence classes of pairs of naturals (with( a, b ) ≡ ( c, d ) ⇐⇒ a + d = b + c ). This leads to Z eqcord , Z eqccard , Z eqcpeano , and Z eqcd . As a consequence of these assumptions: N α ∩ Z α = ∅ .2. As a second option one may understand the integers as equivalence classesof pairs of naturals (with ( a, b ) ≡ ( c, d ) ⇐⇒ a + d = b + c ) and adapt thenaturals to this notation by redefining the natural numbers as the equiv-alence classes of the form [( n, ≡ for n ∈ N α , that is the naturals insidethe integers. Write N eqcα for this set (with eqc for equivalence classes).And as a consequence of these modified assumptions: N eqcα ⊆ Z eqcα .12. Adopt and in addition consider the positive naturals N + α = N α − { } (where the mechanism for deleting 0 differs for the respective cases) adopt Z sα = { } ∪ { , } × N + α ( s for signed) as definition of the integers on topof the naturals. As a result N + α ⊆ N α and N sα ∩ Z sα = { } .4. (Decimal integers.) In the case of decimal integers each of the mentionedoptions may work, but the following is more plausible, and the simplestnotation is reserved for this case: Z d = { } ∪ N + d ∪ − N + d . It then followsthat as sets: N + d ⊆ N d ⊆ Z d .The plurality of options for the integers constitutes a cascade starting witha plurality of options for natural numbers. In no way, however, the survey ofoptions is complete or can be completed.I notice that decimal rationals may be defined as the decimal integers Z d extended with the AQs p/q and − p/q where: (i) p and q are nonzero decimalnaturals, (ii) q >
1, and (iii) p and q have no common divisors beyond 1; withthis definition decimal rationals constitute a subcollecton of AQs while beingquite different from so-called decimal fractions. For rationals this leads to theset of numbers Q d for which N d ⊆ Z d ⊆ Q d is valid. The collections of numbers outlined in the preceding Paragraphs each may beequipped with one or more operations and constants taken form: successor ( S ( ) )addition ( + ), opposite ( − ), multiplication ( · ) and constants 0 and 1. Com-bining a domain with a collection of constants and functions gives rise to the ideaof a datatype, and in the present context I wil speak of arithmetical datatypes,thereby loosely indicating properties expected of these constants and functions.For instance (i) N d (0 , , + , · ) is the datatype of decimal naturals (which alreadycontain 0 and 1) with addition and multiplication, (ii) Z sord (0 , , + , − , · ) is thedatatype of signed integers based on ordinal naturals, (iii) N peano (0 , , +) isthe arithmetical datatype with Peano’s naturals and a constant 1, equippedwith addition. (iv) Z eqccard (0 , , + , − ) represents integers obtained as equivalenceclasses over cardinal naturals, with constants 0 and 1 and operations additionand opposite. These functions are not sets, and are merely defined as collectionsof pairs. The structures N ord (0 , , +) , N peano (0 , , +), and N d ( , , +) have much in com-mon, in fact: N ord (0 , , +) ∼ = N peano (0 , , +) ∼ = N d (0 , , +) i.e. the three struc-tures are isomorphic. N card (0 , , +) is in the same isomorphism class (or ratherisomorphism collection) but a ZF based definition of isomorphism with the otherstructures cannot be used.Similarly Z eqcord (0 , , + , − ) ∼ = Z eqcpeano (0 , , + , − ) ∼ = Z eqcd (0 , , + , − ) ∼ = Z sord (0 , , + , − ) ∼ = Z speano (0 , , + , − ) ∼ = Z sd (0 , , + , − ) ∼ = Z d (0 , , + , − ). An iso-13orphism class of a datatype is an abstract datatype. I will use the followingnotation: N (0 , , +) is the isomorphism class of N d (0 , , +), Z (0 , , + , − ) is theisomorphism class of Z sd (0 , , + , − ), Z (0 , , + , − , · ) is the isomorphism class of Z sd (0 , , + , − cdot ). Working with arithmetical abstract datatypes removes muchof the pluriformity which has been noticed above, leaving the signature, i.e. thelisting of constants and operations as the major remaining parameter.Structuralism (in elementary arithmetic) refers to the viewpoint that isomor-phism classes rather than particular structures (arithmetical abstract datatypesrather than arithmetical datatypes) constitute the fundamental ontology ofnumber systems. Proposition 4.1. (Objective number scepticism). There is no such thing as anatural number. At best there are isomorphism classes of structures for naturalnumbers, assuming the presence of a first element named , a second elementnamed , and a successor function (which may be derived from addition), anumber being a mapping from structures in the isomorphism class producing anelement of the domain of the argument (that is a selection of an element fromeach domain) in such a manner that for each structure equally many successorsteps are needed to arrive at the selected element (this class of numbers allowsan inductive definition). Number realism in each of its forms seems to involve choice between a plu-rality of options. Simply adopting number realism, say for nat , does not createmuch clarity in excess of the fact that having done so a choice for a semanticdomain nat is in order.
In response to objective number scepticism one may insist that structures ratherthan numbers exist be it modulo isomorphism. However, mathematical struc-turalism (realism for mathematical structures, see e.g. [20]) based on ZF settheory is pluralistic just as much as number realism is bound to be. It is notclear to what extent structuralism can be decoupled from explicit mention ofsignatures, as the remaining source of pluriformity for number systems. Doingaway with signatures leads to the following description.
Proposition 4.2. (Structural and pluralistic realism for N ). Assuming ZF settheory as a basis it is plausible to support pluralistic objective structural numberrealism for N : structures for numbers of sort N exist and are unique up toisomorphism. With that understanding of say N an assertion like ,expresses a fact about a chosen structure for N , and indeed about all isomorphicstructures for N , an assertion which is acceptable for all agents who share thechosen structure (or structures). N (0 , , +) consists of all infinite commutative and associative semigroups in ad-ditive notation which are generated by 1. Now the semantics ( J K ) of thenumber denoted by 17 (assuming that one wishes to view 17 as denoting a nu-mer, rather than as a number itself) may be taken to be the mapping (univalentrelation) which assigns to a structure in N (0 , , +) the 17-th element of its do-main. This kind of mapping can be formalised in terms of abstract pointedstructures. N (0 , , c , +) as the isomorphism class of N d (0 , , c , +) with c
17 in-terpreted as 17. Here N d (0 , , c , +) is a so-called pointed structure, in this casewith a constant c
17 pointing to 17.Using abstract pointed arithmetical datatypes as the foundation on an on-tology leads to the following “definition” of the natural number 17: J K apadt = N (0 , , c , +). While one might think of the naturals as a paradigmatic example of an infiniteset, and a proof of concept for the very idea of infinite sets, it is not. It isthe virtue of arithmetical datatypes to sets as domains. By changing one’s settheory, for instance by adopting Ackermann’s set theory as mentioned abovecollection of numbers may be turned into a class, though not into a set.
One may be dissatisfied with the idea that pluralistic structural realism (or itsmeta-class version) is the final word on what a natural number is. Beyond suchideas modern mathematics has a lot to say, however. In the 90’s of the 20thcentury Voudvovsky has initiated a novel approach to the foundations of math-ematics which combines type theory with dependent types (thereby followingde Bruijn and Martin-L¨of), proof checking based on type theory (e.g. followingCoq), constructive logic (Heyting) and mathematics (Brouwer), category the-ory, and homotopy theory. According to [2] so-called homotopy type theory(HTT) overcomes the arbitrariness of ZF set theory as a universal language forencoding mathematics, and opens the door to a more determinate, and thereforemore credible, form of structural realism in mathematics than is provided onthe exclusive and classical foundations of ZF set theory.Moving beyond ZF style axiomatic set theory provides perspective on variousforms of realism which seems to be blocked in set theory.
Perspective 4.1.
Homotopy type theory opens a path towards (univalent, non-pluralistic) structural realism for various number sorts, including N and Z . Perspective 4.2.
Homotopy type theory opens a path towards (univalent, non-pluralistic) realism for various number sorts, including N and Z which will betypes (rather than classes or sets). HTT provides a potential path forward for acquiring an academic CoEA forindividuals who are not satisfied with the lack of determinacy of a pluralisticstructuralist approach within ZF, who consider meta-class realism unattractivebecause meta-classes are uncommon entities, and who prefer not to becomedependant on relatively unknown alternatives for ZF set theory.
Independently of one’s view on semantics, one may or may not accept the ex-istence of syntax. In the context of elementary arithmetic, I will identify theexistence of syntax with the existence of AQs. I will use syntax realism as thelabel for a position which admits the existence of AQs. An attempt to describein more detail what syntax realism amounts to lead to the following Definition,which works conditionally on whether or not semantic realism is adopted.
Definition 5.1. (Syntax realism.) P adopts syntax realism for sort S , herechosen from { pnat, nat, int } , if the following conditions are met:1. P uses a name, say AQ S , for a subtype of AQ.2. P assigns to AQ S as its meaning (extension) a collection AQ S of non-material entities which are referred to as AQs (terms, expressions) of (for)sort S .3. If P assumes some form of realism for sort S (by interpreting S as || S || P )then to each AQ p in AQ S an element J p K of || S || P is assigned.4. The elements of AQ S are referred to as terms for S (alternatively: termsof sort S , expressions for S , or AQs for S ). If the assignment function J − K is surjective AQ S is considered to be complete(from the perspective of P ) for the description of numbers of sort S .Adoption of syntax realism for S may but need not imply that P considers AQ S to be a set on which (mathematical) functions can be defined. Neitheris it always the case that an equality relation is present for AQ S . If present Iwill assume that this equality relation is denoted with (or can be referred to as)= AQ , or if confusion with other sorts may lead to confusion as = S AQ . The varioussorts may be viewed as a type system for AQs which allows polymorphism, thatis AQs having different types at the same time. Polymorphism is the rule ratherthan the exception: 16 laim 5.1. For each agent P who adopts syntax realism for each sort S ∈{ pnat, nat, intt } it is the case that: AQ pnat ⊆ AQ nat ⊆ AQ int . AQs provide names for numbers, and words may provide names for someAQs, though such relations raise questions too. For instance: what is the rela-tion between ‘zero’ and 0, and between ‘one’ and 1? Are one and 1 synonyms,i.e. different names for the same number, or is one a name for 1, just as ‘Einz’is a name for it, though in another language?
Claim 5.2.
Assuming number realism for sort S then syntax realism for sort S is pluralistic. Indeed I believe that whoever adopts syntax realism in addition to numberrealism for sort S , will also believe that pluralism for syntax cannot be avoided.There will always be different ways to denote a given (semantic) entity, and alsodifferent ways to denote semantic objects in a simplest manner, given some con-ventions regarding simplicity of AQs, given a plausible class of AQs. Curiouslyby doing away with number realism syntax pluralism can be avoided at thesame time. By making a distinction between AQs and their meaning, specificAQs become mere options for denoting that meaning. In the absence of numberrealism, however, 2 + 2 = 4 is merely one of many assertions which is derivablein some calculus and is not an assertion about the respective meanings of 2 + 2and 4. Claim 5.3.
Syntax realism is open ended.
Unlike number realism where the entities serving as numbers are clearly de-marcated in each approach, syntax realism need not come with sharp boundarieson what constitutes an expression. In particular syntax realism may come witha context providing definitions and abbreviations allowing the creation of addi-tional expressions. Open endedness involves vagueness. For instance, a contextmay introduce f ( ) as a function of type nat → nat and then one may claimthat f (2) + f (3) + 1 is in AQ nat . Some may object that more information on f isneeded. If f ( ) is introduced a partial function of type nat → nat (for instancethe predecessor function P ( ) under the assumption that it is undefined on 0),is it then valid to claim that f (2) + 1 is an AQ. Doing so or not doing so isdetermined by conventions on how to deal with undefined expressions and suchconventions are not unique. One option for dealing with partiality is to adoptthe convention that whether or not f (2) + 1 is an AQ depends on the existenceof f (2) which must be settled first. If f (2) does not exist f (2) + 1 is not anAQ, and if it exists f (2) + 1 is an AQ. This idea makes syntax dependant onsemantics, a significant complication. Although sumterms do not constitute a set it is intuitively covincing that thecomponents of a sumterm can be selected by means of suitable selection opera-tors. The collection of sumterms not being a set, the sumterm splitting opera-17ions cannot be considered functions, as the domain and the range of functionsare required to be sets, the graph of a function itself being a set.
Definition 5.2. (Sumterm splitting operators.) The pair l s and r s constitutesoperations able to decompose a sumterm in its two parts so that for all AQs X and Y , for which X + Y is a sumterm: l s ( X + Y ) = AQ X and r s ( X + Y ) = AQ Y . For instance with X = AQ Y = AQ X + Y is a sumterm fails as 1 + 2 + 3 is not a sumterm (it is a poly-infix sumterm,however). I assume that if X is not a sumterm we have l s ( X ) = r s ( X ) = 0.The following implications are valid: X = AQ Y = ⇒ l s ( X ) = AQ l s ( Y ) and X = AQ Y = ⇒ r s ( X ) = AQ r s ( Y ). One may claim that a sumterm has threeparts, with the addition operator constituting a third part. The later claim,however, conveys no new information for an entity which is known to be asumterm.The presence of = AQ next to = may be considered overdone and useless,and the distinction between both equality signs may simply be ignored, a lineof thought which is taken in mathematics and education throughout. Thissimplification may be called the “sums are terms” paradigm. I will now assume that the sums are sumterms paradigm has been adopted andthat sum rather than sumterm is used. Instead of sumterm splitting operatorsthere are sum splitting operators, though with the same definition.A naive understanding of sums as (semantic) entities which can be split inparts called summands leads to a paradox. This paradox will be referred toas the sum splitting paradox, which is an instance of phenomenon (AQ relateddecomposition paradox) which may arise for other arithmetical operators justas well.
Proposition 5.1.
Without making a distinction between AQs and values, orbetween = and = AQ , the very presence of sum splitting operators leads to aninconsistency in arithmetic, in particular, it allows to infer .Proof. Assume the presence of l s and l r as in Definition 5.2. Not distinguishingAQs and values one may refer to these operations as sum splitting operations,and assume that the following implications are valid: l s ( X + Y ) = X and r s ( X + Y ) = Y . Now taking X = 1 and Y = 1 yields 1 = l s (1 + 2) = l s (2 + 1) =2. Claim 5.4. As is an unacceptable conclusion, either some of the (implicitor explicit) assumptions for Proposition 5.1 or some part of the argument in theproof of Proposition 5.1 must be dismantled in any sound approach to arithmetic. Ways of dismantling the argument for Proposition 5.1 will be referred to assolutions of the sum splitting paradox.18
Five solutions of the sum splitting paradox
I will distinguish five solutions of the sum splitting paradox each of which removeit as an obstacle for the credibility of an account of elementary arithmetic ofnaturals and integers.Each of these views can be combined with the idea that sum is a role fora value (see Paragraph 1.1 above), rather than a particular kind of values orother entities. It is only in the matter of explaining the notion of a summandthat the viewpoints below differ.
The sumterm solution (for the sum splitting paradox) amounts to making a cleardistinction between syntax and semantics for arithmetic. Using sumterms theissues involved can be discussed in more detail with the effect that what seemsto be a paradox at first sight is in fact a mere a misunderstanding, coming aboutfrom a mere confusion of different notions of equality.Sumterms are among the arithmetical quantities which are not arithmeticalvalues (decimal numbers) at the same time. Now it is claimed that sumtermsinstead of sums which admit decomposition by way of l s and r s . The splittingoperations l s and r s are viewed as operations on AQs. It follows that substi-tution of equals by equals may fail, in the above example l s (1 + 2) = l s (2 + 1)cannot be inferred from 1 + 2 = 2 + 1. The corresponding implication whichcan be maintained instead is 1 + 2 = AQ ⇒ l s (1 + 2) = AQ l s (2 + 1). But1+2 = AQ l s (1 + 2) = AQ l s (2 + 1) which occurs in the proof of the sumsplitting paradox fails and as a consequence the paradox disappears.In the framework of the sumterm solution, the introduction of the functions l s and l r is considered unproblematic on the following grounds: (i) sumtermsare pairs, (ii) the components of a pair can be arbitrary entities (including AQs,even in the absence of a rigorous definition of AQs, which may conceivably foundor constructed outside conventional set theory, and (iii) that pairs can alwaysbe decomposed, a principle which is prior to ZF set theory.The sumterm solution comes with the necessity to distinguish between = and= AQ , though of course not with the commitment to the use of the particularnotation chosen for = AQ in this paper. Assessment.
The sumterm solution constitutes a plausible option for doingaway with the sum splitting paradox. The sumterm solution comes with ahigh price, however, and that is to take the distinction between syntax andsemantics seriously and to allow operations on expressions (AQs) which have nocounterparts in the world of values (numbers).19 .2 Contradiction tolerance based on a foundational spec-ification
Contradiction tolerant solutions for the sum splitting paradox accept the pres-ence of the argument for a contradiction, while making sure that no exten-sive harm is done to the reliability of arithmetic. Paraconsistent logic may beadopted to gain condition tolerance in a very general style. However, I havebeen unable to find a convincing paraconsistent logic for dealing with the sumsplitting paradox. Instead ad hoc strategies can be adopted which take thesubject matter of elementary arithmetic into account. I will distinguish twodifferent options for condition tolerant solutions of the sum splitting paradox.First it is assumed that arithmetic is cast in terms of the equational theory ofan abstract arithmetical datatype, which is specified by means of an algebraicspecification. This specification is given a foundational status, which meansthat its consequences, including negations of non-derivable closed identities,may overrule any conclusion in the form of an equation (such as 1 = 2) whichhas been derived by other means. If a proof is at odds with the implications ofsaid foundational specification of the arithmetical datatype at hand, the proofis rejected and its conclusion is not adopted. A foundational specification whichcan be used in the case of int is given in Paragraph 6.6 below.In order to avoid mistakes and as a method to prevent costly checks ofresults of proofs against the given foundational specification it is advised notto apply substitution of equals for equals at top level to arguments of l s and r s . Meta-theory concerning the observation that this rule of thumb suffices toavoid the derivation of wrong conclusions is considered inessential and need notbe provided.The foundational specification solution has important predecessors, for in-stance ZF set theory may be considered a foundational specification againstthe background of which naive set theory may be used in daily mathematicalpractice in such a manner that the risk of running into a form of the Russelparadox is not completely absent. A similar idea is put forward in Bergstra [5]in the case of fractions instead of sims. The foundational specification (solution)approach rejects the distinction of = and = AQ . This approach is both practicaland theoretically sound and does not involve any sophisticated consideration ofsyntax. It might well serve as the basis for teaching elementary arithmetic. Assessment.
Contradiction tolerance seems to be an attractive feature ofreasoning systems, whereas reasoning systems that have been designed in sucha manner as to minimise the probability of deriving an invalid conclusion comewith seemingly artificial restrictions which give rise to an unnatural look andfeel.The sumterm solution may be adopted as an informal device which helpsto work in such a manner that conclusions drawn in a contradiction tolerantframework based on a foundational specification will not be in contradictionwith that specification. This means that reasoning (in the condition tolerantsetting) must preferably be done in such a manner that it can be provided with20dditional detail (such as type distinctions between value and AQ) so as to leadto reasoning patterns that are sound for the sumterm solution.
Contradiction tolerance can be obtained without adopting a foundational speci-fication by means of reliance on a body of practical experience. Then arithmetic,like any other area of human endeavour is viewed as being informal and faultprone, and only to be trusted to knowledgeable persons, able to avoid a rangeof well-known as well as lesser known mistakes.The introduction of l s and r s may simply be rejected because apparently itleads to problems. But it may be accepted by those who know how to avoidsuch problems. There is nothing special about these matters which suggeststhat avoiding mistakes requires anything else in excess of experience with thesubject at hand.Other wrong inferences may include instances derived from making any ofthe following assumptions: (i) x · x is always even, (ii) x · ( − x ) = 0 (both casesconfusion addition and multiplication), (iii) x · ( y + z ) = ( x · y ) + z (misunder-standing of associativity), and (iv) x + y is always larger than x (failing to takenegative values into account).Arithmetic may be considered as just any practical activity: avoiding pitfallsand making sound steps go hand in hand. For a car driver there are manymistakes which are better avoided, and thinking in terms of such mistakes as wellas in terms of avoidance of these and similar mistakes is helpful. Arithmetic thusconceived consists of many rules of thumb on how to solve certain “problems”and of many guidelines on how to avoid mistakes of various kinds. The ideathat following a logical path of reasoning is sure to lead to valid conclusion andthat therefore purely logical reasoning can (and should) be applied is as distantas it is in playing chess. The idea that arithmetic is a manifestation of a fullyreliable logic applied logic is rejected as lacking both contemporary evidenceand historical justification. Assessment
The idea that arithmetic is just like most areas of human compe-tence, a matter of experience and training, and a matter of adopting successfulpatterns of behaviour while avoiding patterns likely to be less successful, isattractive and cannot be rejected in a principled manner.The situation may be comparable with chess: one may teach a chess studenta classical course on chess with openings, tactics, and strategy, and with asample of end games. The student ends up with a lot of rules of thumb in theirmind, and some ability to apply these guidelines when playing a game. At thesame time a student may be explained that a suitably programmed computermay somehow quickly find out what is the best move while the student hasno clue regarding how the algorithm, which is implemented in the computer athand, works. For the student playing chess is a matter of following guidelinesand avoiding mistakes, even if it is known that a more principled approachexists, i.e. the one followed by an advanced chess playing computer.21n the case of arithmetic a successful fast track introductory course maynot even touch the observation that arithmetic can be done in a waterproofmanner based on rock solid logic without any notion of mistake and avoidanceof mistakes.A disadvantage of perceiving arithmetic, as a topic being taught, as a merepragmatic collection of do’s and don’ts is that it may be unhelpful for the ad-vancement of logical reasoning competence, in that respect a missed opportunityso to say.
One may hold that functions must exist in set theory and that for that reasonany function must have a set of mathematical entities as its domain. Thesums on which l s and r s are supposed to work do not constitute a plausiblecollection of mathematical entities in ZF set theory and for that reason it isinvalid to introduce these functions in the first place. Introduction of sumsplitting operations is at odds with standard conventions and traditions andcan be refuted for that reason. As a consequence Proposition 5.1 disappears.Conventionalism on function definitions refutes the very plausibility of theintroduction of functions l s and r s . Conventionalism on function definitionscomes with the suspicion that sumterms do not constitute a set which can beused in a mathematical argument. Assessment.
Conventionalism on function definitions may be adopted byanyone who considers the incorporation of the syntax of expressions in mathe-matics to be a bridge too far, let alone the incorporation of logic, and to bringabout an extension of elementary arithmetic beyond its natural limits. Conven-tionalism on function definitions leaves the common terminology of summandsunexplained, and will argue that there is always a transition from the formalisedmathematics to informal language the elements of which lack formalisation andthat summand is one of the notions which play an intermediate role and are notconsidered to be in need of mathematical explanation or definition.
Conventionalism on arithmetic signatures denies a participant of school arith-metic the right to introduce functions and in particular objects to the intro-duction of both sumterm splitting operators, because of the absence of anyconvention or tradition supporting such introductions. The architecture (thatis signature) of the language of elementary arithmetic is supposed to be deter-mined in advance. The idea that elementary arithmetic inherits from mathe-matics or logic a systematic methodology for redesigning (or extending) itself isrejected.Conventionalism on arithmetic signatures is less restrictive than convention-alism on function definitions as outlined above, and instead of objecting to the22 ′ ≡ ′ ≡ ′ ≡ ′ ≡ ′ ≡ ′ ≡ ′ ≡ ′ ≡ ′ ≡ DGS : enumeration and successor notation of decimal digitsintroduction of sum splitting operations on methodological grounds this formof conventionalism objects to the the introduction of sum splitting operatorsbecause these change the subject (architecture) of elementary arithmetic in aproblematic manner.The introduction of sum splitting operations l s and r s is considered foreign tothe tradition in and is rejected for that reason. The proper choice of operatorsis considered part of arithmetic, just as with any natural language, and norules of engagement for the introduction of new functions are made explicit. Inparticular as (elementary) arithmetic is considered to be prior to mathematicsas well as being prior to logic, no rules or conventions for the introduction ofnew functions, beyond the use of abbreviations for explicit definitions, are takenon board, and definitely not if these rule only appear in subsequent stages ofthe development of mathematics and logic. Assessment.
Conventionalism on arithmetic signatures is a perfectly validway to deal with the sum splitting paradox. This solution has these disadvan-tages: (i) it leaves the notion of a summand unexplained, (ii) it provides nodefinition of the notion of a sum given the fact that all decimal numbers aresums. N and of Z The datatype DGS (digits) introduces a meta function for successor: with d ranging over the constants 0 , . . . , d ′ stands for 1 , . . . , Z ( dec, + , − ). All elements of N d serve as constants in this specification,which is indicated by writing dec for these constants. The specification containsinfinitely many equations because σ ranges over all of N d . When omitting theopposite operator − , and both equations for it, an initial algebra specificationof N ( dec, +) results. The specification of Table 2 is a simplified version of aspecification in [7].With some work the equations can be given an alternative orientation so thatprecisely the elements of Z d will appear as normal forms and a definition of thearithmetical datatype Z ( dec, + , − ) is obtained. A foundational specification,however, may just as well be an abstract datatype specification.23 nclude: Table DGS ( x + y ) + z = x + ( y + z ) (1) x + y = y + x (2) x + 0 = x (3) x + ( − x ) = 0 (4) − ( − x ) = x (5) d ′ = d + 1 ( for d ∈ { , , , , , , , } ) (6)9 + 1 = 10 (7) σd + 1 = σd ′ ( for d ∈ { , , , , , , , , } ) (8) σ + 1 = τ → σ τ Z ( dec, + , − ); σ ranges over nonempty digit sequences(not starting with 0) There is no way to force anyone to choose between the five solutions for thesum splitting paradox, or to design any other solution for a problem one maynot recognise. Under the assumption that the sum splitting paradox requires asolution, it is easy to find alternatives and modifications of these five solutionsto it, and combinations of these views are options as well. I will understand thefive solutions as building blocks for the design of solutions, the virtue of whichmay be a better proximity to certain intuitions.I wil first discuss a limitation on the sumterm solution. Although adoptingthe distinction between AQs and values with sumterms as a special case ofAQs helps to avoid the problematic fact in Proposition 5.1, it will not preventthat upon the introduction of a more detailed view of AQs similar issues arise.Repeating the same strategy will provoke an infinite regress rather than solve aproblem. This observation is detailed in the following Paragraph.
Stopping an infinite regress may constitute an argument against adopting thesumterm solution. The argument runs thus: (i) suppose sumterms and thesumterm solution of the sum splitting paradox have been adopted. Now givenan AQ, say X , let bp ( X ) be the number of bracket pairs in X . For instance bp (((1 + 2) + (2 + 0)) + 0) = 3.As abstract entities, the expressions 0 and (0) are the same, so 0 = AQ (0),from which it follows that bp (0) = AQ bp ((0)) which implies 1 = bp ((0)) = bp (0) = 0. Apparently a paradox similar to the sum splitting paradox hasarisen. I will refer to such paradoxes as AQ related paradoxes. This one may24e called the bracket pair counting paradox.Apparently adopting the sumterm solution and the introduction of AQs doeseliminate the phenomenon of AQ related paradoxes, while these steps only helpto avoid the sum splitting paradox. A plurality of solutions for the bracket paircounting paradox may be designed just as for the the sum splitting paradox.Assuming that a solution is chosen by introducing a more detailed view, thena further ramification of AQs may be introduced with an equality = bp AQ forbracket pair aware AQ equality so that (i) 0 = bp AQ (0), and (ii) X = bp AQ Y = ⇒ bp ( X ) = bp AQ bp ( Y ) instead of X = AQ Y = ⇒ bp ( X ) = AQ bp ( Y ). Withthese preacautions the bracket pair paradox disappears.At this stage, however, there is further room for AQ related paradoxes. Forinstance counting the number of spaces in an expression with a new operator,say sp ( X ) will produce similar complications. For instance sp (1 + 2) = 2 and sp (1+2) = 0, while 1 + 2 = bp AQ Assessment.
Avoiding an infinite regress is a justified objective. It followsfrom this idea that the sumterm solution to the sum splitting paradox cannotbe justified as a means to do away with AQ related paradoxes in any generalmanner. The sumterm solution can be motivated only by its use to explain in asystematic manner a notion of a summand in relation to a notion of sum, whilethe sumterm solution can not be motivated by the objective to get AQ relatedparadoxes out of the way for once and for all.I hold that commitment to the virtue of avoiding an infinite regress plausiblycomes with an acceptance of either contradiction tolerance or some form ofconventionalism/traditionalism.Avoiding an infinite regress can be combined with the adoption of sum split-ting operations and adopting contradiction tolerance on the basis of a foun-dational specification at the abstraction level of = AQ . The details of such anapproach are non-trivial because substitution has to be defined in a bracket paircounting compliant manner, which necessitates a redevelopment of equationallogic from first principles. I will not develop such details below. Instead I willadopt conventionalism/traditionalism on arithmetic signatures as a justificationfor dismissing the introduction of the bracket pair counting operator bp ( − ).Conventionalism on function definitions and/or on arithmetic signatures willbe increasingly convincing when AQ related paradoxes are met at lower levels25f abstraction. Functions which are introduced for the sole reason of expos-ing a potential contradiction may be rejected as foreign to arithmetic. Indeed,the fact that summands are somehow a well-known notion in arithmetic essen-tially contributes to the justification of the introduction of sumterm splittingoperators. Some combinations and minor adaptations of the above views are attractive. Iwill survey some options for such combinations.
The sumterm solution may be adopted merely as a guideline on how to work onthe basis of the pragmatic contradiction tolerant solution. Using the sumtermsolution in this manner amounts to requiring that each proof which is writtenin the context of a condition tolerant approach allows being refined by replacingzero or more equality signs (=) by in to = AQ signs with the effect that a validproof in the context of the sumterm solution is found. Adopting sumterms, sumsplitting operators, and AQ equality, as a means to avoid invalid reasoning pat-terns in a contradiction tolerant setting, can be considered a heuristic use of thesumterm solution. Alternatively this approach may be labeled delayed contra-diction tolerance. Adopting the sumterm solution as a heuristic device providessignificant guarantees that no equations or negated equations are derived whicheither contradict the foundational specification or which are in contradictionwith outcomes obtained in standard practice. Who opposes contradiction tolerance as a method for providing foundations ofelementary arithmetic may prefer to combine the sumterm solution with eitherone of the forms of conventionalism, thereby accepting both sumterm splittingoperators as being well-defined while rejecting further function introductionssuch as the counting of bracket pairs which was mentioned above. I will referto this idea as delayed conventionalism/traditionalism.
One may portray the use of a foundational specification as being potentially faultprone because axioms may be wrong and so may be proofs based on the axioms.This viewpoint rejects the distinction between both forms of contradiction tol-erance and incorporates the first form (based on a foundational specification)into the second form (doing without a foundational specification), by viewingthe foundational specification as merely one of many ways in which professionalexperience with arithmetic can be documented.26 .2.4 The sumterm solution as a device in the background
One may reject the introduction of sum splitting operations (perhaps motivatedas an instance of either or both forms of conventionalism) and at the same timeadmit that sum splitting operations may be dealt with in the background, that isinvisible for students, and then claim that a choice for a solution of the resultingproblem (i.e. choosing one of the five options or suggesting yet another option)is made in the background as well, in such a manner that premature disclosureto students can be avoided and that it does not matter much which choices havebeen made.
The views outlined above share the property that the same closed identities inarithmetic can be derived and therefore it may be considered unproblematic ifone’s views on the matter change in time. There is no need for anyone to choosebetween the options listed in Paragraph 7.2. An option is to have differentviews, chosen from this listing, at different times, or to claim that each of theseis acceptable and that one leaves the choice to the agents with whom one iscommunicating about arithmetic.
Sumterms as well as sumterm splitting operators have been introduced and it isshown that a naive approach to these matters gives rise to paradoxical resultswhich are referred to as the sum splitting paradox. The sum splitting paradoxis considered an instance of AQ (arithmetical quantity, i.e. expression) relatedparadoxes which arise in cases where (i) semantics is more abstract than syntaxin a context where (ii) certain syntactic differences are brought to the surface,in spite of being abstracted from in the preferred semantic view.A survey is given of policies for avoiding the sum splitting paradox fromoccurring or for making sure that the consequences of its occurrence are accept-able.This work may be considered as belonging to the foundations of school arith-metic, where these foundations are primarily derived from, or biased towards,the theories of datatypes and abstract datatypes in theoretical computer science.Naturals and integers constitute instances of abstract arithmetical datatypes.Several conceptual difficulties arise when working out the details of the per-spective just outlined. First of all syntax cannot be simply defined in advance,and we adopt the idea that AQs (arithmetical quantities) are a vague (fuzzy?)initial approximation of syntax, the elements of which come about most clearlyon the basis of a given signature. In other words a distinction between syntaxand semantics is assumed without either of these being initially defined in anydetail. Assuming that AQs for natural numbers are given at some stage (suchas (91 + 0) + (12 + 3)) one may ask for the meaning of these AQs. Remarkablymathematics, philosophy, and logic have neither produced a stable answer to27hat question nor shown that no such answer exists. The philosophical ontologyof numbers seems to be an unfinished subject.Apparently the educational literature on school mathematics contains verylittle explicit information about the nature (meaning, ontology) of natural num-bers. In [26] the naive Platonic approach (n.P.a.) is advocated which assumesvarious types as given without making any attempt to giving definitions. In thiswork no distinction is made between defining a system of natural numbers up toisomorphism (i.e. an abstract arithmetical datatype for naturals), and definingan arithmetical datatype for naturals. In [27] the same author provides a theoryof definitions in a technical context, claiming that arithmetic and mathematicsconstitute a technical context. This suggestion is then made that definitions (oftechnical notions, say in arithmetic) are useful for the stronger (more interested)students but should not be forced upon students. In the case of natural num-bers this idea suggests that the n.P.a. may give way for the so-called definitionalapproach for students who can appreciate such explanations.After a survey of various options I propose (a certain choice of) projectionsemantics for AQs for naturals and integers, for instance arriving at the digitsequence 106 as the semantics of the AQ 99 + 7 and also as the semantics ofitself. 106 is called a decimal natural number in order to emphasise the fact thatthe details of decimals have not been abstracted away. In projection semanticsfor naturals, decimal natural numbers are not decimal representations of naturalnumbers, but on the contrary no commitment is made to the existence of entitiesthat qualify as natural numbers, while the existence of decimal natural numbersis assumed.Secondly the notion of a sum is chosen to be that of a role name. Summand,however, is not a role name, and neither is sumterm. Remarkably roles playnearly no role in datatype theory, which seems to be a missed opportunity.
Many topics for further work can be formulated. In particular the followingquestions which merit further efforts:(i) To develop a theory of signs for naturals (natsigns) and integers (intsigns).Signs are not abstract entities and the philosophy of objects, including mereol-ogy, will enter the discussion. Recent progress in mereology which may proveindispensable for the mereology of natsigns and intsigns are stage theory andworm theory, see [25].These approaches introduce aspects which seem to be entirely foreign totheoretical computer science because computer science has less focus on com-puters as material entities than on the abstractions which come with analysisand engineering. (ii) To develop a theory of proterms (product terms) wherefactors are components of proterms that can be selected by means of suitablesplitting operations. The situation is similar to the development of sumtermsbut there is an important difference: product need not be understood as a rolename in the arithmetic of naturals and of integers as the products constitute28he complement of the primes and because viewing prime as role name is notplausible, so is viewing non-prime as a role name.(iii) To develop an informal logic of elementary arithmetic which allows in-corporation of AQ related paradoxes and solutions of these.(iv) To analyse the notion of a square. For integers squares are a credibleclass of numbers, so that there is little incentive to view “square” as role, insteadof as a name for a class of entities. Considering the real numbers, however, thesituation changes: being a square is the same as being positive, and the termsquare more plausibly serves a role name.(v) In [10] so-called true fractions are investigated, which are certain elementsof arithmetical datatypes with unconventionally few identifications between dif-ferent fracterms. It remains to be seen whether or not a corresponding notionof a true sum can be defined.
In computer science, in particular in connection with programming and softwareengineering the presence of syntax is so prominent that it is hard to imaginethat in mathematical syntax is much less important. In practical computerscience it is well-known that adding mathematics semantics to programming isdifficult because doing so hardly corresponds to the intuitions of the averageprogrammer. In elementary arithmetic the situation is the other way around:adding an explicit coverage of syntax to semantics oriented thinking goes againstthe intuitions of the average professional, and is quite difficult, if only for thatreason. In this paper I have tried to find some middle ground where bothsyntactic and semantic considerations play a role.The terminology of datatypes and abstract datatypes fits well with my ob-jectives. For instance the decimal natural numbers may be understood as adatatype which implements the abstract datatype of natural numbers. Thisterminology accommodates the observations that: (i) decimal natural numberscan be understood as well-defined abstract entities in a naive set theory, (ii)abstract natural numbers do not exist otherwise than as mappings from thedatatypes in the abstract datatype to elements thereof, which is the mechanismof pointed structure classes of Paragraph 4.4. For (abstract) datatypes andalgebraic specifications thereof I refer to [11, 14, 13].Having thus established that the notions of an arithmetical datatype andan abstract arithmetical datatype are of relevance for elementary arithmetic, itis fitting to notice that conceptualising arithmetic in terms of datatypes andabstract datatypes also leads to novel number systems: common meadows [9],transrationals as in [1] wheels in [24] and [12], and further options surveyedin [4].Contradiction tolerance may be obtained via the use of paraconsistent logics.In [6] an application of paraconsistency in the setting of arithmetical datatypesis developed. An application of condition tolerance via paraconsistency is de-veloped in detail in [8]. For a survey on paraconsistent logics I mention [17]and [18]. Nevertheless, in spite of the presence of a plurality of approaches to29araconsistent reasoning I have not been able to develop a convincing solutionof the sum splitting paradox along the lines of condition-tolerance as providedby paraconsistency.Viewing AQs as names one may take names for a core notion in the de-velopment of elementary arithmetic. That path is taken in [23]. As it turnsout, importing a theory of names into elementary arithmetic is a non-trivialoverhead. Distinguishing syntax and semantics for arithmetic, may be doneand undone repeatedly by the same person. Such steps have been investigatedin [22] where these steps are termed disassociation and association respectively.In [5] fracterms are introduced as a syntactic counterpart of fractions. Thework on sumterms has in part been done in the same style as the work onfracterms, though there are significant differences: in the current paper there isa fairly clear description of what (a) sum is while for fractions no correspondingview is developed in [5]. Unlike for fractions, about which a formidable literatureexists, there seems to be no work in the educational literature where a definitionof a sum of integers is provided in advance of the development of educationalmethods about it. Literature on sums of fractions abounds, e.g. [16]. In thecontext of integer arithmetic, the focus in the literature is always on how tocompute a sum, and not on what a sum is more broadly speaking.
Acknowledgement.
The work on this paper has profited from discussionswith James Anderson (Reading), Inge Bethke (Amsterdam), Kees Middelburg(Amsterdam), Alban Ponse (Amsterdam), Stefan Rollnik (Rostock), John V.Tucker (Swansea), and Dawid Walentek (Utrecht).
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