The logistic construction of the natural numbers Logicism

1 logistic construction of natural numbers

1.1 preliminaries
1.2 definition of natural numbers
1.3 criticism

the logistic construction of natural numbers

the attempt construct natural numbers summarized succinctly bernays 1930–1931. rather use bernays précis, incomplete in details, construction best given simple finite example details found in russell 1919.

in general logicism of dedekind-frege similar of russell, significant (and critical) differences in particulars (see criticisms, below). overall, though, logicistic construction-process [dedekind-frege-russell] different of contemporary set theory. whereas in set theory notion of number begins axiom—the axiom of pairing leads definition of ordered pair —no overt number-axiom exists in logicism. rather, logicism begins construction of numbers primitive propositions include class , propositional function , , in particular, relations of similarity ( equinumerosity : placing elements of collections in one-to-one correspondence) , ordering (using successor of relation order collections of equinumerous classes) . logicistic derivation equates cardinal numbers constructed way natural numbers, , these numbers end of same type —as equivalence classes of classes—whereas in set theory each number of higher class predecessor (thus each successor contains predecessor subset). kleene observes following. (kleene s assumptions (1) , (2) state 0 has property p , n+1 has property p whenever n has property p.)

viewpoint here different of [kronecker s supposition god made integers plus peano s axioms of number , mathematical induction], presupposed intuitive conception of natural number sequence, , elicited principle that, whenever particular property p of natural numbers given such (1) , (2), given natural number must have property p. (kleene 1952:44).

the importance logicism of construction of natural numbers derives russell s contention traditional pure mathematics can derived natural numbers recent discovery, though had long been suspected (1919:4). derivation of real numbers (rationals, irrationals) derives theory of dedekind cuts on continuous number line . while example of how done useful, relies first on derivation of natural numbers. so, if philosophic problems appear in logistic attempt derive natural numbers, these problems sufficient stop program until these fixed (see criticisms, below).

preliminaries

for dedekind, frege , russell, collections (classes) aggregates of things specified proper names, come result of propositions (utterances asserts fact thing or things). russell tore general notion down in following manner. begins terms in sentences decomposes follows:

terms: russell, terms either things or concepts : whatever may object of thought, or may occur in true or false proposition, or can counted one, call term. this, then, widest word in philosophical vocabulary. shall use synonymous words, unit, individual, , entity. first 2 emphasize fact every term one, while third derived fact every term has being, i.e. in sense. man, moment, number, class, relation, chimaera, or else can mentioned, sure term; , deny such , such thing term must false (russell 1903:43)

things indicated proper names; concepts indicated adjectives or verbs: among terms, possible distinguish 2 kinds, shall call respectively things , concepts; former terms indicated proper names, latter indicated other words . . . among concepts, again, 2 kinds @ least must distinguished, namely indicated adjectives , indicated verbs (1903:44).

concept-adjectives predicates ; concept-verbs relations : former kind called predicates or class-concepts; latter or relations. (1903:44)

the notion of variable subject appearing in proposition: shall speak of terms of proposition terms, numerous, occur in proposition , may regarded subjects proposition is. characteristic of terms of proposition of them may replaced other entity without our ceasing have proposition. shall socrates human proposition having 1 term; of remaining component of proposition, 1 verb, other predicate.. . . predicates, then, concepts, other verbs, occur in propositions having 1 term or subject. (1903:45)

in other words, term can place-holder indicates (denotes) 1 or more things can put placeholder. (1903:45).

truth , falsehood: suppose russell point object , utter: object in front of me named emily woman. proposition, assertion of russell s belief tested against facts of outer world: minds not create truth or falsehood. create beliefs . . . makes belief true fact, , fact not (except in exceptional cases) in way involve mind of person has belief (1912:130). if investigation of utterance , correspondence fact , russell discovers emily rabbit, utterance considered false ; if emily female human (a female featherless biped russell likes call humans), utterance considered true .

if russell utter generalization emilys these object/s (entity/ies) must examined, 1 after in order verify truth of generalization. if russell assert emilys women , tipoff utterance entities emily in correspondence concept labeled woman , methodical examination of creatures human names have commence.

classes (aggregates, complexes): class, opposed class-concept, sum or conjunction of terms have given predicate (1903 p. 55). classes can specified extension (listing members) or intension, i.e. propositional function such x u or x v . if take extension pure, our class defined enumeration of terms, , method not allow deal, symbolic logic does, infinite classes. our classes must in general regarded objects denoted concepts, , extent point of view of intension essential. (1909 p. 66)

propositional functions: characteristic of class concept, distinguished terms in general, x u propositional function when, , when, u class-concept. (1903:56)

extensional versus intensional definition of class: 71. class may defined either extensionally or intensionally. say, may define kind of object class, or kind of concept denotes class: precise meaning of opposition of extension , intension in connection. although general notion can defined in two-fold manner, particular classes, except when happen finite, can defined intensionally, i.e. objects denoted such , such concepts. . . logically; extensional definition appears equally applicable infinite classes, practically, if attempt it, death cut short our laudable endeavour before had attained goal.(1903:69)

the definition of natural numbers

the natural numbers derive propositions (i.e. unrestricted) in , other possible worlds, can uttered collection of entities whatsoever. russell makes clear in second (italicized) sentence:

in first place, numbers form infinite collection, , cannot therefore defined enumeration. in second place, collections having given number of terms presumably form infinite collection: presumed, example, there infinite collection of trios in world, if not case total number of things in world finite, which, though possible, seems unlikely. in third place, wish define number in such way infinite numbers may possible; must able speak of number of terms in infinite collection, , such collection must defined intension, i.e. property common members , peculiar them. (1919:13)

to begin, devise finite example. suppose there 12 families on street. have children, not. discuss names of children in these households requires 12 propositions asserting childname name of child in family fn applied collection of households on particular street of families names f1, f2, . . . f12. each of 12 propositions regards whether or not argument childname applies child in particular household. children s names (childname) can thought of x in propositional function f(x), function name of child in family name fn .

to keep things simple 26 letters of alphabet used in example, each letter representing name of particular child (in real life there repeats). notice that, in russellian view these collections not sets, rather aggregates or collections or classes —listings of names satisfy predicates f1, f2, . . .. noted in step 1, russell, these classes symbolic fictions exist aggregate members, i.e. extensions of propositional functions, , not unit-things in themselves.

step 1: assemble classes: whereas following example finite on very-finite propositional function childnames of children in family fn on very-finite street of finite number of (12) families, russell intended following extend propositional functions extending on infinity of , other possible worlds; allow him create numbers (to infinity).

kleene observes russell has set himself impredicative definition have resolve, or otherwise confronted russell paradox. here instead presuppose totality of properties of cardinal numbers, existing in logic, prior definition of natural number sequence (kleene 1952:44). problem appear, in finite example presented here, when russell confronts unit class (cf russell 1903:517).

the matter of debate comes down this: class ? dedekind , frege, class distinct entity of own, unity can identified entities x satisfy propositional function f( ). (this symbolism appears in russell, attributing frege: essence of function left when x taken away, i.e in above instance, 2( ) + ( ). argument x not belong function, 2 make whole (ib. p. 6 [i.e. frege s 1891 function und begriff] (russell 1903:505).) example, particular unity given name; suppose family fα has children names annie, barbie , charles:

[ a, b, c ]fα

this dedekind-frege construction symbolized bracketing process similar to, distinguished from, symbolism of contemporary set theory { a, b, c }, i.e. [ ] elements satisfy proposition separated commas (an index label each collection-as-a-unity not used, be):

[a, b, c], [d], [ ], [e, f, g], [h, i], [j, k], [l, m, n, o, p], [ ], [q, r], [s], [t, u], [v, w, x, y, z]

this notion of collection-or or class-as-object, when used without restriction, results in russell s paradox; see more below impredicative definitions. russell s solution define notion of class elements satisfy proposition, argument being that, indeed, arguments x not belong propositional function aka class created function. class not regarded unitary object in own right, exists kind of useful fiction: have avoided decision whether class of things has in sense existence 1 object. decision of question in either way indifferent our logic (first edition of principia mathematica 1927:24).

russell not waver opinion in 1919; observe words symbolic fictions :

when have decided classes cannot things of same sort members, cannot heaps or aggregates, , cannot identified propositional functions, becomes difficult see can be, if more symbolic fictions. , if can find way of dealing them symbolic fictions, increase logical security of our position, since avoid need of assuming there classes without being compelled make opposite assumption there no classes. merely abstain both assumptions. . . . when refuse assert there classes, must not supposed asserting dogmatically there none. merely agnostic regards them . . .. (1919:184)

and second edition of pm (1927) russell insist functions occur through values, . . . functions of functions extensional, . . . [and] consequently there no reason distinguish between functions , classes . . . classes, distinct functions, loose shadowy being retain in *20 (p. xxxix). in other words, classes separate notion have vanished altogether.

given russell s insistence classes not singular objects-in-themselves, collected aggregates, correct way symbolize above listing eliminate brackets. visually confusing, regards null class, dashed vertical line @ each end of collection used symbolize collection-as-aggregate:

┊a, b, c┊, ┊d┊, ┊┊, ┊e, f, g┊, ┊h, i┊, ┊j, k┊, ┊l, m, n, o, p┊, ┊┊, ┊q, r┊, ┊s┊, ┊t, u┊, ┊v, w, x, y, z┊

step 2: collect similar classes bundles (equivalence classes): these above collections can put binary relation (comparing for) similarity equinumerosity , symbolized here ≈, i.e. one-one correspondence of elements, , thereby create russellian classes of classes or russell called bundles . can suppose couples in 1 bundle, trios in another, , on. in way obtain various bundles of collections, each bundle consisting of collections have number of terms. each bundle class members collections, i.e. classes; each class of classes (russell 1919:14).

take example ┊h,i┊. terms h, cannot put one-one correspondence terms of ┊a,b,c┊,┊d┊,┊┊,┊e,f,g┊, etc. can put in correspondence , ┊j,k┊,┊q,r┊, , ┊t,u┊. these similar collections can assembled bundle (equivalence class) shown below.

┊┊h,i┊, ┊j,k┊, ┊q,r┊, ┊t,u┊┊

the bundles (equivalence classes) shown below.

┊ ┊a, b, c┊, ┊e, f, g┊ ┊
┊ ┊d┊, ┊s┊ ┊
┊ ┊┊, ┊┊ ┊
┊ ┊h, i┊, ┊j, k┊, ┊q, r┊, ┊t, u┊ ┊
┊ ┊ l, m, n, o, p┊, ┊v, w, x, y, z┊ ┊

step 3: define null-class: notice third class-of-classes, ┊ ┊┊, ┊┊ ┊ , special because classes contain no elements, i.e. no elements satisfy predicates created particular class/collection. example: predicates are:

childnames: childname name of child in family fρ .
childnames: childname name of child in family fσ .

these particular predicates cannot satisfied because families fρ , fσ childless. there no terms (names) satisfy these particular predicates. remarkably, class of things, signified fictitious ┊┊, satisfy each of these classes not empty, not exist @ (more or less, russell agnostic-about-class-existence); dedekind-frege exist.

this peculiar non-existent entity ┊┊ nicknamed null class or empty class . not same class of null classes ┊ ┊┊ ┊: class of null classes destined become 0 ; see below. russell symbolized null/empty class ┊┊ Λ. russellian null class? in pm russell says class said exist when has @ least 1 member . . . class has no members called null class . . . α null-class equivalent α not exist . 1 left uneasy: null class exist ? problem bedeviled russell throughout writing of 1903. after discovered paradox in frege s begriffsschrift added appendix 1903 through analysis of nature of null , unit classes, discovered need doctrine of types ; see more unit class, problem of impredicative definitions , russell s vicious circle principle below.

step 4: assign numeral each bundle: purposes of abbreviation , identification, each bundle assign unique symbol (aka numeral ). these symbols arbitrary. (the symbol ≡ means abbreviation or definition of ):

┊ ┊a, b, c┊, ┊e, f, g┊ ┊ ≡ ✖
┊ ┊d┊, ┊s┊ ┊ ≡ ■
┊ ┊┊ ┊ ≡ ♣
┊ ┊h, i┊, ┊j, k┊ ┊, ┊q, r┊, ┊t, u┊ ┊ ≡ ❥
┊ ┊ l, m, n, o, p┊, ┊v, w, x, y, z┊ ┊ ≡ ♦

step 5: define 0 : in order order bundles familiar number-line starting point traditionally called 0 , required. russell picked empty or null class of classes fill role. null class-of-classes ┊ ┊┊ ┊┊ ┊ has been labeled 0 ≡ ♣

step 6: define notion of successor : russell defined new characteristic hereditary , property of classes ability inherit characteristic class (or class-of-classes) i.e. property said hereditary in natural-number series if, whenever belongs number n, belongs n+1, successor of n . (1903:21). asserts natural numbers posterity -- children , inheritors of successor —of 0 respect relation immediate predecessor of (which converse of successor ) (1919:23).

note russell has used few words here without definition, in particular number series , number n , , successor . define these in due course. observe in particular russell not use unit class-of-classes 1 construct successor (in our example ┊ ┊d┊, ┊s┊ ┊ ≡ ■ ) . reason that, in russell s detailed analysis, if unit class ■ becomes entity in own right, can element in own proposition; causes proposition become impredicative , result in vicious circle . rather, states (confusingly): saw in chapter ii cardinal [natural] number defined class of classes, , in chapter iii number 1 defined class of unit classes, of have 1 member, should vicious circle. of course, when number 1 defined class of unit classes, unit classes must defined not assume know meant 1 (1919:181).

for definition of successor, russell use unit single entity or term follows:

remains define successor . given number n let α class has n members, , let x term not member of α. class consisting of α x added on have +1 members. have following definition:
the successor of number of terms in class α number of terms in class consisting of α x x not term belonging class. (1919:23)

russell s definition requires new term (name, thing) added collections inside bundles. keep example abstract abbreviated name smiley ≡ ☺ (on assumption no 1 has ever named child smiley ).

step 7: construct successor of null class: example null class Λ stick smiley face. previous, not obvious how this. predicate:

childnames: childname name of child in family fα .

has modified creating predicate contains term true:

childnames: childname name of child in family fα *and* smiley ;

in case of family no children, smiley term satisfies predicate. russell fretted on use of word *and* here, in barbie , smiley , , called kind of , (symbolized below *&* ) numerical conjunction :

┊ ┊┊ ┊ *&* ☺ → ┊┊☺┊┊

by relation of similarity ≈, new class can put equivalence class (the unit class) defined ■:

┊┊☺┊┊ ≈ ┊d┊,┊s┊ → ┊┊☺┊,┊d┊,┊s┊┊≡ ■, i.e.
0 *&* ☺ → ■,

step 8: every equivalence class, create successor: note smiley-face symbol must inserted every collection/class in particular equivalence-class bundle, relation of similarity ≈ each newly generated class-of-classes must put equivalence class defines n+1:

❥ *&* ☺ ≡┊┊h,i┊, ┊j,k┊, ┊q,r┊, ┊t,u┊┊ *&* ☺ → ┊┊h, i, ☺┊, ┊j, k, ☺┊, ┊q, r, ☺┊, ┊t, u, ☺┊, ┊a, b, c┊, ┊e, f, g┊┊ ≡ ✖, i.e.
❥ *&* ☺ → ✖

and in similar manner, use of abbreviations set above, each numeral successor created:

0
0 *&* ☺ = ■
■ *&* ☺ = ❥
❥ *&* ☺ = ✖
✖ *&* ☺ = ? [no symbol]
 ? *&* ☺ = ♦
♦ *&* ☺ = etc, etc

step 9: order numbers: process of creating successor requires relation . . . successor of . . . , call s , between various numerals , example ■ s 0, ❥ s ■, , forth. must consider serial character of natural numbers in order 0, 1, 2, 3, . . . ordinarily think of numbers in order, , essential part of work of analysing our data seek definition of order or series in logical terms. . . . order lies, not in class of terms, in relation among members of class, in respect of appear earlier , later. (1919:31)

russell applies notion of ordering relation 3 criteria: first, defines notion of asymmetry i.e. given relation such s ( . . . successor of . . . ) between 2 terms x, , y: x s y ≠ y s x. second, defines notion of transitivity 3 numerals x, y , z: if x s y , y s z x s z. third, defines notion of connected : given 2 terms of class ordered, there must 1 precedes , other follows. . . . relation connected when, given 2 different terms of field [both domain , converse domain of relation e.g. husbands versus wives in relation of married] relation holds between first , second or between second , first (not excluding possibility both may happen, though both cannot happen if relation asymmetrical).(1919:32)

he concludes: . . . [natural] number m said less number n when n possesses every hereditary property possessed successor of m. easy see, , not difficult prove, relation less , defined, asymmetrical, transitive, , connected, , has [natural] numbers field [i.e. both domain , converse domain numbers]. (1919:35)

criticism

the problem of presuming extralogical notion of iteration : kleene points out that, logicistic thesis can questioned on ground logic presupposes mathematical ideas in formulation. in intuitionistic view, essential mathematical kernel contained in idea of iteration (kleene 1952:46)

bernays 1930–1931 observes notion 2 things presupposes something, without claim of existence of 2 things, , without reference predicate, applies 2 things; means, simply, thing , 1 more thing. . . . respect simple definition, number concept turns out elementary structural concept . . . claim of logicists mathematics purely logical knowledge turns out blurred , misleading upon closer observation of theoretical logic. . . . [one can extend definition of logical ] however, through definition epistemologically essential concealed, , peculiar mathematics overlooked (in mancosu 1998:243).

hilbert 1931:266-7, bernays, detects extra-logical in mathematics: besides experience , thought, there yet third source of knowledge. if today can no longer agree kant in details, nevertheless general , fundamental idea of kantian epistemology retains significance: ascertain intuitive priori mode of thought, , thereby investigate condition of possibility of knowledge. in opinion happens in investigations of principles of mathematics. priori here nothing more , nothing less fundamental mode of thought, call finite mode of thought: given in advance in our faculty of representation: extra-logical concrete objects exist intuitively immediate experience before thought. if logical inference certain, these objects must surveyable in parts, , presentation, differences, succeeding 1 or being arrayed next 1 , intuitively given us, along objects, neither can reduced else, nor needs such reduction. (hilbert 1931 in mancosu 1998: 266, 267).

in brief: notion of sequence or successor priori notion lies outside symbolic logic.

hilbert dismissed logicism false path : tried define numbers purely logically; others took usual number-theoretic modes of inference self-evident. on both paths encountered obstacles proved insuperable. (hilbert 1931 in mancoso 1998:267) .

mancosu states brouwer concluded that: classical laws or principles of logic part of [the] perceived regularity [in symbolic representation]; derived post factum record of mathematical constructions . . . theoretical logic . . . [is] empirical science , application of mathematics (brouwer quoted mancosu 1998:9).

gödel 1944: respect technical aspects of russellian logicism appears in principia mathematic (either edition), gödel flat-out disappointed:

regretted first comprehensive , thorough-going presentation of mathematical logic , derivation of mathematics [is?] lacking in formal precision in foundations (contained in *1 - *21 of principia) presents in respect considerable step backwards compared frege. missing, above all, precise statement of syntax of formalism (cf footnote 1 in gödel 1944 collected works 1990:120).

in particular pointed out matter doubtful rule of substitution , of replacing defined symbols definiens (russell 1944:120)

with respect philosophy formed these foundations, gödel home in on russell s no-class theory , or gödel call nominalistic kind of constructivism, such embodied in russell s no class theory . . . might better called fictionalism (cf footnote 1 in gödel 1944:119). see more in gödel s criticism , suggestions below.

grattan-guinness: complicated theory of relations continued strangle russell s explanatory 1919 introduction mathematical philosophy , 1927 second edition of principia. set theory, meanwhile had moved on reduction of relation ordered pair of sets. grattan-guinness observes in second edition of principia russell ignored reduction had been achieved own student norbert wiener (1914). perhaps because of residual annoyance, russell did not react @ . 1914 hausdorff provide another, equivalent definition, , kuratowski in 1921 provide 1 in use today.