Editing Logicism (section)

==An example of a logicist construction of the natural numbers: Russell's construction in the ''Principia''==

The logicism of Frege and Dedekind is similar to that of Russell, but with differences in the particulars (see Criticisms, below). Overall, the logicist derivations of the natural numbers are different from derivations from, for example, Zermelo's axioms for set theory ('Z'). Whereas, in derivations from Z, one definition of "number" uses an axiom of that system – the [[axiom of pairing]] – that leads to the definition of "[[ordered pair]]" – no ''overt'' number axiom exists in the various logicist axiom systems allowing the derivation of the natural numbers.  Note that the axioms needed to derive the definition of a number may differ between axiom systems for set theory in any case.  For instance, in ZF and ZFC, the axiom of pairing, and hence ultimately the notion of an ordered pair is derivable from the [[Axiom of Infinity]] and the [[Axiom schema of replacement|Axiom of Replacement]] and is required in the definition of the [[Von Neumann ordinal|von Neumann numerals]] (but not the Zermelo numerals), whereas in [[New Foundations|NFU]] the Frege numerals may be derived in an analogous way to their derivation in the Grundgesetze.

The ''Principia'', like its forerunner the ''Grundgesetze'', begins its construction of the numbers from primitive propositions such as "class", "propositional function", and in particular, relations of "similarity" ("[[bijection|equinumerosity]]": placing the elements of collections in one-to-one correspondence) and "ordering" (using "the successor of" relation to order the collections of the equinumerous classes)".<ref>In his 1903 and in ''PM'' Russell refers to such assumptions (there are others) as "primitive propositions" ("pp" as opposed to "axioms" (there are some of those, too). But the reader is never certain whether these pp are axioms/axiom-schemas or construction-devices (like substitution or ''modus ponens''), or what, exactly. Gödel 1944:120 comments on this absence of formal syntax and the absence of a clearly specified substitution process.</ref> The logicistic derivation equates the [[cardinal number]]s ''constructed'' this way to the natural numbers, and these numbers end up all of the same "type" – as classes of classes – whereas in some set theoretical constructions – for instance the von Neumann and the Zermelo numerals – each number has its predecessor as a [[subset]]. Kleene observes the following. (Kleene's assumptions (1) and (2) state that 0 has property ''P'' and ''n''+1 has property ''P'' whenever ''n'' has property ''P''.)
:"The viewpoint here is very different from that of [Kronecker]'s maxim that 'God made the integers' plus [[Peano's axioms]] of number and [[mathematical induction]]], where we presupposed an intuitive conception of the natural number sequence, and elicited from it the principle that, whenever a particular property ''P'' of natural numbers is given such that (1) and (2), then any given natural number must have the property ''P''." (Kleene 1952:44).

The importance to the logicist programme of the construction of the natural numbers derives from Russell's contention "That all traditional pure mathematics can be derived from the natural numbers is a fairly recent discovery, though it had long been suspected" (1919:4). One derivation of the ''real'' numbers derives from the theory of [[Dedekind cut]]s on the rational numbers, rational numbers in turn being derived from the naturals. While an example of how this is done is useful, it relies first on the derivation of the natural numbers. So, if philosophical difficulties appear in a logicist derivation of the natural numbers, these problems should be sufficient to stop the program until these are resolved (see Criticisms, below).

One attempt to construct the natural numbers is summarized by Bernays 1930–1931.<ref>Cf. ''The Philosophy of Mathematics and Hilbert's Proof Theory'' 1930:1931 in Mancosu, p. 242.</ref>  But rather than use Bernays' précis, which is incomplete in some details, an attempt at a paraphrase of Russell's construction, incorporating some finite illustrations, is set out below:

===Preliminaries===
For Russell, collections (classes) are aggregates of "things" specified by proper names, that come about as the result of propositions (assertions of fact about a thing or things). Russell analysed this general notion. He begins with "terms" in sentences, which he analysed as follows:

For Russell, "terms" are either "things" or "concepts": "Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a ''term''. This, then, is the widest word in the philosophical vocabulary. I shall use as synonymous with it the words, unit, individual, and entity. The first two emphasize the fact that every term is one, while the third is derived from the fact that every term has being, i.e. is in some sense. A man, a moment, a number, a class, a relation, a chimaera, or anything else that can be mentioned, is sure to be a term; and to deny that such and such a thing is a term must always be false" (Russell 1903:43)

"Among terms, it is possible to distinguish two kinds, which I shall call respectively ''things'' and ''concepts''; the former are the terms indicated by proper names, the latter those indicated by all other words . . . Among concepts, again, two kinds at least must be distinguished, namely those indicated by adjectives and those indicated by verbs" (1903:44).

"The former kind will often be called predicates or class-concepts; the latter are always or almost always relations." (1903:44)

"I shall speak of the ''terms'' of a proposition as those terms, however numerous, which occur in a proposition and may be regarded as subjects about which the proposition is. It is a characteristic of the terms of a proposition that anyone of them may be replaced by any other entity without our ceasing to have a proposition. Thus we shall say that "Socrates is human" is a proposition having only one term; of the remaining component of the proposition, one is the verb, the other is a predicate.. . . Predicates, then, are concepts, other than verbs, which occur in propositions having only one term or subject." (1903:45)

Suppose one were to point to an object and say: "This object in front of me named 'Emily' is a woman." This is a proposition, an assertion of the speaker's belief, which is to be tested against the "facts" of the outer world: "Minds do not ''create'' truth or falsehood. They create beliefs . . . what makes a belief true is a ''fact'', and this fact does not (except in exceptional cases) in any way involve the mind of the person who has the belief" (1912:130). If by investigation of the utterance and correspondence with "fact", Russell discovers that Emily is a rabbit, then his utterance is considered "false"; if Emily is a female human (a female "featherless biped" as Russell likes to call humans, following [[Diogenes Laërtius]]'s anecdote about [[Plato]]), then his utterance is considered "true".

"The class, as opposed to the class-concept, is the sum or conjunction of all the terms which have the given predicate" (1903 p.&nbsp;55). Classes can be specified by extension (listing their members) or by intension, i.e. by a "propositional function" such as "''x'' is a ''u''" or "''x'' is ''v''". But "if we take extension pure, our class is defined by enumeration of its terms, and this method will not allow us to deal, as Symbolic Logic does, with infinite classes. Thus our classes must in general be regarded as objects denoted by concepts, and to this extent the point of view of intension is essential." (1909 p.&nbsp;66)

"The characteristic of a class concept, as distinguished from terms in general, is that "''x'' is a ''u''" is a propositional function when, and only when, ''u'' is a class-concept." (1903:56)

"71. Class may be defined either extensionally or intensionally. That is to say, we may define the kind of object which is a class, or the kind of concept which denotes a class: this is the precise meaning of the opposition of extension and intension in this connection. But although the general notion can be defined in this two-fold manner, particular classes, except when they happen to be finite, can only be defined intensionally, i.e. as the objects denoted by such and such concepts. . . logically; the extensional definition appears to be equally applicable to infinite classes, but practically, if we were to attempt it, Death would cut short our laudable endeavour before it had attained its goal."(1903:69)

===The definition of the natural numbers===
In the Prinicipia, the natural numbers derive from ''all'' propositions that can be asserted about ''any'' collection of entities. Russell makes this clear in the second (italicized) sentence below.

:"In the first place, numbers themselves form an infinite collection, and cannot therefore be defined by enumeration. ''In the second place, the collections having a given number of terms themselves presumably form an infinite collection: it is to be presumed, for example, that there are an infinite collection of trios in the world'', for if this were not the case the total number of things in the world would be finite, which, though possible, seems unlikely. In the third place, we wish to define "number" in such a way that infinite numbers may be possible; thus we must be able to speak of the number of terms in an infinite collection, and such a collection must be defined by intension, i.e. by a property common to all its members and peculiar to them." (1919:13)

To illustrate, consider the following finite example: Suppose there are 12 families on a street. Some have children, some do not. To discuss the names of the children in these households requires 12 propositions asserting "''childname'' is the name of a child in family F''n''" applied to this collection of households on the particular street of families with names F1, F2, . . . F12. Each of the 12 propositions regards whether or not the "argument" ''childname'' applies to a child in a particular household. The children's names (''childname'') can be thought of as the ''x'' in a propositional function ''f''(''x''), where the function is "name of a child in the family with name F''n''".<ref>To be precise both ''childname'' = variable ''x'' and family name ''Fn'' are variables. ''Childname''{{'}}s domain is "all childnames", and family name ''Fn'' has a domain consisting of the 12 families on the street.</ref>{{or|date=August 2018}}

Whereas the preceding example is finite over the finite propositional function "''childnames'' of the children in family F''n'''" on the finite street of a finite number of families, Russell apparently intended the following to extend to all propositional functions extending over an infinite domain so as to allow the creation of all the numbers. 

Kleene considers that Russell has set out an [[impredicativity|impredicative]] definition that he will have to resolve, or risk deriving something like the [[Russell paradox]]. "Here instead we presuppose the totality of all properties of cardinal numbers, as existing in logic, prior to the definition of the natural number sequence" (Kleene 1952:44). The problem will appear, even in the finite example presented here, when Russell deals with the unit class (cf. Russell 1903:517).

The question arises what precisely a "class" ''is'' or should be. For Dedekind and Frege, a class is a distinct entity in its own right, a 'unity' that can be identified with all those entities ''x'' that satisfy some propositional function ''F''. (This symbolism appears in Russell, attributed there to Frege: "The essence of a function is what is left when the ''x'' is taken away, i.e in the above instance, 2( )<sup>3</sup> + ( ). The argument ''x'' does not belong to the function, but the two together make a whole (ib. p. 6 [i.e. Frege's 1891 ''Function und Begriff'']" (Russell 1903:505).) For example, a particular "unity" could be given a name; suppose a family Fα has the children with the names Annie, Barbie and Charles:
:{ a, b, c }<sub>Fα</sub>

This notion of collection or class as object, when used without restriction, results in [[Russell's paradox]]; see more below about [[impredicativity|impredicative definitions]]. Russell's solution was to define the notion of a class to be only those elements that satisfy the proposition, his argument being that, indeed, the arguments ''x'' do not belong to the propositional function aka "class" created by the function. The class itself is not to be regarded as a unitary object in its own right, it exists only as a kind of useful fiction: "We have avoided the decision as to whether a class of things has in any sense an existence as one object. A decision of this question in either way is indifferent to our logic" (First edition of ''Principia Mathematica'' 1927:24).

Russell continues to hold this opinion in his 1919; observe the words "symbolic fictions": {{or|date=August 2018}}
:"When we have decided that classes cannot be things of the same sort as their members, that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than ''symbolic fictions''. And if we can find any way of dealing with them as ''symbolic fictions'', we increase the logical security of our position, since we avoid the need of assuming that there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from both assumptions.  . . . But when we refuse to assert that there are classes, we must not be supposed to be asserting dogmatically that there are none. We are merely agnostic as regards them . . .." (1919:184)

And in the second edition of ''PM'' (1927) Russell holds that "functions occur only through their values, . . . all functions of functions are extensional, . . . [and] consequently there is no reason to distinguish between functions and classes . . . Thus classes, as distinct from functions, lose even that shadowy being which they retain in *20" (p. xxxix). In other words, classes as a separate notion have vanished altogether.

'''Step 2: Collect "similar" classes into 'bundles' ''': These above collections can be put into a "binary relation" (comparing for) similarity by "equinumerosity", symbolized here by '''≈''', i.e. one-one correspondence of the elements,<ref>"If the predicates are partitioned into classes with respect to equinumerosity in such a way that all predicates of a class are equinumerous to one another and predicates of different classes are not equinumerous, then each such class represents the ''Number'', which applies to the predicates that belong to it" (Bernays 1930-1 in Mancosu 1998:240.</ref> and thereby create Russellian classes of classes or what Russell called "bundles". "We can suppose all couples in one bundle, all trios in another, and  so on. In this way we obtain various bundles of collections, each bundle consisting of all the collections that have a certain number of terms. Each bundle is a class whose members are collections, i.e. classes; thus each is a class of classes" (Russell 1919:14).

'''Step 3: Define the null class''': Notice that a certain class of classes is special because its classes contain no elements, i.e. no elements satisfy the predicates whose assertion defined this particular class/collection. 

The resulting entity may be called "the null class" or "the empty class". Russell symbolized the null/empty class with Λ. So what exactly is the Russellian null class? In ''PM'' Russell says that "A class is said to ''exist'' when it has at least one member . . . the class which has no members is called the "null class"  . . . "α is the null-class" is equivalent to "α does not exist". The question naturally arises whether the null class itself 'exists'? Difficulties related to this question occur in Russell's 1903 work.<ref name=:0>Cf. sections 487ff (pages 513ff in the Appendix A).</ref> After he discovered the paradox in Frege's ''Grundgesetze'' he added Appendix A to his 1903 where through the analysis of the nature of the null and unit classes, he discovered the need for a "doctrine of types"; see more about the unit class, the problem of [[impredicativity|impredicative definitions]] and Russell's "vicious circle principle" below.<ref name=:0/>

'''Step 4: Assign a "numeral" to each bundle''': For purposes of abbreviation and identification, to each bundle assign a unique symbol (aka a "numeral"). These symbols are arbitrary. 

'''Step 5: Define "0"''' Following Frege, Russell picked the empty or ''null'' class of classes as the appropriate class to fill this role, this being the class of classes having no members.  This null class of classes may be labelled "0"

'''Step 6: Define the notion of "successor"''': Russell defined a new characteristic "hereditary" (cf Frege's 'ancestral'), a property of certain classes with the ability to "inherit" a characteristic from another class (which may be a class of classes) i.e. "A property is said to be "hereditary" in the natural-number series if, whenever it belongs to a number ''n'', it also belongs to ''n''+1, the successor of ''n''". (1903:21). He asserts that "the natural numbers are the ''posterity'' – the "children", the inheritors of the "successor" – of 0 with respect to the relation "the immediate predecessor of (which is the converse of "successor") (1919:23).

Note Russell has used a few words here without definition, in particular "number series", "number ''n''", and "successor". He will define these in due course. ''Observe in particular that Russell does not use the unit class of classes "1" to construct the successor''. The reason is that, in Russell's detailed analysis,<ref>1909 Appendix A</ref> if a unit class becomes an entity in its own right, then it too can be an element in its own proposition; this causes the proposition to become "impredicative" and result in a "vicious circle". Rather, he states: "We saw in Chapter II that a cardinal number is to be defined as a class of classes, and in Chapter III that the number 1 is to be defined as the class of all unit classes, of all that have just one member, as we should say but for the vicious circle. Of course, when the number 1 is defined as the class of all unit classes, ''unit classes'' must be defined so as not to assume that we know what is meant by ''one'' (1919:181).

For his definition of successor,  Russell will use for his "unit" a single entity or "term" as follows:
: "It remains to define "successor". Given any number ''n'' let ''α'' be a class which has ''n'' members, and let ''x'' be a term which is not a member of ''α''. Then the class consisting of ''α'' with ''x'' added on will have ''+1'' members. Thus we have the following definition:
:''the successor of the number of terms in the class α is the number of terms in the class consisting of α together with x where x is not any term belonging to the class''." (1919:23)

Russell's definition requires a new "term" which is "added into" the collections inside the bundles.

'''Step 7: Construct the successor of the null class'''.

'''Step 8: For every class of equinumerous classes, create its successor'''.

'''Step 9: Order the numbers''': The process of creating a successor requires the relation " . . . is the successor of . . .", which may be denoted "''S''", between the various "numerals". "We must now consider the ''serial'' character of the natural numbers in the order 0, 1, 2, 3, . . . We ordinarily think of the numbers as in this order, and it is an essential part of the work of analysing our data to seek a definition of "order" or "series " in logical terms. . . . The order lies, not in the ''class'' of terms, but in a relation among the members of the class, in respect of which some appear as earlier and some as later." (1919:31)

Russell applies to the notion of "ordering relation" three criteria: First, he defines the notion of [[asymmetric relation|asymmetry]] i.e. given the relation such as ''S'' (" . . . is the successor of  . . . ") between two terms ''x'' and ''y'': ''x S y'' ≠ ''y S x''. Second, he defines the notion of [[transitive relation|transitivity]] for three numerals ''x'', ''y'' and ''z'': if ''x S y'' and ''y S z'' then ''x S z''. Third, he defines the notion of [[connected relation|connected]]: "Given any two terms of the class which is to be ordered, there must be one which precedes and the other which follows. . . . A relation is connected when, given any two different terms of its field [both domain and converse domain of a relation e.g. husbands versus wives in the relation of married] the relation holds between the first and the second or between the second and the first (not excluding the possibility that both may happen, though both cannot happen if the relation is asymmetrical).(1919:32)

He concludes: ". . . [natural] number ''m'' is said to be less than another number ''n'' when ''n'' possesses every hereditary property possessed by the successor of ''m''. It is easy to see, and not difficult to prove, that the relation "less than", so defined, is asymmetrical, transitive, and connected, and has the [natural] numbers for its field [i.e. both domain and converse domain are the numbers]." (1919:35)

===Criticism===
'''The presumption of an 'extralogical' notion of iteration''': Kleene notes that "the logicistic thesis can be questioned finally on the ground that logic already presupposes mathematical ideas in its formulation. In the Intuitionistic view, an essential mathematical kernel is contained in the idea of iteration" (Kleene 1952:46)

Bernays 1930–1931 observes that this notion "two things" already presupposes something, even without the claim of existence of two things, and also without reference to a predicate, which applies to the two things; it means, simply, "a thing and one more thing. . . . With respect to this simple definition, the Number concept turns out to be an elementary ''structural concept''  . . . the claim of the logicists that mathematics is purely logical knowledge turns out to be blurred and misleading upon closer observation of theoretical logic. . . . [one can extend the definition of "logical"] however, through this definition what is epistemologically essential is concealed, and what is peculiar to mathematics is overlooked" (in Mancosu 1998:243).

Hilbert 1931:266-7, like Bernays, considers there is "something extra-logical" in mathematics: "Besides experience and thought, there is yet a third source of knowledge. Even if today we can no longer agree with Kant in the details, nevertheless the most general and fundamental idea of the Kantian epistemology retains its significance: to ascertain the intuitive ''a priori'' mode of thought, and thereby to investigate the condition of the possibility of all knowledge. In my opinion this is essentially what happens in my investigations of the principles of mathematics. The ''a priori'' is here nothing more and nothing less than a fundamental mode of thought, which I also call the finite mode of thought: something is already given to us in advance in our faculty of representation: certain ''extra-logical concrete objects'' that exist intuitively as an immediate experience before all thought. If logical inference is to be certain, then these objects must be completely surveyable in all their parts, and their presentation, their differences, their succeeding one another or their being arrayed next to one another is immediately and intuitively given to us, along with the objects, as something that neither can be reduced to anything else, nor needs such a reduction." (Hilbert 1931 in Mancosu 1998: 266, 267).

In brief, according to Hilbert and Bernays, the notion of "sequence" or "successor" is an ''a priori'' notion that lies outside symbolic logic.

Hilbert dismissed logicism as a "false path": "Some tried to define the  numbers purely logically; others simply took the usual [[number theory|number-theoretic]] modes of inference to be self-evident. On both paths they encountered obstacles that proved to be insuperable." (Hilbert 1931 in Mancoso 1998:267).  The incompleteness theorems arguably constitute a similar obstacle for Hilbertian finitism.

Mancosu states that Brouwer concluded that: "the classical laws or principles of logic are part of [the] perceived regularity [in the symbolic representation]; they are derived from the post factum record of mathematical constructions . . . Theoretical logic  . . . [is] an empirical science and an application of mathematics" (Brouwer quoted by Mancosu 1998:9).

With respect to the ''technical'' aspects of Russellian logicism as it appears in ''Principia Mathematica'' (either edition), Gödel in 1944 was disappointed:
:"It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is?] so greatly lacking in formal precision in the foundations (contained in *1–*21 of ''Principia'') that it presents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism" (cf. footnote 1 in Gödel 1944 ''Collected Works'' 1990:120).

In particular he pointed out that "The matter is especially doubtful for the rule of substitution and of replacing defined symbols by their ''definiens''" (Russell 1944:120)

With respect to the philosophy that might underlie these foundations, Gödel considered Russell's "no-class theory" as embodying a "nominalistic kind of constructivism . . . which might better be called fictionalism" (cf. footnote 1 in Gödel 1944:119) – to be faulty. See more in "Gödel's criticism and suggestions" below.

A complicated theory of relations continued to strangle Russell's explanatory 1919 ''Introduction to Mathematical Philosophy'' and his 1927 second edition of ''Principia''. Set theory, meanwhile had moved on with its reduction of relation to the ordered pair of sets. [[Grattan-Guinness]] observes that in the second edition of ''Principia'' Russell ignored this reduction that had been achieved by his own student [[Norbert Wiener]] (1914). Perhaps because of "residual annoyance, Russell did not react at all".<ref>Russell deemed Wiener "the infant phenomenon . . . more infant than phenomenon"; see ''Russell's confrontation with Wiener'' in Grattan-Guinness 2000:419ff.</ref> By 1914 [[Felix Hausdorff|Hausdorff]] would provide another, equivalent definition, and [[Kuratowski]] in 1921 would provide [[Kuratowski ordered pair|the one in use today]].<ref>See van Heijenoort's commentary and Norbert Wiener's 1914 ''A simplification of the logic of relations'' in van Heijenoort 1967:224ff.</ref>