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 and is required in the definition of the
von Neumann numerals (but not the Zermelo numerals), whereas in
NFU the Frege numerals may be derived in an analogous way to their derivation in the . The
Principia, like its forerunner the , begins its construction of the numbers from primitive propositions such as "class", "propositional function", and in particular, relations of "similarity" ("
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)". The logicistic derivation equates the
cardinal numbers
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 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 cuts 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. 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": 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" (1912:13). 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 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. 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". 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
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( )3 + ( ). 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: {{block indent|{ a, b, c }Fα}} This notion of collection or class as object, when used without restriction, results in
Russell's paradox; see more below about
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" (
Principia Mathematica 1st ed. 1927:24). Russell continues to hold this opinion in his 1919; observe the words "symbolic fictions": 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, 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. After he discovered the paradox in Frege's 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
impredicative definitions and Russell's "vicious circle principle" below. 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: 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
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
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: "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-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: 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". By 1914
Hausdorff would provide another, equivalent definition, and
Kuratowski in 1921 would provide
the one in use today. == The unit class, impredicativity, and the vicious circle principle ==