Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic.^{[1]}Bertrand Russell and Alfred North Whitehead championed this theory, created by mathematicians Richard Dedekind and Gottlob Frege.
Dedekind's path to logicism had a turning point when he was able to reduce the theory of real numbers to the rational number system by means of set theory. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of sets; furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real numbers published in the year 1872. This started a period of expansion of logicism, with Dedekind and Frege as its main exponents, which however was brought to a deep crisis with the discovery of the classical paradoxes of set theory (Cantor 1896, Zermelo and Russell 1900-1901). Frege gave up on the project after Russell recognized and communicated his paradox exposing an inconsistency in naive set theory. On the other hand, Russell wrote The Principles of Mathematics in 1903 using the paradox and developments of Giuseppe Peano's school of geometry. Since he treated the subject of primitive notions in geometry and set theory, this text is a watershed in the development of logicism. Evidence of the assertion of logicism was collected by Russell and Whitehead in their Principia Mathematica.^{[2]}
Today, the bulk of modern mathematics is believed to be reducible to a logical foundation using the axioms of Zermelo-Fraenkel set theory (or one of its extensions, such as ZFC), which has no known inconsistencies (although it remains possible that inconsistencies in it may still be discovered). Thus to some extent Dedekind's project was proved viable, but in the process the theory of sets and mappings came to be regarded as transcending pure logic.
Kurt Gödel's incompleteness theorem undermines logicism because it shows that no particular axiomatization of mathematics can decide all statements. ^{[3]}^{[non-primary source needed]} Some believe that the basic spirit of logicism remains valid because that theorem is proved with logic just like other theorems. However, that conclusion fails to acknowledge any distinction between theorems of first-order logic and those of higher-order logic. The former can be proven using the fundamental theorem of arithmetic (see Gödel numbering), while the latter must rely on human-provided models. Tarski's undefinability theorem shows that Gödel numbering can be used to prove syntactical constructs, but not semantic assertions. Therefore, any claim that logicism remains a valid concept relies on the dubious notion that a system of proof based on man-made models is as authoritative as one based on the existence and properties of the natural numbers.^{[]}
Logicism was key in the development of analytic philosophy in the twentieth century.
Ivor Grattan-Guinness states that the French word 'Logistique' was "introduced by Couturat and others at the 1904 International Congress of Philosophy', and was used by Russell and others from then on, in versions appropriate for various languages." (G-G 2000:501).
Apparently the first (and only) usage by Russell appeared in his 1919: "Russell referred several time [sic] to Frege, introducing him as one 'who first succeeded in "logicising" mathematics' (p. 7). Apart from the mis-representation (which Russell partly rectified by explaining his own view of the role of arithmetic in mathematics), the passage is notable for the word which he put in quotation marks, but their presence suggests nervousness, and he never used the word again, so that 'logicism' did not emerge until the later 1920s" (G-G 2002:434).^{[4]}
About same time as Carnap (1929), but apparently independently, Fraenkel (1928) used the word: "Without comment he used the name 'logicism' to characterise the Whitehead/Russell position (in the title of the section on p. 244, explanation on p. 263)" (G-G 2002:269). Carnap used a slightly different word 'Logistik'; Behmann complained about its use in Carnap's manuscript so Carnap proposed the word 'Logizismus', but he finally stuck to his word-choice 'Logistik' (G-G 2002:501). Ultimately "the spread was mainly due to Carnap, from 1930 onwards." (G-G 2000:502).
Symbolic logic: The overt intent of Logicism is to reduce all of philosophy to symbolic logic (Russell), and/or to reduce all of mathematics to symbolic logic (Frege, Dedekind, Peano, Russell). As contrasted with algebraic logic (Boolean logic) that employs arithmetic concepts, symbolic logic begins with a very reduced set of marks (non-arithmetic symbols), a (very-)few "logical" axioms that embody the three "laws of thought", and a couple of construction rules that dictate how the marks are to be assembled and manipulated--substitution and modus ponens (inference of the true from the true). Logicism also adopts from Frege's groundwork the reduction of natural language statements from "subject|predicate" into either propositional "atoms" or the "argument|function" of "generalization"--the notions "all", "some", "class" (collection, aggregate) and "relation".
As perhaps its core tenet, logicism forbids any "intuition" of number to sneak in either as an axiom or by accident. The goal is to derive all of mathematics, starting with the counting numbers and then the irrational numbers, from the "laws of thought" alone, without any tacit (hidden) assumptions of "before" and "after" or "less" and "more" or to the point: "successor" and "predecessor". Gödel 1944 summarized Russell's logicistic "constructions", when compared to "constructions" in the foundational systems of Intuitionism and Formalism ("the Hilbert School") as follows: "Both of these schools base their constructions on a mathematical intuition whose avoidance is exactly one of the principal aims of Russell's constructivism" (Gödel 1944 in Collected Works 1990:119).
History: Gödel 1944 summarized the historical background from Leibniz's in Characteristica universalis, through Frege and Peano to Russell: "Frege was chiefly interested in the analysis of thought and used his calculus in the first place for deriving arithmetic from pure logic", whereas Peano "was more interested in its applications within mathematics". But "It was only [Russell's] Principia Mathematica that full use was made of the new method for actually deriving large parts of mathematics from a very few logical concepts and axioms. In addition, the young science was enriched by a new instrument, the abstract theory of relations" (p. 120-121).
Kleene 1952 states it this way: "Leibniz (1666) first conceived of logic as a science containing the ideas and principles underlying all other sciences. Dedekind (1888) and Frege (1884, 1893, 1903) were engaged in defining mathematical notions in terms of logical ones, and Peano (1889, 1894-1908) in expressing mathematical theorems in a logical symbolism" (p. 43); in the previous paragraph he includes Russell and Whitehead as exemplars of the "logicistic school", the other two "foundational" schools being the intuitionistic and the "formalistic or axiomatic school" (p. 43).
Dedekind 1887 describes his intent in the 1887 Preface to the First Edition of his The Nature and Meaning of Numbers. He believed that in the "foundations of the simplest science; viz., that part of logic which deals with the theory of numbers" had not been properly argued -- "nothing capable of proof ought to be accepted without proof":
Peano 1889 states his intent in his Preface to his 1889 Principles of Arithmetic:
Frege 1879 describes his intent in the Preface to his 1879 Begriffsschrift: He started with a consideration of arithmetic: did it derive from "logic" or from "facts of experience"?
Russell 1903 describes his intent in the Preface to his 1903 Principles of Mathematics:
Dedekind and Frege: The epistemology of Dedekind and Frege is not as well-defined as that of the philosopher Russell, but both seem accepting as a priori the customary "laws of thought" concerning simple propositional statements (usually of belief); these laws would be sufficient in themselves if augmented with theory of classes and relations (e.g. x R y) between individuals x and y linked by the generalization R.
Dedekind's "free formations of the human mind" rebels against the strictures of Kronecker: Dedekind's argument begins with "1. In what follows I understand by thing every object of our thought"; we humans use symbols to discuss these "things" of our minds; "A thing is completely determined by all that can be affirmed or thought concerning it" (p. 44). In a subsequent paragraph Dedekind discusses what a "system S is: it is an aggregate, a manifold, a totality of associated elements (things) a, b, c"; he asserts that "such a system S . . . as an object of our thought is likewise a thing (1); it is completely determined when with respect to every thing it is determined whether it is an element of S or not.*" (p. 45, italics added). The * indicates a footnote where he states that:
Indeed he awaits Kronecker's "publishing his reasons for the necessity or merely the expediency of these limitations" (p. 45).
Leopold Kronecker, famous for his assertion that "God made the integers, all else is the work of man"^{[5]} had his foes, among them Hilbert. Hilbert called Kronecker a "dogmatist, to the extent that he accepts the integer with its essential properties as a dogma and does not look back"^{[6]} and equated his extreme constructivist stance with that of Brouwer's intuitionism, accusing both of "subjectivism": "It is part of the task of science to liberate us from arbitrariness, sentiment and habit and to protect us from the subjectivism that already made itself felt in Kronecker's views and, it seems to me, finds its culmination in intuitionism".^{[7]} Hilbert then states that "mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker . . ." (p. 479).
Russell the realist: Russell's Realism served him as an antidote to British Idealism,^{[8]} with portions borrowed from European Rationalism and British empiricism.^{[9]} To begin with, "Russell was a realist about two key issues: universals and material objects" (Russell 1912:xi). For Russell, tables are real things that exist independent of Russell the observer. Rationalism would contribute the notion of a priori knowledge,^{[10]} while empiricism would contribute the role of experiential knowledge (induction from experience).^{[11]} Russell would credit Kant with the idea of "a priori" knowledge, but he offers an objection to Kant he deems "fatal": "The facts [of the world] must always conform to logic and arithmetic. To say that logic and arithmetic are contributed by us does not account for this" (1912:87); Russell concludes that the a priori knowledge that we possess is "about things, and not merely about thoughts" (1912:89). And in this Russell's epistemology seems different from that of Dedekind's belief that "numbers are free creations of the human mind" (Dedekind 1887:31)^{[12]}
But his epistemology about the innate (he prefers the word a priori when applied to logical principles, cf 1912:74) is intricate. He would strongly, unambiguously express support for the Platonic "universals" (cf 1912:91-118) and he would conclude that truth and falsity are "out there"; minds create beliefs and what makes a belief true is a fact, "and this fact does not (except in exceptional cases) involve the mind of the person who has the belief" (1912:130).
Where did Russell derive these epistemic notions? He tells us in the Preface to his 1903 Principles of Mathematics. Note that he asserts that the belief: "Emily is a rabbit" is non-existent, and yet the truth of this non-existent proposition is independent of any knowing mind; if Emily really is a rabbit, the fact of this truth exists whether or not Russell or any other mind is alive or dead, and the relation of Emily to rabbit-hood is "ultimate" :
Russell and the paradox: In 1902 Russell discovered a "vicious circle" (the so-called Russell's paradox) in Frege's Begriffsschrift and he was determined not to repeat it in his 1903 Principles of Mathematics. In two Appendices that he tacked on at the last minute he devotes 28 pages to both a detailed analysis of Frege's theory contrasted against his own, and a fix for the paradox. But he was not optimistic about the outcome:
"Fictionalism" and Russell's no-class theory: Gödel in his 1944 would disagree with the young Russell of 1903 ("[my premisses] allow mathematics to be true") but would probably agree with Russell's statement quoted above ("something is amiss"); Russell's theory had failed to arrive at a satisfactory foundation of mathematics: the result was "essentially negative; i.e. the classes and concepts introduced this way do not have all the properties required for the use of mathematics" (Gödel 1944:132).
How did Russell arrive in this situation? Gödel observes that Russell is a surprising "realist" with a twist: he cites Russell's 1919:169 "Logic is concerned with the real world just as truly as zoology" (Gödel 1944:120). But he observes that "when he started on a concrete problem, the objects to be analyzed (e.g. the classes or propositions) soon for the most part turned into "logical fictions" . . . [meaning] only that we have no direct perception of them." (Gödel 1944:120)
In an observation pertinent to Russell's brand of logicism, Perry remarks that Russell went through three phases of realism -- extreme, moderate and constructive (Perry 1997:xxv). In 1903 he was in his extreme phase; by 1905 he would be in his moderate phase. In a few years he would "dispense with physical or material objects as basic bits of the furniture of the world. He would attempt to construct them out of sense-data" in his next book Our knowledge of the External World [1914]" (Perry 1997:xxvi).
These constructions in what Gödel 1944 would call "nominalistic constructivism . . . which might better be called fictionalism" derived from Russell's "more radical idea, the no-class theory" (p. 125):
See more in the Criticism sections, below.
The attempt to construct the natural numbers is summarized succinctly by Bernays 1930-1931.^{[13]} But rather than use Bernays' précis, which is incomplete in the details, the construction is best given as a simple finite example together with the details to be found in Russell 1919.
In general the logicism of Dedekind-Frege is similar to that of Russell, but with significant (and critical) differences in the particulars (see Criticisms, below). Overall, though, the logicistic construction-process [Dedekind-Frege-Russell] is very different from that of contemporary set theory. Whereas in set theory the notion of "number" begins from an axiom--the axiom of pairing that leads to the definition of "ordered pair"--no overt number-axiom exists in logicism. Rather, logicism begins its construction of the numbers from "primitive propositions" that include "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)".^{[14]} 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 equivalence classes of classes--whereas in set theory each number is of a higher class than its predecessor (thus each successor contains 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 logicism of the construction of the natural numbers derives from Russell's contention that "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). The derivation of the real numbers (rationals, irrationals) derives from the theory of Dedekind cuts on the continuous "number line". While an example of how this is done is useful, it relies first on the derivation of the natural numbers. So, if philosophic problems appear in the logistic attempt to derive the natural numbers, these problems will be sufficient to stop the program until these are fixed (see Criticisms, below).
For Dedekind, Frege and Russell, collections (classes) are aggregates of "things" specified by proper names, that come about as the result of propositions (utterances about something that asserts a fact about that thing or things). Russell tore the general notion down in the following manner. He begins with "terms" in sentences that he decomposes as follows:
Terms: 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)
Things are indicated by proper names; concepts are indicated by adjectives or verbs: "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).
Concept-adjectives are "predicates"; concept-verbs are "relations": "The former kind will often be called predicates or class-concepts; the latter are always or almost always relations." (1903:44)
The notion of a "variable" subject appearing in a proposition: "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)
In other words, a "term" can be place-holder that indicates (denotes) one or more things that can be put into the placeholder. (1903:45).
Truth and falsehood: Suppose Russell were to point to an object and utter: "This object in front of me named "Emily" is a woman." This is a proposition, an assertion of Russell's belief 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), then his utterance is considered "true".
If Russell were to utter a generalization about all Emilys then these object/s (entity/ies) must be examined, one after another in order to verify the truth of the generalization. Thus if Russell were to assert "All Emilys are women", then the "All" is a tipoff that the utterance is about all entities "Emily" in correspondence with a concept labeled "woman" and a methodical examination of all creatures with human names would have to commence.
Classes (aggregates, complexes): "The class, as opposed to the class-concept, is the sum or conjunction of all the terms which have the given predicate" (1903 p. 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. 66)
Propositional functions: "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)
Extensional versus intensional definition of a class: "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 natural numbers derive from ALL propositions (i.e. completely unrestricted) in this and all other possible worlds, that can be uttered about ANY collection of entities whatsoever. Russell makes this clear in the second (italicized) sentence:
To begin, devise a 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 Fn" applied 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 Fn".^{[15]}
To keep things simple all 26 letters of the alphabet are used up in this example, each letter representing the name of a particular child (in real life there could be repeats). Notice that, in the Russellian view these collections are not sets, but rather "aggregates" or "collections" or "classes"--listings of names that satisfy the predicates F1, F2, . . .. As noted in Step 1, for Russell, these "classes" are "symbolic fictions" that exist only as their aggregate members, i.e. as the extensions of their propositional functions, and not as unit-things in themselves.
Step 1: Assemble ALL the classes: Whereas the following example is finite over the very-finite propositional function " childnames of the children in family Fn'" on the very-finite street of a finite number of (12) families, Russell intended the following to extend to ALL propositional functions extending over an infinity of this and all other possible worlds; this would allow him to create ALL the numbers (to infinity).
Kleene observes that already Russell has set himself up with an impredicative definition that he will have to resolve, or otherwise he will be confronted with his 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 confronts the unit class (cf Russell 1903:517).
The matter of debate comes down to this: what exactly is a "class"? For Dedekind and Frege, a class is a distinct entity all of its own, a "unity" that can be identified with all those entities x that satisfy the propositional function F( ). (This symbolism appears in Russell, attributing it 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 F? has the children with the names Annie, Barbie and Charles:
This Dedekind-Frege construction could be symbolized by a bracketing process similar to, but to be distinguished from, the symbolism of contemporary set theory { a, b, c }, i.e. [ ] with the elements that satisfy the proposition separated by commas (an index to label each collection-as-a-unity will not be used, but could be):
This notion of collection-or 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" (First edition of Principia Mathematica 1927:24).
Russell does not waver from this opinion in his 1919; observe the words "symbolic fictions":
And by the second edition of PM (1927) Russell would insist 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, loose even that shadowy being which they retain in *20" (p. xxxix). In other words, classes as a separate notion have vanished altogether.
Given Russell's insistence that classes are not singular objects-in-themselves, but only collected aggregates, the only correct way to symbolize the above listing is to eliminate the brackets. But this is visually confusing, especially with regards to the null class, so a dashed vertical line at each end of the collection will be used to symbolize the collection-as-aggregate:
Step 2: Collect "similar" classes into bundles (equivalence classes): 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,^{[16]} 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).
Take for example ?h,i?. Its terms h, i cannot be put into one-one correspondence with the terms of ?a,b,c?,?d?,,?e,f,g?, etc. But it can be put in correspondence with itself and with ?j,k?,?q,r?, and ?t,u?. These similar collections can be assembled into a "bundle" (equivalence class) as shown below.
The bundles (equivalence classes) are shown below.
Step 3: Define the null-class: Notice that the third class-of-classes, ? , ? , is special because its classes contain no elements, i.e. no elements satisfy the predicates that created this particular class/collection. Example: the predicates are:
These particular predicates cannot be satisfied because families F_{?} and F_{?} are childless. There are no terms (names) that satisfy these particular predicates. Remarkably, the class of things, signified by the fictitious , that satisfy each of these this classes is not only empty, it does not exist at all (more or less, for Russell the agnostic-about-class-existence); for Dedekind-Frege it does exist.
This peculiar non-existent entity is nicknamed the "null class" or the "empty class". This is not the same as the class of all null classes ? ?: the class of all null classes is destined to become "0"; see below. Russell symbolized the null/empty class with ?. 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" . . . "? is the null-class" is equivalent to "? does not exist". One is left uneasy: Does the null class itself "exist"? This problem bedeviled Russell throughout his writing of 1903.^{[17]} After he discovered the paradox in Frege's Begriffsschrift 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.^{[18]}
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. (The symbol ? means "is an abbreviation for" or "is a definition of"):
Step 5: Define "0": In order to "order" the bundles into the familiar number-line a starting point traditionally called "zero", is required. Russell picked the empty or null class of classes to fill this role. This null class-of-classes ? ? has been labeled "0" ? ?^{[19]}
Step 6: Define the notion of "successor": Russell defined a new characteristic "hereditary", a property of certain classes with the ability to "inherit" a characteristic from another class (or 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 (in our example ? ?d?, ?s? ? ? ? ) . The reason is that, in Russell's detailed analysis,^{[20]} if a unit class ? becomes a 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 (confusingly): "We saw in Chapter II that a cardinal [natural] 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" (name, thing) which is "added into" the collections inside the bundles. To keep the example abstract this will be abbreviated by the name "Smiley" ? ? (on the assumption that no one has ever actually named their child "Smiley").
Step 7: Construct the successor of the null class: For example into the null class ? stick the smiley face. From the previous, it is not obvious how to do this. The predicate:
has to be modified to creating a predicate that contains a term that is always true:
In the case of the family with no children, "Smiley" is the only "term" that satisfies the predicate. Russell fretted over the use of the word *AND* here, as in "Barbie AND Smiley", and called this kind of AND (symbolized below with *&* ) a "numerical conjunction":^{[21]}
By the relation of similarity ?, this new class can be put into the equivalence class (the unit class) defined by ?:
Step 8: For every equivalence class, create its successor: Note that the smiley-face symbol must be inserted into every collection/class in a particular equivalence-class bundle, then by the relation of similarity ? each newly generated class-of-classes must be put into the equivalence class that defines n+1:
And in a similar manner, by use of the abbreviations set up above, for each numeral its successor is created:
Step 9: Order the numbers: The process of creating a successor requires the relation " . . . is the successor of . . .", call it "S", between the various "numerals", for example ? S 0, ? S ?, and so forth. "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)
The problem of presuming the "extralogical" notion of "iteration": Kleene points out 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, detects "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: 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) .
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).
Gödel 1944: With respect to the technical aspects of Russellian logicism as it appears in Principia Mathematic (either edition), Gödel is flat-out 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 formed these foundations, Gödel would home in on Russell's "no-class theory", or what Gödel would call his "nominalistic kind of constructivism, such as that embodied in Russell's "no class theory" . . . which might better be called fictionalism" (cf footnote 1 in Gödel 1944:119). See more in "Gödel's criticism and suggestions" below.
Grattan-Guinness: 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".^{[22]} By 1914 Hausdorff would provide another, equivalent definition, and Kuratowski in 1921 would provide the one in use today.^{[23]}
A benign impredicative definition: Suppose the local librarian wants to catalog (index) her collection into a single book (call it ? for "index"). Her index must list ALL the books and their locations in the library. As it turns out, there are only three books, and these have titles ?, ?, and ?. To form her index-book I, she goes out and buys a book of 200 blank pages and labels it "I". Now she has four books: I, ?, ?, and ?. Her task is not difficult. When completed, the contents of her index I is 4 pages, each with a unique title and unique location (each entry abbreviated as Title.Location_{T}):
This sort of definition of I was deemed by Poincaré to be "impredicative". He opined that only predicative definitions can be allowed in mathematics:
By Poincaré's definition, the librarian's index book is "impredicative" because the definition of I is dependent upon the definition of the totality I, ?, ?, and ?. As noted below, some commentators insist that impredicativity in commonsense versions is harmless, but as the examples show below there are versions which are not harmless. In the teeth of these, Russell would enunciate a strict prohibition--his "vicious circle principle":
A pernicious impredicativity = NOT-?: To create a pernicious paradox, apply input ? to the simple function box F(x) with output ? = 1 - ?. This is the algebraic-logic equivalent of the symbolic-logical ? = NOT-? for truth values 1 and 0 rather than "true" and "false". In either case, when input ? = 0, output ? = 1; when input ? = 1, output ? = 0.
To make the function "impredicative", wrap around output ? to input ?, i.e. identify (equate) the input with (to) the output (at either the output or input, it does not matter):
Algebraically the equation is satisfied only when ? = 0.5. But logically, when only "truth values" 0 and 1 are permitted, then the equality cannot be satisfied. To see what is happening, employ an illustrative crutch: assume (i) the starting value of ? = ?_{0} and (ii) observe the input-output propagation in discrete time-instants that proceed left to right in sequence across the page:
Start with ?_{0} = 0:
Observe that output ? oscillates between 0 and 1. If the "discrete time-instant" crutch (ii) is dropped, the function-box's output (and input) is both 1 and 0 simultaneously.
Fatal impredicativity in the definition of the unit class: The problem that bedeviled the logicists (and set theorists too, but with a different resolution) derives from the ? = NOT-? paradox^{[26]} Russell discovered in Frege's 1879 Begriffsschrift^{[27]} that Frege had allowed a function to derive its input "functional" (value of its variable) not only from an object (thing, term), but from the function's own output as well.^{[28]}
As described above, Both Frege's and Russell's construction of natural numbers begins with the formation of equinumerous classes-of-classes (bundles), then with an assignment of a unique "numeral" to each bundle, and then placing the bundles into an order via a relation S that is asymmetric: x S y ? y S x. But Frege, unlike Russell, allowed the class of unit classes (in the example above [[d]], [[s]]) to be identified as a unit itself:
But, since the class ? or 1 is a single object (unit) in its own right, it too must be included in the class-of-unit-classes as an additional class [?]. And this inclusion results in an "infinite regress" (as Godel called it) of increasing "type" and increasing content:
Russell would make this problem go away by declaring a class to be a "fiction" (more or less). By this he meant that the class would designate only the elements that satisfied the propositional function (e.g. d and s) and nothing else. As a "fiction" a class cannot be considered to be a thing: an entity, a "term", a singularity, a "unit". It is an assemblage e.g. d,s but it is not (in Russell's view) worthy of thing-hood:
This supposes that "at the bottom" every single solitary "term" can be listed (specified by a "predicative" predicate) for any class, for any class of classes, for class of classes of classes, etc, but it introduces a new problem--a hierarchy of "types" of classes.
Classes as non-objects, as useful fictions: Gödel 1944:131 observes that "Russell adduces two reasons against the extensional view of classes, namely the existence of (1) the null class, which cannot very well be a collection, and (2) the unit classes, which would have to be identical with their single elements." He suggests that Russell should have regarded these as fictitious, but not derive the further conclusion that all classes (such as the class-of-classes that define the numbers 2, 3, etc) are fictions.
But Russell did not do this. After a detailed analysis in Appendix A: The Logical and Arithmetical Doctrines of Frege in his 1903, Russell concludes:
In the following notice the wording "the class as many"--a class is an aggregate of those terms (things) that satisfy the propositional function, but a class is not a thing-in-itself:
It is as if Russell-as-rancher were to round up all his critters (sheep, cows and horses) into three fictitious corrals (one for the sheep, one for the cows, and one for the horses) that are located in his fictitious ranch. What actually exists are the sheep, the cows and the horses (the extensions), but not the fictitious "concepts" corrals and ranch.
Ramified theory of types: function-orders and argument-types, predicative functions: When Russell proclaimed all classes are useful fictions he solved the problem of the "unit" class, but the overall problem did not go away; rather, it arrived in a new form: "It will now be necessary to distinguish (1) terms, (2) classes, (3) classes of classes, and so on ad infinitum; we shall have to hold that no member of one set is a member of any other set, and that x ? u requires that x should be of a set of a degree lower by one than the set to which u belongs. Thus x ? x will become a meaningless proposition; and in this way the contradiction is avoided" (1903:517).
This is Russell's "doctrine of types". To guarantee that impredicative expressions such as x ? x can be treated in his logic, Russell proposed, as a kind of working hypothesis, that all such impredicative definitions have predicative definitions. This supposition requires the notions of function-"orders" and argument-"types". First, functions (and their classes-as-extensions, i.e. "matrices") are to be classified by their "order", where functions of individuals are of order 1, functions of functions (classes of classes) are of order 2, and so forth. Next, he defines the "type" of a function's arguments (the function's "inputs") to be their "range of significance", i.e. what are those inputs ? (individuals? classes? classes-of-classes? etc.) that, when plugged into f(x), yield a meaningful output ?. Note that this means that a "type" can be of mixed order, as the following example shows:
This sentence can be decomposed into two clauses: "x won the 1947 World Series" + "y won the 1947 World Series". The first sentence takes for x an individual "Joe DiMaggio" as its input, the other takes for y an aggregate "Yankees" as its input. Thus the composite-sentence has a (mixed) type of 2, mixed as to order (1 and 2).
By "predicative", Russell meant that the function must be of an order higher than the "type" of its variable(s). Thus a function (of order 2) that creates a class of classes can only entertain arguments for its variable(s) that are classes (type 1) and individuals (type 0), as these are lower types. Type 3 can only entertain types 2, 1 or 0, and so forth. But these types can be mixed (for example, for this sentence to be (sort of) true: " z won the 1947 World Series " could accept the individual (type 0) "Joe DiMaggio" and/or the names of his other teammates, and it could accept the class (type 1) of individual players "The Yankees".
The axiom of reducibility: The axiom of reducibility is the hypothesis that any function of any order can be reduced to (or replaced by) an equivalent predicative function of the appropriate order.^{[29]} A careful reading of the first edition indicates that an n^{th} order predicative function need not be expressed "all the way down" as a huge "matrix" or aggregate of individual atomic propositions. "For in practice only the relative types of variables are relevant; thus the lowest type occurring in a given context may be called that of individuals" (p. 161). But the axiom of reducibility proposes that in theory a reduction "all the way down" is possible.
Russell 1927 abandons the axiom of reducibility, and the edifice collapses: By the 2nd edition of PM of 1927, though, Russell had given up on the axiom of reducibility and concluded he would indeed force any order of function "all the way down" to its elementary propositions, linked together with logical operators:
(The "stroke" is Sheffer's inconvenient logical NAND that Russell adopted for the 2nd edition--a single logical function that replaces logical OR and logical NOT).
The net result, though, was a collapse of his theory. Russell arrived at this disheartening conclusion: that "the theory of ordinals and cardinals survives . . . but irrationals, and real numbers generally, can no longer be adequately dealt with. . . .Perhaps some further axiom, less objectionable than the axiom of reducibility, might give these results, but we have not succeeded in finding such an axiom." (PM 1927:xiv).
Gödel 1944 agrees that Russell's logicist project was stymied; he seems to disagree that even the integers survived:
Gödel asserts, however, that this procedure seems to presuppose arithmetic in some form or other (p. 134). He deduces that "one obtains integers of different orders" (p. 134-135); the proof in Russell 1927 PM Appendix B that "the integers of any order higher than 5 are the same as those of order 5" is "not conclusive" and "the question whether (or to what extent) the theory of integers can be obtained on the basis of the ramified hierarchy [classes plus types] must be considered as unsolved at the present time". Gödel concluded that it wouldn't matter anyway because propositional functions of order n (any n) must be described by finite combinations of symbols (cf all quotes and content derived from page 135).
Gödel in his 1944 bores down to the exact place where Russell's logicism fails and offers a few suggestions to rectify the problems. He submits the "vicious circle principle" to reexamination, tearing it apart into three phrases "definable only in terms of", "involving" and "presupposing". It is the first clause that "makes impredicative definitions impossible and thereby destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a good deal of mathematics itself". Since, he argues, mathematics is doing quite well, thank you, with its various inherent impredicativities (e.g. "real numbers defined by reference to all real numbers"), he concludes that what he has offered is "a proof that the vicious circle principle is false [rather] than that classical mathematics is false" (all quotes Gödel 1944:127).
Russell's no-class theory is the root of the problem: Gödel believes that impredicativity is not "absurd", as it appears throughout mathematics. Where Russell's problem derives from is the "constructivistic (or nominalistic^{[30]}) standpoint toward the objects of logic and mathematics, in particular toward propositions, classes, and notions . . . a notion being a symbol . . . so that a separate object denoted by the symbol appears as a mere fiction" (p. 128).
Indeed, this "no class" theory of Russell, Gödel concludes:
He concludes his essay with the following suggestions and observations:
Neo-logicism describes a range of views claiming to be the successor of the original logicist program.^{[31]} More narrowly, it is defined as attempts to resurrect Frege's programme through the use of second order logic as applied to Hume's Principle.^{[32]} This kind of neo-logicism is often referred to as neo-Fregeanism. Two of the major proponents of neo-logicism are Crispin Wright and Bob Hale.^{[33]}