Introduction to Mathematical Philosophy — Chapter-by-Chapter Outline

Author: Bertrand Russell First published: 1919 (George Allen & Unwin, London; The Macmillan Co., New York) Edition covered: Routledge Classics, 1993 (ISBN 0-415-09604-9), with a new introduction by John Slater; text identical to the second edition (1920, George Allen & Unwin) as reprinted by Routledge and by Dover (1993). The book was written in 1918 while Russell served five months in Brixton Prison for anti-war agitation. Chapter titles, order, and pagination verified against the 1919 first-edition scan (Cornell University Library, Internet Archive) and the full-text online edition prepared by Kevin C. Klement (UMass).

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Central thesis

Mathematics is not a mysterious body of truths requiring a special faculty of intuition; it is logic itself, developed beyond the point where the working logician still recognizes his own tools. Russell's project is to walk "backwards" through mathematics — from the familiar series of natural numbers to the more and more abstract logical notions from which they can be defined and deduced — and to show that at the end of that journey mathematics and logic turn out to be one and the same subject. The book is at once an accessible summary of the logicist program of Principia Mathematica and a statement of what remains genuinely uncertain at the frontier of knowledge: whether the axiom of infinity is true, whether the multiplicative axiom (the axiom of choice) holds, and what exactly the doctrine of logical types amounts to.

The argument proceeds in three movements. The first movement (chapters 1–9) reduces the number systems — natural, cardinal, ordinal, rational, real, complex, and transfinite — to classes, relations, and order. The second movement (chapters 10–13) shows that the great concepts of analysis — limit, continuity, infinity — are purely logical and ordinal notions, and confronts the two great unprovable assumptions: the axiom of infinity and the multiplicative axiom, together with the paradoxes that force the theory of types. The third movement (chapters 14–18) supplies the missing foundations — deduction, propositional functions, descriptions, classes — and concludes that the definitions and deductions of mathematics never cross a boundary into a non-logical realm.

We are thus brought face to face with the question: What is this subject, which may be called indifferently either mathematics or logic?

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Chapter 1 — The Series of Natural Numbers

Central question

What are the logical foundations of arithmetic, and can the entire theory of the natural numbers be reduced to a handful of primitive ideas and propositions?

Main argument

Two directions of mathematics Russell opens by distinguishing two ways of pursuing mathematics from its most familiar starting points. The familiar direction is constructive: from integers to fractions, real and complex numbers, from addition to the calculus — toward increasing complexity. The other direction is analytic: it proceeds "by analysing, to greater and greater abstractness and logical simplicity," asking what more general ideas can be found in terms of which the starting point can be defined and deduced. Mathematical philosophy is defined by pursuing this backward direction, and the distinction is one of interest and stage, not of subject matter: early Greek geometers were doing mathematical philosophy when they moved from surveying rules to Euclid's axioms.

Arithmetization and Peano's analysis Russell notes that all traditional pure mathematics, including analytical geometry, may be regarded as consisting wholly of propositions about the natural numbers, which are in turn defined and deduced from the smallest possible set of premisses. This reduction was accomplished by Peano, who showed that the entire theory of the natural numbers could be derived from three primitive ideas and five primitive propositions, in addition to those of pure logic:

• Primitive ideas: 0, number, successor.
• Primitive propositions:
  1. 0 is a number.
  2. The successor of any number is a number.
  3. No two numbers have the same successor.
  4. 0 is not the successor of any number.
  5. Any property which belongs to 0, and also to the successor of any number which has the property, belongs to all numbers. (This is the principle of mathematical induction.)

From these, 1 is defined as the successor of 0, 2 as the successor of 1, and so on; addition is defined recursively by m + 0 = m and m + (n + 1) = successor of (m + n), and multiplication similarly.

Why Peano's system is not final Russell then gives the decisive argument for going beyond Peano to Frege: the three primitive ideas admit of infinitely many different interpretations that all satisfy the five propositions. "0" might mean 100 (taking "numbers" to be the numbers from 100 onward); "number" might mean the even numbers with "successor" meaning "add 2"; "0" might mean 1 with "successor" meaning "half." In general, any endless series with no repetitions, a beginning, and every term reachable from the beginning in finitely many steps — a progression — verifies the Peano axioms, and conversely. So the axioms cannot distinguish the intended numbers from Cleopatra's Needle; they define the class of progressions, not the numbers. Russell wants numbers that "apply in the right way to common objects — we want to have ten fingers and two eyes and one nose" — and that requires a definite logical definition, which the logical theory of arithmetic provides.

Key ideas

• Mathematics can be pursued constructively forward or analytically backward; mathematical philosophy is the backward journey.
• All traditional pure mathematics can be reduced to propositions about natural numbers (the "arithmetisation of mathematics," blocked for centuries by the discovery of incommensurables such as √2).
• Peano reduced arithmetic to three primitive ideas and five primitive propositions, with mathematical induction as the fifth.
• Addition and multiplication are definable by recursion from these primitives.
• Peano's primitives are not categorical: infinitely many interpretations ("0" = 100, "successor" = half, etc.) satisfy the axioms.
• Every progression verifies Peano's axioms, and every series verifying them is a progression.
• The logical (Fregean) theory of numbers is needed to give "0", "number", and "successor" definite meanings; the goal of mathematical philosophy is to postpone undefined terms "as long as possible."

Key takeaway

Peano's axioms capture the formal structure of the natural numbers but leave "0", "number", and "successor" undefined — the genuinely logical foundation, which gives these terms a definite meaning, is Frege's, and supplying it is the task of the next two chapters.

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Chapter 2 — Definition of Number

Central question

What is a number, and how can "the number of a collection" be defined without presupposing counting or a prior notion of number?

Main argument

Frege's answer Russell reports that the question "What is a number?" was correctly answered by Frege in 1884 in Grundlagen der Arithmetik, a short book that attracted almost no attention until Russell rediscovered the definition in 1901. First comes the "grammar of the inquiry": number is what is characteristic of numbers, as man is what is characteristic of men. A plurality is not an instance of number but of some particular number — a trio of men is an instance of 3, and 3 is an instance of number, but the trio is not. The number 3 is something all trios have in common.

Classes, extension, and intension A class may be defined by enumeration (definition by extension) or by a defining property (definition by intension). Definition by intension is logically more fundamental: an extensional definition can always be reduced to an intensional one, while infinite classes cannot even theoretically be enumerated — "at some point we must content ourselves with 'and so on.'"

One-one relations and similarity The decisive logical tool is the one-one relation: a relation is one-one when, if x has it to y, no other term x' has it to y and x does not have it to any y' other than y. Russell's illustration: in a world without polygamy or polyandry, the number of husbands equals the number of wives — we know this without a census, because marriage correlates the two classes term for term. Two classes are similar when there is a one-one relation whose domain is the one class and whose converse domain is the other. Similarity is reflexive, symmetrical, and transitive. Crucially, counting presupposes similarity (to count ten objects is to correlate them with 1–10), so the notion of similarity is logically simpler than counting, applies without an order being imposed, and applies to infinite classes.

The definition Numbers are then bundles of similar classes: the number of a class is the class of all classes similar to it. Hence:

The number of a class is the class of all those classes that are similar to it. Thus the number of a couple will be the class of all couples. In fact, the class of all couples will be the number 2.

And in general: a number is anything which is the number of some class. The definition looks verbally circular but is not: "the number of a given class" is defined first, without using "number" in general. This is the method of defining the class of fathers by first defining what it is to be the father of somebody. Russell concedes the definition has "a certain air of paradox" — the class of couples is indubitable, while a metaphysical number 2 "about which we can never feel sure that it exists" is not — and argues that definiteness and indubitableness are bought at the expense of a little oddity.

Key ideas

• Number is what is characteristic of numbers; a plurality is an instance of a particular number, not of number.
• Extension versus intension: intensional definition is logically fundamental and the only route to infinite collections.
• One-one, one-many, and many-one relations, with the marriage relation as the paradigm of one-one correlation.
• Two classes are similar when a one-one relation correlates them term for term; similarity is reflexive, symmetrical, and transitive.
• Counting presupposes similarity; order is an irrelevant addition to the notion of number.
• The number of a class is the class of all classes similar to it; so 2 = the class of all couples, and a number is anything that is the number of some class.
• Numbers so defined are classes of classes, defined intensionally, and the definition extends to infinite collections.

Key takeaway

A number is a class of classes: 2 just is the class of all couples, and the definition works for finite and infinite collections alike without presupposing counting.

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Chapter 3 — Finitude and Mathematical Induction

Central question

What is a finite number, and how can mathematical induction be used to define the natural numbers without circularity?

Main argument

Replacing "and so on" Given "0" and "successor" alone, any assigned number such as 30,000 can be reached step by step, but the general claim that all numbers can be reached this way needs something more precise than "and so on." The natural proposal — "the process may be repeated any finite number of times" — is circular, because defining "finite" is exactly the problem at hand. The key to the solution is mathematical induction, previously a principle (Peano's fifth proposition) and now to be adopted as a definition.

Hereditary properties and posterity A property is hereditary in the natural-number series when, if it belongs to a number n, it belongs to n + 1. A property is inductive when it is hereditary and belongs to 0. The posterity of a given number, with respect to the relation of immediate predecessor (the converse of successor), is the class of all terms that belong to every hereditary class to which the given number belongs. The posterity of 0 is the precise substitute for the vague idea of "the numbers reachable from 0 by successive steps." This yields the definition:

The "natural numbers" are the posterity of 0 with respect to the relation "immediate predecessor."

Two of Peano's primitive propositions — that 0 is a number and mathematical induction itself — become consequences of this definition rather than assumptions.

Defining 0 and successor The number 0 is the number of terms of the null-class: since the null-class is similar only to itself, 0 is the class whose only member is the null-class. The successor of the number of terms in a class a is the number of terms in the class consisting of a together with any term x not belonging to a. (A class with one member is never identical with that member — a point the theory of classes will explain.)

The definition of finitude Finite numbers are then defined as those that obey mathematical induction starting from 0 — the inductive numbers. "A finite class or cardinal is one which is inductive." This definition yields the ordinary number series and, Russell argues, is what the intuitive notion of finiteness means; the stage is set for asking what happens to collections that do not have an inductive number of terms.

Key ideas

• The "and so on" of the number series must be replaced by a precise definition; "any finite number of times" would be circular.
• Hereditary properties and inductive properties (hereditary and holding of 0) give the mechanism.
• The posterity of a number is the class of terms belonging to every hereditary class containing it.
• The natural numbers are the posterity of 0 under immediate predecessor; induction and "0 is a number" follow from the definition.
• 0 is the class whose only member is the null-class; successor is defined by adding a fresh term to a class.
• Finite (inductive) numbers are those obeying induction from 0; the definition is designed to leave room for non-inductive numbers.

Key takeaway

The natural numbers can be defined purely logically from 0, successor, and the ancestral relation, and a number is finite just when it is reachable from 0 by mathematical induction.

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Chapter 4 — The Definition of Order

Central question

What must a relation be like to generate an order, and in what sense is a series the same thing as a relation?

Main argument

Order is not in the terms The first point is that no set of terms has just one order: "A set of terms has all the orders of which it is capable." The natural numbers can be arranged odds-first or by primes; the order lies not in the class but in a relation among its members. We can no more "arrange" the natural numbers than the starry heavens — what we do is attend to relations that generate arrangements. Hence the definition of order must be sought in relations, not in the nature of the terms to be ordered.

The three properties of serial relations A relation gives rise to order when, for any two terms of the class, one "precedes" and the other "follows." This requires three properties:

1. Asymmetry — if x precedes y, y does not precede x (as with less, earlier, left of; unlike sibling or spouse).
2. Transitivity — if x precedes y and y precedes z, x precedes z (as with ancestor; unlike father).
3. Connexity — of any two terms in the field, one precedes the other (as with integers and moments of time; unlike complex numbers or events).

The three properties are mutually independent: any two can hold without the third. A relation is an aliorelative (a term due to C. S. Peirce) when no term has it to itself; the square of a relation is the relation between x and z when an intermediate y exists; the field of a relation is its domain and converse domain together. The definitions then come together:

A relation is serial when it is an aliorelative, transitive, and connected; or, what is equivalent, when it is asymmetrical, transitive, and connected. A series is the same thing as a serial relation.

A series is not its field: 1, 2, 3 arranged in its six permutations are six different series with one field. Given the ordering relation, the field and the order are determinate. For the natural numbers, "less than" is defined as: m is less than n when n possesses every hereditary property possessed by the successor of m.

Generation of series Series can be generated from relations of consecutiveness, as with the Kings of England (each to his successor), using the ancestral relation — the proper posterity and proper ancestor of a term with respect to a relation R. The round dinner-table of twelve people shows the failure condition: the ancestral relation is connected but not an aliorelative, so no series results (a cyclic order). This method suits the finite; series like the fractions in order of magnitude have no consecutive terms, and only a transitive relation can "leap over" infinitely many intermediate terms. Alternatively, series can be generated from a three-term "between" relation (with seven formal properties), which defines the left-to-right order of points on a line and can be applied to any three-term relation with those purely formal properties. Cyclic order, as on a circle, requires instead a four-term relation of "separation of couples."

Key ideas

• A set of terms has all the orders of which it is capable; order is a matter of relations, not of terms.
• A serial relation is asymmetrical, transitive, and connected; the three properties are mutually independent.
• A series is the same thing as a serial relation; the field is not the series.
• "Less than" among inductive numbers is definable through hereditary properties of the successor.
• Series can be generated from consecutiveness via ancestral relations (proper posterity), subject to the aliorelative condition.
• The "between" relation generates open series; "separation of couples" generates cyclic order.
• Consecutiveness, like counting, is a method for the finite; dense series require transitive relations.

Key takeaway

Order is not intrinsic to terms: a series simply is an asymmetrical, transitive, connected relation over them, and the generation of series from more elementary relations is a general and important subject.

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Chapter 5 — Kinds of Relations

Central question

Which kinds of relations — asymmetrical, one-many, one-one — carry the weight of mathematics, and why?

Main argument

Asymmetry as the most relational property Russell distinguishes the properties that enter into serial relations and argues that asymmetry is the most fundamental. A symmetrical relation (spouse, "same height as") can sometimes be split into asymmetrical ones (husband, wife) when the terms fall into two mutually exclusive classes — but in the important cases (greater, before, to the right of) the domain and converse domain overlap and no such split is possible. More generally: so long as relations are symmetrical, one can formally replace relational propositions by predications (asserting a common or an incompatible predicate); the moment a relation is asymmetrical this is impossible, because both sameness and difference of predicates are symmetrical. "Asymmetrical relations are, we may say, the most characteristically relational of relations" — the point at stake in Russell's rejection of the doctrine of internal relations.

One-many relations and descriptive functions A one-many relation is one to which at most one term has a given term (father, mother, sine of, square of). Phrases of the form "the so-and-so of such-and-such" — "the King of England," "the wife of Socrates," "the father of John Stuart Mill" — describe a term by means of a one-many relation to a given term. All mathematical functions result from one-many relations: "the father of x" is as legitimately a function of x as "the logarithm of x." Such functions are descriptive functions — "the R of x" — which exist when x belongs to the converse domain of the one-many relation R; x is the argument, the term picked out is the value, and the domain of R is the range of values. Formal tricks can reduce any relation to a one-many relation via classes (the "proper ancestry" of n), but classes are logical fictions, so the reduction is no philosophical analysis.

One-one relations and correlation One-one relations are one-many relations whose converses are also one-many; equivalently, the relative product of the relation and its converse implies identity — an R-step followed by a backward R-step brings you back to your starting point. One-one relations correlate two classes term for term: the term from which the relation goes is the referent, the term to which it goes the relatum, and a relation and its converse have opposite senses — the fact that a relation has a sense is part of why order can be generated. Correlations of the inductive numbers with themselves (n → n+1, n → 2n, n → n²) show a one-one relation whose converse domain is a proper part of its domain — a "reflection" of the class into a part of itself, which will occupy the theory of infinity. Permutations of a class (as with three letters) form a group.

Key ideas

• Asymmetry is the most fundamental relational property; symmetrical relations can be replaced by predicates, asymmetrical ones cannot.
• One-many relations underlie all "the so-and-so of such-and-such" phrases and all mathematical functions.
• Descriptive functions ("the R of x") exist exactly when the argument belongs to the converse domain.
• One-one relations are characterized by the relative product with their converse implying identity.
• Relations have a sense; referent and relatum are the two ends of a relation.
• A one-one correlation can "reflect" a class into a proper part of itself — the seed of the theory of infinity.
• Permutations of a class form a group.

Key takeaway

The three great classes of relations — asymmetrical, one-many, one-one — supply respectively order, functions, and correlation, and the last of these can already reflect a class into a part of itself.

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Chapter 6 — Similarity of Relations

Central question

When do two relations have the same structure, and what is a relation-number?

Main argument

Likeness and correlators Just as classes have the same cardinal number when similar, relations can be compared by a new relation, likeness (similarity of relations). The guiding example is a map: when one place is north of another, the corresponding map-place is above it; the structure of the map corresponds to the structure of the country. Formally, a relation S is a correlator of two relations P and Q when S is one-one, has the field of Q for its converse domain, and P is the relative product of S, Q, and the converse of S. Two relations are similar (have likeness) when at least one correlator exists. Similar relations share every property not depending on their actual terms: aliorelation, transitivity, connexity, seriality, one-manyness, and so on.

Relation-numbers The set of all relations similar to a given relation is its relation-number; cardinal numbers are then the special numbers of classes. Applying relation-numbers to series gives serial numbers; what are commonly called ordinal numbers are a sub-class of serial numbers. For finite series the serial number is determined by the cardinal number of terms, but for infinite series it is not: "the length of a series, its relation-number, may vary without change in the cardinal number." An arithmetic of relation-numbers can be developed (sum, product, powers), which does not obey the commutative law but obeys the associative and distributive laws in one form.

Structure The extension of a relation is the class of ordered couples (x, y) such that x has the relation to y; a map of a relation reveals its structure. Two relations have the same structure when the same map does for both — which is exactly likeness, i.e., the same relation-number. "What we defined as the 'relation-number' is the very same thing as is obscurely intended by the word 'structure.'" Since likeness is definable for three-term relations as well, the whole of a geometry can be characterized by the formal properties of its between-relation (Veblen's work is cited): the mathematician need not care what points are, only how they are interrelated. And in general, "what matters in mathematics, and to a very great extent in physical science, is not the intrinsic nature of our terms, but the logical nature of their interrelations." Russell draws the philosophical moral: we know the form of nature much better than its matter — if phenomena have a world behind them, that world has the same structure, and every communicable proposition is true of both worlds or of neither; the "essence of individuality" is irrelevant to science.

Key ideas

• Likeness of relations is defined via one-one correlators that preserve holding between correlates.
• The relation-number of a relation is the class of all relations similar to it; cardinal numbers are the numbers of classes.
• Ordinal numbers are serial numbers, a sub-class of relation-numbers; infinite series can change their serial number without changing their cardinal.
• Relation-arithmetic fails commutativity but keeps associativity and one distributive law.
• Structure = likeness = relation-number; a map of a relation displays its structure.
• Geometry is indifferent to what points are — only the formal properties of relations matter.
• Science and philosophy know the form of nature better than its matter; a "real" world behind phenomena would share its structure.

Key takeaway

What mathematics studies is structure — the relation-number — and two relations with the same structure are mathematically indistinguishable, whatever their terms are.

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Chapter 7 — Rational, Real, and Complex Numbers

Central question

How are fractions, irrationals, and complex numbers to be logically defined, given that each extension of number constructs new objects rather than enlarging the old ones?

Main argument

The error of "special cases" The common idea that each extension of number includes the previous sorts as special cases delayed correct definitions: +1 and −1 are not the integers 1 and −1; a fraction with denominator 1 is not the integer of that name; real numbers are not a sub-class of the rationals plus limits; real numbers are not complex numbers with zero imaginary part. Each extension demands genuinely new logical objects.

Positive and negative integers; fractions +m is the relation of n+m to n, and −m its converse — relations, each as distinct from the cardinal m as from each other. The fraction m/n is the relation holding between x and y when xn = ym, a one-one relation; n/m is its converse. The fraction m/1 is not the cardinal m. The ratio o/n is the "zero of rationals" and m/0 the "infinity of rationals" — the traditional mathematical infinite, "a totally different sort from the true Cantorian infinite." Ratios in order of magnitude form a compact series: between any two fractions lies (m+p)/(n+q), so no two are consecutive, and there are infinitely many between any two. (The claim involves the axiom of infinity, to be discussed later.)

Irrationals and the Dedekind cut No fraction squares to 2 (Russell reproduces Euclid's proof: if m²/n² = 2, then m and n are both even, contradicting lowest terms). Dividing all ratios into those whose square is less than 2 and those whose square is greater gives a Dedekind cut with no maximum in the lower section and no minimum in the upper — a gap. Of the four possible kinds of section, the fourth — an "irrational section" — is the case at hand. What delayed the true theory was the mistaken belief that series must have limits; Dedekind's axiom that every gap must be filled is rejected with Russell's famous remark:

The method of "postulating" what we want has many advantages; they are the same as the advantages of theft over honest toil. Let us leave them to others and proceed with our honest toil.

The honest construction: confine attention to sections whose lower section has no maximum — segments — and define:

• A real number is a segment of the series of ratios in order of magnitude.
• An irrational number is a segment with no boundary (e.g., √2 is the segment consisting of all ratios whose square is less than 2).
• A rational real number is a segment with a boundary (the real number 1 is the class of proper fractions).

The series of segments of any series is Dedekindian (the boundary of a set of segments is their logical sum). Addition and multiplication of reals are defined by forming classes of sums and products of members. Construction "requires no new assumptions, but enables us to proceed deductively from the original apparatus of logic."

Complex numbers Complex numbers are not demanded by geometry in the same imperative way as irrationals but are needed so that every equation has roots (x² + 1 = 0). A complex number is defined as an ordered couple of real numbers, with rules: (x, y) + (x', y') = (x+x', y+y'), and (x, y)(x', y') = (xx' − yy', xy' + x'y). Then the couple (0, 1) is i, and its square is (−1, 0), i.e., −1: the "square root of −1" is constructed, not postulated. Complex numbers of order n generalize the idea via one-many relations with the integers 1 to n as suffixes.

Key ideas

• Each extension of number constructs new logical objects; the "special case" identifications (+1 with 1, m/1 with m, reals inside complexes) are all errors.
• +m and −m are converse relations; the fraction m/n is the relation xn = ym.
• The infinity of rationals (m/0) is not the Cantorian infinite.
• Ratios form a compact series with no consecutive terms.
• √2 is not a ratio; Dedekind cuts with gaps demand construction, not postulation.
• A real number is a segment of the series of ratios; an irrational is a boundaryless segment.
• Complex numbers are ordered couples of reals, with i = (0, 1) and i² = −1.
• The series of segments of any series is Dedekindian, so the reals have no gaps.

Key takeaway

The extensions of number are logical constructions — fractions are relations, irrationals are segments of ratios, and the "imaginary" number i is just the ordered couple (0, 1) — so no new assumptions are needed at any step.

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Chapter 8 — Infinite Cardinal Numbers

Central question

Can there be numbers beyond the finite, and what arithmetic do they obey?

Main argument

A new kind of number The theory of transfinite cardinal numbers was chiefly created by Cantor and is combined here with Frege's logical theory of numbers. Russell first notes the axiom of infinity in its practical form: if n is any inductive number, n is not equal to n + 1 (already involved in Peano's assumptions). The class of inductive numbers itself then has a number that is not an inductive number: however far the series goes, its number exceeds any n. "The first step in understanding infinite numbers consists in realising the mistakenness" of the view that inductive properties must belong to all numbers.

Reflexive classes The most astonishing difference between an inductive number and this new number is that the new number is unchanged by adding 1, subtracting 1, doubling, or halving. A class is reflexive when it is similar to a proper part of itself — the relation n → n+1 reflects the inductive numbers into themselves minus 0, and n → 2n reflects them into the evens. Royce's map of England drawn on England is a "reflexion" of the whole into a part, ad infinitum. Leibniz and others took the part-equal-to-whole result to prove infinite numbers impossible; Russell replies that "the word 'equal' has many meanings," and with "equal" read as "similar" there is no contradiction.

ℵ0 and its arithmetic A progression is a one-one relation with just one term in the domain but not the converse domain, the domain being the posterity of that term — it satisfies Peano's five axioms. The domains of progressions form a cardinal number, the smallest of the infinite cardinals:

The name of the smallest of infinite cardinals is ℵ₀.

Any selection from a progression with no last term is a progression, however sparsely distributed (numbers of the form n^n). Conversely, ratios are countable: arranging them by the sum of numerator and denominator gives a progression, so there are exactly ℵ₀ ratios. But not all infinite collections have ℵ₀ terms. Russell proves Cantor's theorem: the number of sub-classes of a class is always greater than the number of its members. Given any one-one correlation of the members with some sub-classes, form the class β of members correlated with sub-classes they do not belong to; β is not correlated with any member, so the correlation is never complete. Hence 2^n > n even for infinite n, and:

• ℵ₀ + n = ℵ₀, ℵ₀² = ℵ₀, ℵ₀^n = ℵ₀, but 2^ℵ₀ > ℵ₀.
• There is no maximum cardinal: however great n, 2^n is greater.
• 2^ℵ₀ is the number of terms in a Cantorian-continuous series — the number of points in space or instants in time, if space and time are continuous.
• Subtraction and division become ambiguous: ℵ₀ − ℵ₀ may give anything from 0 to ℵ₀ (removing all, some, or the odds), and similarly division; negative numbers and ratios cannot be extended to infinite numbers.

A class or cardinal is finite when inductive, infinite when not; every reflexive class is infinite, but whether every infinite class is reflexive is unknown — it depends on the multiplicative axiom (Chapter XII).

Key ideas

• The number of the inductive numbers is a new number, not inductive; inductive properties cannot be assumed for all numbers.
• A reflexive class is similar to a proper part of itself; reflection is no contradiction once "equal" means "similar."
• Progressions are the series that verify Peano's axioms; their cardinal is ℵ₀, the smallest infinite cardinal.
• ℵ₀ + n = ℵ₀, ℵ₀² = ℵ₀, ℵ₀^n = ℵ₀; ratios are countable.
• Cantor's theorem: 2^n > n even when n is infinite; there is no greatest cardinal.
• 2^ℵ₀ is the cardinal of the continuum (points of space, instants of time).
• Subtraction and division are ambiguous for infinite cardinals; negatives and ratios do not extend to them.
• Finite = inductive; whether infinite = reflexive is an open question.

Key takeaway

Cantor's discovery that the part can be similar to the whole opens the realm of transfinite cardinals, where ℵ₀ + 1 = ℵ₀ but 2^ℵ₀ exceeds ℵ₀, and there is no largest infinite number.

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Chapter 9 — Infinite Series and Ordinals

Central question

How do infinite series behave under rearrangement, and what are the ordinal numbers?

Main argument

Rearrangement changes the serial number An infinite series is a series whose field is infinite. Its most noteworthy feature is that its serial number can be altered by mere rearrangement, whereas a reflexive class keeps its cardinal number when terms are added. Thinning out the progression 1, 2, 3, … by repeatedly moving the first even number to the end gives series with serial numbers ω + 1, ω + 2, …, and finally odds-then-evens, of number 2ω. One serial number is greater than another when any series of the first contains a part of the second, but no series of the second contains a part of the first. Hence ω + 1 > ω, yet adding a term at the beginning of a progression leaves a progression, so 1 + ω = ω while ω + 1 ≠ ω — addition of relation-numbers is not commutative. By further thinning we reach ω², ω³, …, ω^ω, and beyond.

Well-ordered series and ordinals The series so obtained are all well-ordered: a well-ordered series is one in which every sub-class (except the null-class) has a first term. An ordinal number is the relation-number of a well-ordered series — a species of serial number. In well-ordered series a generalized form of mathematical induction, transfinite induction, applies: a transfinitely hereditary property belonging to the first term belongs to the whole series. The series of all ordinals obtainable by thinning a progression is longer than any rearrangement of a progression; its ordinal is ω₁ and its cardinal ℵ₁, and the process continues to ω₂, ℵ₂, and beyond. Whether 2^ℵ₀ equals any of the alephs is unknown — a question connected with the multiplicative axiom.

The formal laws Compact series with ℵ₀ terms exist (arranging integers by a decimal trick gives a compact series of all integers not divisible by 10). Of the formal laws: all hold for transfinite cardinals and finite ordinals; for transfinite ordinals the commutative law fails, the associative law holds, the distributive law holds in one form ((β + γ)α = βα + γα) but not the other, and the exponential laws hold except (αβ)^ν = α^ν β^ν. Russell notes that ordinal transfinite arithmetic was developed by Cantor before cardinal transfinite arithmetic for technical reasons, but cardinals are logically the more fundamental notion.

Key ideas

• Infinite series can change their serial number by rearrangement, unlike cardinals under addition.
• ω + 1 > ω, but 1 + ω = ω: ordinal addition is not commutative.
• ω, 2ω, ω², ω^ω arise by thinning progressions; their series is longer than any rearrangement of a progression.
• A well-ordered series is one in which every sub-class has a first term.
• An ordinal number is the relation-number of a well-ordered series; transfinite induction applies to well-ordered series.
• ω₁ has cardinal ℵ₁, and the aleph hierarchy continues indefinitely.
• The position of 2^ℵ₀ among the alephs is unknown.
• Commutativity fails for transfinite ordinals; associativity and one distributive form survive.

Key takeaway

Infinite series have lengths — ordinals — that depend on arrangement, and since 1 + ω ≠ ω + 1, the arithmetic of ordinals escapes the commutative law entirely.

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Chapter 10 — Limits and Continuity

Central question

What is a limit, and what exactly is meant by the continuity of a series?

Main argument

Limit as an ordinal notion The conception of a limit underlies all higher mathematics; formerly infinitesimals were thought to be involved, but Weierstrass showed that where infinitesimals were supposed to occur, what really occurs is "a set of finite quantities having zero for their lower limit." The notion of limit is not quantitative but purely ordinal: ℵ₀ is the limit of the finite cardinals in order of magnitude, though from a quantitative standpoint finite numbers get no nearer to it — what makes it their limit is that it comes immediately after them in the series.

Definitions Russell lays out the machinery: the minima of a class with respect to a relation P, its maxima (minima of the converse), its sequents (minima of the successors) and precedents, its upper limits (sequents, provided the class has no maximum) and lower limits. In a series, the boundary of a class is the term whose predecessors are the segment defined by the class; a maximum is a boundary belonging to the class, an upper limit a boundary not belonging to it; a class with no boundary has a "gap." Upper limiting-points of a set are the upper limits of sub-sets of it; the limiting-points form the first derivative, and so on.

Grades of continuity The word "continuity" lacked a precise definition until Dedekind and Cantor. Compactness is inadequate (the series of ratios has gaps — two lines might cross in a gap and have no point in common). A series is Dedekindian when every sub-class has a boundary, i.e., there are no gaps; Dedekindian continuity is Dedekindianness plus compactness. But this is still too wide for the purposes of coordinate geometry, which needs every point specifiable as a limit of rational points. Cantor's route: a series is condensed in itself (insichdicht) when every term is the limit of a progression or regression; closed (abgeschlossen) when every progression or regression contained in it has a limit in it; perfect when both hold. Adding a median class — a sub-class with members between any two terms — of ℵ₀ members yields the definition:

A series is "continuous" when (i) it is Dedekindian, (ii) it contains a median class having ℵ₀ terms.

All series with Cantorian continuity are similar (they form one serial number), which is not true of Dedekindian continuity. The chapter closes by contrasting this precise notion with the metaphysician's "continuity" — the absence of distinctness of a thick fog — and asserting that the mathematical conception gives the abstract logical scheme to which empirical material must be brought if it is to be called continuous.

Key ideas

• Limit is a purely ordinal notion; Weierstrass showed the calculus needs no infinitesimals.
• Maxima, sequents, boundaries, and upper limits are definable for any serial relation.
• Upper limiting-points and derivatives structure a series internally.
• Compactness is insufficient for continuity; Dedekindian series are those without gaps.
• Dedekindian continuity = Dedekindian + compact.
• Cantor's perfect series (condensed in itself and closed) plus a countable median class defines Cantorian continuity.
• All Cantorian-continuous series are similar; their cardinal is 2^ℵ₀.
• Continuity of series (space, time) is distinct from continuity of functions (motion).

Key takeaway

Continuity is a precise structural property — a series is continuous when it is gap-free (Dedekindian) and contains a countable median class, as the rationals do in the reals.

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Chapter 11 — Limits and Continuity of Functions

Central question

What is the limit of a function at an argument, and when is a function continuous?

Main argument

The calculus without infinitesimals Wrong views about the calculus have become "firmly embedded in the minds of professional philosophers" since Leibniz; Weierstrass proved that no infinitesimals are involved, but errors "incorporated, e.g., in what Hegel has to say about mathematics, die hard." The rough idea of continuity is that small differences in the argument correspond to small differences in the value, with no sudden jumps. Continuous functions (x², log x, sin x) are the familiar ones, but discontinuous functions are the exceptions only in familiarity: "the place of birth of the youngest person living at time t" and "the integer next below x" are discontinuous; sin(1/x) oscillates infinitely often in any interval containing 0.

The ε–δ definition Define a neighbourhood of x as all numbers from x − ε to x + ε. Then f(x) is continuous at the argument a when, for every positive number σ, however small, there is a positive ε such that for all δ numerically less than ε, the difference f(a + δ) − f(a) is numerically less than σ. The general case is handled through the ultimate section and ultimate upper section of values as the argument approaches a from below (their common part is the ultimate oscillation): the function has a limit when the ultimate oscillation has one term (or none), and four limits in general — lower and upper, from below and from above. The limit exists when all four are equal, and the function is continuous at a when that limit is also the value at a. An equivalent analysis breaks continuity into four conditions of "convergence into classes" as the argument approaches from below and above.

Generalization beyond number None of this essentially involves number. Let P and Q be two (ideally serial) relations and R a one-many relation with domain in the field of P and converse domain in the field of Q: a generalized function, as position-at-time in a time-series. Continuity at the argument a means: given any P-interval containing the value at a, there is a Q-interval containing a (not as end-point) throughout which the function's values lie in the given interval. Motion illustrates the difference between limit and value: in H. G. Wells's story, the limit of the policeman's positions as t approaches the moment of the hero's ejaculation "Go to…" is in contact with the hero, whereas the value at that argument is empty space. The definitions involve infinite classes of finite intervals growing smaller, never infinitesimals: successive bisection of an inch never produces an infinitesimal bit.

Key ideas

• Weierstrass eliminated infinitesimals from the calculus; philosophers' older views persist.
• A function is continuous at a when values can be confined to any prescribed neighbourhood of f(a) by confining arguments to a sufficiently small neighbourhood of a.
• The ultimate section, ultimate upper section, and ultimate oscillation analyze the general behaviour of a function near an argument.
• A function has a limit when all four limits (lower/upper, from below/above) coincide; continuity is limit = value.
• Discontinuous functions are infinitely more numerous than continuous ones; sin(1/x) oscillates infinitely near 0.
• Limit and continuity of functions can be defined for any two series, with no numbers involved.
• Motion is continuous (if at all) as a function; space and time are continuous (if at all) as series.

Key takeaway

The limit of a function and its continuity are definable without infinitesimals and without numbers at all — a function is continuous when its limit at every argument equals its value there.

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Chapter 12 — Selections and the Multiplicative Axiom

Central question

Can we always pick one representative from each of infinitely many classes, and what mathematics depends on this choice?

Main argument

Selections and the definition of multiplication The multiplicative axiom can be enunciated, but not proved, in terms of logic; it is convenient rather than indispensable. Multiplication for a finite number of factors is easy: μ × ν is the number of ordered couples with first term from a class of μ and second from a class of ν. For infinitely many factors, Whitehead's method: given a class κ of classes, a selector from κ is a one-many relation having κ for its converse domain, picking one representative from each member; a selection is the domain of a selector; the multiplicative class is the class of selections. The product of the numbers of the members of κ is then the number of selectors from κ. Exponentiation is defined so as to avoid the axiom.

The axiom and its equivalent forms What cannot be proved is that a product is only zero when one of its factors is zero — that, given a class of non-null classes, there is at least one class containing exactly one term from each. The assumption is:

Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes.

Equivalent forms: a product of cardinals is zero only if a factor is zero; every relation has a one-many sub-relation with any given set of its relata as converse domain; Zermelo's axiom (a selector exists from all non-null sub-classes of a class, first brought to notice in his proof that every set can be well-ordered); Zermelo's theorem (every class can be well-ordered, by counting off representatives one by one with transfinite induction); and the comparability of any two cardinals — if the axiom is false, cardinals μ and ν may exist with μ neither less than, equal to, nor greater than ν (ℵ₁ and 2^ℵ₀ are a possible instance). Its truth or falsehood is at present unknown.

What depends on it The connection of addition and multiplication (the sum of ν mutually exclusive classes of μ terms has μ × ν terms) is unprovable for infinite ν without the axiom. The classic illustration is the millionaire with ℵ₀ pairs of boots and ℵ₀ pairs of socks: with boots we can choose all the right ones; with socks no principle of selection suggests itself, and "an infinite number of arbitrary choices is an impossibility." Without a rule, we do not even know that the socks can be arranged in a progression. The greater part of the theory of ordinals of the second class is unproved without the axiom; and the identification of non-inductive with reflexive cardinals (Chapter VIII's open question) can be proved by using the axiom twice. For most purposes only the restricted assumption is needed: "ℵ₀ is multipliable." The general axiom may yet be shown false — the continuum might prove incapable of being well-ordered — but so far the subject remains "wrapped in obscurity."

Key ideas

• The multiplicative axiom is the proposition that selectors exist for any class of non-null classes; it is enunciable but unprovable in logic.
• Multiplication by infinitely many factors is defined via selectors and selections.
• Equivalent forms: product zero only if a factor zero; Zermelo's axiom; Zermelo's theorem (every class can be well-ordered); comparability of cardinals.
• The boots and socks problem: ℵ₀ pairs of socks may not be countable without a rule of selection.
• Infinite sums of classes with given cardinalities need the axiom; part of ordinal theory is unproved without it.
• Non-inductive implies reflexive only with the axiom; "ℵ₀ is multipliable" suffices for most purposes.
• The truth of the axiom is unknown; it could be falsified by a non-well-orderable dense series.

Key takeaway

The axiom of choice — Russell's multiplicative axiom — cannot be proved, yet without it even the question of how many socks a millionaire owns is undecidable, and large parts of ordinal and cardinal theory collapse.

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Chapter 13 — The Axiom of Infinity and Logical Types

Central question

Can it be proved that there are infinitely many things in the world, and what stands in the way?

Main argument

The axiom and the null-class catastrophe The axiom of infinity is the assumption that if n is any inductive cardinal number, there is at least one class of individuals having n terms. If there were exactly nine individuals, then 10 — defined as the class of classes obtainable by adding a term to a 9-membered class — would be the null-class, and all subsequent inductive cardinals would be identical with it: "arithmetical catastrophes." The axiom of infinity prevents this. For finite arithmetic and for any given ratio, one can ascend the hierarchy of types — individuals, classes of individuals (2^n of them), classes of classes (2^(2^n)), and so on — so that any assigned inductive cardinal or ratio finds non-null classes; even with no individuals at all there would be 1, 2, 4, 16, 65,536 classes at successive stages. The axiom is needed only when we want the whole class of inductive cardinals, or progressions, or the real numbers (which require the compact series of ratios).

The conjurer's argument and confusion of types It might seem that forming the sum of individuals, classes, classes of classes, … ad infinitum manufactures an infinite collection from a finite starting point. Russell detects "an air of hocus-pocus" — "the conjurer who brings things out of the hat" — and the flaw is the confusion of types: we cannot form one class out of objects of different types, because the resulting "impure" class would be a member of itself or of its own sub-classes, generating the contradiction of the greatest cardinal. This leads to the paradox Russell discovered in 1901:

Form now the assemblage of all classes which are not members of themselves. This is a class: is it a member of itself or not? If it is, it is one of those classes that are not members of themselves, i.e. it is not a member of itself. If it is not, it is not one of those classes that are not members of themselves, i.e. it is a member of itself. Thus of the two hypotheses each implies its contradictory.

The solution by the theory of types: the supposition that a class is, or is not, a member of itself is not false but strictly meaningless; a sentence whose symbols mix types "is not false, but strictly devoid of meaning." So the attempt to escape the need for the axiom of infinity breaks down.

No proof, logical or empirical Arguments from Bolzano and Dedekind (the one-one relation of an object to the idea of it) fail: empirically there are no ideas of most objects, and logically "idea" is either identical with the object, or a description (of which there are many), or a psychological fiction. Empirical arguments from space, time, colours, and motion do not prove infinity: quantum theory illustrates that physics can never prove continuity, and a world of small finite jerks would be empirically indistinguishable from continuous motion — as the cinema shows. The conclusion:

From the fact that the infinite is not self-contradictory, but is also not demonstrable logically, we must conclude that nothing can be known a priori as to whether the number of things in the world is finite or infinite. Some of the possible worlds are finite, some infinite, and we have no means of knowing to which of these two kinds our actual world belongs.

Individuals and particulars An individual (or particular) is an object nameable by a proper name — a term that can only occur as subject in propositions. It is to the number of these that the axiom of infinity applies, and whether analysis reaches ultimate subjects, or an endless regress, is unknown.

Key ideas

• The axiom of infinity: for every inductive n, there is a class of individuals with n terms; without it, large inductive cardinals collapse into the null-class.
• Any finite collection can be outgrown by climbing the hierarchy of types, but the whole class of inductive cardinals requires the axiom.
• The attempt to sum all types is the conjurer's trick; the fallacy is confusion of types.
• Russell's paradox (1901): the class of all classes not members of themselves is a member of itself if and only if it is not.
• The theory of types makes self-membership meaningless rather than false.
• Bolzano's and Dedekind's proofs of infinite classes fail; empirical evidence from space, time, and motion is inconclusive (quanta, cinema).
• Whether the number of things in the world is finite or infinite cannot be known a priori.
• Individuals are terms that can only occur as subjects; the axiom applies to them.

Key takeaway

The infinity of the world is an assumption, not a theorem: without the axiom of infinity, 10 might be the null-class, and the conjurer's proof of infinity fails because mixing logical types produces meaningless, not false, statements.

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Chapter 14 — Incompatibility and the Theory of Deduction

Central question

What is deduction, and what is the smallest set of primitive ideas from which the whole theory of deduction can be built?

Main argument

Deduction against intuition Mathematics is a deductive science, and "no appeal to common sense, or 'intuition,' or anything except strict deductive logic, ought to be needed in mathematics after the premisses have been laid down." Kant, seeing that the geometers of his day could not prove their theorems without the figure, invented a theory of mathematical reasoning requiring "intuition"; the whole trend of modern mathematics has been against it. What can be known by mathematical methods is what can be deduced from pure logic; the rest must be ascertained empirically.

The five truth-functions Propositions have truth-values (a term due to Frege). The five fundamental functions of propositions are negation (not-p), disjunction (p or q), conjunction (p and q), incompatibility (p and q are not both true), and implication ("not-p or q" — the widest sense allowing inference of q from p). A function whose truth-value depends only on the truth-values of its arguments is a truth-function; its whole meaning is exhausted by the conditions under which it is true or false. Not all are independent: negation and disjunction suffice (implication is not-p or q, conjunction the negation of incompatibility), and Sheffer showed one primitive idea suffices — incompatibility — with Nicod reducing the primitive propositions to one formal principle and two non-formal ones:

• not-p = p | p ("incompatibility of p with itself")
• p or q = (p | p) | (q | q)
• p implies q = p | (q | q)
• p and q = (p | q) | (p | q)

Formal principles and inference The five formal principles of deduction in Principia Mathematica ("p or p" implies p; q implies "p or q"; "p or q" implies "q or p"; the twist-principle; if q implies r, then "p or q" implies "p or r") have a double use — as premisses of inferences and as establishing that premiss implies conclusion. There is unavoidably something psychological about inference itself: the relation that allows correct inference is logical, but the passage from asserting p to asserting q is a psychological process. Against C. I. Lewis's "strict implication," Russell maintains that mathematics needs no implication beyond the truth-function "not-p or q": whenever formal deducibility holds, we can see that either the premiss is false or the conclusion true, and nothing further need be admitted. The chapter's closing footnote notes that the importance of "tautology" for a definition of mathematics was pointed out by his former pupil Ludwig Wittgenstein.

Key ideas

• Mathematics must be strictly deductive; the Kantian appeal to intuition is obsolete.
• Truth-functions are exhausted by their truth-conditions; negation, disjunction, conjunction, incompatibility, and implication are the five.
• All five reduce to negation and disjunction, and by Sheffer's stroke (incompatibility) to one primitive idea.
• Nicod's single formal principle plus two non-formal principles yield the whole theory of deduction.
• Formal principles of deduction serve both as premisses and as rules.
• Inference is a psychological passage; validity is the logical relation "not-p or q."
• No "strict implication" beyond truth-functions is needed by mathematics.

Key takeaway

The whole theory of deduction rests on one indefinable — incompatibility — from which negation, disjunction, conjunction, and implication are defined, and inference requires nothing beyond the truth-function "not-p or q."

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Chapter 15 — Propositional Functions

Central question

What are propositional functions, and how do "all" and "some" — and with them existence — work?

Main argument

Propositions and propositional functions A proposition is primarily a form of words expressing what is either true or false. A propositional function is an expression containing one or more undetermined constituents which becomes a proposition when values are assigned — "a function whose values are propositions." "x is human" is neither true nor false while x is undetermined; any equation, any expression like "(a + b)² = a² + 2ab + b²," is a propositional function; so are "all A is B" and "lightning is followed by thunder" (the "instances" being values of the function). A propositional function standing alone is "a mere schema, a mere shell, an empty receptacle for meaning."

Always and sometimes There are only two things to be done with a propositional function: assert that it is always true, or that it is sometimes true. "All men are mortal" asserts that the function "if x is human, x is mortal" is always true (a formal implication); "there are unicorns" asserts that "x is a unicorn" is sometimes true; "men exist" means the function "x is a man" is sometimes true. All the propositions of logic are assertions that certain propositional functions are always true — logic mentions no particular things or concepts. The traditional forms are analyzed: "All S is P" = "'φx implies ψx' is always true"; "Some S is P" = "'φx and ψx' is sometimes true"; "No S is P" and "Some S is not P" similarly. This shows how far traditional logic was from the simplest forms, and why "all S is P" does not imply "some S is P" (conversion per accidens fails; Darapti is fallacious) — since universal statements say nothing about the existence of S's.

Existence and modality The fundamental form of existence is derived from "sometimes true": "arguments satisfying φx exist" means φx is sometimes true. "Men exist" is correct; "Socrates exists" is a "mere noise or shape, devoid of significance," since Socrates is a value, not an undetermined argument — the fallacy of "Men are numerous, Socrates is a man, therefore Socrates is numerous" in another guise. Modality likewise reduces to propositional functions: φx is necessary if the function is always true, possible if sometimes true, impossible if never true — as with a ball drawn from a bag of white, mixed, or black balls. "The habit of keeping propositional functions sharply separated from propositions is of the utmost importance, and the failure to do so in the past has been a disgrace to philosophy."

Key ideas

• A propositional function is an expression whose values are propositions; alone it asserts nothing.
• "Instances" and "cases" are values of propositional functions; generalization passes from instances to always-true.
• The two fundamental operations: asserting a function always true or sometimes true.
• "All S is P" is a formal implication; "some S is P" is a joint assertion — and the universal does not imply the particular.
• All logical propositions assert that some propositional function is always true.
• Existence is "sometimes true"; "Socrates exists" is meaningless, not false.
• Necessity, possibility, and impossibility are always/sometimes/never truth of a function.
• The word "is" covers predication, identity, and existence — a disgrace to the human race.

Key takeaway

All general statements are assertions about propositional functions: "all" is "always true," "some" is "sometimes true," existence is the claim that a function is sometimes true, and logic itself is the science of always-true functions.

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Chapter 16 — Descriptions

Central question

What do we really assert when we use the words "a so-and-so" and "the so-and-so"?

Main argument

Indefinite descriptions The word "the" deserves two chapters: "like Browning's Grammarian with the enclitic δε, I would give the doctrine of this word if I were 'dead from the waist down' and not merely in a prison." An indefinite description is a phrase of the form "a so-and-so." "I met a man" does not assert that I met Jones: "no actual man enters into my statement" — the concept enters, and the statement would remain significant, though false, if there were no men at all ("I met a unicorn"). Misled by grammar, logicians have admitted unreal objects — Meinong's golden mountain, the round square — and Russell answers with his famous appeal: "Logic, I should maintain, must no more admit a unicorn than zoology can; for logic is concerned with the real world just as truly as zoology." Hamlet is nothing over and above the thoughts of Shakespeare and his readers; "there is only one world, the 'real' world." The analysis: "an object having the property φ has the property ψ" means "the joint assertion of φx and ψx is not always false." The phrase "a so-and-so" is not a constituent of the propositions in which it appears — that is why such propositions are significant even when there is no such thing.

Names and definite descriptions A name is a simple symbol whose meaning is something that can only occur as subject; a description is a complex symbol whose meaning results from the meanings of its parts. "Scott is the author of Waverley" is not the same proposition as "Scott is Scott": the first is a fact in literary history, the second a trivial truism; descriptions are not interchangeable with names in propositional functions. What distinguishes "the so-and-so" from "a so-and-so" is the implication of uniqueness. "The author of Waverley was Scotch" involves three propositions:

1. "x wrote Waverley" is not always false (at least one person wrote Waverley).
2. "If x and y wrote Waverley, x and y are identical" is always true (at most one).
3. "If x wrote Waverley, x was Scotch" is always true (whoever wrote it was Scotch).

More generally, "the term satisfying φx exists" means: there is a term c such that φx is always equivalent to "x is c"; and "the term satisfying φx satisfies ψx" adds "ψc is true." Descriptions are therefore incomplete symbols: they have no meaning in isolation but only in propositions. Existence can be significantly asserted only of descriptions — "the present King of France does not exist" is significant, while "a exists," where a is a name, is meaningless; "whether Homer existed" is a question because "Homer" is used as an abbreviated description ("the author of the Iliad and the Odyssey").

Primary and secondary occurrences A description has a primary occurrence when the proposition results from substituting it into a propositional function; a secondary occurrence when it forms only part of the proposition. "The present King of France is not bald" is therefore ambiguous: with the description in secondary position it is true ("it is not the case that the present King of France is bald"), with primary position false ("the present King of France is such that he is not bald"). Confusion of the two is "a ready source of fallacies." In mathematics, descriptions appear as descriptive functions — "the R of y" — whose propositions are defined in the same way.

Key ideas

• Indefinite descriptions ("a man") involve only the concept; no actual or unreal object enters the proposition.
• Unreal objects are not admitted: logic is concerned with the real world; Hamlet is nothing beyond the thoughts about him.
• A name is a simple symbol; a description is a complex symbol whose meaning derives from its parts.
• "Scott is the author of Waverley" is not "Scott is Scott"; descriptions are not values of propositional functions.
• Definite descriptions add uniqueness and existence to the indefinite analysis.
• "The term satisfying φx exists" = there is a c such that φx is always equivalent to x = c.
• Descriptions are incomplete symbols; existence can be significantly asserted only of them ("the present King of France," not "a").
• Primary versus secondary occurrences explain the ambiguity of "the present King of France is not bald."

Key takeaway

Descriptions are incomplete symbols: a proposition about "the so-and-so" asserts existence, uniqueness, and the predicate, and never contains the described object as a constituent — which is how we can speak meaningfully of the present King of France.

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Chapter 17 — Classes

Central question

What are classes, if they are not part of the ultimate furniture of the world?

Main argument

Classes as logical fictions The chapter deals with "the" in the plural — the inhabitants of London, the sons of rich men — i.e., classes. "Class" cannot remain a primitive idea: for the reasons of Chapter XIII (Russell's paradox; the proof that the number of classes exceeds the number of individuals), classes are not a species of individuals. They are not heaps or conglomerations: the null-class has no members and cannot be a heap, and a unit class is not identical with its member. They cannot be identified with propositional functions: many functions are formally equivalent (true of the same arguments), and a class must be determined by its membership alone. So:

Classes are in fact, like descriptions, logical fictions, or (as we say) "incomplete symbols."

Russell invokes Occam's razor — "entities are not to be multiplied without necessity" — and adopts agnosticism: "like Laplace, we can say, 'je n'ai pas besoin de cette hypothèse.'"

The five conditions A symbol for a class must satisfy: (1) every propositional function determines a class; (2) formally equivalent functions determine the same class; (3) classes of classes must be possible (cardinal numbers are classes of classes); (4) it must be meaningless for a class to be a member of itself; and (5) — the hardest — it must be possible to make propositions about all classes of any one type, as mathematical induction requires (posterity is defined via all hereditary classes).

Extensional functions and the definition Functions of functions are extensional when their truth-value is unchanged by substituting a formally equivalent function ("all men are mortal") and intensional otherwise ("I believe that all men are mortal" — my beliefs about rational animals may differ). All the functions of functions needed in mathematical logic are extensional: "φx is always true," "φx is sometimes true," and the existence condition for "the term satisfying φx." From any function of a function, whether extensional or not, a derived extensional function is constructible ("there is a function formally equivalent to φx and having the property f"). The definition:

To assert that "the class determined by the function φx has the property f" is to assert that φx satisfies the extensional function derived from f.

The axiom of reducibility Condition (5) forces the issue of types: the functions taking a given argument form an infinite series of types, and no variable can run through them all — defining "a typical Frenchman" as one possessing all qualities possessed by most Frenchmen shows why a totality cannot be used to define its own members (the vicious circle principle; "Napoleon had all the qualities that make a great general"). The technical assumption that repairs this is the axiom of reducibility: there is a type of a-functions such that any a-function is formally equivalent to some function of that type. Russell notes it is a generalized form of Leibniz's identity of indiscernibles, is not logically necessary ("pure logic aims at being true in all possible worlds"), and is therefore a defect, even if empirically true; the theory of classes is consequently less complete than the theory of descriptions. But the reduction of propositions nominally about classes to propositions about their defining functions "must, it would seem, be sound in principle."

Key ideas

• Classes are logical fictions — incomplete symbols — not part of the ultimate furniture of the world.
• Classes are not heaps, not individuals, and not propositional functions (formal equivalence defeats identification).
• Five conditions govern a serviceable class symbol, including classes of classes and the meaninglessness of self-membership.
• Extensional versus intensional functions of functions; all needed ones are extensional.
• To assert a property of a class is to assert that its defining function satisfies the derived extensional function.
• The vicious circle principle and the hierarchy of types block totalities that define their own members.
• The axiom of reducibility is a generalized identity of indiscernibles; it is convenient, not necessary, and philosophically a defect.
• Agnosticism about classes, per Occam's razor and Laplace.

Key takeaway

Classes are logical fictions: statements about them are statements about their defining propositional functions, and the theory rests on the dubiously necessary axiom of reducibility — which is why it is less complete than the theory of descriptions.

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Chapter 18 — Mathematics and Logic

Central question

What is mathematics, and in what sense is it identical with logic?

Main argument

One subject Historically mathematics and logic were entirely distinct studies, but "logic has become more mathematical and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one." The proof of identity is a matter of detail: starting from premisses universally admitted to belong to logic and arriving by deduction at results as obviously mathematical, "there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right" — any proposed boundary in Principia Mathematica is arbitrary. The book's earlier chapters walked backward from the natural numbers to the fundamentals of logic; a synthetic treatment would begin with those fundamentals and reach the numbers last.

Not the science of quantity Mathematics is not the science of number: geometry without co-ordinates has nothing to do with number, and the generalizations of arithmetic (one-one relations, similarity, selections, ancestral relations, series, continuity of functions) dissolve arithmetic into "a set of new deductive systems, in which traditional arithmetic is at once dissolved and enlarged" — whether any one of them belongs to logic or to arithmetic "is entirely arbitrary, and incapable of being decided rationally."

Pure form Logic and mathematics deal with no particular things or properties: "one and one are two, but not that Socrates and Plato are two, because, in our capacity of logicians or pure mathematicians, we have never heard of Socrates and Plato." The syllogism is not "All men are mortal, Socrates is a man, therefore Socrates is mortal" but the formal proposition that the propositional function "if all α's are β's and x is an α, then x is a β" is always true. The form of a proposition is "that, in it, that remains unchanged when every constituent of the proposition is replaced by another"; words like "is" and "than" merely indicate form. Logical constants are what is common among propositions obtainable from each other by term-for-term substitution — incompatibility, and the notion of "there is a term c such that…" from which 1 is derived. All mathematical constants are logical constants or abbreviations defined by means of them. A necessary (not sufficient) criterion: a logical proposition is obtainable from a proposition without variables by turning every constituent into a variable and asserting the result always or sometimes true. The old notion of analytic reappears as tautology — the characteristic of logical propositions — though Russell confesses, "I do not know how to define 'tautology.'"

The frontier The axiom of infinity is a proposition enunciable in logical terms but not assertable by logic; the existence of a world is an accident — "if there were no universe, all general propositions would be true." Logical propositions are known a priori, without study of the actual world. Ordinary language is misleading (grammar makes "ten men" look like "white men"), so logical symbolism is "absolutely necessary to any exact or thorough treatment of our subject." If any student is led by the book into a serious study of mathematical logic, "it will have served the chief purpose for which it has been written."

Key ideas

• Logic and mathematics are one subject: no sharp line can be drawn between them in Principia Mathematica.
• Mathematics is not the science of quantity or number; generalization dissolves arithmetic into many deductive systems.
• Pure mathematics is purely formal: it mentions no particular things or properties.
• The form of a proposition is what remains when all constituents are replaced; logical constants are the residue.
• All mathematical constants are logical constants; incompatibility and "there is a term c" are exemplary.
• A logical proposition is one obtainable by replacing constituents by variables and asserting always or sometimes true.
• The characteristic of logical propositions is tautology — the modern replacement for "analytic."
• The existence of the world is not logically necessary; the axiom of infinity cannot be asserted by logic.
• Ordinary language is misleading; logical symbolism is indispensable.

Key takeaway

Mathematics is logic: both are the study of pure form — propositions built from variables and logical constants — and the backward journey from the natural numbers ends at the frontier of a single subject called indifferently mathematics or logic.

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The book's overall argument

1. Chapter 1 (The Series of Natural Numbers) — establishes that all traditional mathematics reduces to the natural numbers, that Peano's axioms capture their formal structure, and that a genuinely logical foundation must define "0," "number," and "successor."
2. Chapter 2 (Definition of Number) — establishes the Frege–Russell definition: the number of a class is the class of all classes similar to it, so 2 is the class of all couples.
3. Chapter 3 (Finitude and Mathematical Induction) — establishes that the natural numbers are the posterity of 0 under the successor relation and that a number is finite when it is inductive.
4. Chapter 4 (The Definition of Order) — establishes that order lies in relations, not terms: a series is an asymmetrical, transitive, connected relation.
5. Chapter 5 (Kinds of Relations) — establishes the mathematical hierarchy of relations — asymmetrical, one-many, one-one — and derives descriptive functions from one-many relations.
6. Chapter 6 (Similarity of Relations) — establishes relation-numbers and shows that structure is likeness: mathematics studies interrelations, not intrinsic natures.
7. Chapter 7 (Rational, Real, and Complex Numbers) — establishes that fractions are relations, real numbers are segments of the series of ratios, and complex numbers are ordered couples — construction, not postulation.
8. Chapter 8 (Infinite Cardinal Numbers) — establishes the transfinite cardinals: reflexive classes, ℵ₀, Cantor's theorem that 2^n > n, and the ambiguities of infinite subtraction.
9. Chapter 9 (Infinite Series and Ordinals) — establishes the ordinals: rearrangement changes serial numbers, ω + 1 ≠ 1 + ω, well-ordering, and the aleph hierarchy.
10. Chapter 10 (Limits and Continuity) — establishes that limit is a purely ordinal notion and that continuity of series is Dedekindianness plus a countable median class.
11. Chapter 11 (Limits and Continuity of Functions) — establishes the ε–δ analysis of function limits and continuity without infinitesimals, generalizable to any series.
12. Chapter 12 (Selections and the Multiplicative Axiom) — establishes the multiplicative axiom, its equivalent forms (Zermelo's axiom and theorem), and the results that depend on it.
13. Chapter 13 (The Axiom of Infinity and Logical Types) — establishes that infinity cannot be proved, that Russell's paradox forces the theory of types, and that self-membership is meaningless rather than false.
14. Chapter 14 (Incompatibility and the Theory of Deduction) — establishes that the whole theory of deduction reduces to incompatibility as sole primitive, with truth-functions exhausted by truth-conditions.
15. Chapter 15 (Propositional Functions) — establishes that "all" and "some," existence, and modality are operations on propositional functions.
16. Chapter 16 (Descriptions) — establishes that descriptions are incomplete symbols, with "the so-and-so" analyzed into existence, uniqueness, and predicate.
17. Chapter 17 (Classes) — establishes that classes are logical fictions reducible to their defining functions, at the cost of the axiom of reducibility.
18. Chapter 18 (Mathematics and Logic) — establishes the book's thesis: mathematics and logic are one purely formal subject, defined by variables, logical constants, and tautology.

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Common misunderstandings

Misunderstanding: The book proves that mathematics is reducible to logic, once and for all. It argues for the thesis and summarizes the construction of Principia Mathematica, but it is candid about the frontier: the axiom of infinity, the multiplicative axiom, and the axiom of reducibility are assumptions whose truth is unknown, and the definition of "tautology" is unfinished.

Misunderstanding: The Peano axioms define the natural numbers. Russell argues they do not: the three primitive ideas admit infinitely many interpretations (including "0" = 100), and the axioms define the class of progressions, not the numbers. That is precisely why Frege's logical definition is needed.

Misunderstanding: The definition of number as a class of similar classes is circular or trivial. It is not circular: "the number of a class" is defined first, and "a number" is then anything which is the number of some class. The "oddity" that 2 is the class of all couples is the price of definiteness, and the definition provably has all the properties numbers should have.

Misunderstanding: Infinite numbers are contradictory because the part cannot be equal to the whole. "Equal" is ambiguous: with "equal" meaning "similar," a reflexive class (like the inductive numbers correlated with the evens) is simply similar to a proper part of itself, and no contradiction follows.

Misunderstanding: Irrational numbers are limits of series of ratios. The belief that series must have limits "delayed the true theory of irrationals." An irrational is not a limit; it is a segment of the series of ratios — the honest construction that replaces Dedekind's postulate.

Misunderstanding: "The present King of France is not bald" is straightforwardly true or false. The sentence is ambiguous between primary and secondary occurrences of the description: with the description in secondary position it is true; with primary position it is false. Scope, not grammar, settles it.

Misunderstanding: Classes are collections or heaps of things. A class is not a heap (there is no heap of nothing, and a unit class is not its member) and not a thing at all: classes are logical fictions, and statements about them are statements about their defining propositional functions.

Misunderstanding: "Socrates exists" is a false or doubtful proposition. It is not false but meaningless: existence can be significantly asserted only of descriptions, because "exists" means "is sometimes true" of a propositional function. "Men exist" is fine; "Socrates exists" is a category error.

Misunderstanding: The axiom of infinity is proved by the fact that there are infinitely many numbers. The numbers are classes of classes; counting them presupposes the very infinite supply of individuals the axiom asserts. The conjurer's argument that sums over all types manufactures infinity fails by confusion of logical types.

Misunderstanding: Russell's paradox shows that the class of classes not members of themselves is both a member and not a member of itself. The theory of types resolves it differently: the supposition that a class is, or is not, a member of itself is strictly meaningless — a type violation — not a contradiction within a legitimate totality.

Misunderstanding: The book is an elementary introduction in the sense of being easy or popular. It is an introduction in the sense of requiring no prior mathematics or symbolic logic, but it covers the hardest results of the foundations — Cantor's theorem, the axiom of choice, the theory of types — and Russell says in the preface that the method matters more than the results.

Misunderstanding: Continuity of motion and continuity of space are the same kind of continuity. They are different notions: motion is continuous (if at all) as a function, space and time as series. Russell's definitions distinguish them explicitly, and 2^ℵ₀ is the cardinal of a continuous series.

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Central paradox / key insight

The book's most counterintuitive claim is that mathematics and logic are the same subject — that there is no boundary at which logic ends and mathematics begins. The natural numbers with which everyone starts turn out to be a late stage of a backward journey into pure logic, and the humble fact that 2 is the class of all couples is the price of definiteness. Russell adds a second, connected shock: the part can be similar to the whole, so there are infinite numbers with a perfectly consistent — if surprising — arithmetic in which ℵ₀ + 1 = ℵ₀ but 2^ℵ₀ > ℵ₀, and in which ω + 1 differs from 1 + ω.

They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic.

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Important concepts

Cardinal number

The number of a class is the class of all classes similar to it; a cardinal number is anything which is the number of some class. Cardinal numbers are classes of classes — 2 is the class of all couples.

Ordinal number

The relation-number of a well-ordered series — a species of serial number. Ordinal arithmetic is not commutative: 1 + ω = ω but ω + 1 ≠ ω.

Relation-number

The class of all relations similar to a given relation, where likeness is defined by one-one correlators. Relation-numbers are what "structure" obscurely means; serial numbers are their application to series.

Similarity

The relation between two classes when a one-one relation has one as domain and the other as converse domain. Similarity is reflexive, symmetrical, and transitive, and it — not counting — is the logical basis of number.

One-one / one-many / many-one relations

One-one: at most one referent and at most one relatum per term (husband-wife, n → n+1). One-many: at most one term has the relation to a given term (father, sine of). Many-one: the converse. One-one relations correlate classes term for term.

Serial relation

A relation that is asymmetrical (an aliorelative), transitive, and connected. A series is the same thing as a serial relation; order lies in the relation, not the terms.

Inductive number / mathematical induction

A number is inductive when it possesses every property possessed by 0 and by the successors of possessors — i.e., it obeys mathematical induction from 0. Finite classes and cardinals are the inductive ones.

Progression

A one-one relation with just one term in the domain but not the converse domain, the domain being the posterity of that term. Progressions verify Peano's five axioms; their cardinal is ℵ₀.

Posterity / ancestral relation

The posterity of a term x with respect to R is the class of terms belonging to every R-hereditary class to which x belongs. The ancestral relation (proper ancestor) is its transitive closure; it replaces "and so on" with a definition.

Reflexive class

A class similar to a proper part of itself (the inductive numbers and the evens). Reflexive numbers are infinite; whether all infinite classes are reflexive depends on the multiplicative axiom.

ℵ₀ (aleph-null) and the alephs

The smallest infinite cardinal, the number of terms of a progression (inductive numbers, ratios). ℵ₀ + n = ℵ₀, ℵ₀² = ℵ₀, but 2^ℵ₀ > ℵ₀; the series of alephs ℵ₀, ℵ₁, ℵ₂, … continues indefinitely.

Cantor's theorem

The number of sub-classes of a class is always greater than the number of its members: 2^n > n even when n is infinite. Hence there is no greatest cardinal; 2^ℵ₀ is the cardinal of a continuous series.

Axiom of infinity

The assumption that if n is any inductive cardinal, there is at least one class of individuals having n terms. Without it, large inductive cardinals collapse into the null-class; it is needed for the whole number series, progressions, and the reals, and cannot be proved.

Multiplicative axiom (axiom of choice)

The assumption that, given any class of mutually exclusive non-null classes, there is a class with exactly one term in common with each. Equivalent to Zermelo's theorem (every class can be well-ordered) and to the comparability of cardinals; unprovable, and needed for results like counting ℵ₀ pairs of socks.

Axiom of reducibility

The assumption that there is a type of a-functions such that any a-function is formally equivalent to one of that type. It makes the theory of classes usable (mathematical induction, statements about all classes of a type) but is not logically necessary — a defect Russell acknowledges.

Logical types

The hierarchy of individuals, classes of individuals, classes of classes, and so on. Mixing types produces meaningless rather than false statements; self-membership of a class is a type violation. Types resolve Russell's paradox and block the proof of the axiom of infinity.

Propositional function

An expression with undetermined constituents whose values are propositions; "x is human." Two operations: assert always true, or sometimes true. All logic consists of assertions that functions are always true.

Formal implication

The assertion that "φx implies ψx" is always true — the analysis of "all S is P." Universal statements do not imply existence.

Definite description / incomplete symbol

A phrase "the so-and-so," analyzed as existence + uniqueness + predicate; it has no meaning in isolation. Descriptions (and classes) are incomplete symbols: propositions containing them are not what they become when names are substituted.

Primary / secondary occurrence

The scope of a description within a proposition. "The present King of France is not bald" is true with secondary occurrence, false with primary; confusion of the two breeds fallacies.

Truth-function / incompatibility

A function of propositions whose truth-value depends only on their truth-values. Incompatibility (p | q: "not both p and q") is the single primitive from which negation, disjunction, conjunction, and implication are defined, and from which, via Sheffer and Nicod, the whole theory of deduction follows.

Tautology

The characteristic of logical propositions — the modern descendant of "analytic." Russell gestures at it (the mark of propositions expressible in terms of variables and logical constants that are true whatever the world is like) but confesses he cannot yet define it.

Structure / relation-number

The map of a relation: what remains when the terms are changed. Two relations with the same structure are mathematically indistinguishable; geometry, and the whole of mathematics, is about structure.

Dedekind cut / segment

A division of a series into a lower and upper section. Segments are lower sections without a maximum; a real number is a segment of the series of ratios, and an irrational is a boundaryless segment (√2 is the segment of ratios whose square is less than 2).

Compact / Dedekindian / Cantorian continuity

Compact: terms between any two terms. Dedekindian: every sub-class has a boundary (no gaps). Cantorian continuity: Dedekindian plus a median class of ℵ₀ members; all such series are similar, and their cardinal is 2^ℵ₀.

Descriptive function

"The term having the relation R to x," or "the R of x," where R is one-many. All mathematical functions are descriptive functions; they exist when x is in the converse domain of R.

Ultimate oscillation

The common part of the ultimate section and ultimate upper section of a function's values as the argument approaches a value from below; the function has a limit when it has one term, and is continuous when the limit equals the value.

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References and Web Links

Full texts (primary sources — chapter list and all quotations verified against these)

• Project Gutenberg: Introduction to Mathematical Philosophy, ebook 41654 — https://www.gutenberg.org/ebooks/41654
• Internet Archive: 1919 first edition, Cornell University Library scan — https://archive.org/details/cu31924001598188
• Internet Archive: 1920 reprint, University of Toronto scan — https://archive.org/details/introductiontoma00russuoft
• Kevin C. Klement (University of Massachusetts), online edition PDF — http://people.umass.edu/klement/imp/imp.pdf

Edition and bibliographic records

• Open Library: Introduction to mathematical philosophy (work record, listing the 1993 Routledge and Dover editions among others) — https://openlibrary.org/works/OL1088545W/Introduction_to_mathematical_philosophy
• Wikipedia: Introduction to Mathematical Philosophy (background, publication history, and editions) — https://en.wikipedia.org/wiki/Introduction_to_Mathematical_Philosophy

Background and analysis (Stanford Encyclopedia of Philosophy)

• Irvine, Andrew D., "Bertrand Russell," SEP — https://plato.stanford.edu/entries/russell/
• Irvine, Andrew D., and Deutsch, Harry, "Russell's Paradox," SEP — https://plato.stanford.edu/entries/russell-paradox/
• Tennant, Neil, "Logicism and Neologicism," SEP — https://plato.stanford.edu/entries/logicism/

Suggested further reading (by the same author)

• Russell, Principia Mathematica (with A. N. Whitehead), Cambridge University Press, 1910–1913 — the full technical treatment to which this book is the introduction.
• Russell, The Principles of Mathematics, Cambridge University Press, 1903 — the earlier, more controversial defense of the logicist thesis.

A note on the study of this book

The chapters above were written from the full text of the 1919 first edition (Cornell scan, Internet Archive) and the UMass online edition, cross-checked against the SEP entries on Russell, Russell's paradox, and logicism, and against the Wikipedia article for edition facts. No secondary study guide was used as a substitute for the primary text; readers who want supplementary discussion should prefer the SEP entries above, which are peer-reviewed reference works rather than summaries.