The Rise of Mathematical Logic and Set Theory

This collection of excerpts traces the historical development of mathematical logic and set theory, examining the contributions of key figures like Lagrange, Boole, De Morgan, Cantor, Dedekind, Frege, Peano, and Russell. It explores the evolution of ideas such as the algebraic manipulation of functions, the formalization of logic through symbolic systems, the emergence of set theory and transfinite numbers, and the philosophical program of logicism, which aimed to ground mathematics in logic. The text further covers the rise of axiomatic methods, the discovery of paradoxes, and the subsequent attempts to build rigorous foundations for mathematics in the late 19th and early 20th centuries, touching upon the reactions and influences of these developments in various European and American intellectual circles, including discussions of proof theory, type theory, and the philosophical implications of these foundational debates up to the 1930s.

Study Guide: Foundations of Mathematical Thought

Quiz

1. Explain Peano’s use of dots in his notation. What is the hierarchy of scope indicated by the number of dots surrounding logical conjunctions, quantifiers, and connectives?
2. Describe Boole’s “elective symbols” and the three fundamental laws they obeyed. How did these laws relate to properties shared with symbols of quantity?
3. What was Jevons’s concept of the “Universe of Thought”? How did his “Law of infinity” potentially lead to paradox?
4. How did Cantor define a new domain of numbers (B) based on sequences of rational numbers? What criterion did he use to establish equality between numbers in this domain?
5. According to the text, how did Dedekind characterize a “simply infinite” system? What was the significance of the “base-element” in this definition, and to what concept of Cantor’s did it correspond?
6. Outline Frege’s approach to defining a “binary relative.” How did he represent it in terms of “element-pairs” and assign truth values to the “relative coefficient”?
7. Explain Husserl’s phenomenological approach to the concept of number. What two bases did he identify as furnishing the “psychological foundation of the number-concept”?
8. State two of Peano’s axioms for the class N of integers. What role did these axioms play in the formalization of arithmetic?
9. Describe Russell’s distinction between nominal and contextual definitions, particularly as it relates to Peano’s criterion and Russell’s own definition of existence.
10. According to the text, what is a “tautological” proposition in Wittgenstein’s view? How does his concept of analysis relate to Russell’s logical atomism?

Answer Key

1. Peano used dots to reduce the density of brackets, with the number of dots indicating the scope of a logical operation. Logical conjunction had the highest priority and widest scope (both directions), followed by dots after expressions with quantifier brackets, and finally dots around connectives joining propositions.
2. Boole’s elective symbols (like x and y) represented “acts of election” and obeyed three laws: distributivity (x(u + v) = xu + xv), commutativity (xy = yx), and the index law (x^n = x for integer n ≥ 2). Distributivity and commutativity were properties shared with symbols of quantity.
3. Jevons’s “Universe of Thought” was a domain relevant to a logical argument, similar to De Morgan’s relative universe. His “Law of infinity” stated that any quality treated as present could also be treated as absent, potentially leading to paradox by suggesting an unbounded universe with elements outside of it.
4. Cantor defined domain B using sequences of rational numbers (a_n). A sequence had a specific limit b if, for any arbitrary positive number ε, there existed an integer n0 such that the absolute difference between b_n and b was less than ε for all n ≥ n0. Equality (b = b’) was defined based on analogous properties of the absolute difference between corresponding rationals in their sequences.
5. Dedekind characterized a “simply infinite” system N as one for which there exists a similar transformation φ of N such that N appears as the chain of an element (the base-element, not contained in φ(N)). One defining property was that N ≠ φ(N), and this insight corresponded to Cantor’s idea of well-ordering.
6. Frege construed a binary relative ‘a’ extensionally as a class of ordered pairs. It was expressed as the union of its “element-pairs” (i:j), and the “relative coefficient” a_ij (meaning ‘i is an a of j’) was a proposition that took the value 1 if true and 0 if false.
7. Husserl focused on “our grasp of the concept of number” through the intentional act of “abstraction” from diverse entities to form “embodiments.” He identified two psychological foundations: (1) the concept of collective unification and (2) the concept of Something (Etwas), from which numbers were specified as successions of ones.
8. Two of Peano’s axioms for the class N of integers are: (1) 1 belongs to N, and (2) if a belongs to N, then a + 1 belongs to N. These axioms provided a formal basis for defining the natural numbers and the operation of succession.
9. A nominal definition introduces a new symbol with an explicit equivalence, while a contextual definition defines a symbol within the context of a proposition. The text notes that Peano’s criterion was nominal, whereas Russell’s definition of existence was contextual, embedded within a larger proposition.
10. In Wittgenstein’s view, a tautological proposition is one that is true for all possible truth values of its component elementary propositions. He believed there was “one and only one complete analysis of the proposition,” a view resembling Russell’s logical atomism, suggesting a shared idea of breaking down propositions into fundamental components.

Essay Format Questions

1. Compare and contrast the approaches of Boole and De Morgan to the algebra of logic. What were their key innovations and limitations?
2. Discuss the emergence of set theory in the late 19th century, focusing on the contributions of Cantor and Dedekind. What were their central ideas, and what challenges did their work face?
3. Analyze the concept of definition in the development of mathematical logic, considering the distinctions between nominal, contextual, and other types of definitions as discussed in the provided texts.
4. Trace the evolution of Russell’s logical thought as presented in the excerpts, from his early engagement with Cantor to the development of his substitution theory and its eventual problems.
5. Explore the relationship between logic and the foundations of mathematics as reflected in the work of Peano, Frege, and Hilbert. What were their respective goals and methodologies?

Glossary of Key Terms

• Conjunction: A logical connective (often represented by ‘and’ or a dot in these texts) that is true if and only if both of its operands are true.
• Quantifier: A logical symbol (like “for all” or “there exists”) that specifies the quantity of individuals in a domain that satisfy a certain property.
• Proposition: A declarative sentence that is either true or false.
• Functional Equation: An equation where the unknown is a function, rather than a single variable.
• Taylor Expansion: A representation of a function as an infinite sum of terms calculated from the values of the function’s derivatives at a single point.
• Duality: A principle in logic or mathematics where two concepts or statements are related such that interchanging certain elements transforms one into the other.
• Contrary Term: In logic, the negation or opposite of a given term.
• Elective Symbol: Boole’s symbols representing mental operations of selecting or classifying objects.
• Distributive Law: A property of operations where one operation applied to a sum (or union) is equal to the sum (or union) of the operation applied individually to each term.
• Commutative Law: A property of operations where the order of the operands does not affect the result (e.g., a + b = b + a).
• Index Law: In Boole’s algebra, the law that applying an elective symbol multiple times yields the same result as applying it once (x^n = x).
• Moduli: In Boole’s work, the values of a function for specific inputs (e.g., φ(0) and φ(1)).
• Universe of Thought: A domain or context relative to which logical terms and arguments are considered.
• Limit (of a sequence): The value that the terms of a sequence approach as the index increases without bound.
• Difference Quotient: An expression used in the definition of the derivative of a function, representing the average rate of change of the function over a small interval.
• Partition Sums: Sums of the values of a function over subintervals of a partition, used in the definition of the definite integral.
• Everywhere Dense Set: A set such that between any two distinct elements of the set, there is another element of the set.
• Similar Transformation: A one-to-one mapping between two sets that preserves a certain structure or relation.
• Well-Ordering: A total ordering of a set such that every non-empty subset has a least element.
• Denumerable (Countable): A set that can be put into a one-to-one correspondence with the set of natural numbers.
• Undistinguished (m-ads): Collections of m elements where the order or identity of individual elements does not matter for equivalence.
• Distinguished (m-ads): Collections of m elements where the order or identity of individual elements does matter for equivalence.
• Chain: A sequence of elements where each element is related to the next in a specific way.
• Vacuous Term: A term that applies to nothing; an empty set or concept.
• Universe (in logic): The domain of discourse, the collection of all entities under consideration.
• Identity: The relation of being the same.
• Domain (of a function or relation): The set of all possible input values for a function or the set of first elements in the ordered pairs of a relation.
• Elementhood: The relation of being a member of a set or class.
• Cardinality: The number of elements in a set.
• Ordinal Number: A generalization of natural numbers used to describe the order type of well-ordered sets.
• Phenomenology: A philosophical approach that focuses on the study of consciousness and the objects of direct experience.
• Abstraction: The process of forming a general concept by disregarding specific instances or attributes.
• Axiom: A statement that is taken to be true without proof and serves as a starting point for deducing other truths.
• Model Theory: The branch of mathematical logic that studies the relationship between formal theories and their interpretations (models).
• Categoricity: A property of a set of axioms such that all of its models are isomorphic to each other (i.e., they have the same structure).
• Nominal Definition: A definition that introduces a new term by equating it to a combination of already understood terms.
• Contextual Definition: A definition that explains the meaning of a term by showing how sentences containing the term are to be understood.
• Impredicative Property: A property that is defined in terms of a collection that includes the entity being defined.
• Cardinal Number: A number that represents the size of a set.
• Relation: A set of ordered pairs, indicating a connection between elements of two or more sets.
• Tautology: A statement that is always true, regardless of the truth values of its components.
• Logical Atomism: A philosophical view that the world consists of simple, independent facts, and that complex propositions can be analyzed into combinations of elementary propositions corresponding to these facts.
• Axiom of Choice: An axiom in set theory that states that for any collection of non-empty sets, there exists a function that chooses one element from each set.
• Synthetic Judgement: In Kantian philosophy, a judgement where the predicate is not contained in the concept of the subject and adds new information.
• Metalogic (Proof Theory): The study of the properties of logical systems themselves, such as consistency, completeness, and decidability.
• Truth-Function: A function whose output (a truth value) depends only on the truth values of its inputs.
• Logicism: The philosophical view that mathematics can be reduced to logic.
• Formalism: A philosophy of mathematics that treats mathematical statements as formal symbols and their manipulation according to fixed rules, without inherent meaning.
• Constructivism: A philosophy of mathematics that holds that mathematical entities should be constructed rather than merely proven to exist.
• Ordered Pair: A pair of objects where the order matters.

Briefing Document: Themes and Ideas

This briefing document summarizes the main themes, important ideas, and key figures discussed in the provided excerpts from “01.pdf,” focusing on the development of mathematical logic, set theory, and related philosophical concepts during the 19th and early 20th centuries.

Main Themes:

• Evolution of Logical Notation and Systems: The text traces the development of symbolic notations for logic, moving from Peano’s dot system to the use of specialized symbols for logical connectives and quantifiers. It highlights the efforts of figures like De Morgan, Boole, and Schröder to create algebraic systems for logical reasoning.
• Development of Set Theory: A significant portion of the excerpts focuses on the emergence and evolution of set theory, particularly the work of Cantor and Dedekind. Key concepts like denumerability, transfinite numbers (ordinals and cardinals), well-ordering, and the nature of sets (as extensions or intensions) are discussed.
• Formalization of Arithmetic: The attempts to provide a rigorous foundation for arithmetic are a recurring theme. The work of Dedekind and Peano in formulating axioms for natural numbers and exploring the definitions of zero, one, and other number types is examined.
• Paradoxes and the Search for Foundations: The discovery of paradoxes within naive set theory and logic led to significant efforts to resolve them through type theories (Russell), axiomatic set theories, and alternative foundational approaches.
• Influence of Philosophy on Logic and Mathematics: The interplay between philosophical ideas (e.g., phenomenology of Husserl, logicism of Russell, formalism of Hilbert, intuitionism) and the development of mathematical logic and set theory is evident throughout the text.
• Duality and Symmetry in Logical Systems: The concept of duality in logical notations and the symmetry of roles between terms and their contraries are highlighted in the work of De Morgan and Schröder.
• The Nature of Definitions and Existence: The text touches upon different types of definitions (nominal, contextual, implicit, impredicative) and the philosophical implications of defining mathematical objects and asserting their existence.

Most Important Ideas and Facts:

• Peano’s Notation and Axioms: Peano’s system of dots to indicate scope in logical expressions is mentioned as a way to reduce the density of brackets. His axioms for the class of integers (N) are presented in detail, covering properties like succession, identity, and the base element ‘1’.
• Quote: “In addition, to reduce the density of brackets I have made some use of Peano’s systems of dots: the larger their number at a location, the greater their scope.”
• Quote (Examples of Peano’s Axioms): “1. 1 N. … 6. a N . 1 a 1 N.”
• Boole’s Algebra of Logic: Boole’s work on elective symbols and their algebraic properties (distributivity, commutativity, index law $x^n = x$ for integer $n ge 2$) is discussed. His use of moduli (values of a function $phi(x)$ at 0 and 1) to characterize functions satisfying the index law is noted.
• Quote: “From the first of these, it appears that elective symbols are distributie, from the second that they are commutatie; properties which they possess in common with symbols of quantity . . . The third law 3 we shall denominate the index law. It is peculiar to elective”
• Quote (Boole’s expansion): “$phi(x) = phi(0) cdot (1-x) + phi(1) cdot x$” which is represented as “$Phi(x) = Phi(0) + (Phi(1) – Phi(0)) x$”.
• De Morgan’s Contributions: De Morgan’s use of the symbol ‘x’ for the contrary term of X, deploying a symmetry of roles, and his collections of notations displaying duality properties are highlighted.
• Quote: “However, in using the symbol ‘x ’ to represent the contrary term of a term X he deployed a symmetry of roles for X and x, and combinations of them using the dots and brackets of 247.1 , which was rather akin to duality”
• Jevons’s System: Jevons’s inclusion of “All” and “nothing” (term ‘0’) in his system, along with the basic laws $0 cdot 0 = 0$ and $0 + 0 = 0$, and his introduction of a “Universe of Thought” are mentioned. His “Law of infinity” is also noted as bordering on paradox.
• Quote: “All and nothing appeared in Jevons’s system. He defined ‘the term or Žmark 0’ rather thoughtlessly as ‘excluded from thought’ art. 94, where however he did state the basic laws 0.0 0 and 0 0 0”
• Cantor’s Set Theory: Cantor’s concept of “everywhere dense” sets and his argument that the manifold M (denumerably infinite coordinate space over a binary pair) does not have the power of the series of ordinals are mentioned. His footnote regarding the earlier possession of these ideas is also noted, likely in response to du Bois Reymond’s claim of priority for the notion of the everywhere dense set.
• Quote: “‘I now assert, that such a manifold M does not have the power of the series 1, 2, . . . , , . . . ‘”
• Dedekind’s Work on Foundations: Dedekind’s definition of a “simply infinite” system and its similarity to Cantor’s idea of well-ordering are discussed. His treatment of mathematical induction with “metarules” (theorem of complete induction) is also highlighted.
• Quote: “Dedekind characterised a system N as ‘simply infinite’ if ‘there is such a similar transformation $phi$ of N, that N appears Ž .as the chain of an element, which is not contained in $phi(N)$’ and was called the ‘b a s e – e l e m e n t’ 1; thus one of the defining properties was ‘N 1’”
• Kempe’s Contributions: Kempe’s consideration of “heaps” (finite systems of n units) and their properties (discrete, single, independent) is noted, along with his use of graphical representations of units. His anticipation of Dedekind’s term “chains” is also mentioned.
• Schröder’s Algebra of Logic: Schröder’s “identical calculus” and his definition of “identical equality identity” for domains are presented. His work on solving dual pairs of equations involving domains and the interpretation of elementhood within derived manifolds are discussed. The “Negative Postulate” and “Positive Postulate” of his logic are also quoted.
• Quote (Identical Equality): “‘ 1 ’ If a b and b a, then a b”
• Quote (Negative Postulate): “No domain has the property 2 ; all mutually disjoint within the manifold.”
• Quote (Positive Postulate): “Elements are ‘mutually agreeable, so that we are able to think of the manifold as a whole’.”
• Husserl’s Phenomenology of Arithmetic: Husserl’s focus on “our grasp of the concept of number” through the intentional act of “abstraction” to form “embodiments” is discussed. His distinction between “Zahl” and “Anzahl” (cardinal and ordinal) and the “psychological foundation of the number-concept” (collective unification and Something) are mentioned.
• Peano’s Definitions of 0 and 1: Peano’s proposed definitions of 0 and 1 using a relation $supset$ are presented, though their lack of quantification and potential circularity are noted.
• Quote: “‘$s in K . supset . forall s’ s supset s . supset . a subset 0 equiv a$’ and ditto ‘$a subset 1 equiv a$’”
• Russell’s Contextual Definitions and Paradox: The document quotes Russell’s contextual definitions for the existential quantifier ($E!$) and definite descriptions ($iota x phi x$). The paradox that plagued his early substitution theory is presented in detail, involving the substitution of ‘b’ for ‘a’ in a proposition ‘p’.
• Quote (Contextual Definition of $E!$): “$E! iota x phi x . equiv : exists b : forall x . phi x . equiv . x = b$ Df.”
• Quote (Russell’s Paradox setup): “$b a_0(p) . = : exists p, a : a . = . p text{!}_b^a q : neg p text{Df}_0^0$”
• Whitehead’s Notation in Principia Mathematica: Whitehead’s development of notations for domains, converse domains, and fields in the context of relations is mentioned, with examples illustrating potentially redundant notations.
• Quote (Example of Whitehead’s Notation): “$vdash . R”V = D’R = x exists y . xRy$”
• Wittgenstein’s Tractatus and Truth Values: Wittgenstein’s view of tautologies and contradictions as lacking sense and his idea of a unique complete analysis of a proposition are mentioned.
• Ramsey’s Extensionalization of Propositional Functions: Ramsey’s argument for treating propositional functions extensionally, like mathematical functions, is noted as a move away from predicative restrictions.
• Hilbert’s Proof Theory and Metamathematics: Hilbert’s program and the textbook “Founding of theoretical logic” (with Ackermann) are mentioned, highlighting the aim of providing a logical grounding for mathematics.
• Tarski’s Work on Truth-Functions: Tarski’s definition of truth-functions using a new “law of substitution” is presented.
• Quote (Tarski’s Law of Substitution): “$forall p, q, f : p equiv q . supset . f(p) supset f(q)$”
• Carnap’s Views on Logicism and Formalism: Carnap’s perspective on the historical shift from “concept-ranges” to “concept-contents” (praising Frege) and his understanding of logicism as a reduction versus formalism as a common calculus are discussed.
• Whitehead’s Revisions in Principia Mathematica: Whitehead’s revised definition of the ordered pair and his rebuilding of the foundations of relations are mentioned.
• Quote (Whitehead’s Revised Ordered Pair): “$x supset y .= text{Df}. iota^2 x iota^1 y$”

Key Figures Mentioned:

• Peano
• Boole
• De Morgan
• Jevons
• Cantor
• Dedekind
• Kempe
• Schröder
• Husserl
• Russell
• Whitehead
• Wittgenstein
• Ramsey
• Hilbert
• Ackermann
• Tarski
• Carnap

Overall Significance:

These excerpts provide a glimpse into the intense intellectual activity surrounding the foundations of logic and mathematics in the late 19th and early 20th centuries. They highlight the evolution of formal systems, the emergence of new mathematical concepts like transfinite numbers and abstract sets, the challenges posed by logical paradoxes, and the diverse philosophical perspectives that shaped these developments. The discussions around notation, definitions, axioms, and the very nature of mathematical objects demonstrate a profound effort to establish rigor and clarity in these fundamental disciplines.

Mathematical Logic and Set Theory Development: FAQs

Frequently Asked Questions on the Development of Mathematical Logic and Set Theory

1. How did early symbolic systems in logic, such as those by Peano and others, attempt to improve upon traditional logic? Early symbolic systems aimed to reduce the ambiguity and increase the precision of logical expressions by introducing specific symbols and notations for logical connectives, quantifiers, and the scope of these elements. Peano, for instance, used a system of dots to indicate the scope of logical conjunction, quantification, and connectives, aiming for a less dense notation than traditional bracketing. Others explored representing logical relationships through algebraic symbols and operations, drawing parallels between logical laws and algebraic properties like distributivity and commutativity, as seen in Boole’s work with “elective symbols.”

2. What role did the concept of “duality” play in the development of logical notation and thought? The concept of duality, where certain symbols or expressions maintain a symmetrical relationship, emerged in early logical systems. De Morgan’s use of ‘x’ to represent the contrary of a term X illustrated this symmetry. Similarly, the structured use of dots and brackets in some notations hinted at dual relationships between logical operations or concepts. While not always emphasized, these duality properties reflected an underlying structural symmetry in logical reasoning and representation.

3. How did the work of George Boole contribute to the foundation of mathematical logic? Boole’s major contribution was treating logic algebraically, representing logical entities with “elective symbols” that obeyed laws analogous to those of quantity, such as distributivity, commutativity, and a peculiar “index law” (x^n = x for integer n >= 2). He explored how functions of these symbols could be expanded and analyzed, introducing the concept of “moduli” to characterize logical functions. His approach laid the groundwork for applying mathematical techniques to logical reasoning and the development of Boolean algebra.

4. What were some of the early attempts to formalize arithmetic, and what challenges did they face? Several mathematicians, including Dedekind and Peano, made significant early attempts to formalize arithmetic. Dedekind characterized a system of natural numbers as “simply infinite” based on the existence of a similar transformation and a base element, essentially capturing the idea of mathematical induction. Peano developed a set of axioms for the natural numbers. However, these early formalizations sometimes faced challenges related to clarity, such as unclear quantification or potential circularity in definitions, as noted in the analysis of some of Peano’s early definitions of zero and one.

5. How did Georg Cantor’s work on set theory, particularly the concept of infinity, influence the development of mathematical foundations? Cantor’s work on set theory revolutionized mathematics by providing a rigorous framework for understanding different levels of infinity. He introduced concepts like denumerable and non-denumerable sets, developed a theory of transfinite numbers (ordinals and cardinals), and explored the properties of sets like being everywhere dense or perfect. His ideas, although initially met with resistance, became fundamental to the foundations of mathematics, influencing subsequent work in logic and analysis. His “diagonal argument,” for instance, demonstrated the existence of infinities larger than the set of natural numbers.

6. What is “logicism,” and how did figures like Russell attempt to realize this philosophical program? Logicism is the philosophical view that mathematics can be reduced to logic, meaning that mathematical concepts can be defined in terms of logical concepts and mathematical theorems can be derived from logical axioms using purely logical rules of inference. Bertrand Russell was a major proponent of logicism. He, along with Whitehead in Principia Mathematica, attempted to build the entire edifice of mathematics on a foundation of formal logic. This involved developing a comprehensive logical system capable of expressing mathematical entities like numbers and sets. Russell’s work also grappled with paradoxes that arose within naive set theory, leading to the development of type theory as a way to avoid logical contradictions.

7. How did the analysis of paradoxes, such as Russell’s paradox, impact the development of logical systems and the foundations of mathematics? The discovery of paradoxes within seemingly consistent logical and set-theoretic frameworks had a profound impact, exposing fundamental issues in the intuitive understanding of sets and logic. Russell’s paradox, concerning the set of all sets that do not contain themselves, highlighted the dangers of unrestricted set formation. This led to significant efforts to revise and rigorize the foundations of mathematics, resulting in the development of various axiomatic set theories (like Zermelo-Fraenkel set theory) and logical systems (like Russell’s type theory) designed to avoid these contradictions by imposing restrictions on the formation of sets and the application of predicates.

8. What were some alternative perspectives or developments in the foundations of mathematics that emerged alongside logicism and set theory, such as Husserl’s phenomenology of arithmetic? While logicism and set theory were dominant forces, alternative perspectives on the foundations of mathematics also emerged. Edmund Husserl, for example, developed a “phenomenology of arithmetic” that focused on the intentional acts of the mind in grasping the concept of number, rather than just the formal system itself. He emphasized the role of abstraction and collective unification in the formation of number concepts, offering a more psychological or epistemological grounding for arithmetic. This contrasted with the purely logical or set-theoretic approaches of logicism and Cantor’s work.

History and Development of Set Theory

Set theory, or as Georg Cantor referred to his version, ‘Mengenlehre’, is a branch of mathematics that deals with collections of objects. The book from which this information is drawn discusses the history of set theory in the 19th and 20th centuries.

Georg Cantor and the Development of Mengenlehre:

• The set theory introduced is primarily Georg Cantor’s ‘Mengenlehre’, encompassing point set topology, transfinite arithmetic, and the general theory of sets.
• Cantor’s work is divided into phases, including his Acta mathematica phase from 1883 to 1885.
• During this time, there were French translations of his work and unpublished and published ‘communications’ between 1884 and 1885. These communications explored order-types and partial derivatives.
• The period of 1886 to 1897 marked the extension of the Mengenlehre. This involved Cantor’s philosophy of the infinite (1886-1888) and new definitions of numbers.
• Cardinal exponentiation was introduced through Cantor’s diagonal argument in 1891.
• Transfinite cardinal arithmetic and simply ordered sets were developed by 1895, followed by transfinite ordinal arithmetic and well-ordered sets by 1897.
• Cantor clearly separated five distinct but related properties of sets: topology, dimension, measure, size (number of members), and ordering.
• He refined the notion of the infinite into theories of transfinite cardinal and ordinal arithmetic and introduced a range of order-types.
• Cantor considered his Mengenlehre to be an integrated theory.

Cantor’s Set Theory in Contrast with Part-Whole Theory:

• Cantor’s set theory distinguished an object from its unit set, where an object a belongs to a set {a, b, c} and sets {a} and {a, b} are subsets of it. This is in contrast with part-whole theory, where this distinction was not made.
• Figures like De Morgan and Boole employed part-whole analyses of collections, where inclusion was the primary relation. Schröder also consistently used the part-whole theory of classes in his logic.

Parallel Processes and the Reception of Set Theory:

• The period from the 1870s to the 1900s saw parallel developments in set theory, logics, and axiomatics.
• There was a growth of interest in set theory, both as Cantorian Mengenlehre and more generally.
• French initiatives, especially from Borel, and German initiatives from Klein contributed to the spread of these ideas.
• Hilbert also played a role in publicizing set theory around 1900.
• Integral equations and functional analysis provided significant applications for set and measure theory.

Dedekind’s Contribution:

• Dedekind developed his own set theory, with his booklet published in 1888. He used the term ‘System’ for sets, defined as “various things a, b, c . . . comprehended from any cause under one point of view”.
• Dedekind defined union and intersection of systems.
• His concept of ‘part’ between systems blurred the distinction between membership and improper inclusion, unlike Cantor’s more careful approach.

Zermelo’s Axiomatization:

• Zermelo also contributed significantly to set theory, particularly through his axiomatization in 1908.
• His axioms aimed to block paradoxes and included extensionality, basic set construction, power-set, union, infinity, and choice.
• Zermelo’s work was influenced by the concerns of Cantor and Dedekind.
• He provided a proof of the well-ordering theorem in 1904, which involved the axiom of choice.

Key Concepts in Set Theory:

• Cantor handled sets of points, defining a value-set as a “given finite or infinite number of number magnitudes”.
• He distinguished between sets that were ‘countable in the infinite’ and those with the cardinality of the continuum.
• The equality of cardinalities was defined extensionally based on the lack of isomorphism between members.
• Cantor defined disjoint sets, union, and intersection of sets. He also used the terms ‘divisor’ and ‘multiplum’ for set inclusion.
• He defined a set P to be ‘perfect’ when it equaled its derivative P’.
• A set P is ‘dense in itself’ if P’ includes P.
• Cantor defined order-types and considered transfinite numbers as special kinds of order-types. He also explored simply ordered sets and performed operations on their types, such as sum and product.
• He introduced several operations on a set P, including coherence, adherence, inherence, supplement, and remainder.
• The concept of well-ordered set is considered fundamental for the entire theory of manifolds. Cantor believed it was always possible to bring any well-ordered set into a “law of thought”.

Relationship with Other Areas:

• Cantor’s creation of set theory had its origins in the study of the convergence of Fourier series based on Dirichlet’s conditions.
• Set and measure theory found applications in integral equations and functional analysis.
• Peano explicitly worked with the set theory of ‘cl.mus Cantor’, indicating a shift towards Cantorian composition rather than part-whole theory.

Foundational Issues and Paradoxes:

• Cantor’s definition of a set as “each gathering-together into a whole of determined well-distinguished objects of our intuition or of our thought” has been criticized for potentially admitting paradoxes, although it is argued that Cantor formulated it precisely to avoid them.
• By late 1899, Cantor distinguished between “consistent multiplicities” (formerly ‘ready sets’) and “inconsistent multiplicities,” with the totality of all transfinite ordinals (Ω) associated with the latter.
• Russell’s paradox was a significant challenge to naive set theory.

Notation and Symbolism:

• The source provides various symbols used in set theory and logic, such as for equivalence, identity, membership, union, intersection, and inclusion.
• Cantor used specific notations for the union and intersection of sets and introduced symbols for operations on sets like coherence and supplement.
• Peano adopted and sometimes refined notations related to classes and membership, distinguishing individuals from their unit classes.

In conclusion, set theory, originating with Cantor’s Mengenlehre, underwent significant development and faced both support and criticism. It moved from an intuitive basis to more formal axiomatic systems and found crucial applications in various branches of mathematics, fundamentally shaping the understanding of infinity and the foundations of mathematics. The distinction between Cantor’s approach and part-whole theories, along with the emergence of paradoxes, spurred further advancements and different axiomatizations of set theory.

Development of Mathematical Logic

Mathematical logic, also referred to as symbolic logic, has its prehistory in mathematical analysis stemming from Cauchy and Weierstrass, and also has roots in algebra through figures like Boole and De Morgan who adapted algebras to produce mathematicised logic. De Morgan even introduced the expression ‘mathematical logic’ to distinguish a logic growing among mathematicians from that of logicians, emphasizing the proper subordination of the mathematical element.

The sources highlight several key aspects and figures in the development of mathematical logic:

Early Developments and Traditions:

• Algebraic Logic: Boole and De Morgan are considered principal founders of algebraic logic, each adapting different algebras to create their logics. These logics, along with others, largely founded the tradition of algebraic logic. Practitioners in this tradition often handled collections using part-whole theory, where membership was not distinguished from inclusion.
• Mathematical Analysis: A rival tradition to algebraic logic emerged from mathematical analysis, inaugurated by Cauchy and extended by Weierstrass. This laid the groundwork for figures like Cantor and influenced the development of mathematical logic.
• Symbolic Logic: The term ‘symbolic logic’ encompasses both the algebraic and mathematical logic traditions. Occasionally, other traditions like syllogistic logic or Kantian philosophy are also mentioned. Symbolic logic was often viewed as too philosophical by mathematicians and too mathematical by philosophers.

Key Figures and Their Contributions:

• De Morgan: While opining that algebra provided habitual use of logical forms, De Morgan aimed to encompass mathematics as a whole within logic. His work investigated reasoning with reference to the connection of thought and language, including scientific induction justified by probability theory. He explored analogies between logic and algebra and introduced the expression ‘mathematical logic’.
• Boole: Prompted by a dispute between De Morgan and Hamilton, Boole wrote his book Mathematical Analysis of Logic (MAL) in 1847. He treated logic as a normative science and developed an ‘algebra of logic’ with ‘elective symbols’ and laws.
• Peano: Peano and his school developed the ‘Logic of Algebra’. Peano believed mathematics to be pure logic, with all its propositions in the form “If one supposes A true, then B is true”. He formalized analysis and developed a symbolism to represent propositions concisely. Peano explicitly worked with Cantor’s set theory and considered mathematical logic as a tool for analyzing ideas and reasoning in mathematics.
• Russell: Russell, along with Whitehead, aimed to provide a “complete investigation of the foundations of every branch of mathematical thought” in Principia Mathematica. They sought to deduce pure mathematics from logical foundations, with the propositional and predicate calculi providing deduction and set theory furnishing the “stuff”. Russell’s logicism posited that all mathematical constants are logical constants. However, the logicism of Principia Mathematica faced complexities and criticisms. Russell also identified ‘contradictions’ and proposed the Vicious Circle Principle as a remedy. His work heavily involved the theory of types.

Relationship with Other Fields:

• Algebra and Arithmetic: De Morgan noted many analogies between logic and algebra, and to a lesser extent, arithmetic. Boole also saw connections between logical operations and arithmetical ones. Peano explored the logistic of arithmetic.
• Set Theory: Cantor’s Mengenlehre is seen as a foundation for mathematics that mathematical logic aimed to explicate logicistically. Russell’s logic included set theory as the “stuff” for mathematical deductions. Zermelo’s logic was intertwined with his set theory.
• Philosophy: Symbolic logic has a complex relationship with philosophy, often seen as both too mathematical and too philosophical. Logicism, a school within mathematical logic, contends with other philosophical schools like metamathematics, intuitionism, and phenomenology.

Foundational Issues and Developments:

• Logicism: The idea that mathematics is reducible to logic was a central theme, particularly with Peano and Russell. However, the exact nature and success of this reduction were debated.
• Axiomatization: Hilbert’s growing concern with axiomatics influenced the development of logic. Zermelo’s axiomatization of set theory in 1908 was also a significant development.
• Paradoxes: The emergence of paradoxes in set theory and logic spurred further developments and refinements. Russell’s paradox was a significant challenge.

Evolution and Influence:

• Mathematical logic, particularly through Principia Mathematica, became well established, including the logic of relations.
• The field saw various developments, including the use of truth-tables and the reduction of connectives.
• Different national contexts also played a role, with the U.S.A. showing more sympathy towards symbolic logic than Britain in some periods.

In summary, mathematical logic emerged from dual roots in algebra and mathematical analysis, evolving through the work of key figures like De Morgan, Boole, Peano, and Russell. It aimed to provide a logical foundation for mathematics, particularly through the program of logicism, and engaged deeply with set theory and philosophical considerations. The development of mathematical logic also involved addressing foundational issues and paradoxes, leading to a rich and evolving field.

Foundations of Mathematics: Traditions and Schools of Thought

The sources discuss the foundations of mathematics from various perspectives and across different historical periods. The pursuit of mathematical foundations has involved various “traditions” and “schools” of thought.

Early Traditions:

• Algebraic Logic: Figures like Boole and De Morgan are considered principal founders, adapting different algebras to create their logics. Boole, prompted by a dispute involving De Morgan, developed an ‘algebra of logic’ in his Mathematical Analysis of Logic (MAL). De Morgan, who introduced the expression ‘mathematical logic’, aimed to encompass mathematics as a whole within logic, exploring analogies between logic and algebra. He believed that algebra provided habitual use of logical forms.
• Mathematical Analysis: A tradition rival to algebraic logic emerged from Cauchy and was extended by Weierstrass, laying the groundwork for others like Cantor. Cauchy inaugurated mathematical analysis based on the theory of limits.

These two traditions together constitute symbolic logic. However, symbolic logic was often seen as too philosophical by mathematicians and too mathematical by philosophers.

Logicism:

• Logicism is presented as a “school” in contention with metamathematics, intuitionism, and phenomenology. It posits that mathematics is reducible to logic.
• Peano believed mathematics to be pure logic, with all its propositions in the form “If one supposes A true, then B is true”. His school developed the ‘Logic of Algebra’ and formalized analysis, developing a symbolism to represent propositions concisely. Peano explicitly worked with Cantor’s set theory and considered mathematical logic a tool for analyzing mathematical ideas and reasoning.
• Russell, along with Whitehead, aimed to provide a “complete investigation of the foundations of every branch of mathematical thought” in Principia Mathematica. They sought to deduce pure mathematics from logical foundations, with propositional and predicate calculi providing deduction and set theory furnishing the “stuff”. Russell’s logicism posited that all mathematical constants are logical constants. The “definition of Pure Mathematics” in Principia Mathematica and earlier works emphasized propositions of the form ‘p implies q’ containing variables and logical constants.
• However, the logicism of Principia Mathematica faced complexities, including the axiom of reducibility, which distanced them from strict logicism. Russell also identified ‘contradictions’ and proposed the Vicious Circle Principle as a remedy, with his work heavily involving the theory of types.
• Carnap‘s work helped to popularize the term “logicism”. However, logicism eventually faced criticisms and was even described as having been converted into “a reduction of mathematics to set theory”, which was deemed “unsatisfactory” by some.

Set Theory (Mengenlehre):

• Cantor‘s Mengenlehre (set theory) is presented as a foundation for mathematics that mathematical logic aimed to explicate logicistically. Cantor’s work involved the development of cardinals (‘Machtigkeiten’) and their arithmetic, with Cantor considering them epistemologically prior to ordinals. He also addressed the definition and generality of well-ordering.
• Russell’s logic included set theory as the “stuff” for mathematical deductions. Zermelo also intertwined logic with his set theory.
• By the late 1890s, the range and status of Mengenlehre were being reviewed, with an emphasis on its foundational and general features.

Formalism:

• Hilbert‘s growing concern with axiomatics influenced the development of logic. He emphasized the independence, completeness, and consistency of axiomatic systems, as well as the decidability of mathematical questions. Hilbert’s formalism posited that consistency implied existence.
• Hilbert outlined his approach to the foundations of arithmetic at the International Congress of Mathematicians in 1904.

Intuitionism:

• Brouwer is primarily associated with the ‘intuitionistic’ philosophy of mathematics. Intuitionism is presented as a contrasting view to logicism and formalism.

Axiomatization:

• Hilbert‘s different axiom systems for Euclidean geometry are mentioned. Zermelo’s axiomatization of set theory in 1908 was also significant. Peano also laid out axioms in a fully symbolic manner.

Paradoxes:

• The emergence of paradoxes in set theory and logic spurred further developments and refinements. Russell’s paradox was a significant challenge. The application by Whitehead and Russell to the Royal Society to fund Principia Mathematica acknowledged the role of their logical principles in making “mathematical contradictions… vanish”.

In conclusion, the foundations of mathematics have been explored through various approaches, with logicism, set theory, formalism, and intuitionism being prominent schools of thought. The interplay between logic and mathematics, the development of symbolic systems, the axiomatization of mathematical theories, and the resolution of paradoxes have been central to this ongoing search for the fundamental roots of mathematics.

Logicism: Its Core, Impact, and Historical Trajectory

The influence of Logicism, the philosophical and foundational program asserting that mathematics is reducible to logic, was significant and multifaceted, as detailed throughout the sources.

Core Tenets and Key Figures:

• Logicism, particularly as championed by Russell and Whitehead in Principia Mathematica (PM), aimed to provide a complete investigation of the foundations of every branch of mathematical thought by deducing pure mathematics from logical foundations.
• Peano also held a logicist view, believing mathematics to be pure logic, and his school’s work significantly influenced Russell. Russell explicitly characterized logicism in terms of ‘pure mathematics’.
• While Frege is also a key figure in the history of logicism, the sources note that his work was little read during his lifetime. However, his logic and logicism did influence figures like Russell.
• Russell’s logicism posited that all mathematical constants are logical constants, with propositional and predicate calculi providing deduction and set theory furnishing the “stuff” for mathematical deductions.

Initial Impact and Reception:

• Principia Mathematica became well established, including the logic of relations. Its publication led to a wide range of reactions regarding both its logical calculus and its logicist thesis.
• The application by Whitehead and Russell to fund PM mentioned the role of their logical principles in making “mathematical contradictions… vanish”.
• Early reviews and engagements with logicism varied across countries. Peano, despite his own logicist leanings, contrasted his use of ‘logic-mathematics’ as an ‘instrument’ with its role in PM ‘for science in itself’.
• In Britain, figures like Jourdain offered long complimentary reviews of Russell’s work emphasizing the role of logic. However, others like Johnson presented a more heterogeneous view of logic, not fully engaging with logicism.
• American reactions were also diverse. Some, like Sheffer, admired the project of PM but questioned the presupposition of logic in its foundation. Others, like Lewis, explored alternative logics and saw PM as potentially just one among many possible logics.
• In France, early reactions were polarized between figures like Couturat, who was a strong advocate for ‘logistique’ (mathematical logic with mathematical intent), and mathematicians like Poincaré, who stressed the role of intuition in mathematics. Later, French attitudes became more neutral.
• German-speaking reactions were varied, with neo-Kantians expressing their views. Carnap later played a significant role in popularizing the term “logicism”.

Influence on Other Fields and Ideas:

• Logicism competed with other philosophies of mathematics such as formalism (associated with Hilbert) and intuitionism (associated with Brouwer and Weyl). Godel’s work particularly affected logicism and formalism.
• The development of set theory by Cantor was seen as a foundation for mathematics that logicism aimed to explicate logicistically. Russell’s logic included set theory as its “stuff”.
• Logicism had a profound influence on the relationship between logic and epistemology, particularly through Russell’s work. His book Our Knowledge of the External World had a significant impact on the Vienna Circle.
• The emphasis on reduction, a key feature of logicism, aimed to reduce mathematics to arithmetic and then to logic. This influenced discussions about the foundations of mathematics and the nature of mathematical objects.

Challenges and Criticisms:

• The discovery of paradoxes, such as Russell’s paradox in set theory, posed a significant challenge to the logical foundations sought by logicism. Russell’s theory of types was developed as a remedy.
• The axiom of reducibility in Principia Mathematica was a point of complexity and criticism, moving it away from strict logicism.
• Wittgenstein criticized Russell’s mixing of logic and logicism.
• Godel’s incompleteness theorem in 1931 had a major impact on logicism and formalism, demonstrating inherent limitations in formal systems. This theorem forced a re-framing of many fundamental questions. Godel himself noted imprecisions in PM compared to Frege.

Evolution and Fate of Logicism:

• Russell himself revised Principia Mathematica, influenced by Wittgenstein.
• The term “logicism” gained wider currency in the 1920s and 1930s, partly through the work of Carnap.
• Despite its initial ambitions, no authoritative position within or outside logicism emerged. After 1931, many central questions had to be re-framed.
• By the later periods covered in the sources, logicism was increasingly seen as part of history. While PM remained a source for basic notions in mathematical logic, the central tenets of logicism faced significant challenges.
• Some modern versions of logicism have been proposed in recent years, and figures from its history are still invoked in contemporary philosophical discussions. However, these often involve modernized interpretations of the older ideas.

Overall Influence:

• Logicism significantly spurred the development of mathematical logic and its techniques.
• It played a crucial role in highlighting foundational issues in mathematics and prompting rigorous investigation.
• Logicism contributed to philosophy the centrality of the distinction between theory and metatheory.
• Despite its ultimate challenges, logicism’s pursuit of reducing mathematics to logic profoundly shaped the landscape of 20th-century philosophy of mathematics and continues to be a point of reference in discussions about the foundations of mathematics.

The sources indicate that while logicism as a comprehensive program faced significant obstacles and is no longer the dominant view, its influence on the development of logic, the study of mathematical foundations, and the relationship between logic and philosophy remains undeniable.

A History of Philosophy and Logic

The sources extensively discuss the history of philosophy, especially as it intertwines with the development of logic and the foundations of mathematics. The narrative often presents philosophical movements and figures in their historical context, tracing their influence and reactions to new developments.

Several sections of the sources explicitly engage with the history of logic:

• Early Developments: The text begins by noting the “prehistory” related to the mathematical aspects of logic in the early 19th century in France and their adoption in England, mentioning the development of new algebras. It also discusses the emergence of ‘logique’ and its connection to the French doctrine of ‘ideologie’ in the late 18th century, highlighting figures like Condillac and his views on the analysis and synthesis of ideas. Condillac’s approach, which showed uninterest in traditional syllogistic logic, is noted as rather novel and influential on French science.
• Symbolic Logic: The rise of symbolic logic, encompassing algebraic logic (with figures like Boole and De Morgan) and the tradition of mathematical analysis (from Cauchy and Weierstrass), is presented as a historical development. The source mentions that De Morgan introduced the expression ‘mathematical logic’.
• Influence of Kant and Hegel: The sources discuss the influence of Kantian philosophy and its reception, particularly by Russell and his followers, who generally found Kant wanting in logic and mathematics. The neo-Hegelian movement and its dominance at the end of the century, particularly in England with the young Russell, are also noted. Russell’s initial enthusiasm for Bradley’s neo-Hegelian philosophy and his eventual rejection of the tenet that relations were internal are discussed in a historical sequence.
• Bolzano’s Influence: The work of Bolzano, particularly his Wissenschaftslehre, is mentioned as an important influence on logic, with his concern for ‘deducibility’ and ‘objective truths’ highlighted.
• Peano and the Peano School: The emergence of Peano and his school is presented as a crucial historical development influencing Russell’s logicism. The Paris Congress of Philosophy in 1900 is highlighted as a key moment where Russell recognized Peano’s superiority, particularly in a discussion with Schroder on ‘the’. The impact of Peano’s work on Russell’s understanding of order and relations is also noted.
• Husserl’s Phenomenological Logic: The early career of Edmund Husserl, his background as a follower of Weierstrass and Cantor, and the development of his phenomenological logic in his works of 1891 and 1900-1901 are traced historically. His critique of psychologism in the ‘Prolegomena to pure logic’ is mentioned.
• Hilbert’s Early Proof and Model Theory: The source outlines Hilbert’s growing concern with axiomatics and his work on different axiom systems for Euclidean geometry between 1899 and 1905. His advocacy for Cantor’s Mengenlehre and his own ideas on proof theory around 1900 are also situated historically. The allied emergence of model theory in the USA around 1900 is also mentioned.
• Reactions to Logicism: Chapter 7 of the source is dedicated to the reactions to mathematical logic and logicism between 1904 and 1907, covering German philosophers, mathematicians, Peanists, and American philosophers. Chapter 8 then discusses the influence and place of logicism from 1910 to 1930, examining the transitions of Whitehead and Russell from logic to philosophy, American reactions, and the engagement of figures like Wittgenstein and Ramsey.
• Russell’s Philosophical Development: Russell’s journey from neo-Hegelianism towards his ‘Principles of Mathematics’ is described historically, including his changing views on relations and the absolute nature of space and time. His early engagement with Whitehead from 1898 and their joint discovery of the Peano school are presented as pivotal historical moments leading to logicism.
• Paradoxes and Their Impact: The discovery of Russell’s paradox in set theory in 1901 is noted as a crucial event that compromised the new foundations and spurred Russell to collect other paradoxes. The development of the theory of types in Principia Mathematica is presented as a response to these paradoxes.
• The Influence of Principia Mathematica: The reception of Principia Mathematica after its publication is discussed over several chapters, detailing British, American, French, and German reactions across different periods. The influence of PM on figures like Wittgenstein and Carnap is also noted historically.
• Development of Metatheory: The emergence of the distinction between theory and metatheory is presented as a key contribution of mathematical logic to philosophy, with the full distinctions being effected by Godel and Tarski in the 1930s.
• The Fate of Logicism: The concluding chapter reflects on the history of logicism, noting that while it competed with other philosophies, no single authoritative position emerged, and after 1931, many questions had to be re-framed. The survey in Chapters 8 and 9 aims to show the variety of positions held and uses made of mathematical logic beyond the traditional narrative of three competing philosophies.

In essence, the sources adopt a historical approach to discussing philosophy, particularly in the context of logic and mathematics. They trace the lineage of ideas, the interactions between key figures, the development of different schools of thought, and the impact of significant events like the discovery of paradoxes. This historical lens is crucial for understanding the evolution and influence of logicism and its place within the broader landscape of philosophical thought. The author explicitly states that the book lays its main emphasis on the logical and mathematical sides of this history.

By Amjad Izhar
Contact: amjad.izhar@gmail.com
https://amjadizhar.blog

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