The Analytic Tradition in Philosophy, Volume 1

The Founding Giants

Scott Soames

• Publisher: Princeton University Press
• Year: 2014
• ISBN:
  9781400850457
• Topic: Philosophie
• Subtopic: Philosophie analytique

This is the first of five volumes of a definitive history of analytic philosophy from the invention of modern logic in 1879 to the end of the twentieth century. Scott Soames, a leading philosopher of language and historian of analytic philosophy, provides the fullest and most detailed account of the analytic tradition yet published, one that is unmatched in its chronological range, topics covered, and depth of treatment. Focusing on the major milestones and distinguishing them from the dead ends, Soames gives a seminal account of where the analytic tradition has been and where it appears to be heading.Volume 1 examines the initial phase of the analytic tradition through the major contributions of three of its four founding giants—Gottlob Frege, Bertrand Russell, and G. E. Moore. Soames describes and analyzes their work in logic, the philosophy of mathematics, epistemology, metaphysics, ethics, and the philosophy of language. He explains how by about 1920 their efforts had made logic, language, and mathematics central to philosophy in an unprecedented way. But although logic, language, and mathematics were now seen as powerful tools to attain traditional ends, they did not yet define philosophy. As volume 1 comes to a close, that was all about to change with the advent of the fourth founding giant, Ludwig Wittgenstein, and the 1922 English publication of his Tractatus, which ushered in a "linguistic turn" in philosophy that was to last for decades.

Part 1

• FREGE •

CHAPTER 1

Foundations of Logic, Language, and Mathematics

1. Overview

2. The Language of Logic and Mathematics

3. Sense, Reference, Compositionality, and Hierarchy

4. Frege’s Logic

5. Frege’s Philosophy of Mathematics

5.1. Critique of Naturalism, Formalism, and Psychologism

5.2. Critique of Kant

5.3. Frege’s Definition of Number

5.3.1. Numerical Statements Are about Concepts

5.3.2. But Numbers Are Objects

5.3.3. Objects and Identity

5.3.4. The Number of F’s, Zero, Successor, and the Numerals

5.3.5. The Natural Numbers

6. The Logicist Reduction

6.1. The Axioms of Logic and Arithmetic

6.2. Informal Proofs of the Arithmetical Axioms

6.3. Arithmetical Operations

6.4. Further Issues

1. OVERVIEW

The German philosopher-logician Gottlob Frege was born in 1848, graduated with a PhD in Mathematics from the University of Gottingen in 1873, and earned his Habilitation in Mathematics from the University of Jena in 1874, where he taught for 43 years until his retirement in 1917, after which he continued to write on issues in philosophical logic and the philosophy of mathematics until his death in 1925. While he is now recognized as one of the greatest philosophical logicians, philosophers of mathematics, and philosophers of language of all time, his seminal achievements in these areas initially elicited little interest from his contemporaries in mathematics. Though he did attract the attention of, and have an important influence on, four young men—Bertrand Russell, Edmund Husserl, Rudolf Carnap, and Ludwig Wittgenstein—who were to become giants in twentieth-century philosophy, it took several decades after his death before the true importance of his contributions became widely recognized.

Frege’s main goal in philosophy was to ground the certainty and objectivity of mathematics in the fundamental laws of logic, and to distinguish both logic and mathematics from empirical science in general, and from the psychology of human reasoning in particular. His pursuit of this goal can be divided into four interrelated stages. The first was his development of a new system of symbolic logic, vastly extending the power of previous systems, and capable of formalizing the notion of proof in mathematics. This stage culminated in his publication of the Begriffsschrift (Concept Script) in 1879. The second stage was the articulation of a systematic philosophy of mathematics, emphasizing (i) the objective nature of mathematical truths, (ii) the grounds for certain, a priori knowledge of them, (iii) the definition of number, (iv) a strategy for deriving the axioms of arithmetic from the laws of logic plus analytical definitions of basic arithmetical concepts, and (v) the prospect of extending the strategy to higher mathematics through the definition and analysis of real, and complex, numbers. After the virtual neglect of the Begriffsschrift by his contemporaries—due in part to its forbidding technicality and idiosyncratic symbolism—Frege presented the second stage of his project in remarkably accessible, and largely informal, terms in Die Grundlagen der Arithmetik (The Foundations of Arithmetic), published in 1884. In addition to being among the greatest treatises in the philosophy of mathematics ever written, this work is one of the best examples of the clarity, precision, and illuminating insight to which work in the analytic tradition has come to aspire. The third stage of the project is presented in a series of ground-breaking articles, starting in the early 1890s and continuing at irregular intervals throughout the rest of his life. These articles include, most prominently, “Funktion und Begriff” (“Function and Concept”) in 1891, “Über Begriff und Gegenstand” (“On Concept and Object”) in 1892, “Über Sinn und Bedeutung” (“On Sense and Reference”) in 1892, and “Der Gedanke” (“Thought”) in 1918. In addition to elucidating the fundamental semantic ideas needed to understand and precisely characterize the language of logic and mathematics, this series of articles contains important insights about how to extend those ideas to natural languages like English and German, thereby providing the basis for the systematic study of language, thought, and meaning. The final stage of Frege’s grand project is presented in his treatise Grundgesetze der Arithmetik (Basic Laws of Arithmetic), volumes 1 and 2, published in 1893 and 1903 respectively. In these volumes, Frege meticulously and systematically endeavors to derive arithmetic from logic together with definitions of arithmetical concepts in purely logical terms. Although, as we shall see, his attempt was not entirely successful, the project has proven to be extraordinarily fruitful.

The discussion in this chapter will not strictly follow the chronological development of Frege’s thought. Instead, I will begin with his language of logic and mathematics, which provides the starting point for developing his general views of language, meaning, and thought, and the fundamental notions—truth, reference, sense, functions, concepts, and objects—in terms of which they are to be understood. With these in place, I will turn to a discussion of the philosophical ideas about mathematics that drive his reduction of arithmetic to logic, along with a simplified account of the reduction itself. The next chapter will be devoted to critical discussions of Frege’s most important views, including the interaction between his philosophy of language and his philosophy of mathematics. In what follows I refer to Frege’s works under their English titles—with the exception of the Begriffsschrift, the awkwardness of the English translation of which is prohibitive.

2. THE LANGUAGE OF LOGIC AND MATHEMATICS

I begin with the specification of a simple logical language which, though presented in a more convenient symbolism than the one Frege used, is a direct descendant of his. The first step is to specify how the formulas and sentences of the language are constructed from the vocabulary of the language. After that, we will turn to Fregean principles for understanding the language.

THE SYMBOLIC LANGUAGE L_F

Vocabulary

Names of objects: a, b, c, …

Function signs: f( ), g( ), h( ), f′( , ), g′( , ) h′( , , ), … These stand for functions from objects to objects. Function signs are sorted into 1-place, 2-place, …, and n-place. One-place function signs combine with a single name (or other term) to form a complex term, 2-place function signs combine with a pair of names (or other terms) to form a complex term, and so on. Standardly, the terms follow the function sign, but in the case of some 2-place function signs—like ‘+’ and ‘×’ for addition and multiplication—the function symbol is placed between the terms.

Predicate signs: ( ) = ( ), P( ), Q( , ), R( , , ) … Predicate signs are sorted into 1-place, 2-place, etc. An n-place predicate sign combines with n terms to form a formula.

Terms

Individual variables (ranging over objects) are terms: x, y, z, x′, y′, z′, …

Names of objects are terms: a, b, c, …

Expressions in which an n-place function sign is combined with n terms are terms: e.g., if a and b are terms, f and h are 1-place function signs, and g is a 2-place function sign, then ^⌈f(a)^⌉, ^⌈g(a,b)^⌉, ^⌈h(f(a))^⌉, and ^⌈g(a,f(b))^⌉ are terms.

Definite descriptions are terms: If Φv is a formula containing the variable v, then ^⌈the v Φv^⌉ is a term.

Nothing else is a term.

Formulas

An atomic formula is the combination of an n-place predicate sign with n terms. Standardly the terms follow the predicate sign, but in the case of some 2-place predicate signs—like ‘( ) = ( )’ for identity—the terms are allowed to flank predicate sign.

Other (non-atomic) formulas

If Φ and Ψ are formulas, so are ^⌈~Φ^⌉, ^⌈(Φ v Ψ)^⌉, ^⌈(Φ & Ψ)^⌉, ^⌈(Φ → Ψ)^⌉ and ^⌈(Φ ↔ Ψ)^⌉. If v is a variable and Φ(v) is a formula containing an occurrence of v, ^⌈∀v Φ(v)^⌉ and ^⌈∃v Φ(v)^⌉ are also formulas. (Parentheses can be dropped when no ambiguity results.)

^⌈~Φ^⌉, which is read or pronounced ^⌈not Φ^⌉, is the negation of Φ; ⌈(Φ ∨ Ψ)^⌉, read or pronounced ^⌈either Φ or Ψ^⌉, is the disjunction of Φ and Ψ; ^⌈(Φ & Ψ)^⌉, read or pronounced ^⌈Φ and Ψ^⌉, is the conjunction of Φ and Ψ; ^⌈(Φ → Ψ)^⌉, read or pronounced ^⌈if Φ, then Ψ^⌉, is a conditional the antecedent of which is Φ and the consequent of which is Ψ; ^⌈(Φ ↔ Ψ)^⌉, read or pronounced ^⌈Φ if and only if Ψ^⌉, is a biconditional connecting Φ and Ψ; ^⌈∀v Φ(v)^⌉, read or pronounced ^⌈for all v Φ(v)^⌉, is a universal generalization of Φ(v); and ^⌈∃v Φ(v)^⌉, which is read or pronounced ^⌈at least one v is such that Φ(v)^⌉, is an existential generalization of Φ(v). ∀v and ∃v are called “quantifiers.”

Sentences

A sentence is a formula that contains no free occurrences of variables. An occurrence of a variable is free iff it is not bound.

An occurrence of a variable in a formula is bound iff it is within the scope of a quantifier, or the definite description operator, using that variable.

The scope of an occurrence of a quantifier ∀v and ∃v, or of the definite description operator, the v, is the quantifier, or description operator, together with the (smallest complete) formula immediately following it. For example, ∀x (Fx → Gx) and ∃x (Fx & Hx) are each sentences, since both occurrences of ‘x’ in the formula attached to the quantifier are within the scope of the quantifier. Note, in these sentences, that (i) Fx does not immediately follow the quantifiers because ‘(‘intervenes, and (ii) (Fx is not a complete formula because it contains ‘(‘without an accompanying ‘)’. By contrast, (∀x Fx → Gx) and (∀x (Fx & Hx) → Gx) are not sentences because the occurrence of ‘x’ following ‘G’ is free in each case. The generalization to ‘the x’ is straightforward.

Frege’s representational view of language provides the general framework for interpreting L_F. On this view, the central semantic feature of language is its use in representing the world. For a sentence S to be meaningful is for S to represent the world as being a certain way—which is to impose conditions the world must satisfy if it is to be the way S represents it to be. Since S is true iff (i.e., if and only if) the world is the way S represents it to be, these are the truth conditions of S. To sincerely accept, or assertively utter, S is, very roughly, to believe, or assert, that these conditions are met. Since the truth conditions of a sentence depend on its grammatical structure plus the representational contents of its parts, interpreting a language involves showing how the truth conditions of its sentences are determined by their structure together with the representational contributions of the words and phrases that make them up. There may be more to understanding a language than this—even a simple logical language like L_F constructed for formalizing mathematics and science—but achieving a compositional understanding of truth conditions is surely a central part of what is involved.

With this in mind, we apply Fregean principles to L_F. Names and other singular terms designate objects; sentences are true or false; function signs refer to functions that assign objects to the n-tuples that are their arguments; and predicates designate concepts—which are assignments of truth values to objects (i.e., functions from objects to truth values). A term that consists of an n-place function sign f together with n argument expressions designates the object that the function designated by f assigns as value to the n-tuple of referents of the argument expressions. Similarly, a sentence that consists of an n-place predicate P plus n names is true iff the names designate objects o₁, …, o_n and the concept designated by P assigns these n objects (taken together) the valu...