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Overview
This book surveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in each. John Burgess considers every proposed fix, each with its distinctive philosophical advantages and drawbacks. These systems range from those barely able to reconstruct the rudiments of arithmetic to those that go well beyond the generally accepted axioms of set theory into the speculative realm of large cardinals. For the most part, Burgess finds that attempts to fix Frege do less than advertised to revive his system. This book will be the benchmark against which future analyses of the revival of Frege will be measured.
Product Details
| ISBN-13: | 9780691122311 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 07/25/2005 |
| Series: | Princeton Monographs in Philosophy , #48 |
| Edition description: | New Edition |
| Pages: | 272 |
| Product dimensions: | 5.50(w) x 8.50(h) x (d) |
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Read an Excerpt
Fixing Frege
Chapter One
Frege, Russell, and After
The great logician Gottlob Frege wrote three books, each representing a stage in a grand program for providing a logical foundation for arithmetic and higher mathematics. His Begriffsschrift (1879) introduced a comprehensive system of symbolic logic. The first half of his Grundlagen (1884) offered a devastating critique of previous accounts of the foundations of arithmetic, while the second half offered an outline of his own ingenious proposed foundation. The two volumes of the Grundgesetze (1893, 1903) filled in the technical details of his outline using his logical symbolism, and extended the project from arithmetic, the theory of the natural numbers, to mathematical analysis, the theory of the real numbers. Unfortunately, just as the second volume of the Grundgesetze was going to press Bertrand Russell discovered a contradiction in Frege's system.
Subsequent work in logic and foundations of mathematics largely bypassed the Grundgesetze until a couple of decades ago, when philosophers and logicians took a new look at Frege's inconsistent system, and recognized that more can be salvaged from it than had previously been thought. In these last years amended and paradox-freeversions of Frege's system have been produced; substantial portions of classical mathematics have been developed within such systems; and a number of workers have claimed philosophical benefits for such an approach to the foundations of mathematics.
The thought underlying the present monograph is that however wonderful the philosophical benefits of Frege-inspired reconstructions of mathematics, the assessment of the ultimate significance of any such approach must await a determination of just how much of mathematics can be reconstructed, without resort to ad hoc hypotheses, on that approach. What is under-taken in the pages to follow is accordingly a survey of various modified Fregean systems, attempting to determine the scope and limits of each. The present work, though entirely independent of Burgess and Rosen (1997), is thus in a sense a companion to the survey of various nominalist strategies in the middle portions of that work. As in that earlier survey, so in the present one, familiarity with intermediate-level logic is assumed. Boolos, Burgess, and Jeffrey (2002) contains more than enough background material, but neither that nor any other specific textbook is presupposed.
Every strategy, if it is to be consistent, must involve some degree of departure from Frege; but some of the approaches to be surveyed here stay much closer to Frege's own strategy than do others. It is sometimes suggested that the closer one stays to Frege, the greater the philosophical benefits. It is not my aim in the present work to argue for or against such claims. What I do insist is that any philosophical gains must be weighed against mathematical losses. For the survey to follow shows that some approaches yield much more of mathematics than others, and it often seems that the less one keeps of Frege, the more one gets of mathematics. Nonetheless, even in the last system to be considered here, which yields all of orthodox mathematics and more also, there remains one small but significant ingredient of Fregean ancestry.
As a necessary preliminary to the survey of attempts to repair Frege's system, that system must itself be reviewed. The underlying logic of the Begriffsschrift, the assumption added thereto in the Grundgesetze for purposes of developing mathematics, that development itself, the paradox Russell found in the system, and Russell's own attempts to repair it, must each be briefly examined, and the mathematical and philosophical goals a modified Fregean project, or for that matter any present-day foundational program, might set itself must be briefly surveyed.
1.1 Frege's Logic
While Frege is honored as a founder of modern logic, his system will not look at all familiar to present-day students of the subject. To begin with, Frege uses a non-linear notation that no subsequent writer has found it convenient to adopt, and that will not be encountered in any modern textbook. Since the present work is anything but an historical treatise, the notation will be ruthlessly modernized when Frege's system is expounded here.
But even when the notation is modernized, Frege's higher-order logic has a grammar that, though still simpler by far than the grammar of German or English or any natural language, is appreciably more complex than the grammar of the first-order logic of present-day textbooks. Nonetheless, after Frege's unfamiliar underlying grammatical and ontological assumptions have been expounded, only a few further explanations should be required to enable the reader familiar with first-order logic to understand higher-order logic.
Let us begin, then, with Frege's grammar. For Frege there are two grammatical categories or grammatical types of what he calls saturated expressions. The first, here to be called N, or the category of names, includes proper names such as "Plato," but also singular definite descriptions such as "the most famous student of Socrates" that are free from indexicals and designate an object (which may be a person or place rather than a "thing" in a colloquial sense), independently of context. The second, here to be called S, or the category of sentences, includes declarative sentences such as "Plato was the most famous student of Socrates" that are free from indexicals and have a truth-value, true or false, independently of context.
In addition there are many types of unsaturated expressions, with one or more gaps that, if filled in with an expression or expressions of appropriate type(s), would produce an expression of type N or S. Those that, thus filled in, would produce expressions of type S rather than type N will be of most interest here, and these may be called predicates in a broad sense. In a notation derived from the much later writers Kasimierz Ajdukiewicz and Yehoshua Bar-Hillel, an expression with k blanks in it, that if filled in with expressions of types [T.sub.1] ..., [T.sub.k] will produce an expression of type S, may be said to be a predicate of type S/[T.sub.1] ... [T.sub.k]. The simplest case is that of predicates in the narrow sense of expressions of type S/N. Some other cases are shown in table A at the back of the book.
The label relational predicate will be used for the two-place, three-place, and many-place types S/NN, S/NNN, and so on, with the number of places being mentioned explicitly when it is greater than two. The label higher predicate will be used for the second-level, third-level, and higher-level types S/(S/N), S/(S/(S/N)), and so on, with the number of the level being mentioned explicitly if it is higher than the second. Similarly with the label higher relational predicate, which even with just two places and even at just the second level covers a variety of types, including not only S/(S/N)(S/N) as shown in the table, but also, for instance, S/(S/NN)(S/NN), and such mixed types as S/(S/N)(S/NN) and even S/N(S/N). It is an instructive exercise to look for natural language examples illustrating such possibilities.
In Frege (1892), which after his three books is its author's most famous work, Frege introduced a distinction between the sense expressed and the referent denoted by an expression. The reader will not go far wrong who thinks of what Frege calls the "sense" of an expression of whatever type as roughly equivalent to what other philosophers would call its "meaning." What the "referent" of an expression is to be understood to be varies from grammatical type to grammatical type.
In the case of a proper name or singular definite description of type N, the referent is the object designated, the thing bearing the name or answering to the description. Clearly two expressions of type N, for instance, "the most famous student of Socrates" and "the most famous teacher of Aristotle," can have different senses even though they have the same referent-in this instance, Plato. Expressions with different senses but the same referent provide different "modes of presentation" of the same object.
The sense of a sentence of type S Frege calls a thought, and the reader will not go far wrong who thinks of what Frege calls a "thought" as roughly equivalent to what other philosophers call a "proposition." The referent of a sentence of type S Frege takes to be simply its truth-value. Obviously two sentences of type S, for instance, "Plato is the most famous student of Socrates" and "Plato is the most famous teacher of Aristotle," or "Plato is a featherless biped" and "Plato is a rational animal," can have different senses, though they have the same referent-in these instances, the truth-value true.
So much for the referents of complete or saturated expressions. As for the referents of incomplete or unsaturated expressions of types N/ or S/ , they are supposed to be incomplete, like the expressions themselves, containing gaps that when appropriately filled in will produce an object or a truth-value. The referents of expressions of type N/ ... Frege calls functions, and the referents of expressions of type S/ ..., that is, the referents of predicates, he calls concepts. Corresponding to the different grammatical types of predicates are different ontological types of concepts, including concepts of the narrowest, first-level, one-place kind, but also relational concepts and higher concepts.
For a concept of type S/N, if filling it in with a certain object produces the truth-value true, then the object is said to fall under the concept. For instance, assuming for the sake of example that Plato may truly be called wise, since the concept denoted by "... is wise," which is to say the concept of being wise, when filled in by the object denoted by "Plato," which is to say the object Plato, produces the truth-value denoted by "Plato is wise", which is to say the truth-value true, it follows that the object Plato falls under the concept of being wise. The same terminology is used in connection with other types of predicates: since Socrates taught Plato, the pair Socrates and Plato, in that order, fall under the relational concept of having taught; since Plato is an example of someone who is wise, the concept of being wise falls under the higher concept of being exemplified by Plato.
Two concepts are called coextensive if they apply to the same items, or in other words, if whatever falls under either falls under the other. Now the crucial difference between the referents of predicates, which is to say concepts, and the senses of predicates is that, according to Frege, coextensive concepts are the same. Thus if every featherless biped is a rational animal and vice versa, then though the senses of "... is a featherless biped" and "... is a rational animal" are different, the concept of being a featherless biped and the concept of being a rational animal are the same.
It sounds odd to say so, and the degree of oddity is a measure of the degree of departure of Frege's technical usage of "concept" from the ordinary usage of "concept," which tends to suggest the sense rather than the referent of a predicate. The label "concept" was in fact chosen by Frege well before he recognized the importance of systematically distinguishing sense and reference. By hindsight it seems he might have done well, after recognizing the importance of that distinction, to revise his terminology. I was tempted to substitute "classification" for "concept" in the foregoing short exposition, but have stuck with "concept" because it is still used by most of the writers with whose views I will be concerned.
In Fregean terminology, then, to the grammatical categories of names, sentences, and predicates there correspond the ontological categories of objects, truth-values, and "concepts." The formal language of Frege's higher-order logic is more complex than the formal languages of the first-order logic expounded in present-day textbooks in that it makes provision for predicates of all types, denoting concepts of all types.
So much for the grammar behind the logic. Turning to logic itself, modern textbooks introduce the student to the notions of a first-order language, with the symbol = for identity, and usually other non-logical symbols (n-place relation symbols, including 0-place ones or sentence symbols, and n-place function symbols, including 0-place ones or constants). Also introduced are rules of formation, and the notion of a term (built up from variables and constants using function symbols), atomic formula (obtained by putting terms in the places of relation symbols), and formula (built up from atomic formulas using the logical operators ~, &, v, -> , [left and right arrow], [for all], [there exists]), along with the ancillary notions of free and bound variables, and open and closed formulas (with some variables free and with all variables bound, respectively).
Also introduced are certain rules of deduction, which may take widely different forms in different books-the different formats for deductions used in different books going by such names as "Hilbert-style" and "Gentzen-style" and "Fitchstyle"-but which lead in all books to equivalent notions of what it is for one formula to be deducible from others, and hence to equivalent notions of a theory, consisting of all the formulas, called theorems of the theory, that are deducible from certain specified formulas, called the non-logical axioms of the theory.
Familiarity with all these notions must be presupposed here, but certain conventions that are not covered in all textbooks may be briefly reviewed. To begin with, it proves convenient in practice, when writing out formulas, to make certain departures from what in principle ought to be written. In particular, for the sake of conciseness and clarity certain defined symbols are added to the primitive or official symbols. The simplest case is the introduction of [not equal to] for distinctness by the usual definition, as follows:
(1) x [not equal to] y [left and right arrow] ~ x = y
What it means to say that [not equal to] is "defined" by (1) is that officially [not equal to] isn't part of the notation at all, and that the left side of (1) is to be taken simply as an unofficial abbreviation for the right side of (1). The "definition" (1) is thus not a substantive axiom, but merely an abbreviation for a tautology of the form p [left and right arrow] p. The slash notation for negation may also be used with certain non-logical relation symbols when their shape permits.
Another slightly less simple case, often not covered in textbooks, is that of [there exists]! for unique existence, with two of the several equivalent usual definitions being as follows:
(2a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(2b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Here the first and second conjuncts of the conjunction on the right in (2a) are called the existence and uniqueness clauses, respectively.
(Continues...)
Excerpted from Fixing Frege by John P. Burgess Copyright © 2005 by Princeton University Press. Excerpted by permission.
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Table of Contents
Acknowledgments ix
CHAPTER 1: Frege, Russell, and After 1
CHAPTER 2: Predicative Theories 86
CHAPTER 3: Impredicative Theories 146
Tables 215
Notes 227
References 241
Index 249
What People are Saying About This
I suspect that this will become a must-read among those working in the philosophy of mathematics and the foundations of mathematics. It will most certainly be a significant contribution to the field.
— Stewart Shapiro, Ohio State University
"This book is without match. I suspect that it will be used both as a general introduction to the subject and as a source on particular topics for years to come."—Kit Fine, New York University"I suspect that this will become a must-read among those working in the philosophy of mathematics and the foundations of mathematics. It will most certainly be a significant contribution to the field."—Stewart Shapiro, Ohio State University
This book is without match. I suspect that it will be used both as a general introduction to the subject and as a source on particular topics for years to come.
— Kit Fine, New York University