The Principles of Mathematics Revisited by Jaakko Hintikka

[ Table of Contents | Quotes | Reviews | Errata ]

Jaakko Hintikka (Boston University, Philosopy Department).
The Principles of Mathematics Revisited.
Cambridge University Press, 1996, paperback edition 1998.
ISBN 0-521-49692-6 (1996 hardcover) [See this book at Amazon.com]
ISBN 0-521-62498-3 (1998 paperback) [See this book at Amazon.com]

Table of Contents

Introduction vii
1 The functions of logic and the problem of truth definition 1
2 The game of logic 22
3 Frege's fallacy foiled: Independence-friendly logic 46
4 The joys of independence: Some uses of IF logic 72
5 The complexities of completeness 88
6 Who's afraid of Alfred Tarksi? Truth definitions for IF first-order languages 105
7 The liar belied: Negation in IF logic 131
8 Axiomatic set theory: Fraenkelstein's monster? 163
9 IF logic as a framework for mathematical theorizing 183
10 Constructivism reconstructed 211
11 The epistemology of mathematical objects 235
Appendix (by Gabriel Sandu) 254
References 271
Index of names 281
Index of subjects and titles 285

Quotes

The following quotes are not (intended as) a replacement for buying the book and reading it in its entirety. On the contrary, I hope the quotes wet your appetite.

Introduction

p. vii
`The title of this work is modeled on Bertrand Russell's 1903 book The Principles of Mathematics.'

`In this book, I am hoping in the same spirit to prepare the ground for the next revolution ... in the foundations of mathematics.'

p. viii
`The dogmas that are ripe for rejection include the following commonplaces: The basic part of logic, the true elementary logic, is ordinary first-order logic, at least if you are a classicist.'

p. ix
`... I am trying to wake my fellow philosophers of mathematics from their skeptical slumbers and to point out to them a wealth of new constructive possibilities for the foundations of mathematics. This will be done by developing a new and better basic logic to replace ordinary first-order logic.'

p. x
`... it turns out that in one perfectly natural sense of the word[,] mathematics can in principle be done on the first-order level. In general my liberated first-order logic can do much more in mathematics than philosophers have recently thought of as being possible.'

`The strategy of this logician's liberation movement does not rely on complicated technical results, but on a careful examination of several of the key concepts we employ in logic and mathematics, including the following concepts: quantifier, scope, logical priority and logical dependence, completeness, truth, negation, constructivity, and knowledge of objects as distinguished from knowledge of facts.'

`... I am not developing my ideas here in the form of a mathematical or logical treatise, with full formal details and explicit proof. This work is a philosophical essay ...'

The Functions of Logic and the Problem of Truth Definition

p. 7
`... first-order logic ... very nearly looks like a logician's paradise.'

`... storm clouds begin to gather as soon as logicians venture beyond the enchanted land of first-order logic. And they have to do so, for unfortunately first-order logic soon turns out to be too weak for most mathematical purposes. Its resources do not suffice to characterize fully such crucial concepts as mathematical induction, well-ordering, finiteness, cardinality, power set, and so forth. First-order logic is thus insufficient for most purposes of actual mathematical theorizing.'

p. 8
`... here is the other, unjustly underemphasized, second function of logical concepts in mathematics. They are relied on essentially in the very formulation of mathematical theories. We can make the axioms of a typical mathematical theory say what they say only by using suitable logical concepts, such as quantifiers and logical connectives.

p. 9
`Many interesting phenomena in the foundations of mathematics become easier to understand in the light of the tension which there often is in evidence between this descriptive function and the deductive function of logic in mathematics.'

p. 16
`... the apparent dependence of Tarski-type truth definitions on set theory is in my view one of the most disconcerting features of the current scene in logic and in the foundations of mathematics. I am sorely tempted to call it ``Tarski's curse''.'

The Game of Logic

p. 23
`The deep idea in Wittgenstein is ... that descriptive meaning [of language] itself has to be mediated ... by language games.'

p. 29
`GTS [Game-Theoretical Semantics] is little more than a systemization of the mathematicians' time-honored ways of using quantifiers and of thinking of them.'

p. 33
`... in the rest of this book, it will turn out that ordinary first-order logic is a fool's paradise in that it offers a poor and indeed misleading sample of the variety of things that can happen in logic in general.'

p. 40
`GTS [Game-Theoretical Semantics] resoundingly vindicates this controverial axiom [of choice].'

Frege's Fallacy Foiled: Independence-Friendly Logic

p. 47
`... the real source of the expressive power of first-order logic lies not in the notion of quantifier per se, but in the idea of a dependent quantifier.'

`One important service which game-theoretical semantics performs is to explain the nature of the dependence in question. In game-theoretical terms, this dependence is informational dependence.'

p. 50
`We ... extend our familiar traditional first-order logic into a stronger logic which allows for informational independence where the received Frege-Russell notation forbids it. The result is a logic which will be called independence-friendly (IF) first-order logic.'

p. 51
`... IF first-order logic is a true Mafia logic: It is a logic you cannot refuse to understand.'

p. 54
`... the traditional notion of scope has two entirely different functions when it is applied to quantifiers.'
p. 66
`By means of IF first-order logic we can overcome most of the limitations of ordinary first-order logic ...'

The Joys of Independence: Some Uses of IF Logic

p. 73
`The ubiquity of the phenomenon of informational independence in natural language ... has been hidden by the fact that informational independence is not indicated in natural languages by any uniform syntactical device.'
p. 74
`... the mathematical notion of uniformity [of convergence e.g.] is closely related to the idea of informational independence.'

p. 78
`... Ramsey Theory is shot through with informationally independent quantifiers.'

p. 81
`... a parallel combination of architectures corresponds, not always to an ordinary first-order formula, bu (always) to an IF first-order formula. In this interesting sense, IF first-order logic is the logic of parallel processing.'

p. 83
`... we obtain striking evidence no only of the acceptability of the axiom of choice but also of its character as a purely logical principle. Indeed, the rule of functional instantiation can be considered a generalization of the axiom of choice ... [and] a generalization of the usual rule of existential instantiation in terms of individuals.'

The Complexities of Completeness

p. 90
`I do not think it is any exaggeration to say that the community of philosophers and mathematicians has still not managed to cope with the true import of Gödel's [incompleteness] discoveries and to draw the right consequences from them for the future development of the foundations of mathematics.'

p. 95
`... Gödel's incompleteness result casts absolutely no shadow on the notion of truth. All that it says is that the whole set of arithmetical truths cannot be listed, one by one, by a Turing machine.'

p. 96
`... in a sense Gödel's real message was merely the deductive incompleteness of elementary arithmetic. As such, his famous result affects only the prospects of our deductive mastery of elementary arithmetic and not our ability to deal with this branch of mathematics axiomatically or descriptively, unless such axiomatic treatments are constrained to use only ordinary first-order logic.'

Who's Afraid of Alfred Tarksi? Truth Definitions for IF First-Order Languages

p. 107
`It seems to me that the principle of compositionality was one of the most important presuppositions of Tarski's work on the concept of truth.'

`... truth can be defined compositionally only in conjunction with some other semantical concept like satisfaction.'

p. 109
`One of the most important kinds of impacts that IF first-order logic has on our ideas about logic and language, is that it shows once and for all the utter futulity of trying to abide by the pinciple of compositionality in our linguistic and logical theorizing.'

The Liar Belied: Negation in IF Logic

p. 133
`... in IF first-order languages, the law of excluded middle inevitably fails.'

p. 134
`The kind of failure of the principle of excluded middle [in IF logic] is an unavoidable combinatorial consequence of the way quantifiers and other concepts interact with each other. It has nothing to do with the limitations of human knowledge.'

p. 144
`... the liar paradox has more to do with our concept of negation than with our concept of truth.'

p. 154
`Our results ... show ... that in any sufficiently rich language there will be two different notions of negation present. Or if you prefer a different formulation, our ordinary concept of negation is intrinsically ambinguous.'

Axiomatic Set Theory: Fraenkelstein's Monster?

p. 163
`In this chapter, it will be argued that [the] priviliged role of axiomatic first-order set theory becomes extremely dubious in the light of the insights that the game-theoretical approach has yielded ...'

p. 176
`... axiomatic set theory cannot be true to its own intended interpretation. When we try to construct the liar paradox for axiomatic set theory, the liar turns out to be axiomatic set theory itself.'

p. 178
`... the usual techniques of axiomatic set theory are in a sense incompatible with IF first-order logic.'

p. 181
`... a major new way of improving existing techniques of set theoretical conceptualization and argumentation ... consists in departing radically from Cantor's requirement that set membership must be well-defined in the sense that the law of excluded middle does not apply to membership: besides definite members and nonmembers there are also potential elements which are neither.'

IF logic as a framework for mathematical theorizing

p. 198
`... the additional expressive power of IF first-order logic ... is combinatorial in nature, rather than set-theoretical. The overt novelty of IF logics is a greater freedom of the various dependence relations between quantifiers and connectives.'

Constructivism Reconstructed

The Epistemology of Mathematical Objects

p. 242
`... a rough-and-ready distinction can be made between constructivistic and intuitionistic approaches to the foundations of mathematics. The former emphasize the role of what a mathematician can in fact do, while the latter emphasize the role of what a mathematician can know.'

p. 243
`... one important difference between classical and intuitionistic mathematicians is that the former are satisfied with knowing mathematical truths while the latter also want to know mathematical objects.'

Reviews

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