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This chapter summaries the key thesis of the book: that, given the Sense/Circumstance/World framework in the philosophy of language, we can show that mathematics expresses objective truths, but not by dint of delineating a mind-independent reality. Neo-formalism is held to incorporate sound aspects of rival views, such as classic formalism; neo-logicism, in the idea that stipulation of axioms grounds mathematical truth; constructivism, insofar as the constructivist links truth with provability; strict finitism to the extent that the strict finitist emphasizes that the ontology of mathematics can include only a finite corpus of concrete tokens. Neo-formalism even incorporates elements of platonism, insofar as it upholds the objectivity of mathematical truth. However neo-formalism, in rejecting the platonistic realm of an external mathematical ontology as a mythological projection of human activity, avoids the crippling metaphysical and epistemological problems of platonism.

This chapter lays out the central theme of the book, the interaction between Frege’s understanding of conceptual analysis and his understanding of logic. It also sketches the contents of the succeeding chapters.

Frege and Russell were particularly important influences on the early Wittgenstein, and this chapter presents a brief sketch of some themes in their work. Two themes in particular are emphasized: (1) the notion that philosophical problems can be exposed as pseudo-problems through logical analysis of propositions; and (2) the question of the nature of logical truth. Later chapters in the book argue that Wittgenstein develops in radical ways the thought of Frege and Russell on these themes.

If our ability to count depends on the structure of space and time, this may explain its applicability to the things we intuit in space and time, but it will also limit its applicability to just those things: we shall, in short, be left without the ability to count anything that is not made up of spatio-temporal elements. This evidently did not bother Kant, but it has seemed implausible to many subsequent thinkers. The most notable of these was Frege, who published Die Grundlagen der Arithmetik (The Foundations of Arithmetic), a book arguing against Kant's conclusion, in 1884. Frege was the first to put forward the thesis, now known as logicism, that logic is capable of grounding mathematical truths without thereby rendering them wholly trivial.

According to Gottlob Frege, his logicism died when it was discovered that the underlying theory of extensions is inconsistent. The neo-logicist attempts to found mathematics on other abstraction principles, such as the so-called Hume’s principle that two concepts have the same number if and only if they are equinumerous. This chapter discusses the state of neo-logicism, responding to various objections that have been raised against it.

Roughly, logicism is the view that mathematics is logic. This chapter identifies several distinct logicist theses, and shows that their truth-values can be established on minimal assumptions. There is also a discussion of “Neo-Logicism.”

This chapter addresses the question of what led Wittgenstein to study philosophy with Russell in Cambridge. Since the narrative of Wittgenstein's life before 1911 is well summarized in available biographies, the focus will be on a few points including Wittgenstein's early life, his experiences as a research student at Manchester, his interest in the philosophy of mathematics, logicism, and Russell's paradox.

The chapter gives some background on the mathematical achievements of Cantor and Dedekind, and describes how Russell aimed to show that all mathematics can be derived from a purely logical starting point (logicism). But it also describes Russell’s discovery of what he always called ‘the contradiction’, which we call ‘Russell’s paradox’, which presented a serious obstacle to that derivation. It gives Russell’s first reactions, namely the early theory of types sketched in 1903, the ‘zigzag’ theory, and the theory of limitation of size. The last is compared with the set theory of Zermelo-Fraenkel that is common nowadays.

Relativism is distinguished from pluralism; mathematical theses which are true in some systems but not others are held to express different propositions which different truth values in each. Comparisons are drawn with if-thenism, postulationism, and deductivism. A number of objections are tackled: that proofs need not be formal, that mathematicians believe theses without having proofs, that axiom systems can be inadequate and incomplete, that any consistent sentence counts as a mathematical truth, since it is provable from some system. In response, further comparisons are drawn with neo-logicism, and the role of abstraction principles such as Hume's principle discussed. The superiority of neo-formalism is urged, in the treatment for example of the ‘Julius Caesar problem’ and the problem of pairwise inconsistent abstraction principles. Finally Gödel's incompleteness theorem and the problem of true, but unprovable, Gödel sentences, is raised. A quick response by the neo-formalist is rejected, and the difficulty set aside until the last chapters.

This chapter articulates a major theme in Russell's thought: his conception of logic and of the philosophy of logic. It begins by raising the question of the philosophical significance that logicism, the reduction of mathematics to logic, had for Russell when he first developed that doctrine. The answer is that it was part of a complex argument against Kant and post-Kantian Idealism. For this argument to work, logic must be thought of as made up of absolute and unconditioned truths. A certain conception of logic is thus implicit in the philosophical use that Russell makes of logicism. The chapter articulates this conception and contrasts it with a widely held modern conception according to which the central notion is truth in an interpretation, rather than truth tout court; the notion of an interpretation is alien to Russell's thought. It is argued that given his general conception of logic, it is natural, perhaps inevitable, that logic will be higher-order logic, equivalent to set theory. Russell's use of logicism, however, is cast in doubt by the need to accommodate the paradox that bears his name. The theory of types undermines his conception of logic as consisting of universal and unconditioned truths. The infinitude of objects can no longer be proved, but is taken as an explicit assumption when needed; this threatens the idea that it is indeed mathematics which is being reduced to logic. The magnificent intellectual achievement of Principia Mathematica is thus, cut off from the philosophical motivations that lay behind Russell's initial formulation of logicism.

Numbers have many puzzling features. Their properties are mostly essential to them, but they exist in all possible worlds. Number theory seems a priori, yet it makes existence claims and existence (setting aside the Cogito) is not supposed to be a priori knowable. If-thenism can perhaps explain the felt a priority, but it makes numerical truth relative where it seems absolute. A figuralist solution is proposed: ‘2 + 3 = 5’ seems necessary, a priori, and absolute because it has a logical truth as its assertive content. A rule that associates logical truths with each arithmetical truth is given, and also a rule that associates a logical truth (modulo concrete combinatorics) with each truth about hereditarily finite impure sets. The view that emerges takes something from Frege and something from Kant; one might call it Kantian logicism. The view is Kantian because it sees mathematics as arising out of our representations. Numbers and sets are ‘there’ because they are inscribed on the spectacles through which we see other things. It is logicist because the facts seen through our numerical spectacles are facts of first-order logic.

This chapter explains the role of conceptual analysis in Frege’s logicist project. It is explained that Frege employs conceptual analyses in order to reduce arithmetical truths to complexes of relatively simple components in order to facilitate the proof of those truths. The importance of the adequacy of the conceptual analyses is emphasized, and Frege’s view of that adequacy is explored in a preliminary way by looking carefully at a selection of Frege’s analyses in Begriffsschrift, Grundlagen, and Grundgesetze. Some of the difficulties surrounding Frege’s conception of analysis are explained, and it is pointed out that a clear conception of Frege’s understanding of analysis requires a clear understanding of his notion of thought, the subject of Chapter 2.

This chapter provides a broad overview of the philosophy of mathematics and the philosophy of logic. It gives brief coverage to the various issues and positions, such as platonism or realism, varieties of nominalism, or anti-realism, logicism, intuitionism, empiricism, and structuralism.

The common thread running through the logicism of Frege, Dedekind, and Russell is their opposition to the Kantian thesis that our knowledge of arithmetic rests on spatio-temporal intuition. Our critical exposition of the view proceeds by tracing its answers to three fundamental questions: (1) What is the basis for our knowledge of the infinity of the numbers? (2) How is arithmetic applicable to reality? (3) Why is reasoning by induction justified?

Mathematics poses a difficult problem for methodological naturalists, those who embrace scientific method, and also for ontological naturalists who eschew non-physical entities such as Cartesian souls. Mathematics seems both essential to science but also committed to abstract non-physical entities while methodologically it seems to have no place for experiment or empirical confirmation. The chapter critically reviews a number of responses naturalists have made including logicism, Quinean radical empiricism, and Penelope Maddy’s variant thereof and suggests some further problems both for ontological and for methodological naturalists.

The notion of operation plays a pivotal role in the symbolism of the Tractatus Logico-Philosophicus: on the one hand, truth-functions are based on truth-operations; on the other, numbers are exponents of an operation. Considering that operations seem to be so central, it is amazing to notice how little is understood of Ludwig Wittgenstein's remarks: not enough attention has been paid in the past to the curious piece of symbolism of 6.01. What Wittgenstein says about the notion of operation very much resembles informal explanations of the notion of operator. Two differences with the set-theoretic notion of function were mentioned in Chapter 1: firstly, an operator is defined by describing how it transforms an input without defining the set of inputs, that is without defining its domain. Secondly, there is no restriction on the domain of some operators.