The introduction motivates the book’s arguments by showing how mental illness stigma remains pervasive despite greater awareness of mental health issues and more resources directed at mental health treatment and destigmatization. The forms of mental illness stigma most commonly expressed are stigma against people with severe mental illness who are perceived as homeless, and internalized stigma that people with mental illness project onto themselves. Mental illness stigma arises as a reaction to the violation of social norms of what a human being should be in the Western world in the twenty-first century. I give an account of stigma as the devaluing and discrediting of a person based on possessing a social trait that is seen as violating social norms, constituting a relationship of power. Components of stigma include labeling, stereotyping, prejudice, moral distancing, social exclusion, status loss, dehumanization, microaggressions, discrimination, and epistemic injustice. The chapter ends with a description of the book’s scope, methodology, and chapter outline.

In Chapter 3, we return to several questions asked at the end of Chapter 1: what, if any, are the grounds for the introduction of the rules we currently have in our mathematics books? Are they adopted arbitrarily? Why these rules, and not others? How does the introduction of such rules proceed? And so on. However, the chapter looks only at the ‘basic’ rules, such as the simplest addition rules (e.g., 2 + 2 = 4), the rules in the multiplication table (e.g., 3 × 2 = 6), and other simple multiplications, e.g., 21 × 36 = 756. These rules are ‘basic’ insofar as they emerge by ‘hardening’ certain empirical regularities displayed by people’s behavior in arranging and counting manageable physical objects; so, this is in essence a genealogical story. This chapter also serves as preparation for the discussion in the next chapter, where we tackle the possible challenges to these rules (by ‘skeptics’), with the aim of dissolving the problem of mathematical skepticism: what if we have always been wrong about our basic arithmetic?

Conceiving of mathematics as a network of rules may prompt worries about their arbitrariness. In the previous chapter, it was argued that these concerns can be appeased by Wittgenstein’s genealogical-anthropological approach, where the origin of these basic rules is found in human practices, which in turn presuppose large-scale and long-term consensus of action. Chapter 4 continues this discussion by raising, and answering, additional questions, e.g., about the conditions in which the basic rules can be modified, or even dropped altogether and replaced by other rules. It makes the case that the problem-question ‘what if we have always been wrong about basic arithmetic, e.g., that 25 × 25 = 625?’, i.e., the problem of mathematical skepticism – can be eliminated as well. The last section takes up, and mostly rejects, an idea often mentioned in this context, that Wittgenstein was a social constructivist about mathematics.

Chapter 1 is mainly devoted to articulating Wittgenstein’s normativism. This label reflects what I consider to be Wittgenstein’s fundamental insight: that mathematical sentences actually serve to formulate norms, or rules; furthermore, as rules, they lack truth-values. (However, in my reading, he also accepts that these sentences can make truth-apt assertions.) The chapter also sketches the two doctrines relevant to the subsequent discussion – platonism and conventionalism – and highlights the eliminativist aspect of normativism while emphasizing its therapeutic consequences: by eliminating problems, one achieves ‘peace of mind.’

Chapter 5 examines central aspects of Wittgenstein’s view on mathematical proof, as integrated within his normativist eliminativism. It elucidates and defends the claim that “the proof creates a new concept.” His (‘interactionist’) conception of proof is spelled out, as a clarification of the idea that proofs “change the grammar of our language.” The main outcome is an explication as to why the puzzle generated by the seemingly absolute power of a mathematical proof – to produce an unbreakable link between the (concepts appearing in the) premises and the (concepts in the) conclusion of a mathematical argument – vanishes. The chapter appeals to examples from both advanced and ‘pure’ mathematics. An important concern will be expounding on Wittgenstein’s point that proofs ensure the objectivity of (higher) mathematics, since for him proofs justify changes in the network of mathematical norms. Advanced, new rules are introduced on the basis of proofs, making the management of the network a nonarbitrary process. The chapter also argues for Wittgenstein’s priority over Lakatos in advocating interactionism.

The Introduction sets the stage for the project. It situates the idea of the book in the scholarly landscape, by explicating the main directions of the present account and how it relates to other accounts. It briefly introduces some of the central positions appearing in the text (e.g., ‘normativism,’ ‘eliminativism,’ ‘(non)revisionism’), presents the methodological commitments of the work, and gives summaries of the chapters.

Chapter 2 further articulates the normativist view, but its central task is to demonstrate its effectiveness; that is, to illustrate the eliminativist aspect of normativism by showing how normativists dissolve three central problems in the philosophy of mathematics. It first deals with the problem of the specialness of mathematical propositions, traditionally considered necessarily true. Then it tackles the so-called Benacerraf dilemma, the (supposed) difficulty of knowing mathematical truths – but before that, it clarifies the notion of a pseudo-problem as understood by Wittgenstein. In the last sections, the chapter dissolves a third problem, the traditional ontological question ‘do mathematical objects exist?’ More concretely, it tackles three incarnations of realism: traditional platonism, Fregean logicist realism, and the species of realism due to Quine and defended on the basis of indispensability considerations.

Chapter 7 returns to a key issue from the previous chapter: should Cantor’s diagonal construction (CD) be considered a genuine real number? To answer, we need first to clarify the distinction underlying Wittgenstein’s take on Cantor’s proof, between ‘intensionalism’ and ‘extensionalism.’ The main message here is, again, ultimately therapeutic-liberatory, and not revisionist (only reluctantly so): we are not forced to accept that CD is a genuine real, and hence to say that there are more reals than naturals. We may say this, and introduce a new concept of a real, but we don’t have to. That CD is a genuine irrational is not an epochal discovery, but an invention, a technical stipulation. A background difficulty in this chapter concerns the overarching philosophical problem Wittgenstein grapples with in RFM II. I propose that this is the question, ‘What is the significance of Cantor’s discovery of the hierarchy of infinities?’ – and I argue that he saw it as ill-posed. Since mathematicians are inventors and not discoverers (RFM I-168), Cantor did not discover anything; so, the question is based on a confused presupposition and should be discarded.

Chapter 8 shows how Wittgenstein’s normativism integrates his thoughts on the Frege–Russell definition of number, and on the previously discussed topic of the infinite. The key concept under scrutiny is the notion of one-to-one correspondence; the main issue is the ‘grammar’ of the sentence ‘Sets A and B can be one-to-one correlated.’ I will contend that, for Wittgenstein, it has a normative status, despite its assertoric form; such sentences formulate permissions, not assert facts. And this means that the question, ‘Can sets A and B be one-to-one correlated?’ is fundamentally incomplete. What is missing is a specification of the permissible methods for achieving the correlation. We begin by discussing Wittgenstein’s discontent with the logicist definition of number, and note that the motivation to address this issue is traceable to his overall therapeutic concerns. His reservations about this key component of the logicist project are due to his conviction such a definition may not be needed, so the effort to find it, and thus answer the venerable question ‘What is a number?’, can be spared.

This final chapter provides a summary of the book, together with some additional clarifications. Along the way, I will highlight several open questions and difficulties faced by the present account – and, if the account offered here is accurate, by Wittgenstein’s own position.

One aim of Chapter 6 is to tie together several strands of argument developed in the previous chapters. First, we shall return to the “urge” (to assimilate, to conflate the normative and the assertoric) signaled in the first two chapters. Here we will explore the hypothesis that the urge ultimately amounts to our overwhelming desire, and temptation, to experience what I call in Chapter 1 ‘metaphysical awe.’ Second, following some remarks from Lectures and Conversations on Aesthetics and Religious Belief, we will take one’s succumbing to such temptation as an indication of a seriously flawed ‘style of thinking.’ However, the main aim of the chapter is to connect the hypothesis above with a reading of Wittgenstein’s controversial views on Cantor’s proof. It is argued that what primarily concerns Wittgenstein are the assimilationist tendencies embodied in transfinite set theory, and its encouragement of a sensationalist style of thinking (i.e., its serving as a source of metaphysical awe). Toward the end of this chapter, I begin examining his often decried ‘revisionism’ – the idea that he proposes revisions to mainstream mathematical results, including Cantor’s.