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During his internal realist period, Putnam attack "metaphysical realism". In so doing, he drew inspiration both from Carnap's methodological solipsism and from Dummett's antirealism. However, Putnam came to argue ('Between Scylla and Charybdis') that both of these latter positions collapse into metaphysical solipsism. When the charge of solipsism is construed sufficiently broadly, it is exactly right.

Tennenbaum's Theorem yields an elegant characterisation of the standard model of arithmetic. Several authors have recently claimed that this result has important philosophical consequences: in particular, it offers us a way of responding to model-theoretic worries about how we manage to grasp the standard model. We disagree. If there ever was such a problem about how we come to grasp the standard model, then Tennenbaum's Theorem doesn't help. We show this by examining a parallel argument, from a simpler model-theoretic result.

There are important analogies between brain-in-vat scepticism and Skolemite scepticism (as noted by Putnam, Tymoczko and AW Moore). So: can Skolemite scepticism be defeated by a Putnam-style brain-in-vat argument? Pursuing this question helps to uncover a few disanalogies between our two varieties of scepticism. Nonetheless, it seems that we can still use a Putnam-style argument to defend the use of "full" second-order logic.

Metaphysicians sometimes worry that their favourite disputes are superficial, thanks to conceptual relativity. At the moment, the spectre of conceptual relativity wears the clothing of quantifier variance. I find this decidedly odd. Advocates of quantifier variance employ "cookie-cutter" metaphors, and these are essentially incompatible with conceptual relativity. Putnam pointed this out back in 1987, and advocates of quantifier variance would do well to remember it. However, Putnam's own relationship with conceptual relativity is equally strange. Putnam endorsed conceptual relativity whilst rejecting cookie-cutter metaphors; but he also advocated an interpretationist philosophy of mind. To avoid the apparent inconsistency, Putnam proposed that an interpreter might be at home in several different conceptual schemes. Unfortunately, this did not help.

In the patchwork of Hilary Putnam's various positions, the longest thread is his attack on positivism. Putnam also attacks justificationism; that is, the philosophical project of explaining truth in terms of justification. He claims that it is insufficiently distant from positivism, and so likewise cannot refute the charge of solipsism. His most recent justificationist target (2007) is Dummett's antirealism. During his internal realist period, Putnam had regarded Dummett as a comrade-in-arms against metaphysical realism. Dummett (2007) and Putnam now talk past each other. However, when Putnam's criticisms of Dummett are placed alongside Carnap's self-criticism, we obtain a fundamental problem for Dummettian antirealism.

Putnam used a permutation argument to attack the correspondence theory of truth and to argue for internal realism. This argument consists in reshuffling the relationship between our words (like "Tabby") and the world (maybe Tabby, maybe Tabby*), whilst preserving the truth-values of all the sentences of our best theory. But what fixes the truth-values of those sentences in the first place? For Putnam, the answer was: how things seem to me to be. Strangely, though, for Putnam, my own survival is part of how things seem to me to be. Solipsism looms.

Published as 'Dadaism: restrictivism as militant quietism'.

Published as 'Hyperloops do not threaten the notion of an effective procedure'.

Putnam's responded to the problem of referential underdetermination by advocating internal realism. But might the correct response to the problem of referential underdetermination sometimes be fictionalism? Fictionalists think that we do not literally refer to the entities of our favourite theories; so one might expect that fictionalists face no problems concerning referential underdetermination. In fact, we can ask fictionalists two kinds of questions about referential underdetermination. First, we might ask how reference is fixed within the context of a fiction: how it is that, when we pretend that 2 exists, we are able to pretend that '2' genuinely refers to it. Second, we might ask how we are able to refer to one fiction rather than another: how it is that we are all engaged in the same pretence, say. I show that there is a coherent version of fictionalism that can answer both questions. However, it is not clear that this version of fictionalism is really any different from Putnam's internal realism.

Traditional versions of objects-realism face a problem of referential underdetermination. The problem arises from the fact that, if a theory has any models, it has more than one model. Realistic structuralism (also known as ante rem structuralism) has been thought to solve this problem. Realistic structuralists think that our theories describe structures, which are genuine objects, and that structures contain positions, which are also genuine objects. By positing a single, real structure as the referent-base of our theory, rather than multiple different models, we may be able to obtain the fixity of reference. I want to dash these hopes; at least, for realistic structuralism in the philosophy of mathematics. Realistic structuralism does not solve problems of referential underdetermination. Reference and correspondence are as indeterminate for realistic structuralists as they are for more traditional realists.

Published as 'SAD computers and two versions of the Church-Turing Thesis'.

Michael Dummett has argued that certain mathematical concepts are indefinitely extensible. The purpose of this paper is to ask what he means. The main problem is that thinking about indefinitely extensible concepts seems to force us to make sense of an 'indefinite totality' of objects falling under that concept. This prompts the question: What is an indefinite totality? I shall show that there can be no formal theory of indefinite totalities. I shall then review various informal explications of the notion of an indefinite totality, and shall find them all lacking. My conclusion is that thinking about indefinite totalities will not help us to understand indefinite extensibility.

Putnam's responded to the problem of referential underdetermination by advocating internal realism. But might the correct response to the problem of referential underdetermination sometimes be fictionalism? Fictionalists think that we do not literally refer to the entities of our favourite theories; so one might expect that fictionalists face no problems concerning referential underdetermination. In fact, we can ask fictionalists two kinds of questions about referential underdetermination. First, we might ask how reference is fixed within the context of a fiction: how it is that, when we pretend that 2 exists, we are able to pretend that '2' genuinely refers to it. Second, we might ask how we are able to refer to one fiction rather than another: how it is that we are all engaged in the same pretence, say. I show that there is a coherent version of fictionalism that can answer both questions. However, it is not clear that this version of fictionalism is really any different from Putnam's internal realism.

Published as 'Spotty scope and our relation to fictions'.

Published as 'SAD computers and two versions of the Church-Turing Thesis'.