# Tim Button

Cambridge University
Faculty of Philosophy
PhD, 2011
Areas of Specialization
Areas of Interest
•  178
##### Wand/set theories
Here is a template for introducing mathematical objects: “Objects are found in stages. For every stage S: (1) for any things found before S, you find at S the bland set whose members are exactly those things; (2) for anything, x, which was found before S, you find at S the result of tapping x with any magic wand (provided that the result is not itself a bland set); you find nothing else at S.” This Template has rich applications, it realizes John Conway’s (1976) Mathematicians’ Liberation Moveme…Read more
•  189
##### The iterative conception of function and the iterative conception of set
Hilary Putnam once suggested that “the actual existence of sets as ‘intangible objects’ suffers… from a generalization of a problem first pointed out by Paul Benacerraf… are sets a kind of function or are functions a sort of set?” Sadly, he did not elaborate; my aim, here, is to do so on his behalf. There are well-known methods for treating sets as functions and functions as sets. But these do not raise any obvious philosophical or foundational puzzles. For that, we first need to provide a full-…Read more
•  254
##### Too clever by halving with Daniel Rothschild and Levi Spectre
We argue against the halving response to Sleeping Beauty. First, we outline an appealing constraint on probability assignments: the Principle of Irrelevant Information. Roughly, this says: if you don’t know whether C, but you would assign probability p to X regardless of whether C or not-C, then you should assign p to X. This Principle is deeply plausible, but we show that it contradicts halving. Second, we show that halving either violates solid statistical reasoning or draws absurd distinction…Read more
•  112
##### Naturalism, Realism, and Normativity, by Hilary Putnam, edited by Mario de Caro (review) Philosophy 92 (2): 305-15. 2017.
Hilary Putnam’s Realism with a Human Face began with a quotation from Rilke, exhorting us to ‘try to love the questions themselves like locked rooms and like books that are written in a very foreign tongue’. Putnam followed this advice throughout his life. His love for the questions permanently changed how we understand them. In Naturalism, Realism, and Normativity – published only a few weeks after his death – Putnam continued to explore central questions concerning realism and perception, from…Read more
•  82
##### Other minds and God: Russell and Stout on James and Schiller In Maria Baghramian & Sarin Marchetti (eds.), Pragmatism and the European Traditions: Encounters with analytic philosophy and phenomenology before the great divide, Routledge. pp. 86-109. 2018.
In 1907–8, Russell and Stout presented an objection against James and Schiller, to which both James and Schiller replied. In this paper, I shall revisit their transatlantic exchange. Doing so will yield a better understanding of Schiller’s relationship to a worryingly solipsistic brand of phenomenalism. It will also allow us to appreciate a crucial difference between Schiller and James; a difference which James explicitly downplayed.
•  81
##### Review of: Ontology After Carnap, by Stephan Blatti and Sandra Lapointe (eds.) (review) Notre Dame Philosophical Reviews. 2016.
Ontology after Carnap focusses on metaontology in the light of recent interest in Carnap’s ‘Empiricism, Semantics and Ontology’. That paper is at the centre of things, as it is where Carnap formulates his internal/external dichotomy. If you haven’t already encountered the dichotomy, then neither Ontology after Carnap, nor this review, is for you. My aim in this review is to try to tease out some of the book’s themes, thereby giving some sense of contemporary neo-Carnapianism.
•  309
##### Realistic structuralism's identity crisis: A hybrid solution Analysis 66 (3). 2006.
Keränen (2001) raises an argument against realistic (ante rem) structuralism: where a mathematical structure has a non-trivial automorphism, distinct indiscernible positions within the structure cannot be shown to be non-identical using only the properties and relations of that structure. Ladyman (2005) responds by allowing our identity criterion to include 'irreflexive two-place relations'. I note that this does not solve the problem for structures with indistinguishable positions, i.e. positio…Read more
•  284
##### A fictionalist theory of universals with Robert Trueman In Peter Fritz & Nicholas K. Jones (eds.), Higher-Order Metaphysics, Oxford University Press. forthcoming.
Universals are putative objects like wisdom, morality, redness, etc. Although we believe in properties (which, we argue, are not a kind of object), we do not believe in universals. However, a number of ordinary, natural language constructions seem to commit us to their existence. In this paper, we provide a fictionalist theory of universals, which allows us to speak as if universals existed, whilst denying that any really do.
•  171
##### Symmetric relations, symmetric theories, and Pythagrapheanism Philosophy and Phenomenological Research. 2022.
It is a metaphysical orthodoxy that interesting non-symmetric relations cannot be reduced to symmetric ones. This orthodoxy is wrong. I show this by exploring the expressive power of symmetric theories, i.e. theories which use only symmetric predicates. Such theories are powerful enough to raise the possibility of Pythagrapheanism, i.e. the possibility that the world is just a vast, unlabelled, undirected graph.
•  121
##### Forallx Adelaide with Antony Eagle and P. D. Magnus
An open source open access logic textbook
•  267
##### Mathematical Internal Realism In Sanjit Chakraborty & James Ferguson Conant (eds.), Engaging Putnam, De Gruyter. pp. 157-182. 2022.
In “Models and Reality” (1980), Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. Here, I will develop that sketch. By combining Putnam’s model-theoretic arguments with Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the Skolem-Gödel Antinomy. In brief: our mathematical concepts are perfectly precise; however, these perfectly precise mathematical concepts are manifested and acquired via a formal theory, which is und…Read more
•  386
##### Against Cumulative Type Theory with Robert Trueman Review of Symbolic Logic 15 (4): 907-49. 2022.
Standard Type Theory, STT, tells us that b^n(a^m) is well-formed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed type-restrictions: according to CTT, b^β(a^α) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT ’s type-restrictions are unjustifiable, the type-restrictions i…Read more
•  233
##### Level theory, part 1: Axiomatizing the bare idea of a cumulative hierarchy of sets Bulletin of Symbolic Logic 27 (4): 436-460. 2021.
The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: 'Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S.' Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories due to Scott, M…Read more
•  172
##### Level theory, part 2: Axiomatizing the bare idea of a potential hierarchy Bulletin of Symbolic Logic 27 (4): 461-484. 2021.
Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following bar-bones story introduces the idea of a potential hierarchy of sets: 'Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.' Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-…Read more
•  145
##### Level Theory, Part 3: A Boolean Algebra of Sets Arranged in Well-Ordered Levels Bulletin of Symbolic Logic 28 (1): 1-26. 2022.
On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; a…Read more
•  58
##### Forall x: Dortmund with Simon Wimmer, P. D. Magnus, Aaron Thomas-Bolduc, Richard Zach, J. Robert Loftis, and Robert Trueman
forall x: Dortmund is an adaptation and German translation of forall x: Calgary. As such, it is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity, the syntax of truth-functional (propositional) logic and truth-table semantics, the syntax of first-order (predicate) logic with identity and first-order interpretations, formalizing German in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with so…Read more
•  1729
##### Para Todxs: Natal - uma introdução à lógica formal with P. D. Magnus, Robert Loftis, Robert Trueman, Aaron Thomas Bolduc, Richard Zach, Daniel Durante, Maria da Paz Nunes de Medeiros, Ricardo Gentil de Araújo Pereira, Tiago de Oliveira Magalhães, Hudson Benevides, Jordão Cardoso, Paulo Benício de Andrade Guimarães, and Valdeniz da Silva Cruz Junior PPGFIL-UFRN. 2022.
Livro-texto de introdução à lógica, com (mais do que) pitadas de filosofia da lógica, produzido como uma versão revista e ampliada do livro Forallx: Calgary. Trata-se da versão de 13 de outubro de 2022. Comentários, críticas, correções e sugestões são muito bem-vindos.
•  150
##### Review of: Hilary Putnam on Logic and Mathematics, by Geoffrey Hellman and Roy T. Cook (eds.) (review) Mind 129 (516): 1327-1337. 2019.
Putnam’s most famous contribution to mathematical logic was his role in investigating Hilbert’s Tenth Problem; Putnam is the ‘P’ in the MRDP Theorem. This volume, though, focusses mostly on Putnam’s work on the philosophy of logic and mathematics. It is a somewhat bumpy ride. Of the twelve papers, two scarcely mention Putnam. Three others focus primarily on Putnam’s ‘Mathematics without foundations’ (1967), but with no interplay between them. The remaining seven papers apparently tackle unrelate…Read more
•  227
##### Review of: Reading Putnam, by Maria Baghramian (ed.) (review) Mind 123 (490): 569-575. 2014.
Reading Putnam consists largely of papers from the fantastic ‘Putnam @80’ conference (organised by Maria Baghramian in 2007) together with replies from Hilary Putnam. Given the diversity of Putnam’s work, the papers in this collection cover many different topics. This makes the collection difficulty to read but, ultimately, extremely rewarding. In this review, I focus on the contributions from Michael Devitt, Charles Parsons, Richard Boyd, Ned Block, Charles Travis and John McDowell, together wi…Read more
•  408
##### Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics with Sean Walsh Philosophia Mathematica 24 (3): 283-307. 2016.
This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
•  291
##### The Philosophical Significance of Tennenbaum’s Theorem with P. Smith Philosophia Mathematica 20 (1): 114-121. 2012.
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 does not help. We show this by examining a parallel argument, from …Read more
•  993
##### Forall x: Calgary. An Introduction to Formal Logic with P. D. Magnus, Robert Trueman, Richard Zach, and Aaron Thomas-Bolduc Open Logic Project. 2023.
forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), symbolizing English in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with some advanced topics such as modal logic, soundness, and fu…Read more
•  246
##### Wittgenstein on Solipsism in the 1930s: Private Pains, Private Languages, and Two Uses of ‘I’ Royal Institute of Philosophy Supplement 82 205-229. 2018.
In the early-to-mid 1930s, Wittgenstein investigated solipsism via the philosophy of language. In this paper, I want to reopen Wittgenstein's ‘grammatical’ examination of solipsism.Wittgenstein begins by considering the thesis that only I can feel my pains. Whilst this thesis may tempt us towards solipsism, Wittgenstein points out that this temptation rests on a grammatical confusion concerning the phrase ‘my pains’. In Section 1, I unpack and vindicate his thinking. After discussing ‘my pains’,…Read more
•  178
##### Grades of Discrimination: Indiscernibility, Symmetry, and Relativity Notre Dame Journal of Formal Logic 58 (4): 527-553. 2017.
There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discr…Read more
•  55
##### Philosophy and Model Theory with Sean P. Walsh Oxford University Press. 2018.
Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found on…Read more
•  217
##### Knot and Tonk: Nasty Connectives on Many-Valued Truth-Tables for Classical Sentential Logic Analysis 76 (1): 7-19. 2016.
Prior’s Tonk is a famously horrible connective. It is defined by its inference rules. My aim in this article is to compare Tonk with some hitherto unnoticed nasty connectives, which are defined in semantic terms. I first use many-valued truth-tables for classical sentential logic to define a nasty connective, Knot. I then argue that we should refuse to add Knot to our language. And I show that this reverses the standard dialectic surrounding Tonk, and yields a novel solution to the problem of ma…Read more
•  114
##### Hyperloops do not threaten the notion of an effective procedure Lecture Notes in Computer Science 5635 68-78. 2009.
This paper develops my (BJPS 2009) criticisms of the philosophical significance of a certain sort of infinitary computational process, a hyperloop. I start by considering whether hyperloops suggest that "effectively computable" is vague (in some sense). I then consider and criticise two arguments by Hogarth, who maintains that hyperloops undermine the very idea of effective computability. I conclude that hyperloops, on their own, cannot threaten the notion of an effective procedure.
•  1963
##### The Limits of Realism Oxford University Press UK. 2013.
Tim Button explores the relationship between words and world; between semantics and scepticism. A certain kind of philosopher – the external realist – worries that appearances might be radically deceptive. For example, she allows that we might all be brains in vats, stimulated by an infernal machine. But anyone who entertains the possibility of radical deception must also entertain a further worry: that all of our thoughts are totally contentless. That worry is just incoherent. We cannot, then, …Read more
•  310
##### Dadaism: Restrictivism as Militant Quietism Proceedings of the Aristotelian Society 110 (3pt3): 387-398. 2010.
Can we quantify over everything: absolutely, positively, definitely, totally, every thing? Some philosophers have claimed that we must be able to do so, since the doctrine that we cannot is self-stultifying. But this treats restrictivism as a positive doctrine. Restrictivism is much better viewed as a kind of militant quietism, which I call dadaism. Dadaists advance a hostile challenge, with the aim of silencing everyone who holds a positive position about ‘absolute generality’
•  473
##### Review of: Truth by Analysis: Games, Names, and Philosophy, by Colin McGinn (review) Analysis 73 (3): 577-580. 2013.
In Truth by Analysis (2012), Colin McGinn aims to breathe new life into conceptual analysis. Sadly, he fails to defend conceptual analysis, either in principle or by example.