# Indeterminacy in Mathematics

Edited by Rafal Urbaniak (Uniwersytetu Gdanskiego, Uniwersytetu Gdanskiego)
Assistant editors: Pawel Pawlowski, Sam Roberts
 Summary A sentence C is independent of a theory T iff neither C, nor the negation of C is derivable from T. A theory is negation-complete iff no sentence in its language is independent of it. Some of key results in metamathematics are independence theorems. According to arithmetical incompleteness theorem, no consistent (recursively axiomatizable) extension of a relatively weak arithmetic is negation-complete. Another important independence result is the independence of the Conituum Hypothesis of the axioms of standard set theory. (There are numerous other examples in analysis, combinatorics, group theory and set theory.) Independence results seem to have impact on philosophical views on mathematical truth and mathematical knowledge. Are sentences independent of mainstream theories determinately true or false and why? If yes, how can we know, which is it? If no, what philosophical views about mathematics are consistent with this view and how are they motivated?
 Key works Gödel 1931, Gödel 1940,  Gödel 1947, .Cohen 1963, Feferman manuscript and Feferman et al 1999. For an in-depth study of arithmetical incompletness, see Franzén 2003.
 Introductions A great introduction to arithmetical incompleteness theorems and related issues is Smith 2007. A more advanced book is Lindstrom 2002. Franzén 2005 is invaluable. See also Feferman manuscript and Feferman manuscript. As for set-theoretic indeterminacy, see Koellner 2010 and references therein.
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1. It's currently fashionable to take Putnamian model theoretic worries seriously for mathematics, but not for discussions of ordinary physical objects and the sciences. But I will argue that (under certain mild assumptions) merely securing determinate reference to physical possibility suffices to rule out nonstandard models of our talk of numbers. So anyone who accepts realist reference to physical possibility should not reject reference to the standard model of the natural numbers on Putnamian model theoretic grounds.

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2. Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in numerical (...)

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3. Divergent Potentialism: A Modal Analysis With an Application to Choice Sequences.Ethan Brauer, Øystein Linnebo & Stewart Shapiro - forthcoming - Philosophia Mathematica.
Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripke semantics for intuitionistic logic, we overcome those (...)

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4. Mathematical Internal Realism.Tim Button - 2022 - In Sanjit Chakraborty & James Ferguson Conant (eds.), Engaging Putnam. De Gruyter. pp. 157-182.
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 understood in terms of a computable system of proof, (...)

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5. Arithmetic is Determinate.Zachary Goodsell - 2022 - Journal of Philosophical Logic 51 (1):127-150.
Orthodoxy holds that there is a determinate fact of the matter about every arithmetical claim. Little argument has been supplied in favour of orthodoxy, and work of Field, Warren and Waxman, and others suggests that the presumption in its favour is unjustified. This paper supports orthodoxy by establishing the determinacy of arithmetic in a well-motivated modal plural logic. Recasting this result in higher-order logic reveals that even the nominalist who thinks that there are only finitely many things should think that (...)

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6. The Price of Mathematical Scepticism.Paul Blain Levy - 2022 - Philosophia Mathematica 30 (3):283-305.
This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. -/- Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, choice and consistency should all be aligned.

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7. Computational Indeterminacy and Explanations in Cognitive Science.Philippos Papayannopoulos, Nir Fresco & Oron Shagrir - 2022 - Biology and Philosophy 37 (6):1-30.
Computational physical systems may exhibit indeterminacy of computation (IC). Their identified physical dynamics may not suffice to select a unique computational profile. We consider this phenomenon from the point of view of cognitive science and examine how computational profiles of cognitive systems are identified and justified in practice, in the light of IC. To that end, we look at the literature on the underdetermination of theory by evidence and argue that the same devices that can be successfully employed to confirm (...)

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8. Pluralities as Nothing Over and Above.Sam Roberts - 2022 - Journal of Philosophy 119 (8):405-424.
This paper develops an account of pluralities based on the following simple claim: some things are nothing over and above the individual things they comprise. For some, this may seem like a mysterious statement, perhaps even meaningless; for others, like a truism, trivial and inferentially inert. I show that neither reaction is correct: the claim is both tractable and has important consequences for a number of debates in philosophy.

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9. Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?Nicolas Gisin - 2021 - Erkenntnis 86 (6):1469-1481.
It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is (...)

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10. Carnap and Beth on the Limits of Tolerance.Benjamin Marschall - 2021 - Canadian Journal of Philosophy 51 (4):282–300.
Rudolf Carnap’s principle of tolerance states that there is no need to justify the adoption of a logic by philosophical means. Carnap uses the freedom provided by this principle in his philosophy of mathematics: he wants to capture the idea that mathematical truth is a matter of linguistic rules by relying on a strong metalanguage with infinitary inference rules. In this paper, I give a new interpretation of an argument by E. W. Beth, which shows that the principle of tolerance (...)

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11. Categoricity by convention.Julien Murzi & Brett Topey - 2021 - Philosophical Studies 178 (10):3391-3420.
On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order PA and Zermelo’s quasi-categoricity (...)

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12. The Key to Complexity.Ilexa Yardley - 2021 - Https://Medium.Com/the-Circular-Theory/.
Complexity is dependent on the circular-linear relationship between an individual and a group, meaning we cannot use 'observation' to tell us what we need to know (to explain complexity).

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13. Descriptivism About the Reference of Set-Theoretic Expressions: Revisiting Putnam’s Model-Theoretic Arguments.Zeynep Soysal - 2020 - The Monist 103 (4):442-454.
Putnam’s model-theoretic arguments for the indeterminacy of reference have been taken to pose a special problem for mathematical languages. In this paper, I argue that if one accepts that there are theory-external constraints on the reference of at least some expressions of ordinary language, then Putnam’s model-theoretic arguments for mathematical languages don’t go through. In particular, I argue for a kind of descriptivism about mathematical expressions according to which their reference is “anchored” in the reference of expressions of ordinary language. (...)

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14. A Metasemantic Challenge for Mathematical Determinacy.Jared Warren & Daniel Waxman - 2020 - Synthese 197 (2):477-495.
This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss a particularly popular approach (...)

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15. Supertasks and Arithmetical Truth.Jared Warren & Daniel Waxman - 2020 - Philosophical Studies 177 (5):1275-1282.
This paper discusses the relevance of supertask computation for the determinacy of arithmetic. Recent work in the philosophy of physics has made plausible the possibility of supertask computers, capable of running through infinitely many individual computations in a finite time. A natural thought is that, if supertask computers are possible, this implies that arithmetical truth is determinate. In this paper we argue, via a careful analysis of putative arguments from supertask computations to determinacy, that this natural thought is mistaken: supertasks (...)

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16. The Semantic Plights of the Ante-Rem Structuralist.Bahram Assadian - 2018 - Philosophical Studies 175 (12):1-20.
A version of the permutation argument in the philosophy of mathematics leads to the thesis that mathematical terms, contrary to appearances, are not genuine singular terms referring to individual objects; they are purely schematic or variables. By postulating ‘ante-rem structures’, the ante-rem structuralist aims to defuse the permutation argument and retain the referentiality of mathematical terms. This paper presents two semantic problems for the ante- rem view: (1) ante-rem structures are themselves subject to the permutation argument; (2) the ante-rem structuralist (...)

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17. Concrete Mathematical Incompleteness: Basic Emulation Theory.Harvey Friedman - 2018 - In Roy Cook & Geoffrey Hellman (eds.), Hilary Putnam on Logic and Mathematics. Springer Verlag.
there are mathematical statements that cannot be proved or refuted using the usual axioms and rules of inference of mathematics.
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18. Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - Philosophia Mathematica 24 (3):283-307.
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.

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19. Can the Cumulative Hierarchy Be Categorically Characterized?Luca Incurvati - 2016 - Logique Et Analyse 59 (236):367-387.
Mathematical realists have long invoked the categoricity of axiomatizations of arithmetic and analysis to explain how we manage to fix the intended meaning of their respective vocabulary. Can this strategy be extended to set theory? Although traditional wisdom recommends a negative answer to this question, Vann McGee (1997) has offered a proof that purports to show otherwise. I argue that one of the two key assumptions on which the proof rests deprives McGee's result of the significance he and the realist (...)

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20. The Search for New Axioms in the Hyperuniverse Programme.Claudio Ternullo & Sy-David Friedman - 2016 - In Andrea Sereni & Francesca Boccuni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Berlin: Springer. pp. 165-188.
The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (...)

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21. Models and Computability.W. Dean - 2014 - Philosophia Mathematica 22 (2):143-166.
Computationalism holds that our grasp of notions like ‘computable function’ can be used to account for our putative ability to refer to the standard model of arithmetic. Tennenbaum's Theorem has been repeatedly invoked in service of this claim. I will argue that not only do the relevant class of arguments fail, but that the result itself is most naturally understood as having the opposite of a reference-fixing effect — i.e., rather than securing the determinacy of number-theoretic reference, Tennenbaum's Theorem points (...)

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22. What is Absolute Undecidability?†.Justin Clarke-Doane - 2013 - Noûs 47 (3):467-481.
It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)

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23. The Metamathematics of Putnam’s Model-Theoretic Arguments.Tim Button - 2011 - Erkenntnis 74 (3):321-349.
Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.

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24. Math Anxiety.Aden Evens - 2000 - Angelaki 5 (3):105 – 115.
This article presents an explication of the references to the history of the calculus in the first few pages of Chapter 4 of Deleuze's _Difference and Repetition_. In those pages, Deleuze uses anachronistic readings of the calculus to explain his theory of ontogenesis, beginning with the differential, dx, that is strictly nothing by itself but that establishes singular points in relation to other differentials. He builds from the differential to power series, showing a corresponding process of determination in the ontogenesis (...)

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25. Set Theory and the Continuum Problem.Raymond Smullyan - 1996 - Clarendon Press.
A lucid, elegant, and complete survey of set theory, this three-part treatment explores axiomatic set theory, the consistency of the continuum hypothesis, and forcing and independence results. 1996 edition.

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26. On the Most Open Question in the History of Mathematics: A Discussion of Maddy.Adrian Riskin - 1994 - Philosophia Mathematica 2 (2):109-121.
In this paper, I argue against Penelope Maddy's set-theoretic realism by arguing (1) that it is perfectly consistent with mathematical Platonism to deny that there is a fact of the matter concerning statements which are independent of the axioms of set theory, and that (2) denying this accords further that many contemporary Platonists assert that there is a fact of the matter because they are closet foundationalists, and that their brand of foundationalism is in radical conflict with actual mathematical practice.

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27. The Independence of the Continuum Hypothesis II.Paul Cohen - 1964 - Proc. Nat. Acad. Sci. USA 51 (1):105-110.

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28. The Independence of the Continuum Hypothesis.Paul Cohen - 1963 - Proc. Nat. Acad. Sci. USA 50 (6):1143-1148.

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29. Principal Doctrines of Epicurus.Irfan Ajvazi - manuscript
Epicurean philosophy, as Epicurus's teachings became known, was used as the basis for how the community lived and worked. At the time, founding a school and teaching a community of students was the main way philosophical ideas were developed and transmitted. Greek philosopher Aristotle (384–322 BCE), for instance, founded a school in Athens called the Lyceum. Epicurus and his disciples believed either there were no gods or, if there were, the gods were so remote from humans that they were not (...)

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30. The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.