Fictionalism in the philosophy of mathematics

2018 ◽  
Author(s):  
Mark Colyvan
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Discourse ◽  
Literary Fiction ◽  
Scientific Theories ◽  
Adequate Account ◽  
Mathematical Realism ◽  
Applicability Of Mathematics ◽  
Coming To Know ◽  
In Virtue Of ◽  
Error Theories

Fictionalism in the philosophy of mathematics is the view that mathematical statements, such as ‘8+5=13’ and ‘∏ is irrational’, are to be interpreted at face value and, thus interpreted, are false. Fictionalists are typically driven to reject the truth of such mathematical statements because these statements imply the existence of mathematical entities, and according to fictionalists there are no such entities. Fictionalism is a nominalist (or antirealist) account of mathematics in that it denies the existence of a realm of abstract mathematical entities. It should be contrasted with mathematical realism (or Platonism) where mathematical statements are taken to be true, and moreover are taken to be truths about mathematical entities. Fictionalism should also be contrasted with other nominalist philosophical accounts of mathematics that propose a reinterpretation of mathematical statements, according to which the statements in question are true but no longer about mathematical entities. Fictionalism is thus an error theory of mathematical discourse: at face value mathematical discourse commits us to mathematical entities; and although we normally take many of the statements of this discourse to be true, in doing so we are in error (cf. error theories in ethics). Although fictionalism holds that mathematical statements implying the existence of mathematical entities are strictly speaking false, there is a sense in which these statements are true - they are true in the story of mathematics. The idea here is borrowed from literary fiction, where statements like ‘Bilbo Baggins is a hobbit’ is strictly speaking false (because there are no hobbits), but true in Tolkien’s fiction The Hobbit. Fictionalism about mathematics shares the virtue of ontological parsimony with other nominalist accounts of mathematics. It also lends itself to a very straightforward epistemology: there is nothing to know beyond the human-authored story of mathematics. And coming to know the various fictional claims requires nothing more than knowledge of the story in question. The most serious problem fictionalism faces is accounting for the applicability of mathematics. Mathematics, unlike Tolkien’s stories, is apparently indispensable to our best scientific theories and this, according to some, suggests that we ought to be realists about mathematical entities. It is fair to say that there are serious difficulties facing all extant philosophies of mathematics, and fictionalism is no exception. Despite its problems fictionalism remains a popular option in virtue of a number of attractive features. In particular, it endorses a uniform semantics across mathematical and nonmathematical discourse and it provides a neat answer to questions about attaining mathematical knowledge. The major challenge for fictionalism is to provide an adequate account of mathematics in applications.

scholarly journals Putnam, Kuhn e a relatividade conceitual

2017 ◽  
Vol 20 (2) ◽  
pp. 285
Author(s):  
Robinson Guitarrari
Keyword(s):  
Critical Role ◽  
Conceptual Structure ◽  
Scientific Development ◽  
Conceptual Scheme ◽  
Scientific Theories ◽  
Metaphysical Realism ◽  
Conceptual Schemes ◽  
Close Relationship ◽  
Formal Requirements ◽  
In Virtue Of

http://dx.doi.org/10.5007/1808-1711.2016v20n2p285 The understanding of conceptual relativity in Putnam’s and Kuhn’s writings should not be restricted to the claim that the existence is relative to, but not in virtue of, a conceptual scheme. This approach does not capture some significant differences between their positions about the notion of conceptual scheme. We understand that the thesis of conceptual relativity contains a statement about the close relationship between conceptual scheme and content, and another claim about the differences between conceptual schemes. Based on these two formal requirements, we propose a reconstruction of the Putnam’s treatment of it and show how it can be understood from Kuhn’s perspective of scientific development. We defend that, although both fulfill a critical role against metaphysical realism, they are applied to distinct domains: while Putnam’s conceptual relativity is in the record of the conceptual structure of scientific theories and presupposes a choice between cognitively equivalent conceptual schemes, Kuhn considers the field of the dynamics of development of science. Thus, we note relevant scientific cases of conceptual relativity that do not involve semantic incommensurability.

The anatytic conception of truth and the foundations of arithmetic

2000 ◽  
Vol 65 (1) ◽  
pp. 33-102 ◽  
Author(s):  
Peter Apostoli
Keyword(s):  
Philosophy Of Mathematics ◽  
Logical Truth ◽  
Numerical Identity ◽  
Hume’S Principle ◽  
Contextual Definition ◽  
Definition Of ◽  
Axiomatic Set Theory ◽  
In Virtue Of ◽  
Second Order Logic ◽  
Foundations Of Arithmetic

Until very recently, it was thought that there couldn't be any current interest in logicism as a philosophy of mathematics. Indeed, there is an old argument one often finds that logicism is a simple nonstarter just in virtue of the fact that if it were a logical truth that there are infinitely many natural numbers, then this would be in conflict with the existence of finite models. It is certainly true that from the perspective of model theory, arithmetic cannot be a part of logic. However, it is equally true that model theory's reliance on a background of axiomatic set theory renders it unable to match Frege's Theorem, the derivation within second order logic of the infinity of the number series from the contextual “definition” of the cardinality operator. Called “Hume's Principle” by Boolos, the contextual definition of the cardinality operator is presented in Section 63 of Grundlagen, as the statement that, for any concepts F and G,the number of F s = the number of G sif, and only if,F is equinumerous with G.The philosophical interest in Frege's Theorem derives from the thesis, defended for example by Crispin Wright, that Hume's principle expresses our pre-analytic conception of assertions of numerical identity. However, Boolos cites the very fact that Hume's principle has only infinite models as grounds for denying that it is logically true: For Boolos, Hume's principle is simply a disguised axiom of infinity.

Hilbertian Structuralism and the Frege-Hilbert Controversy†

2019 ◽  
Vol 27 (3) ◽  
pp. 335-361
Author(s):  
Fiona T Doherty
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Objects ◽  
Structuralist Approach ◽  
Complex Roots ◽  
In Virtue Of

ABSTRACT This paper reveals David Hilbert’s position in the philosophy of mathematics, circa 1900, to be a form of non-eliminative structuralism, predating his formalism. I argue that Hilbert withstands the pressing objections put to him by Frege in the course of the Frege-Hilbert controversy in virtue of this early structuralist approach. To demonstrate that this historical position deserves contemporary attention I show that Hilbertian structuralism avoids a recent wave of objections against non-eliminative structuralists to the effect that they cannot distinguish between structurally identical but importantly distinct mathematical objects, such as the complex roots of $-1$.

scholarly journals The Honest Weasel A Guide for Successful Weaseling

Disputatio ◽  
2020 ◽  
Vol 12 (56) ◽  
pp. 41-69
Author(s):  
Patrick Dieveney
Keyword(s):  
Good News ◽  
Scientific Theories ◽  
Mathematical Realism ◽  
Mathematical Objects ◽  
Indispensability Arguments ◽  
Take Back ◽  
Controversial Nature

AbstractIndispensability arguments are among the strongest arguments in support of mathematical realism. Given the controversial nature of their conclusions, it is not surprising that critics have supplied a number of rejoinders to these arguments. In this paper, I focus on one such rejoinder, Melia’s ‘Weasel Response’. The weasel is someone who accepts that scientific theories imply that there are mathematical objects, but then proceeds to ‘take back’ this commitment. While weaseling seems improper, accounts supplied in the literature have failed to explain why. Drawing on examples of weaseling in more mundane contexts, I develop an account of the presumption against weaseling as grounded in a misalignment between two types of commitments. This is good news to the weasel’s opponents. It reinforces that they were right to question the legitimacy of weaseling. This account is also beneficial to the weasel. Uncovering the source of the presumption against weaseling also serves to draw out the challenge that the weasel must meet to override this presumption—what is required to be an ‘honest weasel’.

INDISPENSABILITY ARGUMENTS AND INSTRUMENTAL NOMINALISM

2012 ◽  
Vol 5 (4) ◽  
pp. 687-709 ◽  
Author(s):  
RICHARD PETTIGREW
Keyword(s):  
Possible Worlds ◽  
Philosophy Of Mathematics ◽  
Scientific Theories ◽  
Mathematical Objects ◽  
Modal Operators ◽  
Indispensability Arguments ◽  
Everyday Reasoning ◽  
Do So

In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that mathematical objects exist on the grounds that we make indispensable reference to such objects in our best scientific theories (Quine, 1981a; Putnam, 1979a) and in our everyday reasoning (Ketland, 2005). I wish to defend a particular objection to such arguments called instrumental nominalism. Existing formulations of this objection are either insufficiently precise or themselves make reference to mathematical objects or possible worlds. I show how to formulate the position precisely without making any such reference. To do so, it is necessary to supplement the standard modal operators with two new operators that allow us to shift the locus of evaluation for a subformula. I motivate this move and give a semantics for the new operators.

Realism in the philosophy of mathematics

2018 ◽  
Author(s):  
Patricia A. Blanchette
Keyword(s):  
Human Activities ◽  
Philosophy Of Mathematics ◽  
Mathematical Truth ◽  
Mathematical Realism ◽  
Mathematical Objects ◽  
Natural Position ◽  
Potential Problems ◽  
The Subject ◽  
Important Form ◽  
Clear Relation

Mathematical realism is the view that the truths of mathematics are objective, which is to say that they are true independently of any human activities, beliefs or capacities. As the realist sees it, mathematics is the study of a body of necessary and unchanging facts, which it is the mathematician’s task to discover, not to create. These form the subject matter of mathematical discourse: a mathematical statement is true just in case it accurately describes the mathematical facts. An important form of mathematical realism is mathematical Platonism, the view that mathematics is about a collection of independently existing mathematical objects. Platonism is to be distinguished from the more general thesis of realism, since the objectivity of mathematical truth does not, at least not obviously, require the existence of distinctively mathematical objects. Realism is in a fairly clear sense the ‘natural’ position in the philosophy of mathematics, since ordinary mathematical statements make no explicit reference to human activities, beliefs or capacities. Because of the naturalness of mathematical realism, reasons for embracing antirealism typically stem from perceived problems with realism. These potential problems concern our knowledge of mathematical truth, and the connection between mathematical truth and practice. The antirealist argues that the kinds of objective facts posited by the realist would be inaccessible to us, and would bear no clear relation to the procedures we have for determining the truth of mathematical statements. If this is right, then realism implies that mathematical knowledge is inexplicable. The challenge to the realist is to show that the objectivity of mathematical facts does not conflict with our knowledge of them, and to show in particular how our ordinary proof-procedures can inform us about these facts.

The Definition of Religion, Super-empirical Realities and Mathematics

2016 ◽  
Vol 58 (1) ◽  
Author(s):  
Andrea Sauchelli
Keyword(s):  
Necessary Conditions ◽  
Philosophy Of Mathematics ◽  
Precise Definition ◽  
Mathematical Realism ◽  
Sufficient And Necessary Conditions ◽  
Definition Of Religion ◽  
And Mathematics ◽  
Definition Of

SummaryProviding a precise definition of “religion” – or an analysis in terms of sufficient and necessary conditions of the concept of religion – has proven to be a difficult task, more so in light of the diverse types of practices considered religious by scholars. Here, I discuss Kevin Schilbrack’s recent definition of “religion”, elaborate it and raise several objections, one of which is based on a specific theory in philosophy of mathematics: mathematical realism.

Philosophy of Mathematics

2016 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Discourse ◽  
Mathematical Objects ◽  
Ontological Realism ◽  
Truth Value ◽  
And Mathematics

This article examines a number of issues and problems that motivate at least much of the literature in the philosophy of mathematics. It first considers how the philosophy of mathematics is related to metaphysics, epistemology, and semantics. In particular, it reviews several views that account for the metaphysical nature of mathematical objects and how they compare to other sorts of objects, including realism in ontology and nominalism. It then discusses a common claim, attributed to Georg Kreisel that the important issues in the philosophy of mathematics do not concern the nature of mathematical objects, but rather the objectivity of mathematical discourse. It also explores irrealism in truth-value, the dilemma posed by Paul Benacerraf, epistemological issues in ontological realism, ontological irrealism, and the connection between naturalism and mathematics.

The Expression of Will View II

2021 ◽  
pp. 116-125
Author(s):  
Tom Dougherty
Keyword(s):  
Speech Acts ◽  
Speech Act ◽  
Adequate Account ◽  
Directive Speech ◽  
In Virtue Of

This chapter elaborates the Expression of Will View as a disjunctive view, in so far as it allows that there are two ways that someone can consent by expressing their will. First, they can deliberately perform a ‘directive’ speech act. Examples of these speech acts include requests, invitations, and orders. Second, someone can consent by expressing that they are giving another person permission. While it is possible to simultaneously give consent in both ways, there are some situations in which someone consents in only one of these two ways. Since the Expression of Will View allows that consent can be given in either of these two ways, it is a disjunctive view. Although a disjunctive view is less cohesive and simple, we need to endorse a disjunctive view to have an extensionally adequate account. But although the view is disjunctive, it has some unity in virtue of the fact that it has the Interpersonal Justification Argument as an underlying rationale.

Objectivity

2018 ◽  
Author(s):  
Alexander Miller
Keyword(s):  
Ice Cream ◽  
The Other ◽  
Philosophical Debate ◽  
Other Hand ◽  
Widespread Agreement ◽  
Physical Objects ◽  
In Virtue Of ◽  
Error Theories

Objectivity is one of the central concepts of metaphysics. Philosophers distinguish between objectivity and agreement: ‘Ice-cream tastes nice’ is not objective merely because there is widespread agreement that ice-cream tastes nice. But if objectivity is not mere agreement, what is it? We often think that some sorts of claim are less objective than others, so that a different metaphysical account is required of each. For example, ethical claims are often held to be less objective than claims about the shapes of middle-sized physical objects: ‘Murder is wrong’ is held to be less objective than ‘The table is square’. Philosophers disagree about how to capture intuitive differences in objectivity. Those known as expressivists say that ethical claims are not, strictly speaking, even apt to be true or false; they do not aim to record facts but, rather, express some desire or inclination on the part of the speaker. Others, dubbed subjectivists, say that ethical statements are in some sense about human desires or inclinations. Unlike the expressivist, the subjectivist views ethical claims as truth-apt, but as being true (when they are true) in virtue of facts about human desires or inclinations. Error-theorists and fictionalists argue that (atomic, positive) ethical claims are systematically and uniformly false, but that they may be regarded as trading in ‘useful fictions’. Some philosophers, referred to as antirealists, disagree with expressivism, subjectivism, error-theories and fictionalism, and attempt to find different ways of denying objectivity. Quietists, on the other hand, think that there are no interesting ways of distinguishing discourses in point of objective status and that philosophical debate about differences in objective status is in some sense misguided.

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