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.

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’.

Quine and the Web of Belief

2009 ◽  
Author(s):  
Michael D. Resnik
Keyword(s):  
Ontological Commitment ◽  
Philosophy Of Mathematics ◽  
Scientific Theories ◽  
Indispensability Arguments ◽  
Ontological Relativity ◽  
Systematic Philosophy ◽  
Web Of Belief ◽  
The Relationship ◽  
The Web

This article focuses on Quine's positive views and their bearing on the philosophy of mathematics. It begins with his views concerning the relationship between scientific theories and experiential evidence (his holism), and relate these to his views on the evidence for the existence of objects (his criterion of ontological commitment, his naturalism, and his indispensability arguments). This sets the stage for discussing his theories concerning the genesis of our beliefs about objects (his postulationalism) and the nature of reference to objects (his ontological relativity). Quine's writings usually concerned theories and their objects generally, but they contain a powerful and systematic philosophy of mathematics, and the article aims to bring this into focus.

Language, Meaning, and Information

2014 ◽  
Author(s):  
Scott Soames
Keyword(s):  
Modal Logic ◽  
Philosophy Of Language ◽  
Possible Worlds ◽  
Philosophy Of Mathematics ◽  
Scientific Study ◽  
Bertrand Russell ◽  
Gottlob Frege ◽  
Possible Worlds Semantics ◽  
Language Meaning

This chapter is a case study of the process by which the attempt to solve philosophical problems sometimes leads to the birth of new domains of scientific inquiry. It traces how advances in logic and the philosophy of mathematics, starting with Gottlob Frege and Bertrand Russell, provided the foundations for what became a rigorous and scientific study of language, meaning, and information. After sketching the early stages of the story, it explains the importance of modal logic and “possible worlds semantics” in providing the foundation for the last half century of work in linguistic semantics and the philosophy of language. It argues that this foundation is insufficient to support the most urgently needed further advances. It proposes a new conception of truth-evaluable information as inherently representational cognitive acts of certain kinds. The chapter concludes by explaining how this conception of propositions can be used to illuminate the notion of truth; vindicate the connection between truth and meaning; and fulfill a central, but so far unkept, promise of possible worlds semantics.

6. Necessity and possibility: what will be must be?

2017 ◽  
pp. 37-44
Author(s):  
Graham Priest
Keyword(s):  
Possible Worlds ◽  
Modal Operators

We often say that something must be so (e.g. ‘It must be going to rain’). We also have ways of saying that, though something may not be the case, it could be (e.g. ‘It could rain tomorrow’). ‘Necessity and possibility: what will be must be?’ considers the notion of things that may happen, things that could happen, and modal operators (it is possible for something to be the case—so it is not necessarily the case that it is false). The argument for fatalism illustrates modal operators. Fatalism is the view that whatever happens must happen. We suppose that every situation comes furnished with a bunch of possibilities, or possible worlds.

Representability in second-order propositional poly-modal logic

2002 ◽  
Vol 67 (3) ◽  
pp. 1039-1054 ◽  
Author(s):  
G. Aldo Antonelli ◽  
Richmond H. Thomason
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Second Order ◽  
Order Logic ◽  
Multiple Modalities ◽  
Modal Operators ◽  
Second Order Systems ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Propositional System

AbstractA propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.In this paper we generalize this framework by allowing multiple modalities. While this does not affect the undecidability of K, B, T, K4 and S4, poly-modal second-order S5 is dramatically more expressive than its mono-modal counterpart. As an example, we establish the definability of the transitive closure of finitely many modal operators. We also take up the decidability issue, and, using a novel encoding of sets of unordered pairs by partitions of the leaves of certain graphs, we show that the second-order propositional logic of two S5 modalitities is also equivalent to full second-order logic.

Descartes on the Eternal Truths and Essences of Mathematics: An Alternative Reading

Vivarium ◽  
2016 ◽  
Vol 54 (2-3) ◽  
pp. 204-249
Author(s):  
Helen Hattab
Keyword(s):  
Rene Descartes ◽  
Philosophy Of Mathematics ◽  
Ontological Status ◽  
Divine Simplicity ◽  
Historical Evidence ◽  
Eternal Truths ◽  
Mathematical Objects ◽  
Divine Essence ◽  
René Descartes ◽  
Alternative Reading

René Descartes is neither a Conceptualist nor a Platonist when it comes to the ontological status of the eternal truths and essences of mathematics but articulates a view derived from Proclus. There are several advantages to interpreting Descartes’ texts in light of Proclus’ view of universals and philosophy of mathematics. Key passages that, on standard readings, are in conflict are reconciled if we read Descartes as appropriating Proclus’ threefold distinction among universals. Specifically, passages that appear to commit Descartes to a Platonist view of mathematical objects and the truths that follow from them are no longer in tension with the Conceptualist view of universals implied by his treatment of the eternal truths in the Principles of Philosophy. This interpretation also fits the historical evidence and explains why Descartes ends up with seemingly inconsistent commitments to divine simplicity and God’s efficient creation of truths that are not merely conceptually distinct from the divine essence.

scholarly journals IMAGINE THERE'S NO (PLATONIC) HEAVEN

Think ◽  
2014 ◽  
Vol 14 (39) ◽  
pp. 73-75
Author(s):  
Matteo Plebani
Keyword(s):  
Scientific Theories ◽  
Mathematical Objects

Some people think that numbers and other mathematical entities exist. They believe in a platonic heaven of ideal mathematical objects, as some (other) people like to put it. This may seem a very strange thing to believe in: after all, we cannot see numbers, nor touch them, nor smell them. So why should one believe that they exist? Because, as Putnam and Quine used to say, numbers are indispensable to science: it seems almost impossible to state our best scientific theories without mentioning numbers or other mathematical objects.

scholarly journals What Counts as Evidence for a Logical Theory?

2019 ◽  
Vol 16 (7) ◽  
pp. 250 ◽  
Author(s):  
Ole Thomassen Hjortland
Keyword(s):  
Classical Logic ◽  
A Priori ◽  
Scientific Theories ◽  
Alternative Account ◽  
Logical Theory ◽  
Crucial Question ◽  
Indispensability Arguments ◽  
Timothy Williamson ◽  
Graham Priest ◽  
Number Of Authors

Anti-exceptionalism about logic is the Quinean view that logical theories have no special epistemological status, in particular, they are not self-evident or justified a priori. Instead, logical theories are continuous with scientific theories, and knowledge about logic is as hard-earned as knowledge of physics, economics, and chemistry. Once we reject apriorism about logic, however, we need an alternative account of how logical theories are justified and revised. A number of authors have recently argued that logical theories are justified by abductive argument (e.g. Gillian Russell, Graham Priest, Timothy Williamson). This paper explores one crucial question about the abductive strategy: what counts as evidence for a logical theory? I develop three accounts of evidential confirmation that an anti-exceptionalist can accept: (1) intuitions about validity, (2) the Quine-Williamson account, and (3) indispensability arguments. I argue, against the received view, that none of the evidential sources support classical logic.

scholarly journals The Normative Pluriverse

2020 ◽  
Vol 18 (2) ◽  
Author(s):  
Matti Eklund
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Objects ◽  
Normative Realism ◽  
Important Form ◽  
The Way

According to a certain pluralist view in philosophy of mathematics, there are as many mathematical objects as there can coherently be. Recently, Justin Clarke-Doane has explored what consequences the analogous view on normative properties would have. What if there is a normative pluriverse? Here I address this same question. The challenge is best seen as a challenge to an important form of normative realism. I criticize the way Clarke-Doane presents the challenge. An improved challenge is presented, and the role of pluralism in this challenge is assessed.

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