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Fictionalism in the philosophy of mathematics
Stanford Encyclopedia of Philosophy (2008)
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La conclusión principal de este artículo es que las actitudes proposicionales son procesos físicos del cerebro que pertenecen al lenguaje. Por lo tanto, no pueden ser manifestaciones directas de nuestros estados mentales, ni pueden ser representaciones directas, sino que deben ser entendidas como representaciones de segundo orden. En efecto, las actitudes proposicionales no sirven para explicar nuestra actividad mental básica. Por otra parte, deberán entenderse como atajos del lenguaje usados para referirse a estados mentales, eventos o procesos. 

This dissertation is centered around a set of apparently conflicting intuitions that we may have about mathematics. On the one hand, we are inclined to believe that the theorems of mathematics are true. Since many of these theorems are existence assertions, it seems that if we accept them as true, we also commit ourselves to the existence of mathematical objects. On the other hand, mathematical objects are usually thought of as abstract objects that are nonspatiotemporal and causally inert. This makes (...) 

This paper is an attempt to convince antirealists that their correct intuitions against the metaphysical inflationism derived from some versions of mathematical realism do not force them to embrace nonstandard, epistemic approaches to truth and existence. It is also an attempt to convince mathematical realists that they do not need to implement their perfectly sound and judicious intuitions with the antiintuitive developments that render fullblown mathematical realism into a view which even Gödel considered objectionable. I will argue for the following (...) 

This is a survey of contemporary work on ‘fictionalism in metaphysics’, a term that is taken to signify both the place of fictionalism as a distinctive anti‐realist metaphysics in which usefulness rather than truth is the norm of acceptance, and the fact that philosophers have given fictionalist treatments of a range of specifically metaphysical notions. 

Contemporary philosophers of mathematics are deadlocked between two alternative ontologies for numbers: Platonism and nominalism. According to contemporary mathematical Platonism, numbers are real abstract objects, i.e. particulars which are nonetheless “wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal.” While this view does justice to intuitions about numbers and mathematical semantics, it leaves unclear how we could ever learn anything by mathematical inquiry. Mathematical nominalism, by contrast, holds that numbers do not exist extramentally, which raises difficulties about how mathematical statements could be (...) 

Moral error theorists argue that moral thought and discourse are systematically in error, and that nothing is, or can ever be, morally permissible, required or forbidden. I begin by discussing how error theorists arrive at this conclusion. I then argue that if we accept a moral error theory, we cannot escape a pressing problem – what should we do next, metaethically speaking? I call this problem the ‘what now?’ problem, or WNP for short. I discuss the attempts others have made (...) 

Fictionalists propose that some apparently factstating discourses do not aim to convey factual information about the world, but rather allow us to engage in a fiction or pretense without incurring ontological commitments. Some philosophers have suggested that using mathematical, modal, or moral discourse, for example, need not commit us to the existence of mathematical objects, possible worlds, or moral facts. The mental fictionalist applies this reasoning to our mental discourse, suggesting that we can use ‘belief’ and ‘desire’ talk without committing (...) 

This chapter argues that mental fictionalism can only be a successful account of our ordinary folkpsychological practices if it can in some way preserve its original function, namely its explanatory aspect. A too strong commitment to the explanatory role moves fictionalism unacceptably close to the realist or eliminativist interpretation of folk psychology. To avoid this, fictionalists must degrade or dispense with this explanatory role. This motivation behind the fictionalist movement seems to be rather similar to that of Sellars when he (...) 

Mathematical pluralism notes that there are many different kinds of pure mathematical structures—notably those based on different logics—and that, qua pieces of pure mathematics, they are all equally good. Logical pluralism is the view that there are different logics, which are, in an appropriate sense, equally good. Some, such as Shapiro, have argued that mathematical pluralism entails logical pluralism. In this brief note I argue that this does not follow. There is a crucial distinction to be drawn between the preservation (...) 

There is a wide range of realist but nonPlatonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other nonabstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...) 

There has been much discussion of the indispensability argument for the existence of mathematical objects. In this paper I reconsider the debate by using the notion of grounding, or noncausal dependence. First of all, I investigate what proponents of the indispensability argument should say about the grounding of relations between physical objects and mathematical ones. This reveals some resources which nominalists are entitled to use. Making use of these resources, I present a neglected but promising response to the indispensability argument—a (...) 

Religious fictionalism holds that religious sentences are false, that religious practitioners accept rather than believe religious sentences, and that it is justifiable for them to act on religious sentences. I develop an alternative to religious fictionalism, which I call “religious practicalism.” It holds that we do not know whether religious sentences are true or false, that religious practitioners believe rather than merely accept religious sentences, and that it is justifiable for them to act on religious sentences. I argue that religious (...) 

Can fictionalists have faith? It all depends on how we disambiguate ‘fictionalists’ and on what faith is. I consider the matter in light of my own theory. After clarifying its central terms, I distinguish two fictionalists – atheistic and agnostic – and I argue that, even though no atheistic fictionalist can have faith on my theory, agnostic fictionalists arguably can. After rejecting Finlay Malcolm's reasons for thinking this is a problem, I use his paradigmatic agnostic fictionalist as a foil to (...) 







This paper, written in Romanian, compares fictionalism, nominalism, and neoMeinongianism as responses to the problem of objectivity in mathematics, and then motivates a fictionalist view of objectivity as invariance. 

A problem for Aristotelian realist accounts of universals (neither Platonist nor nominalist) is the status of those universals that happen not to be realised in the physical (or any other) world. They perhaps include uninstantiated shades of blue and huge infinite cardinals. Should they be altogether excluded (as in D.M. Armstrong's theory of universals) or accorded some sort of reality? Surely truths about ratios are true even of ratios that are too big to be instantiated  what is the truthmaker (...) 

In this entry I will offer a survey of the contemporary debate on fic tionalism, which is a distinctive antirealist view about certain regions of discourse that are valued for their usefulness rather than their truth. 

Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is. I raise two objections against mathematical realism. First, the abstract world is queer in that it allows for contradictory states of affairs. Second, mathematical realism does not have a theoretical resource to explain why a sentence about a tricle is true or false. A tricle is an (...) 

How can we acquire a grasp of cardinal numbers, even the first very small positive cardinal numbers, given that they are abstract mathematical entities? That problem of cognitive access is the main focus of this paper. All the major rival views about the nature and existence of cardinal numbers face difficulties; and the view most consonant with our normal thought and talk about numbers, the view that cardinal numbers are sizes of sets, runs into the cognitive access problem. The source (...) 

I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and fictionalism. 

The ‘indispensability argument’ for the existence of mathematical objects appeals to the role mathematics plays in science. In a series of publications, Joseph Melia has offered a distinctive reply to the indispensability argument. The purpose of this paper is to clarify Melia’s response to the indispensability argument and to advise Melia and his critics on how best to carry forward the debate. We will begin by presenting Melia’s response and diagnosing some recent misunderstandings of it. Then we will discuss four (...) 

