Putnam rejects having argued in the terms of the argument known in the literature as “the Quine-Putnam indispensability argument”. He considers that mathematics contribution to physics does not have to be interpreted in platonist terms but in his favorite modal variety (Putnam 1975; Putnam 2012). The purpose of this paper is to consider Putnam’s acknowledged argument and philosophical position against contemporary so called in the literature ‘fictionalist’ views about applied mathematics. The conclusion will be that the account of the applicability of mathematics that stems from Putnam‘s acknowledged argument can be assimilated to so-called ‘fictionalist’ views about applied mathematics.

Crucial to Hilary Putnam’s realism in the philosophy of mathematics is to maintain the objectivity of mathematics without the commitment to the existence of mathematical objects. Putnam’s indispensability argument was devised as part of this conception. In this paper, I reconstruct and reassess Putnam’s argument for the indispensability of mathematics, and distinguish it from the more familiar, Quinean version of the argument. Although I argue that Putnam’s approach ultimately fails, I develop an alternative way of implementing his form of realism about mathematics that, by using different resources than those Putnam invokes, avoids the difficulties faced by his view.

This paper focuses on Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. Putnam’s 1971 book Philosophy of Logic came one year later than Quine’s homonymous volume. In the first section, I compare these two Philosophies of Logic which exemplify realist-nominalist viewpoints in a most conspicuous way. The next section examines Putnam’s views on modality, moving from the modal qualification of his intuitive conception to his official generalized non-modal second-order set-theoretic concept of logical truth. In the third section, I emphasize how Putnam´s “mathematics as modal logic” departs from Quine’s “reluctant Platonism”. I also suggest a complementary view of Platonism and modalism showing them perhaps interchangeable but underlying different stages of research processes that make up a rich and dynamic mathematical practice. The final, more speculative section, argues for the pervasive platonistic conception enhancing the aims of inquiry in the practice of the working mathematician.

When considering mathematical realism, some scientific realists reject it, and express sympathy for the opposite view, mathematical nominalism; moreover, many justify this option by invoking the causal inertness of mathematical objects. The main aim of this note is to show that the scientific realists’ endorsement of this causal mathematical nominalism is in tension with another position some (many?) of them also accept, the doctrine of methodological naturalism. By highlighting this conflict, I intend to tip the balance in favor of a rival of mathematical nominalism, the mathematical realist position supported by the ‘Indispensability Argument’ —but I do this indirectly, by showing that the road toward it is not blocked by considerations from causation.

Two conceptions of the nature of mathematical objects are contrasted: the conception of mathematical objects as preconceived objects (Yablo 2010), and heavy duty platonism (Knowles 2015). It is argued that some theses defended by friends of the indispensability argument are in harmony with heavy duty platonism and in tension with the conception of mathematical objects as preconceived objects.

This paper discusses Baker’s Enhanced Indispensability Argument (EIA) for mathematical realism on the basis of the indispensable role mathematics plays in scientific explanations of physical facts, along with various responses to it. I argue that there is an analogue of causal explanation for mathematics which, of several basic types of explanation, holds the most promise for use in the EIA. I consider a plausible case where mathematics plays an explanatory role in this sense, but argue that such use still does not support realism about mathematical objects.

In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez (2000), I propose one particular conceptual metaphor, the Process → Object Metaphor (POM), as a key element in understanding the development of mathematical thinking.

This paper considers two models of embodied mathematical cognition (a modular model and a dynamic model), and analyses the image of mathematics that they support.

In the first part of the article, a semiotic reading of the embodied approach to mathematics will be presented. From this perspective, the role of the sensorimotor in mathematics will be considered, by looking at some work in experimental psychology on the segmentation of formulas and at an analysis of the practice of topology as involving manipulative imagination. According to the proposed view, representations in mathematics are cognitive tools whose functioning depends on pre-existing cognitive abilities and specific training. In the second part of the paper, the view of cognitive tools as props in games of “make-believe” will be discussed; to better specify this claim, the notion of affordance will be explored in its possible extension from concrete objects to representations.

Frege writes in Numbers and Arithmetic about kindergarten-numbers and “an a priori mode of cognition” that they may have “a geometrical source.” This resembles recent findings on arithmetical cognition. In my paper, I explore this resemblance between Gottlob Frege’s later position concerning the geometrical source of arithmetical knowledge, and some current positions in the literature dedicated to arithmetical cognition, especially that of Stanislas Dehaene. In my analysis, I shall try to mainly see to what extent (Frege’s) logicism is compatible with (Dehaene’s) intuitionism.

The use of unrealistic assumptions in Economics is usually defended not only for pragmatic reasons, but also because of the intrinsic difficulties in determining the degree of realism of assumptions. Additionally, the criterion used for evaluating economic models is associated with their ability to provide accurate predictions. This mode of thought involves —at least implicitly— a commitment to the existence of unvarying invariant factors or regularities. Contrary to this, the present paper presents a critique to the use of invariant knowledge in economics. One reason for this analysis lies in the fact that economic phenomena are not compatible with the logic of invariance, but with the logic of "possibility trees" or "open-ended results". The other reason is that the use of invariant knowledge may entail both external validity problems and negative exposures to a "black swan". Alternatively, an approach where models are understood as possible scenarios is proposed. It is argued that the realism of (substantive) assumptions is crucial here, since it helps to ascertain the degree of resemblance between the different models and the target system.

In “Study of Concepts”, Peacocke puts forward an argument for non-conceptualism derived from the possession conditions of observational concepts. In this paper, I raise two objections to this argument. First, I argue that if non-conceptual perceptual contents are scenario contents, then perceptual experiences cannot present perceivers with the circumstances specified by the application conditions of observational concepts and, therefore, they cannot play the semantic and epistemic roles Peacocke wants them to play in the possession conditions of these concepts. Second, I argue that if non-conceptual perceptual contents are protopropositions, then Peacocke’s account of the possession conditions of observational concepts falls into the Myth of the Given.

The clock hypothesis is taken to be an assumption independent of special relativity necessary to describe accelerated clocks. This enables to equate the time read off by a clock to the proper time. Here, it is considered a physical system—the light clock—proposed by Marzke and Wheeler. Recently, Fletcher proved a theorem that shows that a sufficiently small light clock has a time reading that approximates to an arbitrary degree the proper time. The clock hypothesis is not necessary to arrive at this result. Here, one explores the consequences of this regarding the status of the clock hypothesis. It is argued in this work that there is no need for the clock hypothesis in the special theory of relativity.

This is a discussion of Delia Fara’s theory of vagueness, and of its solution to the sorites paradox, criticizing some of the details of the account, but agreeing that its central insight will be a part of any solution to the problem. I also consider a wider range of philosophical puzzles that involve arguments that are structurally similar to the argument of the sorites paradox, and argue that the main ideas of her account of vagueness helps to respond to some of those puzzles.

Names are referring expressions and interact with the determiner system only exceptionally, in stark contrast with count nouns. The-predicativists like Sloat, Matushansky, and Fara claim otherwise, maintaining that syntactic data indicates that names belong to a special syntactic category which differs from common count nouns only in how they interact with ‘the’. I argue that the-predicativists have incorrectly discerned the syntactic facts. They have bypassed a large range of important syntactic data and misconstrued a critical data point on which they ground the-predicativism. The right data offers new compelling syntactic grounds for referentialism.

This paper is a tribute to Delia Graff Fara. It extends her work on failures of meta-rules (conditional proof, RAA, contraposition, disjunction elimination) for validity as truth-preservation under a supervaluationist identification of truth with supertruth. She showed that such failures occur even in languages without special vagueness-related operators, for standards of deductive reasoning as materially rather than purely logically good, depending on a context-dependent background. This paper extends her argument to: quantifier meta-rules like existential elimination; ambiguity; deliberately vague standard mathematical notation. Supervaluationist attempts to qualify the meta-rules impose unreasonable cognitive demands on reasoning and underestimate her challenge.

The aim of this paper is to offer a version of the so-called conversational hypothesis of the ontogenetic connection between language and mindreading (Harris 1996, 2005; Van Cleave and Gauker 2010; Hughes et al. 2006). After arguing against a particular way of understanding the hypothesis (the communicative view), I will start from the justificatory view in philosophy of social cognition (Andrews 2012; Hutto 2004; Zawidzki 2013) to make the case for the idea that the primary function of belief and desire attributions is to justify and normalize deviant patterns of behaviour. Following this framework, I elaborate upon the idea that development of folk psychological skills requires the subjects to engage in conversationally mediated joint and cooperative activities in order to acquire the conceptual capacity of ascribing propositional attitudes. After presenting the general version of the hypothesis, I present several testable sub-hypotheses and some psychological studies that give empirical plausibility to the hypothesis.