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When the traditional distinction between a mathematical concept and a mathematical intuition is tested against examples taken from the real history of mathematics one can observe the following interesting phenomena. First, there are multiple examples where concepts and intuitions do not well fit together; some of these examples can be described as “poorly conceptualised intuitions” while some others...
This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted...
This paper sets out to adduce visual evidence for Kroneckerian finitism by making perspicuous some of the insights that buttress Kronecker’s conception of arithmetization as a process aiming at disclosing the arithmetical essence enshrined in analytical formulas, by spotting discrete patterns through algorithmic fine-tuning. In the light of a fairly tractable case study, it is argued that Kronecker’s...
In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical...
This paper explores Simon Stevin’s l’Arithmétique of 1585, where we find a novel understanding of the concept of number. I will discuss the dynamics between his practice and philosophy of mathematics, and put it in the context of his general epistemological attitude. Subsequently, I will take a close look at his justificational concerns, and at how these are reflected in his inductive, a postiori...
The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”,...
In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a...
This essay argues that propositions are made true by facts. A proposition is the sense expressed by a statement (sentence token used to make a truth claim). Facts are positive or negative constitutive properties of the domain of discourse (usually the actual world). The presence of horses is a positive constitutive property of the world; the absence of unicorns is a negative one. This notion of constitutive...
Although a number of truth theorists have claimed that a deflationary theory of ‘is true’ needs nothing more than the uniform implication of instances of the theorem ‘the proposition that p is true if and only if p’, reflection shows that this is inadequate. If deflationists can’t support the instances when replacing the biconditional with ‘because’, then their view is in peril. Deflationists sometimes...
The truth-condition theory of meaning is, naturally, thought of an as explanatory theory whose explananda are the meaning facts. But there are at least two deductive arguments that purport to establish the truth of the theory irrespective of its explanatory virtues. This paper examines those arguments and concludes that they succeed.
Philosophical semantics requires an ontology that includes negative as well as positive states of affairs as truth-makers and truth-breakers. Theories that try to do without negative states of affairs while interpreting propositional truth as positive correspondence with existent states of affairs are inherently inadequate and incomplete. A semantics and ontology of negative states of affairs can...
This essay is a reflection on the idea of truth-making and its applications. I respond to a critique of my 1986 paper on truth-making and discuss some key principles at play in the Truth-maker Program as it has emerged over the past 25 years, paying special attention to negative and general truths. I maintain my opposition to negative and general facts, but give an improved account of how to do without...
Section 1 discerns ambiguity in the word “truth”, observing that the term is used most naturally in reference to truth-bearers rather than truth-makers. Focusing on truths-as-truth-bearers, then, it would appear that alethic realism conflicts with metaphysical realism as naturalistically construed. Section 2 discerns ambiguity in the purporting of truth (as in assertion), conjecturing that all expressions,...
After arguing that truth-making is properly construed as a partnership between truth bearers and truth-makers, I focus on two prominent arguments against the category of fact as one of the key relata in the truth-making relation. After rejecting those arguments, I go on to examine a more difficult issue, one that might force us to appreciate more fully the robust role that thought has in “creating”...
The commonplace view about metaphorical interpretation is that it can be characterized in traditional semantic and pragmatic terms, thereby assimilating metaphor to other familiar uses of language. We will reject this view, and propose in its place the view that, though metaphors can issue in distinctive cognitive and discourse effects, they do so without issuing in metaphorical meaning and truth,...
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