César Frederico dos Santos

Logical theories are usually seen as true or false in relation to the phenomenon they aim to describe, namely, validity. An alternative view suggests that the laws governing validity are ``legislated-true,'' making logical theories conventional. In this chapter, I propose an intermediate standpoint in which logics exhibit a blend of descriptive and conventional aspects, balanced to fulfill their primary purpose, which is not seen as theoretical, but practical: assisting us in generating and identifying valid inferences. From this perspective, logics are cognitive tools, human creations designed primarily to facilitate the performance of cognitive operations. This viewpoint aligns logic more closely with technology than science and supports a pluralistic approach: the diversity of alternative technical solutions for issues related to the creation and recognition of valid inferences allows for the coexistence of multiple logics.

This chapter critically examines the dominance of aprioristic methodologies in the philosophy of arithmetic, particularly in debates concerning the existence of numbers. It challenges the assumption that purely a priori methods–such as conceptual analysis, appeal to theoretical virtues, or reliance on logical and semantic principles, can yield conclusive answers about the ontological status of mathematical entities. Instead, it argues that such methods, while useful for hypothesis formation, lack the epistemic resources to confirm or refute ontological claims. The chapter defends the position that the existence of numbers is a worldly matter that cannot be settled by aprioristic reflection alone.

This chapter addresses the paradox of how human beings could have developed number concepts in the absence of numeral systems–systems that are, in turn, typically required for such concepts to emerge. Known as “the origins problem,” this puzzle challenges developmental accounts that emphasize the dependence of number concept acquisition on pre-existing symbolic systems. The chapter begins with Pelland’s formulation of the problem and explores a novel resolution grounded in Dutilh Novaes’s theory of re-semantification. It argues that early numeral systems could have emerged through semantic shifts in pre-existing symbolic practices among anumeric peoples. Drawing on linguistic and anthropological evidence, the chapter proposes a historically plausible account of how humans may have first constructed numeral systems–and with them, the conceptual foundations for number.

This chapter proposes an alternative, a posteriori approach to the ontological inquiry into numbers, grounded in the view that the fundamental phenomenon to be explained in an investigation of the metaphysics of mathematics is a class of human experiences. Humans do mathematics and use it to help them solve practical and theoretical problems. When investigating the metaphysics of mathematics, we are looking for a set of entities that can account for the main features displayed by these human experiences. This investigation can follow a “reverse engineering” strategy: starting with the assumption that certain humans have numerical competence, go back to a time when they did not, and observe what happened in the meantime. This involves answering two questions: what does a human being experience during cognitive development so as to acquire numerical competence, and what, in the history of the human species, enabled us to acquire it? Hopefully, data about our experiences as individuals and as a species will suggest a hypothesis about what entities underlie our number concepts.

This chapter explores how number concepts are acquired, emphasizing that such concepts are not innate nor derived from direct perceptual experience. Building on earlier discussions, it advances the hypothesis that number concepts emerge through engagement with numerals as initially de-semanticized symbols, whose meanings are shaped by structured practices like counting. The chapter begins by clarifying what it means to possess number concepts, then reviews empirical findings from numerical cognition and developmental psychology. It highlights that while quantical abilities show some correlation with numerical cognition, exposure to number talk and symbolic practices plays a far more critical role. Studies on children’s counting development further support this view, showing that understanding numerical meaning arises only after mastering the syntactic rules of counting.

This chapter advances a nominalistic and empirically grounded account of the semantics and epistemology of arithmetic, challenging the traditional view that the truth and objectivity of arithmetical statements require the existence of numbers as abstract objects. Building on the discussion from previous chapters, it argues that numerical reference, truth, and epistemic features such as objectivity, necessity, and apriority can be explained by the cognitive and social practices of counting and calculation. Drawing on a Peircean framework of reference, the chapter proposes that numerals can function referentially even in the absence of ontologically robust numerical entities. The core thesis is that the epistemic roles ascribed to numbers are instead fulfilled by procedural structures inherent in arithmetical practices. This approach preserves the standard interpretation of arithmetic while avoiding ontological commitment to abstract mathematical objects.

This chapter investigates the second developmental stage in number concept acquisition–calculation–and its role in the ontogenetic and historical reification of numbers. While early numerical competence is grounded in the mastery of number words and counting, calculation introduces new cognitive and linguistic structures that promote treating numerals as referring to abstract objects. The chapter argues that the widespread tendency to speak of numbers as objects arises not from their ontological status but from the projection of object-based linguistic frameworks onto numerical discourse. This reification is traced through developmental, historical, and philosophical perspectives, offering a cognitive account of how numbers come to be treated as independent entities within mathematics and everyday language.

This chapter explores the hypothesis that numerical competence arises from interactions with symbolic cultural tools rather than being an innate human ability. While research in numerical cognition has shown that infants and non-human animals possess inborn abilities to discriminate quantities, these abilities differ fundamentally from the symbolic numerical skills exhibited by adults. The chapter examines the debate between nativist and externalist perspectives, ultimately proposing that inborn quantity-related capacities should be characterized as instances of “quantical cognition”–a form of non-symbolic, non-numerical processing. Building on prior discussions, the chapter provides arguments and evidence to support the claim that quantical cognition, while foundational, does not itself constitute numerical thought.

This books offers a novel account of the nature of numbers firmly grounded in results from numerical cognition and the philosophy of mathematics. Drawing on empirical data on the human experience of what we call “numbers,” the author shows that numbers do not exist as abstract objects, but that the idea that they do is a useful cognitive tool. Contrary to the platonist view, according to which arithmetic is true of a realm of abstract entities, the nominalistic account presented in this book shows arithmetic to be true of descriptions of structural properties of techniques such as counting and calculating procedures. This book is of interest to both philosophers and cognitive scientists who want to have a deeper understanding of what numbers are.

In Maddy’s philosophy, mathematics is autonomous, i.e., it is not subordinated to either science or philosophy. Mathematics establishes and pursues its own goals and must be judged on its own terms. This leads Maddy to admit, in Naturalism in Mathematics (1997) and also in Second Philosophy (2007), that, even if mathematicians choose to pursue a goal that could seem improper from the philosophical or scientific point of view, there is nothing to be done except accepting the new state of affairs. In Defending the Axioms (2011), nevertheless, Maddy changes her position regarding this issue. She claims to have found the basis from which to assess the adequacy of mathematical goals. From this basis, if mathematicians choose to pursue what seems to be an improper goal, the philosopher could claim that they are going astray. In this paper, I will review Maddy’s positions in these books; and, especially regarding Defending the Axioms, I will sustain that the institution of a permanent parameter for the judgment of mathematical goals goes against the alleged autonomy of mathematics and other important traits of her philosophy. Keywords: Penelope Maddy, autonomy of mathematics, mathematical ends.

The field of numerical cognition provides a fairly clear picture of the processes through which we learn basic arithmetical facts. This scientific picture, however, is rarely taken as providing a response to a much‐debated philosophical question, namely, the question of how we obtain number knowledge, since numbers are usually thought to be abstract entities located outside of space and time. In this paper, I take the scientific evidence on how we learn arithmetic as providing a response to the philosophical question of how we obtain number knowledge. I reject the view that numbers are abstract entities located outside of space and time and, alternatively, derive from the scientific evidence a novel account of the nature of numbers. In this account, numbers are reifications of the counting procedure and arithmetic statements are seen as describing the functioning of counting and calculation techniques.

No Discurso sobre as Ciências e as Artes, seu primeiro discurso, Rousseau defende a polêmica tese de que o progresso das ciências e das artes, contrariamente ao que pretendia o Iluminismo, estava contribuindo mais para a degeneração dos costumes e da sociedade do que para seu aperfeiçoamento. O Primeiro Discurso foi escrito em 1749, há quase 300 anos. Nesse período, a ciência e a nossa compreensão sobre ela mudaram profundamente. Mais importante, nesse período surgiu da ciência algo imprevisto para Rousseau no Primeiro Discurso: a tecnologia moderna. A proposta deste trabalho é, pois, revisitar as críticas de Rousseau à ciência com um olhar contemporâneo, buscando avaliar o quanto daquelas críticas ainda faz sentido nos panoramas científico e social atuais. Para tanto, classificamos esquematicamente as críticas de Rousseau em dois grupos: as que acusam a inutilidade das ciências e as que acusam como nociva a sofisticação que as ciências produzem na sociedade. Defenderemos que o surgimento da tecnologia moderna tornou obsoletas as críticas quanto à inutilidade ao mesmo tempo que potencializou as críticas quanto à sofisticação. Assim, uma parte importante de suas críticas à ciência pode ser recuperada, vestida em nova roupagem e aplicada justamente na crítica à tecnologia.

In 1939, the influential psychophysicist S. S. Stevens proposed definitional distinctions between the terms ‘number,’ ‘numerosity,’ and ‘numerousness.’ Although the definitions he proposed were adopted by syeveral psychophysicists and experimental psychologists in the 1940s and 1950s, they were almost forgotten in the subsequent decades, making room for what has been described as a “terminological chaos” in the field of numerical cognition. In this paper, I review Stevens’s distinctions to help bring order to this alleged chaos and to shed light on two closely related questions: whether it is adequate to speak of a number sense and how philosophers can make sense of the claim by cognitive scientists that numbers are perceptual entities. Moreover, I offer further support to Stevens’s distinction between numerosity and numerousness by showing that they are relational properties that emerge through different relationships between agents and their environments. The final conclusion is that, by adopting Stevens’s distinctions, numbers do not need to be seen as perceptual entities, since the so-called number sense is better described as a sense of numerousness.

Anti-exceptionalists about logic claim that logical methodology is not different from scientific methodology when it comes to theory choice. Two anti-exceptionalist accounts of theory choice in logic are abductivism and predictivism. These accounts have in common reliance on pre-theoretical logical intuitions for the assessment of candidate logical theories. In this paper, I investigate whether intuitions can provide what abductivism and predictivism want from them and conclude that they do not. As an alternative to these approaches, I propose a Carnapian view on logical theorizing according to which logical theories do not simply account for pre-theoretical intuitions, but rather improve on them. In this account, logical theories are ameliorative, rather than representational.

In the literature on enculturation—the thesis according to which higher cognitive capacities result from transformations in the brain driven by culture—numerical cognition is often cited as an example. A consequence of the enculturation account for numerical cognition is that individuals cannot acquire numerical competence if a symbolic system for numbers is not available in their cultural environment. This poses a problem for the explanation of the historical origins of numerical concepts and symbols. When a numeral system had not been created yet, people did not have the opportunity to acquire number concepts. But, if people did not have number concepts, how could they ever create a symbolic system for numbers? Here I propose an account of the invention of symbolic systems for numbers by anumeric people in the remote past that is compatible with the enculturation thesis. I suggest that symbols for numbers and number concepts may have emerged at the same time through the re-semantification of words whose meanings were originally non-numerical.