This article examines how experimental psychology experienced a revolution as cognitive science replaced behaviorism in the mid-20th century. This transition in the scientific account of human nature involved making normal what had once been normative: borrowing ideas of democratic thinking from political culture and conceptions of good thinking from philosophy of science to describe humans as active, creatively thinking beings, rather than as organisms that simply respond to environmental conditions. Reflexive social and intellectual practices were central to this process as (...) cognitive scientists used anti-positivist philosophy of science simultaneously to justify their own work as valid and also as a model of human thinking. In the process, the normative philosophy of science or ‘good academic thinking’ that cognitive scientists used to reshape the discipline of psychology and characterize themselves became, at the same time, the descriptive model of human nature. (shrink)

How can anyone be rational in a world where knowledge is limited, time is pressing, and deep thought is often an unattainable luxury? Traditional models of unbounded rationality and optimization in cognitive science, economics, and animal behavior have tended to view decision-makers as possessing supernatural powers of reason, limitless knowledge, and endless time. But understanding decisions in the real world requires a more psychologically plausible notion of bounded rationality. In Simple heuristics that make us smart (Gigerenzer et al. 1999), we (...) explore fast and frugal heuristics – simple rules in the mind's adaptive toolbox for making decisions with realistic mental resources. These heuristics can enable both living organisms and artificial systems to make smart choices quickly and with a minimum of information by exploiting the way that information is structured in particular environments. In this précis, we show how simple building blocks that control information search, stop search, and make decisions can be put together to form classes of heuristics, including: ignorance-based and one-reason decision making for choice, elimination models for categorization, and satisficing heuristics for sequential search. These simple heuristics perform comparably to more complex algorithms, particularly when generalizing to new data – that is, simplicity leads to robustness. We present evidence regarding when people use simple heuristics and describe the challenges to be addressed by this research program. Key Words: adaptive toolbox; bounded rationality; decision making; elimination models; environment structure; heuristics; ignorance-based reasoning; limited information search; robustness; satisficing; simplicity. (shrink)

From ancient times to the present, the discovery and presentation of new proofs of previously established theorems has been a salient feature of mathematical practice. Why? What purposes are served by such endeavors? And how do mathematicians judge whether two proofs of the same theorem are essentially different? Consideration of such questions illuminates the roles that proofs play in the validation and communication of mathematical knowledge and raises issues that have yet to be resolved by mathematical logicians. The Appendix, in (...) which several proofs of the Fundamental Theorem of Arithmetic are compared, provides a miniature case study. (shrink)

In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.

In Making Sense of Life, Keller emphasizes several differences between biology and physics. Her analysis focuses on significant ways in which modelling practices in some areas of biology, especially developmental biology, differ from those of the physical sciences. She suggests that natural models and modelling by homology play a central role in the former but not the latter. In this paper, I focus instead on those practices that are importantly similar, from the point of view of epistemology and cognitive science. (...) I argue that concrete and abstract models are significant in both disciplines, that there are shared selection criteria for models in physics and biology, e.g. familiarity, and that modelling often occurs in a similar fashion. (shrink)

In current philosophical research, there is a rather one-sided focus on the foundations of proof. A full picture of mathematical practice should however additionally involve considerations about various methodological aspects. A number of these is identified, from large-scale to small-scale ones. After that, naturalism, a philosophical school concerned with scientific practice, is looked at, as far as the translations of its epistemic principles to mathematics is concerned. Finally, we call for intensifying the interaction between both dimensions of practice and epistemology.

Are there arguments in mathematics? Are there explanations in mathematics? Are there any connections between argument, proof and explanation? Highly controversial answers and arguments are reviewed. The main point is that in the case of a mathematical proof, the pragmatic criterion used to make a distinction between argument and explanation is likely to be insufficient for you may grant the conclusion of a proof but keep on thinking that the proof is not explanatory.

What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians' appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.

Relational reasoning is a hallmark of human higher cognition and creativity, yet it is notoriously difficult to encourage in abstract tasks, even in adults. Generally, young children initially focus more on objects, but with age become more focused on relations. While prerequisite knowledge and cognitive resource maturation partially explains this pattern, here we propose a new facet important for children's relational reasoning development: a general orientation to relational information, or a relational mindset. We demonstrate that a relational mindset can be (...) elicited, even in 4‐year‐old children, yielding greater than expected spontaneous attention to relations. Children either generated or listened to an experimenter state the relationships between objects in a set of formal analogy problems, and then in a second task, selected object or relational matches according to their preference. Children tended to make object mappings, but those who generated relations on the first task selected relational matches more often on the second task, signaling that relational attention is malleable even in young children. (shrink)

This study applied a method of assisted introspection to investigate the phenomenology of mathematical intuition arousal. The aim was to propose an essential structure for the intuitive experience of mathematics. To achieve an intersubjective comparison of different experiences, several contemporary mathematicians were interviewed in accordance with the elicitation interview method in order to collect pinpoint experiential descriptions. Data collection and analysis was then performed using steps similar to those outlined in the descriptive phenomenological method that led to a generic structure (...) that accounts for the intuition surge in the experience of mathematics which was found to have four irreducible structural moments. The interdependence of these moments shows that a perceptualist view of intuition in mathematics, as defended by Chudnoff, is relevant to the characterization of mathematical intuition. The philosophical consequences of this generic structure and its essential features are discussed in accordance with Husserl’s philosophy of ideal objects and theory of intuition. (shrink)

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

This paper investigates an important aspect of mathematical practice: that proof is required for a finished piece of mathematics. If follows that non-deductive arguments — however convincing — are never sufficient. I explore four aspects of mathematical research that have facilitated the impressive success of the discipline. These I call the Practical Virtues: Permanence, Reliability, Autonomy, and Consensus. I then argue that permitting results to become established on the basis of non-deductive evidence alone would lead to their deterioration. This furnishes (...) us with a partial rational justification for mathematicians strict insistence on proof. (shrink)