‘Number sense’ is a short‐hand for our ability to quickly understand, approximate, and manipulate numerical quantities. My hypothesis is that number sense rests on cerebral circuits that have evolved specifically for the purpose of representing basic arithmetic knowledge. Four lines of evidence suggesting that number sense constitutes a domain‐specific, biologically‐determined ability are reviewed: the presence of evolutionary precursors of arithmetic in animals; the early emergence of arithmetic competence in infants independently of other abilities, including language; the existence of a homology (...) between the animal, infant, and human adult abilities for number processing; and the existence of a dedicated cerebral substrate. In adults of all cultures, lesions to the inferior parietal region can specifically impair number sense while leaving the knowledge of other cognitive domains intact. Furthermore, this region is demonstrably activated during number processing. I postulate that higher–level cultural devel‐opments in arithmetic emerge through the establishment of linkages between this core analogical representation and other verbal and visual representations of number notations. The neural and cognitive organization of those representations can explain why some mathematical concepts are intuitive, while others are so difficult to grasp. Thus, the ultimate foundations of mathematics rests on core representations that have been internalized in our brains through evolution. (shrink)

A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.

Contemporary psychological research that studies how people apply mathematics has largely viewed mathematics as a computational tool for deriving an answer. The tacit assumption has been that people first understand a situation, and then choose which computations to apply. We examine an alternative assumption that mathematics can also serve as a tool that helps one to construct an understanding of a situation in the first place. Three studies were conducted with 6th‐grade children in the context of proportional situations because early (...) proportional reasoning is a premier example of where mathematics may provide new understanding of the world. The children predicted whether two differently‐sized glasses of orange juice would taste the same when they were filled from a single carton of juice made from concentrate and water. To examine the relative contributions and interactions of situational and mathematical knowledge, we manipulated the formal features of the problem display (e.g., diagram vs. photograph) and the numerical complexity (e.g., divisibility) of the containers and the ingredient ratios. When the problem was presented as a diagram with complex numbers, or “realistically” with easy numbers, the children predicted the glasses would taste different because one glass had more juice than the other. But, when the problem was presented realistically with complex numbers, the children predicted the glasses would taste the same on the basis of empirical knowledge (e.g., “Juice can't change by itself”). And finally, when the problem was presented as a diagram with easy numbers, the children predicted the glasses would taste the same on the basis of proportional relations. These complex interactions illuminate how mathematical and empirical knowledge can jointly constrain the construction of a new understanding of the world. We propose that mathematics helped in the case of successful proportional reasoning because it made a complex empirical situation cognitively tractable, and thereby helped the children construct mental models of that situation. We sketch one aspect of the mental models that are constructed in the domain of quantity—a preference for specificity—that helps explain the current findings. (shrink)

This article argues that certain philosophically devised quality control parameters should guide approaches to interdisciplinary education. We sketch the kind of reflections we think are necessary in order to produce epistemologically responsible curricula. We suggest that the two overarching epistemic dimensions of levels of analysis and basic viewpoints go a long way towards clarifying the structure of interdisciplinary validity claims. Through a discussion of how best to teach basic ideas about numeracy in Mind, Brain, and Education, we discuss what it (...) means for an interdisciplinary curriculum to respect both the minds of students and the complexity of the subject matter. (shrink)

Human knowledge is a phenomenon whose roots extend from the cultural, through the neural and the biological and finally all the way down into the Precambrian “primordial soup.” The present paper reports an attempt at understanding this Greater System of Knowledge (GSK) as a hierarchical nested set of selection processes acting concurrently on several different scales of time and space. To this end, a general selection theory extending mainly from the work of Hull and Campbell is introduced. The perhaps most (...) drastic change from previous similar theories is that replication is revealed as a composite function consisting of what is referred to as memory and synthesis. This move is argued to drastically improve the fit between theory and human-related knowledge systems. The introduced theory is then used to interpret the subsystems of the GSK and their interrelations. This is done to the end of demonstrating some of the new perspectives offered by this view. (shrink)

We discuss the nature of reality in the ontological context of Penrose’s math-matter-mind triangle. The triangle suggests the circularity of the widespread view that math arises from the mind, the mind arises out of matter, and that matter can be explained in terms of math. Non-physicists should be wary of any claim that modern physics leads us to any particular resolution of this circularity, since even the sample of three theoretical physicists writing this paper hold three divergent views. Some physicists (...) believe that current physics has already found the basic framework for a complete description of reality, and only has to fill in the details. Others suspect that no single framework, from physics or other sources, will ever capture reality. Yet others guess that reality might be approached arbitrarily closely by some form of future physics, but probably based on completely different frameworks. We will designate these three approaches as the fundamentalist, secular and mystic views of the world, as seen by practicing physicists. We present and contrast each of these views, which arguably form broad categories capturing most if not all interpretations of physics. We argue that this diversity in the physics community is more useful than an ontological monoculture, since it motivates physicists to tackle unsolved problems with a wide variety of approaches. (shrink)

Infinity, in various guises, has been invoked recently in order to ‘explain’ a number of important questions regarding observable phenomena in science, and in particular in cosmology. Such explanations are by their nature speculative. Here we introduce the notions of relative infinity, closure, and economy of explanation and ask: to what extent explanations involving relative or real constructed infinities can be treated as reasonable?