I take issue in this talk with AI formalizations of context, primarily the formalization by McCarthy and Buvac, that regard context as an undefined primitive whose formalization can be the same in many different kinds of AI tasks. In particular, any theory of context in natural language must take the special nature of natural language into account and cannot regard context simply as an undefined primitive. I show that there is no such thing as a coherent theory of context simpliciter---context pure and simple---and that context in natural language is not the same kind of thing as context in KR. In natural language, context is constructed by the speaker and the interpreter, and both have considerable discretion in so doing. Therefore, a formalization based on pre-defined contexts and pre-defined `lifting axioms' cannot account for how context is used in real-world language.

The first part of this paper highlights some key aspects of the differences in the use of mathematical tools in physics and in biology. Scientific knowledge is viewed as a network of interactions, more than as a hierachically organized structure where mathematics would display the essence of phenomena. The concept of "unity" in the biological phenomenon is then discussed. In the second part, a foundational issue in mathematics is revisited, following recent perspectives in the physiology of action. The relevance of the historical formation of mathematical concepts is also emphasized. Part I: Reflections on Mathematics in Biology Introduction: hierarchies of disciplines. When hearing biologists about working methods in their discipline, one may often appreciate traces of the emotions of a scientific experience of great 1 In Proceedings of the International Symposium on Foundations in Mathematics and Biology: Problems, Prospects, Interactions, Invited lecture, Pontifical Lateran Univer...

analyze the biological foundations of human cognition. A crucial component of their arguments is a simple but profound aphorism: Everything said is said by someone. It follows from this that any concept, idea, belief, definition, drawing, poem, or piece of music, has to be produced by a living human being, constrained by the peculiarities of his or her body and brain. The entailment is straightforward: without living human bodies with brains, there are no ideas — and that includes mathematical ideas. This chapter deals with the structure of mathematical ideas themselves, and with how their inferential organization is provided by everyday human cognitive mechanisms such as conceptual metaphor. The Cognitive Study of Ideas and their Inferential Organization The approach to Mathematical Cognition we take in this chapter is relatively new, and it differs in important ways from (but is complementary to) the ones taken by many of the authors in this Handbook. In order to avoid potential misunderstandings regarding the subject matter and goals of our piece, we believe that it is important to clarify these differences right upfront. The differences reside mainly on three fundamental aspects:

This article gives a brief introduction to a new discipline called the cognitive science of mathematics (Lakoff & Núñez, 2000), that is, the empirical and multidisciplinary study of mathematics (itself) as a scientific subject matter. The theoretical background of the arguments is based on embodied cognition, and on relatively recent findings in cognitive linguistics. The article discusses Mathematical Idea Analysis—the set of techniques for studying implicit (largely unconscious) conceptual structures in mathematics. Particular attention is paid to everyday cognitive mechanisms such as image schemas and conceptual metaphors, showing how they play a fundamental role in constituting the very fabric of mathematics. The analyses, illustrated with a discussion of some issues of set and hyperset theory, show that it is (human) meaning what makes mathematics what it is: Mathematics is not transcendentally objective, but it is not arbitrary either (not the result of pure social conventions). Some implications for mathematics education are suggested. Have you ever thought why (I mean, really why) the multiplication of two negative numbers yields a positive one? Or why the empty class is a subclass of all

The learning of mathematics is of universal interest across all OECD countries. In modern cultures, all children are expected to master up to at least 15 years of formal instruction in mathematics. Innumeracy has serious negative consequences for society. Mathematics is important not only for its own sake but also because it allows citizens to operate more effectively in their day-to-day lives. People need to balance checkbooks, make payments on items, and plan for future retirement. The topics studied by professional mathematicians extend far beyond everyday arithmetic or algebra. For example, some mathematicians devote themselves to the study of topology, others are interested in analysis or non-Euclidian geometries. The limitations of brain-imaging technologies, however, require researchers to simplify the cognitive demands facing subjects and to focus on simpler (but more common) mathematics topics. As we have seen in the chapter on reading, brain-imaging technologies can model relatively simple tasks. Thus, even though researchers who study the relationship between brain learning and mathematics hope someday to find brain activity correlates for all of mathematics, the current focus is on the learning of small whole numbers,

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presented in the OED, first by looking to its etymology (from Latin stilus, an instrument for writing on wax tablets), tracing that image back to the origins of cuneiform; thence by tracking the disparity between this word and stylus, which proves to be related to a different Indo-European root. These roots, and others with st- initials, are systematically presented, along with their modern descendants, and we see that the entire ontology of style recapitulates an ancient and powerful embodied image – the Standing Man – that illustrates the sacramental nature of writing. When I began this study, I wondered what a linguist might have to say about metaphor that would be appropriate and interesting to readers of a journal with a name like Style. In this context, it quickly became obvious to me that my own ordinary understanding of what was meant by style was deficient – deficient enough, anyway, that I decided to perform an act of linguistic research: I looked up style in the dictionary. And thereby I found my topic – or rather my topics, because there are

Cognitive research on metaphoric concepts of time has focused on differences between moving Ego and moving time models, but even more basic is the contrast between Ego- and temporal-reference-point models. Dynamic models appear to be quasi-universal cross-culturally, as does the generalization that in Ego-reference-point models, FUTURE IS IN FRONT OF EGO and PAST IS IN BACK OF EGO. The Aymara language instead has a major static model of time wherein FUTURE IS BEHIND EGO and PAST IS IN FRONT OF EGO; linguistic and gestural data give strong confirmation of this unusual culture-specific cognitive pattern. Gestural data provide crucial information unavailable to purely linguistic analysis, suggesting that when investigating conceptual systems both forms of expression should be analyzed complementarily. Important issues in embodied cognition are raised: how fully shared are bodily grounded motivations for universal cognitive patterns, what makes a rare pattern emerge, and what are the cultural entailments of such patterns?

I take issue in this talk with AI formalizations of context, primarily the formalization by McCarthy and Buvač, that regard context as an undefined primitive whose formalization can be the same in many different kinds of AI tasks. In particular, any theory of context in natural language must take the special nature of natural language into account and cannot regard context simply as an undefined primitive. I show that there is no such thing as a coherent theory of context simpliciter—context pure and simple—and that context in natural language is not the same kind of thing as context in KR. In natural language, context is constructed by the speaker and the interpreter, and both have considerable discretion in so doing. Therefore, a formalization based on pre-defined contexts and pre-defined ‘lifting axioms ’ cannot account for how context is used in real-world