Revised models of papers provided via philosophers, historians of technology, and mathematicians at a multidisciplinary symposium on buildings in Mathematical Theories, held on the collage of the Basque state (UPV/EHU) in Donostia/San Sebastian (Basque nation, Spain), September 1990. The 23 papers are geared up inside of 4 large parts: structural dimensions; dimensions of applicability; old dimensions; and worldwide dimensions of knowledge--information, implementation, and intertheoretic family.
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Extra info for The Space of Mathematics: Philosophical, Epistemological, and Historical Explorations (Foundations of Communication and Cognition, Library Edition)
Broadly speaking there are two kinds of variable quantity, the extensive and the intensive. Again speaking broadly, the extensive quantities are "quantity of space" and the intensive quantities are "ratios" between extensive ones. For example, mass and volume are extensive (measures), while density is intensive (function). Although Maxwell managed to get extensive quantities accepted within the particular science of thermodynamics, and although Grassmann demonstrated their importance in geometry, there is still a reluctance to give them status equal to that of functions and differential forms; in particular the use of the absurd terminology "generalized function" for such distributions as Categories of Space and of Quantity 19 the derivative of the Dirac measure has created a lot of confusion, for as Courant in effect observed, they are not intensive quantities, generalized or not.
E. to subobjects); indeed one of the two basic axioms of topos theory is that subobjects are representable by (indicator maps to) the truth-value space. On the other hand the great variety of useful extensive logic has been little studied (at least Categories of Space and of Quantity 25 as logic). In practice logic is not really a starting-point but rather the study of supports and roots of non-idempotent quantity: for example, the inhabited part of the world is the part where population exists, yet population (unlike the indicator of the part) is a non-idempotent quantity; distributions have supports and a pair of functions determines the ("root-") space of their agreement as well as the ("open") subspace of their disagreement.
A new feature (probably distinguishing homotopy categories from the other distributive categories which do contain "becoming"-parameterizers and "quantitative" objects, although an axiomatic definition is unclear to me) is the appearance of homotopy groups, extensive quantity-types finer than homology and (co-) representable (by spheres). Note that the definition of "point" when applied in a homotopy category in fact means "component". 3. "Unity and Identity of Opposites" as a Specific Mathematical Structure; Philosophical Dimension Not only should considerations of the above sort provide a useful guide to the learning and application of mathematics, but also the investigation of a given spatial category can be partly guided by philosophical principles.
The Space of Mathematics: Philosophical, Epistemological, and Historical Explorations (Foundations of Communication and Cognition, Library Edition)