Abstract structure

An abstract structure is an abstraction that might be of the geometric spaces or a set structure, or a hypostatic abstraction that is defined by a set of mathematical theorems and laws, properties and relationships in a way that is logically if not always historically independent of the structure of contingent experiences, for example, those involving physical objects. Abstract structures are studied not only in logic and mathematics but in the fields that apply them, as computer science and computer graphics, and in the studies that reflect on them, such as philosophy (especially the philosophy of mathematics). Indeed, modern mathematics has been defined in a very general sense as the study of abstract structures (by the Bourbaki group: see discussion there, at algebraic structure and also structure). An abstract structure may be represented (perhaps with some degree of approximation) by one or more physical objects - this is called an implementation or instantiation of the abstract structure. But the abstract structure itself is defined in a way that is not dependent on the properties of any particular implementation. An abstract structure has a richer structure than a concept or an idea. An abstract structure must include precise rules of behaviour which can be used to determine whether a candidate implementation actually matches the abstract structure in question, and it must be free from contradictions. Thus we may debate how well a particular government fits the concept of democracy, but there is no room for debate over whether a given sequence of moves is or is not a valid game of chess (for example Kasparovian approaches). A sorting algorithm is an abstract structure, but a recipe is not, because it depends on the properties and quantities of its ingredients. A simple melody is an abstract structure, but an orchestration is not, because it depends on the properties of particular instruments. Euclidean geometry is an abstract structure, but the theory of continental drift is not, because it depends on the geology of the Earth.

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Abstraction (mathematics)

Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. Two of the most highly abstract areas of modern mathematics are and model theory.

Philosophy

Philosophy (love of wisdom in ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, values, mind, and language. It is a rational and critical inquiry that reflects on its own methods and assumptions. Historically, many of the individual sciences, like physics and psychology, formed part of philosophy. But they are considered separate academic disciplines in the modern sense of the term.

Logic

Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language.

Geometric Learning: Leveraging differential geometry for learning and control

Bernardo Fichera

In this thesis, we concentrate on advancing high-level behavioral control policies for robotic systems within the framework of Dynamical Systems (DS). Throughout the course of this research, a unifying thread weaving through diverse fields emerges, and tha ...

EPFL2024

Metamodeling is raising more and more interest in the field of language engineering. While this approach is now well understood for the definition of abstract syntaxes, the formal definition of concrete syntaxes is still a challenge. Concrete syntaxes are ...

2006