On Abstraction (1980-1989)

"Mathematical reality is in itself mysterious: how can it be highly abstract and yet applicable to the physical world? How can mathematical theorems be necessary truths about an unchanging realm of abstract entities and at the same time so useful in dealing with the contingent, variable and inexact happenings evident to the senses?" (Salomon Bochner, "The Role of Mathematics in the Rise of Science", 1981)

"Today abstraction is no longer that of the map, the double, the mirror, or the concept. Simulation is no longer that of a territory, a referential being or substance. It is the generation by models of a real without origin or reality: A hyperreal. The territory no longer precedes the map, nor does it survive it. It is nevertheless the map that precedes the territory - precession of simulacra - that engenders the territory." (Baudrillard Jean, "Simulacra and Simulation", 1981)

"[…] a mathematician's ultimate concern is that his or her inventions be logical, not realistic. This is not to say, however, that mathematical inventions do not correspond to real things. They do, in most, and possibly all, cases. The coincidence between mathematical ideas and natural reality is so extensive and well documented, in fact, that it requires an explanation. Keep in mind that the coincidence is not the outcome of mathematicians trying to be realistic - quite to the contrary, their ideas are often very abstract and do not initially appear to have any correspondence to the real world. Typically, however, mathematical ideas are eventually successfully applied to describe real phenomena […]"(Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)

"Language is the most formless means of expression. Its capacity to describe concepts without physical or visual references carries us into an advanced state of abstraction." (Ian Wilson, "Conceptual Art", 1984)

"Theoretical scientists, inching away from the safe and known, skirting the point of no return, confront nature with a free invention of the intellect. They strip the discovery down and wire it into place in the form of mathematical models or other abstractions that define the perceived relation exactly. The now-naked idea is scrutinized with as much coldness and outward lack of pity as the naturally warm human heart can muster. They try to put it to use, devising experiments or field observations to test its claims. By the rules of scientific procedure it is then either discarded or temporarily sustained. Either way, the central theory encompassing it grows. If the abstractions survive they generate new knowledge from which further exploratory trips of the mind can be planned. Through the repeated alternation between flights of the imagination and the accretion of hard data, a mutual agreement on the workings of the world is written, in the form of natural law." (Edward O Wilson, "Biophilia", 1984)

"A central problem in teaching mathematics is to communicate a reasonable sense of taste - meaning often when to, or not to, generalize, abstract, or extend something you have just done." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"There is no agreed upon definition of mathematics, but there is widespread agreement that the essence of mathematics is extension, generalization, and abstraction [… which] often bring increased confidence in the results of a specific application, as well as new viewpoints."  (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"In mathematics itself abstract algebra plays a dual role: that of a unifying link between disparate parts of mathematics  and that of a research subject with a highly active life of its own." (Israel N Herstein, "Abstract Algebra", 1986)

"A mental model is a data structure, in a computational system, that represents a part of the real world or of a fictitious world. It is assumed that there can be mental models of abstract realms, such as that of mathematics, but little more will be said about them. A model-theoretic semanticist is free to think of the entities in his model as actual items in the world.[...] Mental model is an appropriate term for the mental representations that underlie everyday reasoning about the world. To understand the everyday world is to have a theory of how it works." (Alan Granham, "Mental Models as Representations of Discourse and Text", 1987)

"Metaphor [is] a pervasive mode of understanding by which we project patterns from one domain of experience in order to structure another domain of a different kind. So conceived metaphor is not merely a linguistic mode of expression; rather, it is one of the chief cognitive structures by which we are able to have coherent, ordered experiences that we can reason about and make sense of. Through metaphor, we make use of patterns that obtain in our physical experience to organise our more abstract understanding. " (Mark Johnson, "The Body in the Mind", 1987)

"The essence of modeling, as we see it, is that one begins with a nontrivial word problem about the world around us. We then grapple with the not always obvious problem of how it can be posed as a mathematical question. Emphasis is on the evolution of a roughly conceived idea into a more abstract but manageable form in which inessentials have been eliminated. One of the lessons learned is that there is no best model, only better ones."  (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)

"Probabilities are summaries of knowledge that is left behind when information is transferred to a higher level of abstraction." (Judea Pearl, Probabilistic Reasoning in Intelligent Systems: Network of Plausible, Inference, 1988)

"Western culture’s world-view appears to be dominated by material objects. […] One of the ways mathematics has gained its power is through the activity of objectivising the abstractions from reality. Through its symbols (letters, numerals, figures) mathematics has taught people how to deal with abstract entities, as if they were objects." (Alan J Bishop, "Mathematics education in its cultural context", Educational Studies in Mathematics 19, 1988)

"[…] a model is the picture of the real - a short form of the whole. Hence, a model is an abstraction or simplification of a system. It is a technique by which aspects of reality can be 'artificially' represented or 'simulated' and at the same time simplified to facilitate comprehension." (Laxmi K Patnaik, "Model Building in Political Science", The Indian Journal of Political Science, Vol. 50, No. 2, 1989)

"As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, and even experimentation as a means of discovering truth. "(National Research Council, "Everybody Counts", 1989)

"Modeling in its broadest sense is the cost-effective use of something in place of something else for some [cognitive] purpose. It allows us to use something that is simpler, safer, or cheaper than reality instead of reality for some purpose. A model represents reality for the given purpose; the model is an abstraction of reality in the sense that it cannot represent all aspects of reality. This allows us to deal with the world in a simplified manner, avoiding the complexity, danger and irreversibility of reality." (Jeff Rothenberg, "The Nature of Modeling. In: Artificial Intelligence, Simulation, and Modeling", 1989)