On Concepts IX

"[In mathematics] we behold the conscious logical activity of the human mind in its purest and most perfect form. Here we learn to realize the laborious nature of the process, the great care with which it must proceed, the accuracy which is necessary to determine the exact extent of the general propositions arrived at, the difficulty of forming and comprehending abstract concepts; but here we learn also to place confidence in the certainty, scope and fruitfulness of such intellectual activity." (Hermann von Helmholtz, "Über das Verhältnis der Naturwissenschaften zur Gesammtheit der Wissenschaft", 1896)

"Former ages thought in terms of images of the imagination, whereas we moderns have concepts. Formerly the guiding ideas of life presented themselves in concrete visual form as divinities, whereas today they are conceptualized. The ancients excelled in creation; our own strength lies rather in destruction, in analysis." (Johann Wolfgang von Goethe, 1806)

"'You cannot base a general mathematical theory on imprecisely defined concepts. You can make some progress that way; but sooner or later the theory is bound to dissolve in ambiguities which prevent you from extending it further.' Failure to recognize this fact has another unfortunate consequence which is, in a practical sense, even more disastrous: 'Unless the conceptual problems of a field have been clearly resolved, you cannot say which mathematical problems are the relevant ones worth working on; and your efforts are more than likely to be wasted.'" (Edwin T Jaynes, "Foundations of Probability Theory and Statistical Mechanics", 1967)

"To master a concept means to be able to recognize it, that is, to be able to determine whether or not any given situation belongs to the set that characterizes this concept." (Valentin F Turchin, "The Phenomenon of Science: A cybernetic approach to human evolution", 1977)

"A conceptual system is an integrated system of concepts that supports a coherent vision of some aspect of the world. A conceptual system is personal; it is a 'way of seeing', that is, a 'way of knowing'. [...] You cannot do mathematics or science without a conceptual system but such systems are not objective and permanent. They are subject to change and development. Therefore we cannot claim that the reality that we experience and work with in science is independent of the mind of the scientist." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"A mathematical concept, then, is an organised pattern of ideas that are somehow interrelated, drawing on the experience of concepts already established. Psychologists call such an organised pattern of ideas a ‘schema’. " (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Facts and concepts only acquire real meaning and significance when viewed through the lens of a conceptual system. [...] Facts do not exist independently of knowledge and understanding for without some conceptual basis one would not know what data to even consider. The very act of choosing implies some knowledge. One could say that data, knowledge, and understanding are different ways of describing the same situation depending on the type of human involvement implied - 'data' means a de-emphasis on the human dimension whereas 'understanding' highlights it." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"The problem of teaching is the problem of introducing concepts and conceptual systems. In this crucial task the procedures of formal mathematical argument are of little value. The way we reason in formal mathematics is itself a conceptual system - deductive logic - but it is a huge mistake to identify this with mathematics. [...] Mathematics lives in its concepts and conceptual systems, which need to be explicitly addressed in the teaching of mathematics." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)