On Concepts II

"We construct concepts when we represent them in intuition a priori, without experience, or when we represent in intuition  the object which corresponds to our concept of it. - The mathematician can never apply his reason to mere concepts, nor the philosopher to the construction of concepts. - In mathematics  the reason is employed in concreto, however, the intuition is not  empirical, but the object of contemplation is something a priori." (Immanuel Kant, "Logic", 1800)

"This is the greatest degree of impoverishment; the image, deprived little by little of its own characteristics, is nothing more than a shadow. It has become that transitional form between image and pure concept that we now term ‘generic image’, or one that at least resembles the latter. The image, then, is subject to an unending process of change, of suppression and addition, of dissociation and corrosion." (Théodule-Armand Ribot, "Essay on the Creative Imagination" , 1900)

"But surely it is self-evident that every theory is merely a framework or scheme of concepts together with their necessary relations to one another, and that the basic elements can be constructed as one pleases." (Gottlob Frege, "On the Foundations of Geometry and Formal Theories of Arithmetic" , cca. 1903-1909)

"Thus, the object presents itself in the image as having to be apprehended in a multiplicity of synthetic acts. Due to this fact, and because its content retains a sensible opacity, like a phantom, because it does not involve either essences or generating laws but only an irrational quality, it gives the impression of being an object of observation: from this point of view the image appears to be more like a perception than a concept." (Jean-Paul Sartre, "The Psychology of Imagination", 1940)

"A useful concept may be a barrier to the acceptance of a better one if long-entrenched in the minds of scientists." (James B Conant, "On Understanding Science", 1947)

"Science is an interconnected series of concepts and schemes that have developed as a result of experimentation and observation and are fruitful of further experimentation and observation."(James B Conant, "Science and Common Sense", 1951)

"Science is the creation of concepts and their exploration in the facts. It has no other test of the concept than its empirical truth to fact." (Jacob Bronowski, "Science and Human Values", 1956)

"A mental model is not normally based on formal definitions but rather on concrete properties that have been drawn from life experience. Mental models are typically analogs, and they comprise specific contents, but this does not necessarily restrict their power to deal with abstract concepts, as we will see. The important thing about mental models, especially in the context of mathematics, is the relations they represent. […]  The essence of understanding a concept is to have a mental representation or mental model that faithfully reflects the structure of that concept. (Lyn D. English & Graeme S. Halford, "Mathematics Education: Models and Processes", 1995)

"An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner, "The Disciplined Mind", 1999)

"When the words are used without mental image or concrete objects, we label them as metaphor. […] While concepts are being internalised, language is not only appropriated but metaphorised." (Lynne Cameron, "Metaphor in the construction of a learning environment", 2008)