On Abstraction (Unsourced)

"A teacher who cannot explain any abstract subject to a child does not himself thoroughly understand his subject; if he does not attempt to break down his knowledge to fit the child's mind, he does not understand teaching." (Fulton J Sheen, "Life Is Worth Living")

"All abstract sciences are nothing but the study of relations between signs." (Denis Diderot)
"I think that science may be styled the knowledge of universals, or abstract wisdom; and art is science reduced to practice - or science is reason, and art the mechanism of it - and may be called practical science. Science, in fine, is the theorem, and art the problem." (Laurence Sterne)

"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." (Hermann Weyl)

"[…] it is only through Mathematics that we can thoroughly understand what true science is. Here alone can we find in the highest degree simplicity and severity of scientific law, and such abstraction as the human mind can attain." (Auguste Comte)

"The more abstract the truth you wish to teach, the more you need to seduce the senses to it." (Friedrich Nietzsche)

"The most dangerous thing about an academic education is that it enables my tendency to over-intellectualize stuff, to get lost in abstract thinking instead of simply paying attention to what’s going on in front of me." (David F Wallace)

"The primary service of modern mathematics is that it alone enables us to understand the vast abstract permanences which underlie the flux of things, without requiring us to regard its self-consistent abstractions as more than specific limited instruments of thought." (George D Birkhoff)

"The spirit of modern mathematics is based on mathematical logic, mathematical linguistics, the study of formal systems and on abstract algebra." (Grigore C Moisil)

"There exists, among mathematicians, a deep-seated and strong belief which sustains them in their abstract studies, namely that none of their problems can remain without any answer." (Gheorghe Ţiţeica)

"Unrestricted abstraction tends to divert attention from whole areas of application whose very discovery depends on the features that the abstract point of view rules out as being accidental." (Mark Kac)

"Mathematics originates in the mind of an individual, as it doubtless originated historically in the collective life of mankind, with the recognition of certain recurrent abstract features in common experience, and the development of processes of counting, measuring, and calculating, by which order can be brought into the manipulations of these features. It originated in this manner, indeed; but already at a very early stage it begins to transcend the practical sphere and its character undergoes a corresponding change. Intellectual curiosity progressively takes charge, despite the fact that practical considerations may for long continue to be the main source of interest and may indeed never cease to stimulate the creation of new concepts and new methods. As mathematics breaks from its early dependence on practical utility, its ‘immediate’ significance is at the same time lost and the goal is to discover what it is that makes 'emancipated' mathematics valid. (Geoffrey T Kneebone)