What Is Abstraction?
Mathematics is often said to be especially difficult because it deals in abstractions. But, as a matter of fact, abstraction is an absolutely indigenous to everyday human reasoning. Indeed, abstract and the ability for abstraction are innate properties of the human mind. [Devlin, p. 11] wonders
The etymology of the word is found in The Words of Mathematics by S. Schwartzman, although, we humans use the process of abstraction in everyday life, not just in mathematics.
The Harper Collins Dictionary of Mathematics tells us
The Brafman siblings [Sway, pp. 74-75] quote psychologist Franz Epting, an expert in understanding how people construct meaning in their experiences.
Sometimes ignoring things does make a tremendous difference, of course, and may lead to disastrous results; sometimes it does not. But sometimes, especially in mathematics, ignoring things bolsters comprehension by focusing on the essential. Let's consider a (not necessarily mundane) example of the sort often encountered in high school math texts. [Sawyer, pp. 32-34] asks the reader to imagine two railway stations A and B connected by a single track-line. Assuming that at about the same time a train leaves A for B and another leaves B for A and there are no safety devices along the line, one expects an imminent collision. The details (the length of trains, whether they are passenger or freight, country outline) are irrelevant. One may only hope that the visibility is good, the train engineers are attentive, responsible and do not text messages to friends while in motion, and the line does not make sharp turns. If all this is true then there may be a chance to avoid the disaster. What you actually see with your mind's eye depends on the previous experience, whether you survived or witnessed a train collision or watched one on a TV news program. But in any event, actually seeing the details is not necessary to make one shudder at the mere thought of the possibility of collision. In so far as abstraction leads to omitting details, the result is similar to having to reason with incomplete information which is quite a common practice. So, how mathematics may be perceived difficult just because it deals in abstraction. The reason I think (see also [Devlin, pp. 121-122]) is implicit in the previous sentence. Much of the abstraction in mathematics has indeed originated in the practical experience as is the case of counting, for example. But the nature of the mathematical abstraction is in that, once created - admittedly by leaving out non essential details - it acquires an independent existence becoming an entity in its own right and fully defined at that. To deal with mathematical obstraction/objects no longer requires omitting details. Just the opposite is true: to deal successfully with mathematical obstraction/objects one needs to account for as much information as is built in into their definition. An analogy from a novel by Anthony Powell may serve to explain the difference:
The idea of a straight line may have originated historically from the repeated observation of the trace left by a tight powdered thin rope that was clinched on a flat surface, or some other similar experiences and observations. But in modern mathematics, the straight line is that object that may cross another one of the same sort in at most one point and which is uniquely determined by any two points it is incident with. And there is all there is to it. The straight line so defined is as true as Powell's novel and, in a sense, is truer than any image of rectilinear real world objects we may have in our minds. However strange this may sound, humans are better wired for dealing with incomplete information and getting approximate, plausible results than dealing with complete information that requires exactitude of reasoning. As a relevant aside, abstraction is one of the pillars of the object-oriented programming - an universally adopted methodology of designing and writing modern software. References
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