What Is Abstraction?
Matter is as much an abstract concept as truth, good or evil.
P. D. Ouspensky
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
To my mind, a limitation in coping with abstraction presents the greatest barrier to doing mathematics. And yet, as I shall show, the human brain acquired this ability when it acquired language, which everyone has. Thus the reason most people have trouble with mathematics is not that they don't have the ability but that they cannot apply it to mathematical abstractions.
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.
abstract (adjective): the first element is from Latin abs "away from, off," from the Indo-European root apo- "away," as seen in English of, off, and ebb. The second element is from Latin tractus, past participle of trahere "to draw, drag, haul." The Indo-European root is tragh- "to draw, drag." An Indo-European variant dhragh- is the source of native English draw as well as drag, borrowed from the Old Norse. An abstract principle is one which has been drawn away from any specific examples of that principle. Modern abstract algebra, for example, is far removed from the physical problems that originally gave rise to the methods and manipulations of elementary algebra.
abstraction (noun): the process of formulating a generalized concept of a common property by disregarding the differences between a number of particular instances ...
The Brafman siblings [Sway, pp. 74-75] quote psychologist Franz Epting, an expert in understanding how people construct meaning in their experiences.
"We use diagnostic labels to organize and simplify. But any classification that you come up with," cautioned Epting, "has got to work by ignoring a lot of other things - with the hope that the things you are ignoring don't make a difference."
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:
People think that because a novel's invented, it isn't true. Exactly the reverse is the case. Biography and memoirs can never be wholly true, since they cannot include every conceivable circumstance of what happened. The novel can do that.
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.
- O. and R. Brafman, Sway: The Irresistable Pull of Irrational Behavior, Doubleday, 2008
- K. Devlin, The Math Gene, Basic Books, 2000
- A. Powell, Books Do Furnish a Room, in 4th Movement in A Dance to the Music of Time, University of Chicago Press, 1995
- W. W. Sawyer, Mathematician's Delight, Dover, 2007 (from the 1943 Penguin edition)
- S. Schwartzman, The Words of Mathematics, MAA, 1994
Copyright © 1996-2018 Alexander Bogomolny