9/17/99

1. Abstraction is at once a weakness and a powerful strength. The ability to think abstractly is probably the defining characteristic of the human species. In this essay, I will try to explain these two ideas.

2. In my studies of mathematics, both as an undergraduate and as a graduate student, I was never particularly adept at solving the problems, proving the theorems, or getting top grades. I was, however very good at catching on to the concepts and understanding the immense power and majesty of the subject.

3. I remember Professor Grimm, just as we were about to embark on a six-week intensive effort to define what numbers are, turning to the class and saying "A few of you will catch on to what we are going to talk about and you will discover what mathematics is all about. The rest of you should make sure you change your major if you are currently enrolled as a math major." Fortunately, I was among the few that caught on, and I have been in awe of mathematics ever since.

4. There were many new concepts revealed to me along the way. For example, I learned that it is not only OK, but normal and common for mathematicians to consider and talk about propositions that they know to be untrue. For most people, to talk about ideas that they know are false is something to be avoided and to do so may even be considered a sin. In mathematics, it is a powerful weapon in the arsenal of reasoning.

5. There are many other such examples. Among them is the concept of abstraction which is the subject of this essay. In a nutshell, abstraction is talking about talking. More precisely it is thinking about thinking. This, of course, doesn't convey the meaning at all to anyone who doesn't already understand the concept. It is too abstract, so to speak.

6. As I already intimated, it probably takes a study in a course like Professor Grimm's in order to catch on to the concept, but nonetheless I am going to give it a shot in this essay. Abstraction is the notion of classifying things. That is, to put bunches of things into classes, or groups of similar things. Mathematicians typically call these classes or groups ‘sets' and they call the things inside them ‘elements'. The terms ‘class' and ‘group' are reserved for very special sets with specific properties and I am not aware of any use of the term ‘bunch' in mathematics.

7. In ordinary language, we use terms like these interchangeably without causing much of a problem. For example, the idea of a mammal is an abstraction which puts a certain bunch of animals in a group called mammals, (actually the biological Class of Mammalia) and leaves the other animals out.

8. Typically, mathematicians try to avoid specific examples when going through the mathematical process. For example, if one is describing something about triangles, it is considered bad form to actually draw a triangle on the blackboard. This is because there is a perceived trap of implying that the discussion applies only to this particular triangle, and not to others. If the triangle on the board was, for example, a right triangle, the trap would be the assumption that the discussion applies only to right triangles and not to obtuse ones. This reason might have some validity, but I have suspected that the more common reason was that the instructor used the rule to obfuscate the discussion in order to maintain his (or her) superiority and maintain the students' inferiority.

9. Whatever the case, when mathematicians talk about sets, they deliberately try not to describe them or the elements they contain any more specifically than absolutely necessary for the discussion. This oftentimes makes the discussion seem detached, groundless, and hard to grasp. The power of this, however, is that the mathematical truths (or consequences, depending on how you look at it), may be much more profound, or far-reaching than the mathematician might at first realize.

10. In our example of the concept of a mammal, in ordinary discussion it would be typical to first describe ‘mammal' as a member of a higher set by saying something like ‘mammals are animals' and then saying something to distinguish mammals from all other animals, like ‘mammals are more or less covered with hair, give birth to live young, feed them mother's milk'. This is the typical pattern or paradigm of a definition of a term that we use in everyday life; we first place the term in a general category that is well known, and then describe specific characteristics that distinguish it from other members of this category. More succinctly (and sort of mathematically) we would define X by saying that X is a P which has properties Q.

11. Don't let that algebra stuff throw you for a loop. I threw it in to illustrate another example of abstraction. Actually, at this point, we may already be in the trap I mentioned earlier. Our example of mammals might have implied that abstraction only deals with the classification of objects by defining a hierarchy of categories containing the objects. This is only one example.

12. Another example is that we can take a set of elements (a bunch of things), like words, and not only put them in a set, but give that set a name, like X. That's exactly what I did in paragraph 10. I was talking about how to define a ‘term', and at the end of the paragraph, when I summed up, I used the name ‘X' to stand for one arbitrary member of the set of ‘terms' that we might try to define. (In mathematics, the word ‘arbitrary' is used very precisely and means ‘any old one, and not any one in particular'). This is just a simple substitution of the symbol ‘X' for any old word that we want to define. This kind of substitution is very common and makes the language of mathematics a lot shorter. The result of doing a lot of this is that you get the seeming gobbledygook that fills the math books. Don't let it scare you.

13. Now, if you understood paragraph 12, go back and read it again. (If you didn't understand it, also go back and read it again, but this time try to understand it.) Once you have understood the paragraph, read it again and see if it isn't an example of talking about talking. Actually, it is really writing about writing, but in writing this, I am imagining that I am talking to you and you are listening so let's go with what is in my imagination. (Just another type of abstraction.). What paragraph 12 is all about, is describing what is going on when we have a particular type of conversation. That is, a conversation involving the definition of a word.

14. Now, don't let these latest examples of abstraction limit you either. Abstraction involves more than just classifying objects, substituting symbols for other things, or making analogies between one form of reality and another. There is also the idea of levels of abstraction.

15. To illustrate this, consider that the exercise of paragraph 13 was to get an idea of the notion of talking about talking. Abstraction is not limited to this. For example, we can talk about talking about talking. (Or read about writing about writing). As an example of this, go back and look at paragraphs 11 through 14 one more time.) Some of the time I was talking about some ideas, but at other times, I was talking about the paragraphs themselves. I referred to them by number, asked you to re-read them, and so on. At these latter times, the discussion went up a notch in the levels of abstraction to that of talking about talking about talking.

16. Similarly, the last sentence of paragraph 15 was talking about talking about talking about talking. Similarly, the previous sentence of this paragraph was talking about talking about talking about talking about talking. Similarly, the previous.... I think that's enough.

17. Then there is the notion of what can be talked about in principle as opposed to what can be done in actual practice. For example, in paragraph 16, we could talk about continuing those sentences which start with ‘Similarly' indefinitely. That is we could have an infinite number of such sentences which would declare that there are an infinite number of levels of abstraction. Mathematicians talk about this kind of thing without caring a whit about whether or not the idea is true or false in the absolute sense. They would say that in principle, we can continue the sequence of sentences indefinitely. In reality, of course we cannot because we would run out of paper, ink, time, energy, atoms, and even space before we could complete the job.

18. So far, we have only looked at what abstraction is. Now let me try to illustrate the power of what abstraction can do. I think it deserves most of the credit for all the success of our technological world. Abstraction is involved in virtually all aspects of the human activity that has brought us our present civilization. Language itself is a form of abstraction where we substitute spoken and written words for things and concepts in the real world. Mathematics, as we have discussed, derives much of its power from abstraction. Science, which is based on experiment, observation, and mathematical reasoning, derives much of its power from abstraction as well. The design, engineering, and manufacturing of all of our technological machines follows directly from the discoveries of science.

19. It might surprise you to know that earlier in this (the 20th) century, we, as a species, have been learning more about our universe as a result of mathematics than by direct observation. For most of the time since Aristotle, scientists have learned mostly by observing nature directly. They would either just look at what was going on in the natural world, or they would deliberately rig up an experiment, let it run, and then watch how nature behaved in that more or less controlled environment.

20. The turning point could be marked by the discovery of the planet Neptune in 1846. This was the first planet which was proved to exist by mathematics before it was actually observed in the sky. Since then, most all new discoveries in the physical sciences are made by mathematical calculations before they are seen, if they ever are, in nature. Mathematics itself made a major leap in this century when Kurt Goedel introduced a very abstract line of reasoning where the elements of the sets he considered were the very statements that mathematicians use in their reasoning, as well as sets of these sets, and so on.

21. I think the power of abstraction speaks for itself when you consider all that has been developed so far in our technological society. But what about my opening comment that abstraction is a weakness?

22. In another nutshell, what I mean by this is that abstraction is in a way a simplification. Classification can be seen as generalization, and as you may have heard, all generalizations are false, including this one. Or, as Murphy put it, for every problem there is a simple elegant solution - which doesn't work.

23. As I mentioned earlier, I am inclined to look for the general concepts in a subject, as opposed to mastering a lot of detail. In a way, this gives me a sort of superficial knowledge. If I try to describe some aspect of a particular subject in the presence of an expert in that field, then my descriptions typically look foolish, if not false, in the face of detailed, or isolated counter-examples to a more general observation that I might make. This is what I mean by the weakness of simplification.

24. Physics has been fortunate, in a way, that there is so much uniformity in nature. As far as we know, each and every type of elementary particle is identical. Not just similar, as humans, or cheetahs, are, but identical like mathematical points are. It is also fortunate, and surprising if you think about it, that so much of nature's behavior follows simple mathematical laws. As a result of this uniformity, physicists have been able to come up with simple, general formulas that describe the behavior of a vast array of physical systems. These formulas are powerful simplifications which aren't seen as a weakness at all.

25. So not all abstract generalities are a sign of weakness. In social and biological sciences, we don't have such strong patterning and similarity of elements as we have in the physical sciences, so it's a lot harder to make generalizations which are believable and incontestible. Nonetheless, if there are true generalizations to be made, they would lend the same kind of power to the subject that they do in physics. It's just harder to find them.