26.1 METHODS
In the prologue (Chapter 1) we discussed the explosive growth of knowledge, especially in mathematics, and the rapidity of change. This leads to the hopelessness of trying to equip the student with all the specific results that may be needed. Since mathematics is the language of science, it is equally true that the explosive growth of the applications of specific results of mathematics to science presents serious problems. Instead of presenting mainly results, we chose to emphasize the methods of mathematics. The examples used were chosen either because they illustrated the point or else were both illustrative and useful (probably!). Methods that apply in science generally were specifically mentioned when appropriate.
We adopted a deliberate policy of including as many examples as we could of the general methods used in mathematics and also commonly occurring in applications. Many points of importance in the examples were not mentioned explicitly in the text. For example, we regularly estimated the area or volume of an integral by independent means. The hidden purpose was to inculcate in your mind the idea that mathematics is not just the manipulation of abstract symbols but is also often connected with measurable things in the real world. Sometimes we used Monte Carlo methods of estimating the integrals to further emphasize the usefulness of random processes; randomness is something to be used constructively rather than always avoided! The teaching of science is slowly coming to this realization.
We also tried to teach you to read mathematical equations by frequently putting the results in words. The purpose was to give you practice in “listening to what equations have to say.” Hence many of the results were stated in words as well as in equations. We also frequently stated things in both algebraic and geometric language to give alternative views of the same thing. This, too, is a useful habit to acquire.
Again, dimensional analysis was not explicitly discussed, but was used implicitly many times. For example, in the discussion of the mean value theorem, in the discussion of van der Waals’ equation, and in many other places we used the methods of dimensional analysis.
26.2 METHODS OF MATHEMATICS
As earlier stated, the few widely agreed upon methods of mathematics are: Extension Generalization Abstraction
Extension
Generalization
Abstraction
These are all somewhat the same thing. Their importance is that only by such an approach can you hope to master the large mass of specific details that arise in the use of mathematics (as well as in mathematics itself!). There have been many examples and exercises with the label “extend,” “generalize,” or “abstract.” As we advanced in the book, we deliberately made things more abstract to develop your abilities in this direction. Making everything as easy as possible would have defeated this goal of learning to do mathematics.
But we also, by example and by specific statement, gave many rules such as: instead of proving a = b it is often better to prove a – b = 0. The way you go about doing mathematics is important. If you expect to cover a lot of ground, an efficient approach is essential! Another word for the same thing is “style.” We have thought about style throughout the book, a style that will be useful to you not only in mathematics but in science and engineering generally. It is hoped that you have both consciously and unconsciously absorbed a good style. It is one thing to take over another’s style, but is more important for you to develop your own.
We also gave examples of how to find and prove theorems. Theorems do not arise in a vacuum and then require a proof. Experience shows that often the hypotheses of the theorem come from the proof you finally find, hence the name proof-driven theorems [L]. It is only after you have found some kind of a proof that you know what you have to assume for the hypotheses of the theorem. Of course, you then generalize and extend the theorem you have just found to try to find the limits that it can be extended to, and which are the conditions you first assumed but you now see can be removed. You rarely find the final theorem the first time. It is important to realize this; otherwise, you are apt to become discouraged when you look at the final polished results that are published in books and journals. The published material is the result of hours and hours of rethinking, finding alternative proofs, and final polishing. Unfortunately, many mathematicians have the subconscious standard of beauty that correlates mathematical elegance with surprise. They arrange the final presentation in as surprising a form as they can, and you have little chance of understanding how they originally found the results. They also seem to prefer to give you the results rather than the methods.
Another powerful tool for doing mathematics is reasoning by analogy and similarity, and we have exploited it repeatedly. For example, area and probability are closely related. Symmetry is another tool of mathematics that we have repeatedly illustrated.
You have seen that in the calculus the infinite is handled by dropping back to the finite case and ultimately taking the limit. You saw this potential infinity in the following:
1. Numbers: the limit of the partial sums
2. Missing values: limit of function
3. Tangent line: limit of secant line
4. Integral: limit of a sum
5. Indeterminate forms: limit of function values
6. Improper integrals: limit of finite case
7. Infinite series: limit of partial sums
8. Fourier series: limit of partial sums
It is the same simple idea in each case, in spite of the surface diversity. It could be claimed that the calculus is the systematic use of the limit method for dealing with infinity.
Similarly, you saw the method of undetermined coefficients, beginning with mathematical induction and going through the whole book; if you can find the form of the answer, then by imposing the conditions of the problem you can determine the coefficients of the form. It is a slow but sure method of doing many problems. There is often a clever method of solving a problem, but the method of undetermined coefficients saves learning or devising a lot of isolated tricks. Yes, it is often inelegant, but it is a tool having broad applications.
The book also included a lot of material on linear independence, because it is a fundamental tool of mathematics. We also gave examples of proving the impossibility of doing something. It is important to be able to prove that something is impossible to do, rather than that you are merely stupid.
You have seen abstraction, extension, and generalization many times. It is an attitude you need to develop to the point where it is almost automatic; when a specific problem is solved, how general is the method and what other kinds of similar problems can you now do? It is for this reason we have repeatedly advocated that you do a few problems but study them carefully. A general approach will solve more specific problems than you can possibly do one at a time.
If you now review the book, you should see how comparatively few ideas are involved, how things that once seemed confusing are now obvious and clear. The deliberate use of the spiral of learning, which tends to encourage “chunking” of ideas, means that you can finally see the simplicity behind the face of complexity. The book was written mainly by thinking and only occasionally was a “copy from some other source” used. Thus you see that the author “knows” comparatively little; he mainly constructed the details from general principles he had learned over a lifetime of practice. Many items were recalled from having pondered what the mathematics was that had proved to be successful in some applications and how far the same methods could be extended to other situations. It is necessary, apparently, for individuals to make the abstractions for themselves if the abstraction is to become useful later on. What you learn for yourself is often what turns out to be most useful in later life. Thus many of the abstractions have not been specifically labeled, and you have been encouraged to find them yourself so they will be your own.
26.3 APPLICATIONS
We have shown how the things you have learned are applicable in many places, within mathematics itself as well as in other fields. The main fields of application we used for the calculus, probability and statistics, were chosen for their importance in modem science (as compared with the classical application of the calculus to mechanics), as well as the range of applications generally in life. In neither case was a complete course attempted; only those parts were used that illustrated the use of the mathematics that you had just learned.
The case histories were included to give you a slight understanding of the actual use of mathematics in practice. They were, of course, only small parts of much larger units of thought, but reality is too large to encompass completely in a single book.
The rich variety of applications we have given should prove the point that mathematics is indeed the language of science and engineering, as well as increasingly of other fields as they become more precise in their statements of results. A few of the examples were chosen for mere teaching purposes, but an effort was made to select useful results even when we did not explain their use at that time. The attempt to teach so many different ideas naturally gave you trouble in mastering things. Since you will likely take later courses in both probability and statistics, the concepts will get further reinforcement. The exposure here will hopefully facilitate their later complete mastery. We took the risk of including too much variety for the beginner to digest.
Many of the illustrations are useful in other fields. For example, many of the specific integrals were chosen because they arise in many places. One such case is the integral
which arises in a key result in probability theory. Similarly, as you go on in mathematics, probability, and statistics, you can expect to meet other specific examples we have given.
26.4 PHILOSOPHY
Unlike almost all mathematics courses, we have deliberately engaged your attention on the philosophical foundations of mathematics. We have not tried to give you the answers to many of the points raised, but rather it was intended that they be sufficiently disturbing to you that you would think through your own position as to the bases of mathematics and its applications. If you have skipped such items, it is your loss. It is hoped that you have seen through any glib answers that you may have been given and have been willing to think for yourself. Thinking for yourself is perhaps the single most important habit you could have learned from this book.
To encourage thinking for yourself, we have gone out of the way to produce examples of the use of mathematics that seem to contradict “common sense”; surfaces of revolution whose volume was finite but whose surface was infinite, lines with peculiar lengths, optimization problems with no shortest path, and the like. The mathematics you have learned must be tempered with “common sense” when it is applied and action taken based on the results. There is a definite gap between the theories of mathematics and the various fields of applications. In a sense, society has already exploited most of the applications where the mathematical model is close to our observations of the real world; most (but not all!) new applications will have larger gaps between the mathematical model and what is observed.
Mathematics is both useful and intellectually interesting in its own right, and we have often engaged in what might be called “metamathematics,” the study of why mathematics is the way it is. It is an interesting topic of endless fascination. To name but one example, why are the sin and cos functions of trigonometry not exact duals of each other? Why do you usually find more cosines than you find sines as you scan material involving them? We showed how trigonometric identities can be reduced to rational functions with complex coefficients, another example of the partial equivalence of different fields.
Pure mathematicians are people who find mathematics is their main interest, mathematicians have an interest in both mathematics and in its applications, while applied mathematicians care mainly about applications. There is room for a wide variety of interests in the coming world in which you can use what you have learned and extend known results to get new results. Much of mathematics and its applications still remains to be discovered (created if you prefer).
It is hoped that you have seen how mathematics is used and have a somewhat accurate impression of its role in science and engineering as well as in other fields. It is also hoped that you have some idea of your future needs for mathematics in your coming career.
This book is a serious attempt to prepare you to push forward the frontiers of knowledge, rather than merely fill you full of past results. The author believes in the approach based on regeneration rather than retrieval of knowledge. It is hoped that the earlier parts of the book, which displayed the methods of mathemetics, prepared you for the later more rapid discussion of many of the applications. It is the methods of the mathematics that produce the results, and, because of the vastness of current knowledge, learning the methods of mathematics appears to be the only hope for preparing you to play your part in future developments.