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1.1 THE IMPORTANCE OF MATHEMATICS
You live in an age that is dominated by science and engineering. Whether you like it or not, they have significant effects on your life. And it seems probable that in the near future their effects will become even greater than they are now. Thus if you wish to be effective in this world and to achieve the things you want, it is necessary to understand both science and engineering (and these require mathematics).
Long ago Pythagoras (died about 492 B.C.) said:
Number is the measure of all things.
(I am told that the strict translation is simply "everything is number.") Galileo (1564–1642) similarly said:
Mathematics is the language of science.
Mathematics clearly plays a fundamental role in the older sciences, such as astronomy, physics, and chemistry, and is of increasing importance in the other "hard" sciences. But mathematics is also rapidly invading all the biological sciences, especially such fields as genetics and molecular biology. Even in the humanities we find that questions of authorship and style are being decided by applying statistical tests to the written material. For a long time business administration has been using more and more mathematics as people have tried to understand both the workings of the large, complex organizations they have to manage and the competition between such organizations. Mathematics is similarly needed in modern government administration. The social sciences are also heavily dependent on the statistical approach to many of their problems. Indeed, one may say
that Science is a habit of the mind as well as a way of life, and that mathematics is an aspect of culture as well as a collection of algorithms. (C. B. Boyer, 1906–1976)
It appears, therefore, that mathematics, in one form or another, will invade most fields of knowledge as we try to make them more reliable. There is an inevitability of this happening. But it has long been observed that the mathematics that is not learned in school is very seldom learned later, no matter how valuable it would be to the learner. Any unwillingness to learn mathematics today can greatly restrict your possibilities tomorrow.
This is not an assertion that all of mathematics will be useful; all that can be done is to look at both the past and the present, and then make educated guesses as to future needs for mathematics. This book covers what is believed to be the most useful applications (on the average). The three fields, calculus, probability, and statistics are all in constant use. Mathematicians in the past have tended to avoid the latter two, but probability and statistics are now so obviously necessary tools for understanding many diverse things that we must not ignore them even for the average student.
Calculus is the mathematics of change. The mathematics you have learned up to this point has served mainly to describe static (unchanging) situations; the calculus handles dynamic (changing) situations. Change is characteristic of the world. As Heraclitus (sixth to fifth century B.C.) said,
You cannot step in the same river twice.
Everything is in a state of becoming and flux.
Probability is the mathematics of uncertainty. Not only do we constantly face situations in which there is neither adequate data nor an adequate theory, but many modern theories have uncertainty built into their foundations. Thus learning to think in terms of probability is essential.
Statistics is the reverse of probability (glibly speaking). In probability you go from the model of the situation to what you expect to see; in statistics you have the observations and you wish to estimate features of the underlying model. There is, of course, much more to statistics than this.
This book is not mainly about the results obtained in mathematics; rather it is concerned with mathematics itself. There is simply too much mathematics in current use, let alone what will be in use in the near future, to try to cover all the applications of mathematics. Instead of concentrating on the results, we will concentrate more on the methods from which the results follow. Thus this book is fundamentally different from the other books on mathematics you have studied. Most mathematics books are filled with finished theorems and polished proofs, and to a surprising extent they ignore the methods used to create mathematics. It is as if you were merely walked through a picture gallery and never told how to mix paints, how to compose pictures, or all the other "tricks of the trade." Of course it would be simpler if I could tell you all the things you need to know about mathematics, but this approach seems to be hopeless for the coming years. Showing you the methods for doing mathematics covers a wider range of applications, but it does leave more creativity to you when you need some specific result. You will be left more able to do mathematics, to create mathematics as you need it, but less able to recall some specific result you happen to need. In short, in the face of almost infinite useful knowledge, we have adopted the strategy of "information regeneration rather than information retrieval." This means, most importantly, you should be able to generate the result you need even if no one has ever done it before you—you will not be dependent on the past to have done everything you will ever need in mathematics.
I have also chosen to raise many questions about the relationship between the mathematical models developed in the text and the physical world in which you live. I have not attempted to supply you with all the answers; rather it is up to you to think about them and come to your own opinions of how far you can trust the results of applying mathematics in the real world. If you are to go very far in your chosen field, it is doubtful if you can long avoid some new applications of mathematics; yet, as you will see, not all applications give sensible answers!
The assumptions and definitions of mathematics and science come from our intuition, which is based ultimately on experience. They then get shaped by further experience in using them and are occasionally revised. They are not fixed for all eternity. In many applications it is essential that you be able to trace the effects of various assumptions and definitions on any conclusions you draw—perhaps the particular mathematics you used was inappropriate for your case! New applications of mathematics will, from time to time, require new assumptions and altered definitions, and it is the intent of the text to prepare you to make them when needed, but naturally we cannot tell what they will be.
It is not claimed that one course will make you a great mathematician able to create all of mathematics for yourself; all that can be done is to start you down the path of learning to create mathematics. In a very real sense, all we can do is coach you; you must have both some talent and the willingness to practice what is being taught. If you expect to continue learning all your life, you will be teaching yourself much of the time. You must learn to learn, especially the difficult topic of mathematics.
1.2 THE UNIQUENESS OF MATHEMATICS
At first glance much of mathematics seems arbitrary, but, at least for most useful mathematics, this is not so. To study the essential uniqueness of mathematics, I asked many expert scientists and engineers the following question: "If we ever find ourselves in two-way communication with a distant world, will they have essentially the same mathematics as we do?" The answers were all a definite "Yes." They generally reasoned along the following lines:
1. The physical phenomena we see in space resemble those we see on Earth.
2. From this we infer that the same laws of physics apply everywhere.
3. Since mathematics is the language of science, it too must be essentially the same.
Their arguments were much more detailed and complete than this, but at present we are in no position to explain the technical details they used, nor the breadth of their arguments.
The meaning of the word "essentially" needs some explanation. In Euclidean geometry (approximately 300 B.C.), for example, the Greeks apparently chose to ignore orientation. They said that a left-handed and a right-handed triangle could be congruent (Figure 1.2-1). To carry out the proof, they allowed the flopping over in three dimensions of a triangle, even though a triangle is a two-dimensional figure. As a result of this choice, when they came to three-dimensional geometry they could not get the important theorem that in three dimensions there are only two orientations, the left-handed and the right-handed spirals. The distant intelligent beings we are imagining might have chosen to include orientation in their geometry. But it is claimed that this is not an essential difference. Again, Ptolemy (second century A.D.) used the chords of the double angle where we now use sines (Figure 1.2-2), but this is hardly an essential difference; it is merely a notational difference, although an important one in practice.
Although the major users of mathematics are almost all in agreement that mathematics is essentially unique, we need to consider the idea often expressed by pure mathematicians that
Mathematics is the free creation of the human mind.
In a sense the users of mathematics are saying that mathematics, like science itself, is discovered rather than created. The pure mathematicians are saying that mathematics resembles the other arts in the sense that creation is a personal thing. In science creativity is comparatively impersonal; if one scientist does not discover something then another one will. But if Shakespeare had not written the play King Lear, we would probably have no closely similar play.
There may be less conflict between these two extreme opinions than appears; perhaps they are talking about different things. The users are talking about the mathematics they have found to be useful, while the pure mathematician may be talking about the mathematics now being created.
Another explanation of the difference is that those who daily work closely with the real world tend to believe that our senses, while occasionally deceived, report fairly accurately what is out there. On the other hand, those who work more with their imaginations tend to believe that our senses are rather unreliable, and the world out there could be very different from what we think it is. All are agreed that we cannot know with absolute certainty.
The mathematical results in this book are user oriented; they are the kinds that have been found to be useful in helping us to understand the universe in which we live. Textbooks in the past, especially at the calculus level, have concentrated on physical science applications. This was appropriate both because they are the historical source of much of the mathematics and because they are usually easier to understand. But the applications of mathematics of interest to the typical student are of far wider range than this. We will do a little in this direction of selecting suitable applications, but we are forced to minimize them lest they get in the way of the essential part of the book—how you do mathematics. The applications are often illustrated in the case histories.
However, we do not neglect the beauty of mathematics; it often makes the subject matter much more attractive and hence easier to master. The student should often find beauty in this book and gradually learn to do mathematics in attractive ways. As Edna St. Vincent Millay (1892–1950) wrote,
Euclid alone has looked on beauty bare.
1.3 THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS
There is a universality about mathematics; what was created to explain one phenomenon is very often later found to be useful in explaining other, apparently unrelated, phenomena. Theories that were developed to explain some poorly measured effects are often found to fit later, much more accurate measurements. Furthermore, from measurements over a limited range the theory is often found to fit a far wider range. Finally, and perhaps most unreasonably, quite regularly from the mathematics alone new pehnomena, previously unknown and unsuspected, are successfully predicted. This universality of mathematics could, of course, be a reflection of the way the human mind works and not of the external world, but most people believe it reflects reality.
This remarkable effectiveness is difficult to explain unless one makes some kind of assumption, such as that there exists, in some sense, both a physical and a logical universe to be discovered, and that these two universes are somehow intimately related. Mathematics is then seen to be a description of the logical structure of the universe.
A second explanation is that the useful mathematics, being based on long experience, follows the scientific approach. Only after being well established were the postulates (or axioms) carefully abstracted. Currently, mathematicians make no distinction between axiom and postulate; even Archimedes (287?–212 B.C.) did not bother to distinguish between them. Mathematics based on arbitrary postulates, or possibly picked either for their elegance or else for the elegance of what follows from them, would seem to have little chance of being successfully used in explaining the universe.
The postulational approach is widely used in mathematics and is very valuable when used properly. There is an innate elegance about the postulational approach. But it has been well said, and repeated by several well-known mathematicians, that
A book which starts off with axioms should be preceded by another volume explaining how and why these axioms have been chosen, and with what end in view.
We will carefully observe this rule and examine closely any assumptions we lay down.
Yet it must be acknowledged that an eminent physicist, the Nobel laureate P. A. M. Dirac (1902–), has said
I want to emphasize the necessity for a sound mathematical basis for any fundamental physical theory. Any philosophical ideas that one may have play only a subordinate role. Unless such ideas have a mathematical basis they will be ineffective.
Apparently, the elegance of mathematics should not be dismissed lightly.
When a theory is sufficiently general to cover many fields of application, it acquires some "truth" from each of them. Thus there is a positive value for generalization in mathematics that may not be apparent to the beginner. This is one of the many reasons this book emphasizes the processes of extension, generalization, and abstraction. They often bring increased confidence in the results of a specific application, as well as new viewpoints. Notice that we are mainly interested in the processes, and we are not interested in presenting mathematics in its most abstract form. On the contrary, we will often begin with concrete forms and then exhibit theprocess of abstraction.
It is necessary to note that science and mathematics do not explain everything. In more than 2000 years they have added little to our understanding of such things as Truth, Beauty, and Justice. There may be definite limits to the applicability of the scientific method.
1.4 MATHEMATICS AS A LANGUAGE
As Galileo said,
Mathematics is the language of science.
Not only must you learn to think in the language of mathematics, you also need to read it. To do this, you must learn the alphabet, the vocabulary, and the grammar. In this language there are no syllables, and the individual characters tend to be the words, while the equations tend to be the sentences. Much of mathematics consists of rewriting a sentence in another logically equivalent form.
This book is filled with strange symbols that you need to recognize easily. For example, you have already met in school the Greek lowercase letter π (pi). Because the individual letters often play the role of words, we need a rich alphabet of symbols in mathematics, and for this reason mathematicians have been driven to the use of the Greek alphabet, both upper- and lowercase. It would be foolish to try to avoid the Greek alphabet because it is in constant use in mathematics and its applications. Thus you should learn to recognize these symbols, and for this purpose they are given in Appendix C. Learn to recognize them rapidly so that when they occur in the middle of some difficult passage you need not be distracted by, "What symbol is that?" There are also other strange symbols such as the elongated S (∫), which is an operation called "integration." Thus the language has both processes (verbs) and things (nouns).
Excerpted from Methods of Mathematics Applied to Calculus, Probability, and Statistics by Richard W. Hamming. Copyright © 1985 Richard W. Hamming. Excerpted by permission of Dover Publications, Inc..
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Posted August 11, 2007