Part 1
Elements
What is Mathematics?
The book of nature is written in the language of mathematics.
Johannes Kepler, Astronomer
The Queen of Sciences
Mathematics is the queen of sciences and arithmetic the queen of mathematics.
Carl Friedrich Gauss, Mathematician
In second-grade classes, I try to explain the importance of numbers. I tell the children a story of a king who hated numbers so much that he forbade their use throughout his kingdom. Together, we try to imagine a world free of numbers, and discover that life in it is very limited. Since it is forbidden to mention a childâs age, children of all ages enter the first grade. You cannot pay for your groceries, nor can you set up an appointment, since you are not allowed to mention the number of hours and minutes.
This is only an illustration of the importance of mathematics in our lives. As civilization and technology advance, our lives become more and more dependent on mathematics. Steven Weinberg, a Nobel laureate in physics, dedicated two chapters of his book Dreams of a Final Theory to subjects beyond physics: mathematics and philosophy. He writes that time and again he is surprised to discover how useful mathematics is, and how futile is philosophy.
To understand why this is so, one must understand what is mathematics. This is not a simple question. Even professional mathematicians find it difficult to answer. Bertrand Russell said of mathematicians that they âdonât know what they are doing.â (His judgment of philosophers was even harsher: A philosopher in his eyes is âa blind man in a dark room looking for a black cat that isnât there.â) This is true in at least one sense: Most mathematicians do not bother to ask themselves what it is, exactly, that they are doing.
To try answering this question, we will start with a simple example: What is the meaning of 3 + 2 = 5?
In the first grade, I ask the children to examine how many pencils there are when you add 3 pencils to 2 pencils. They know that âadditionâ means âjoining.â Therefore they join 3 pencils and 2 pencils and count: 5 pencils. Now I ask, âHow many buttons are there when you add 3 buttons and 2 buttons?â â5 buttons,â they answer, without missing a beat. âHow do you know?â I insist. âWe know from the previous question.â âBut the previous question was about pencils. Maybe itâs different with buttons?â They laugh. But not because the question is pointless. On the contrary. It contains the secret of mathematics â abstraction. It does not matter if the objects in question are pencils, buttons or apples. The answer is the same in all cases. This is why we can abstractly say: 3 + 2 = 5.
This is an elementary, but characteristic, example: Mathematics abstracts thinking processes. Obviously, every thought is abstract to a certain degree. But mathematics is unique in that it abstracts the most elementary processes of thought. In the example of 3 + 2 = 5, the process involved is the joining of objects: 3 objects and 2 objects. One can ask many questions about these objects: Are they pencils or apples? Are they in your hand or on a table? And if they are on a table, how are they arranged? Mathematics ignores all these details, and asks a question that relates not to the various details, but only to the fact that these are objects that are joined: the resulting amount. That is, how many objects are there?
Abstract thought is the secret of manâs domination of his environment. The power of abstractions lies in the fact that they enable us to cope efficiently with the world. In other words, they save effort. They enable going beyond the boundaries of the âhere and nowâ â something discovered here and now can be used in another place and at another time. If 3 pencils and 2 pencils equal 5 pencils, the same will be true for apples, and it will also be true tomorrow. A one-time effort provides information about an entire world.
If abstractions in general are useful, then all the more so is mathematics, which takes abstractions to their limit. Therefore, it is not surprising that mathematics is so useful and practical.
Should Everyone Learn Mathematics?
People, on learning that I am a mathematician, often react with a thin smile, barely hiding a grimace of agony: âMathematics wasnât one of my strong subjects.â For so many people learning mathematics is such a tormenting experience that each generation asks the same question â what for? Why is this torture necessary? Shouldnât most people just give up on the attempt to learn mathematics? Nowadays, when a calculator can instantly perform mathematical operations, what is the point of learning the multiplication table or long division?
One answer is that mathematics is the key to all professions demanding knowledge of the exact sciences, and there are many of those. But mathematics is important not only for understanding reality. It offers much more than that â it teaches abstract thought, in an accurate and orderly way. It promotes basic habits of thought, such as the ability to distinguish between the essential and the inessential, and the ability to reach logical conclusions. These are some of the most significant assets that schooling can provide.
The question still remains unanswered â why is it so difficult? Must mathematics be a cause of suffering? A currently popular answer is ânoâ â the problem lies in the teaching. Common opinion is that many children considered to be âlearning disabledâ are actually âteaching disabled.â But it canât be that simple. Blaming the teachers is too simplistic, and unreasonable. Anyone who claims that for hundreds and thousands of years mathematics teachers have been doing a bad job, must explain why this is so and why it isnât so in other subjects.
The special problem in teaching mathematics lies in the difficulty of conveying abstractions. You can tell people the name of the capital of Chile, but you canât abstract for them. This is a process each person must accomplish on his or her own. One must mentally pass through all the stages from the concrete to the abstract. The teacherâs role in this process is to guide the student so that he experiments with the principles in the correct order. This is not a simple art that is easy to come by. But neither is it impossible. One of the purposes of this book is to relay some of the principles along the path of such âmidwiferyâ teaching, as Socrates put it.
The Three Mathematical Ways of Economy
I didnât have time to write you a short letter, so I wrote a long one.
Blaise Pascal, Mathematician
Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself.
George PÂŽolya, Mathematician
The true virtue of mathematics (and not many know this) is that it saves effort. This is true of any abstraction, but mathematics has turned economy of thought into an art form. It has three ways to economize: order, generalization and concise representation.
Order
Carl Friedrich Gauss was the greatest mathematician of the 19th century. One of the most famous stories in the history of mathematics tells of how his talent came to light when he was seven years old. One day, his teacher, looking for a break, gave the class the task of summing up all the numbers between 1 and 100. To his surprise, young Carl Friedrich returned after a few minutes, or perhaps even seconds, with the answer: 5050.
How did the seven-year-old accomplish this? He looked at the sum he was supposed to calculate, 1 + 2 + 3 + ··· + 98 + 99 + 100, and added the first and last terms: 1 and 100. The result was 101. Then, he added the second number to the one before last, that is, 2 and 99, and again the result was 101. Then 3 and 98, which yielded 101 again. He arranged all 100 terms in 50 pairs, the sum of each equaling 101. Their sum was thus 50 times 101, or 5050.
What little Gauss discovered here was order. He found a pattern in what seemed to be a disorganized sum of numbers, and the entire situation changed â suddenly matters became simple.
Imagine a phone-book arranged by a random order, or an unknown order. To find a phone number, you would have to go through each and every name. The order introduced into the phone-book, and the fact that we are familiar with it, saves a great deal of effort. A relatively small effort invested in alphabetical arrangement is returned many times over.
Or, think how much easier it is to live in a familiar city than in a strange one. A local knows where to find the supermarket or the laundromat. Knowing the order of the world around us provides us with orientation. Science, and mathematics in particular, has taken upon itself to discover the order of the universe, so that we may adjust our actions to it.
Generalization
There are many jokes about the nature of mathematics and mathematicians. The following is probably the best known of them all. I make a point of telling it to my students in every course I teach, since it is not only the most familiar, but also the most useful. It illustrates the principle of mathematical practice: Something once done does not require redoing.
How can you tell the difference between a mathematician and a physicist? You ask: Suppose you have a kettle in the living room. How do you boil water? The physicist answers: I take the kettle to the kitchen, fill it with water from the tap, place it on the stove and light the fire. The mathematician gives the same answer. Then you ask: Suppose you have a kettle in the kitchen. How do you boil water now? The physicist says: I fill the kettle with water from the tap, place it on the stove and light the fire. The mathematician answers: I take the kettle to the living room, and this problem has already been solved!
This brings economizing of thought ad absurdum, by placing it before true economization.
âThis has already been solvedâ was also the answer we heard from the children who said they did not need to check how many buttons are 3 buttons and 2 buttons, since they had done the same with pencils. It appears, whether overtly or hidden, in each mathematical proof, and in every mathematical argument. âWe have already done this, and now we can use it.â In fact, this idea lies behind every abstraction: What we discover now will also be valid in other situations.
Proof in Stages: Induction
There is a mathematical process that is based entirely on the principle of âthis has already been done.â It is called âMathematical Induction.â A certain point is established in stages, with each stage relying on its predecessor, that is, on the fact that the previous case âhas already been solved.â
We will encounter this process several times throughout the book but will not mention it explicitly. For example, the decimal system is inductive: First, ten single units are collected to equal a new entity called a âten.â Then ten tens are collected to equal a new entity called a âhundred,â and so forth. Yet another example is calculations: All algorithms used to calculate arithmetical operations are based on induction.
Concise Representation
The third mathematical economy is in representation. We are so used to the way numbers and mathematical propositions are represented, we forget that methods of representation were not always so sophisticated, and it was not so long ago, relatively speaking, that mathematical notation was much more cumbersome.
Letâs begin with the representation of numbers. Up until about three thousand years ago, numbers were represented directly â â4â was represented by four markings, for example, four lines. This is a good idea for small numbers, but impractical for larger ones. Using the decimal system, we can now represent huge numbers concisely: A âmillionâ only requires seven digits.
The second type of economy is in the representation of propositions. A âmathematical propositionâ is the equivalent of a sentence in spoken language. Up until a little over two thousand years ago, mathematical propositions were phrased in words, for instance: âThree and two is five.â Then, a very useful tool was invented: the formula. Its originator was probably Diophantus of Alexandria, who lived during the 3rd century B.C. Formulas are not only shorter, they are also more accurate and uniform, and allow systematic handling.
Historical Note
The notation we currently use developed slowly and gradually. Its current form was only established ...