We have studied Euclid's Elements as an example of a pure science, the science of geometry, but we saw also that Books VII, VIII, and IX deal with the science of arithmetic, or (as we would now call it) with the science of algebra. If we wish to see such a pure science of number developed even at great length, we should examine the work which became standard among the Greeks, Romans, and medievals, both Mohammedan and Christian, the Introduction to Arithmetic of Nicomachus of Gerasa, who lived about one hundred years after the birth of Our Lord. It deals with that basic study of numbers which we today call Number Theory.
The term "algebra" was introduced to indicate a growing interest in the art of calculation, rather than a demonstrative study of the properties of numbers. just as there is a science of logic which seeks to prove the rules of logic (see page 568 ff.) and an art of logic which is concerned with using these rules, so the science of number is concerned with proving the properties of numbers, while the art of calculation is concerned with applying a knowledge of these properties to the solution of particular problems.
Today, however, algebra includes both the art of numerical problem-solving and the science of numbers, but the elementary study of algebra usually lays emphasis on its problem-solving aspect. Now that we have analyzed geometry as a science, we need to review algebra from the same point of view and see how it might be set up as a pure science.
The axioms of algebra and geometry are the same, since an axiom is a principle common to several sciences. Algebra and geometry are very closely related; but they are, nevertheless, distinct sciences and hence they have different definitions and postulates.
The definitions of algebra we have already discussed in Chapter I of this Part. We must have at least nominal definitions of the unit, of every natural number, of all the operations of algebra, of the operational numbers, and of the various properties and combinations of numbers, such as "odd," "even," "sum," " ratio," etc.
It is not so easy, however, to decide on the postulates of algebra. Each postulate will state the existence of one of the things which we have nominally defined (in other words, it will be a real definition), or it will be a truth immediately evident from these definitions. One of the first mathematicians to attempt to state the postulates of algebra in a rigorous form in accordance with modern methods was the Italian mathematician, Giuseppe Peano (1858-1932), who proposed the following five postulates:
- 0 is a number.
- The successor of any number is a number.
- No two numbers have the same successor.
- 0 is not the successor of any number.
- If s is a class to which 0 belongs and also the successor of every number belonging to s, then every number belongs to s. (This is called the principle of mathematical induction.)
This list raises many difficulties, particularly because it uses zero to define number, instead of the other way around (see page 320). A more satisfactory statement of essentially the same principles is as follows:
- Every integer has a successor.
- Every integer has, at most, one successor.
- There is an integer, which we shall call the unit, which has no predecessor.
- The only class of integers which contains the unit and each of its successors is the complete class of positive integers.
The weakness of these postulates is that they leave "integer" as an "undefined term," whose meaning is determined only by the postulates and their application. Rather the meaning of the postulates should be derived from the meaning of integer.
Preferable to these are the postulates proposed by Aristotle:
- 1. There exist mathematical units.
- Definition: A mathematical unit is a corporeal thing capable of existing
- as a substance, which is abstractly considered as indivisible and without position. (See page 315).
- 2. There exist mathematical numbers.
- Definition: A mathematical number is a discrete quantity, that is, a whole
- divisible into units.
The two definitions are nominal, but they are made real by the postulates, which assert the existence of the thing defined. These two postulates are sufficient because all the theorems of algebra can be proved to follow from the definition of numbers, and, as we have already shown (pp. 317 ff.), all the numbers can be constructed by means of the unit.
In arithmetic or algebra the fundamental theorems are to prove the following:
- 1. That addition is valid, and both commutative and associative.
- 2. That subtraction is valid, but only if a lesser number is taken from a greater.
- 3. That multiplication is valid, both commutative and associative, and also distributive with respect to addition and subtraction.
- 4. That division is valid, when the divisor is smaller than the dividend and commensurate with it.
- 2. That subtraction is valid, but only if a lesser number is taken from a greater.
By commutative we mean that if we perform an operation with numbers taken in one order, the same operation with the same numbers taken in the reverse order will give the same result. Thus 6 + 4 + 2 = 2 + 4 + 6 and 6 × 4 × 2 = 2 × 4 × 6, but 6 - 4 - 2 2 - 4 - 6 and 6 ÷ 2 2 ÷ 6. By associative we mean that if an operation is performed on three or more numbers, the result is independent of the manner in which they are grouped. Thus 6+ (4+2) = (6 + 4) +2 and 6 (4 × 2) = (6 × 4) × 2; but 6 - (4 - 2) (6 - 4) - 2 and 12 ÷ (6 ÷ 2) (12 ÷ 6) ÷ 2. By distributive with respect to addition and subtraction, we mean that a (b+c-d) = ab+ac-ad. By commensurate we mean that the divisor will "go into" the dividend an integral number of times without a remainder.
The commutative, associative, and distributive laws are of fundamental importance in calculation, as is obvious, but they are not of fundamental scientific importance, since they have more to do with notation than with the properties of numbers themselves. But the validity of addition, subtraction, multiplication, and division is fundamental scientifically, since only if these laws are valid can we demonstrate the existence of constructed numbers.
We will not prove these theorems here formally, but it is obvious that we can prove the result of any operation of addition to be a number by showing that it is equivalent to a collection of units. Thus 3 + 4 = (1 + 1 + 1) + (1 + 1 + 1 + 1). Since our Postulate 2 tells us that a whole composed of units is a number, and since we can recognize a unit by Postulate 1, this proof is valid. Since addition is valid, its converse (subtraction) can be shown to be valid in the same way. Multiplication and its converse (division) can be shown to be abbreviations of addition, since 4 × 3 = 3 + 3 + 3 + 3.
Inasmuch as we have shown in Chapter I that all the artificial numbers (0, negative numbers, fractions, irrationals, imaginary, and complex numbers) can be defined as natural numbers with some operation, and these operations are fundamentally reducible to the above four, there is no difficulty about proving the validity of the entire complex number system.
The addition of these operational numbers makes it possible to prove that the four laws of operation may be extended to all elements in the complex number system. For example, if we admit negative numbers, we can subtract any number from any other number. If we also admit fractions and irrationals, we can divide any number by any other, etc.
Since all of ordinary algebra is limited to the complex number system, it is evident that we can reduce every problem in this algebra to our two postulates, just as in geometry every theorem can be reduced to a small set of postulates.
Students often do not realize that every solution to an equation is a theorem, since the answer is correct only when it can be proved to have been obtained by operations which are valid according to the postulates. If these laws of operation are violated, then the result is not proved to be true, although by accident it may be true.
Thus when we write 3 × 4 - 2 = 10, we are asserting the theorem that "three fours minus a two are equal to ten, because of the laws of multiplication and subtraction." Thus we are giving a reason, or middle term, for our conclusion. When we write "a - 2
6, therefore a = 8," we are saying: "If a number minus 2 is equal to 6, then that number is 8 because of the law of subtraction." If we have simultaneous equations such as "a × b = 6 and a - b = 1, therefore a = 3, and b = 2," we are saying: "If two numbers multiplied equal 6, and if the lesser subtracted from the greater equals 1, then the numbers are respectively 3 and 2 because of the laws of multiplication and subtraction."
It will be noticed that many algebraic theorems are conditional. They state that if a certain relation (function) holds between two quantities, then these quantities are such and such, or belong to such and such classes. This means that we begin with a definition of the number in terms of its properties, and then seek to discover what number has these properties. For example, if I say, "4 x2 = 36," I am saying: "What number has the property of equaling 36 when squared and multiplied by 4?"
But in number theory, which is the more fundamental part of the science of number, we may begin with the number and seek to prove its properties. For example, we might ask whether 65,537 is a prime number (which it is).
The Pythagoreans were fond of the saying, "The essences of all things are numbers." Hence they liked to show the close connection between magnitudes and numbers. They spoke of "triangular," "square," and "cubic" numbers, and they were fascinated with the problems of both incommensurable magnitudes and irrational numbers. Finally, they took a special delight in the study of the five regular or Pythagorean solids, because there seemed to be some mysterious connection between them and the integers.
Plato and his followers were very Pythagorean in this respect. Since number is more abstract than magnitude, they considered arithmetic a science superior to geometry, and wondered if perhaps the lower might not be replaced by the higher science. In opposition to this, Aristotle argued that the science of number and the science of magnitude are not merely a perfect and an imperfect form of the same knowledge, but are specifically distinct, although related, sciences which cannot be reduced to one science.
In general the Greek mathematicians adhered to Aristotle's teaching and kept geometry and arithmetic unmixed. Nevertheless, in Euclid's Elements we see that they very well perceived how close the relation between the two disciplines is. The Elements is devoted principally to geometry, and yet it culminates in the last book in a discussion of the Pythagorean solids as if this were the supreme geometrical study. Furthermore, Book V is devoted to the topic of ratio and proportion because this is a topic common to both arithmetic and geometry. Then after Books VII-IX which are devoted to pure arithmetic, Euclid deals in Book X with the problem of incommensurables, so closely related to the problem of irrational numbers. Thus Euclid seems always to be about to combine the two sciences, without ever actually doing so. If he had been thoroughly Aristotelian lie would have divided his work into two distinct parts. The first would have been devoted to arithmetic, since Aristotle agrees with Plato in holding that this science is more perfect and abstract, but the second would have been a geometry independent of arithmetic.
The mixture of the two sciences went further in trigonometry (the science of measuring triangles). Ptolemy, the astronomer of Alexandria in Egypt, who lived in the second century after our Lord, introduced the use of a numerical measurement of angles into degrees, derived from the Babylonian division of the circle into 3600 (see page 306). Then mathematicians began to make tables showing the sine, cosine, tangent, and cotangent ratios between the sides of triangles in terms of numbers.
Thus trigonometry is a mixed science, developed from the study of ratio and proportion common to both geometry and arithmetic, but making the study of geometry easier by reducing proportions between lines to proportions between numbers which are precise and easy to manipulate.
The real step was not taken, however, until René Descartes (1596-1650), whose discovery of analytical, co-ordinate, or cartesian geometry*
[*It is called "cartesian" from Descartes' name (Des Cartes). It is called "co-ordinate" because it is based on the use of co-ordinates by which positions in space are measured numerically. It is called "analytic" because it analyzes magnitudes into numbered parts which are easier to manipulate. The student should not confuse the terms "analytic geometry" and "synthetic geometry" (i.e., Euclid's pure geometry) with the terms "analytic proof" and "synthetic proof," which are used in all types of mathematics (see page 338).]was one of the most important factors in producing our modern culture. When we think of "modern times" and contrast them with "medieval times," we are usually thinking about two aspects of modern life. On the one hand, we see the immense growth of modern science and engineering which has transformed our way of traveling, communicating, and producing our food and shelter. On the other hand, we think of the lack of unity in thought and religion and the widespread sense of uncertainty which makes today's world so different from the unified world of the middle ages. Strangely, this single mathematician, Descartes, is largely responsible for both results, although he certainly did not anticipate any such outcome. His discovery made possible the rapid advance of mathematics and the whole system of modern technology based on it, but his confusion of the two sciences also led men to doubt the possibility of certain truth. The last outcome was especially ironical, because Descartes thought that his methods would make mathematics and all branches of knowledge much more certain than they had been. He did not realize that long ago Aristotle had pointed out that those who confuse the branches of mathematics are sure to end by casting doubt on the possibility of a certain knowledge of the material world. Aristotle was keenly aware of this danger because some of his fellow-students in the school of Plato had already fallen into this error, and were teaching that truth is to be found only in some other world.
The essential feature of Descartes' discovery was this: It is possible to represent a point or position on a surface by two numbers, or in a space by three numbers, by means of two or three measurements made on arbitrary scales, showing the distances between that point and other arbitrarily chosen lines called co-ordinates. From this it seems that every geometrical magnitude can be perfectly described in terms of numbers, and the relations between these magnitudes can then be reduced to relations between numbers. In this way the science of geometry which deals with the relations of magnitudes can be reduced to the science of algebra which deals with the relations of numbers.
Since numbers are so much easier to manipulate and use in calculation than magnitudes are, this opened the way to the marvelous modern advance in mathematics and the engineering sciences based on them. On the other hand, since it seemed to show that the principles of geometry, formerly considered to be immediately evident, are only a kind of approximation, it casts doubts on all human knowledge. If we cannot trust our senses which show us that magnitudes are not numbers, then we must correct our senses by our reason. This was what Descartes was inclined to do. But then if our reasoning does not rest on the evidence of our senses, where does it get its certitude? From "innate ideas" given by God, said Descartes. This had been Plato's answer, and, as in Plato's time, men quickly pointed out that there is no proof that we have innate ideas. Hence the modern world has drawn the conclusion, which Descartes would have viewed with horror, that since we can have certitude neither from the evidence of our senses nor from the unsupported assumptions of our reason, therefore certitude is impossible.
Both Descartes and the sceptics went too far and fell into error. Descartes' discovery is a perfectly valid one, if we do not claim too much for it; nor does it destroy the independence of geometry and the certitude of the first principles of geometry.
We have seen in Chapter I that number, or discrete quantity, is derived from an abstract consideration of magnitude (the continuum). Physical reality is made up of magnitudes, or things whose parts can be divided again and again ad infinitum. The parts of magnitude have both divisibility and position, since one part is joined to the next. Yet these parts are numberable in an abstract way. By leaving out of consideration the divisibility of the parts and their position, we can consider each actual part as a unit, and taken together they form a number.
Nevertheless, this divisibility of a magnitude is potentially infinite, so that no matter how small we make the unit of magnitude, that unit is still divisible. Since the unit of number is indivisible, we can never reduce magnitude to number. The error has been repeatedly made in history of supposing that a magnitude is made up of infinitesimals, that is, of parts so small that they cannot be further divided. Others have made even a worse mistake by holding that a magnitude is made up of points. If an infinitesimal could not be divided, then it would be of a different nature than a magnitude, which is divisible; but a whole and its parts must have the same. nature if they are uniform. A mathematical line or solid is conceived by us as something perfectly uniform. Hence just as a whole line or solid is divisible, so must its parts be ever divisible. If a magnitude were made up of Points, the same difficulty would follow, since even an infinity of points could not fill the smallest space.
The proper solution is not to make contradictory guesses like those which introduce such unknown entities as infinitesimals and actual infinities, but to stick to the known facts. The fact is that the world is made up of physical magnitudes which can be divided beyond any limit we know of. Furthermore, the abstract quantities which we can imagine are potentially divisible ad infinitum. We can mentally divide any quantity into units as small as we wish, but we can never exhaust its divisibility. Hence the numbering of magnitude is only approximate, but it can be made as accurate as we wish. Thus it is a limit, which we can approach but never reach, just as the circle is the limit of an inscribed polygon of n sides. The polygon can never be made into a circle, but it can be made to approximate it as closely as we like.
- Hence Descartes' method is of very great importance for two reasons:
- 1. It permits us to reduce cumbersome geometrical problems to approximate algebraic problems which are much easier to manipulate, arid this approximation can be made as perfect as we wish to make it.
- 2. It permits us to represent algebraic problems graphically, that is, by a diagram which helps our imagination.
- 1. It permits us to reduce cumbersome geometrical problems to approximate algebraic problems which are much easier to manipulate, arid this approximation can be made as perfect as we wish to make it.
Thus by elevating geometry to algebra we get greater intellectual precision; by lowering algebra to geometry, we get greater imaginative vividness.
We make an error in this matter only when we claim that the approximation is more than an approximation.
Another way of saying that analytical geometry is an approximation is to say that it is an analogy. Magnitudes and numbers are simply two different species of quantity, just as a man and a dog are two different species of animals. Nevertheless, there is some similarity between a man and a dog, and between magnitudes and numbers. We have learned (page 49 f.) that when two things are essentially different but have some similarity, then they can be called analogous, and that we can apply the same name to them by analogy (see page 59). Although magnitudes and numbers are essentially different (since magnitudes have divisible parts and numbers have indivisible parts), nevertheless there is a similarity between them. We can compare a point to a number on a scale, and we can graph an equation by a curve, because of a similarity of relations. In effect, we are stating a proportion, or comparison of relations (see pp. 347 f.) as follows:
In the first ratio, the relation is one of distance: A is a certain distance from B. In the second ratio, the relation is one of order: a and b occupy a certain position in a number series. Obviously the distance of A from B is an essentially different kind of "distance" than that from 3 to 10.
If such comparisons are mistaken for identities, so that we think that a point is a number, then we are equivocating and thinking in a confused fashion, just as if we were to make the following analogy:
and conclude that the kind of "distance" between intelligence and stupidity is the same kind of distance as between Chicago and Timbuctoo.
This is the reason why the Greeks were fearful of mixing geometry and arithmetic. They realized that reasoning which rests on equivocation cannot be scientific demonstration, but only dialectic (see page 162). Euclid's geometry was a true science, but in the eyes of Aristotle this analytic geometry of Descartes would have appeared to be only dialectic.
This is certainly the case unless we take great pains in the presentation of analytic geometry to remove equivocation. This can be done in two ways:
- 1. We may use analytic geometry merely for the purposes of giving a graphic
representation to our algebraic reasoning. In this case the diagram is merely a help
to our imagination and does not actually enter into the algebraic argument, any
more than a diagram of a syllogism enters into syllogistic reasoning itself.
Actually this is the chief function of analytic geometry today. In this case our advance in knowledge has been a development of pure algebra, and analytic geometry has merely served as a convenient instrument. On this score Descartes' discovery was not really of much theoretical importance; rather it was a technical help, like the improvement of mathematical notation when Roman numerals were replaced by Arabic numerals.
- 2. We may use the methods of analytic geometry to give us approximate solutions of geometry problems, The results in this case are still dialectical, since they do not give us the proper reasons for our conclusions; but equivocation is removed because we make explicit the fact that our conclusion is approximate. Furthermore, so clear are the objects of mathematics that in this case we can make our approximation as close as we please, approaching the limit to any degree of accuracy required.
Thus analytic geometry can be understood in two ways: 1) as pure geometry algebraically expressed; 2) as pure algebra graphically expressed. In both cases the expression is approximate and dialectical. Descartes, we may conclude, gave to mathematics a very powerful technical instrument, but he did not really disprove the separate and irreducible character of the two sciences of geometry and arithmetic.
TO PHYSICAL QUANTITY
ANOTHER KIND OF APPLIED MATHEMATICS
Analytic geometry is an application of one branch of pure mathematics to another. We apply algebra which is more abstract to assist us in understanding geometry which is less abstract, and we illustrate algebra by geometrical figures to assist us in picturing more abstract numerical relations.
There is another and very different way in which mathematics is applied.
This is produced when mathematical quantities are used to assist us in understanding physical quantities, which are not abstract at all. This procedure most properly deserves the name applied mathematics, since in this case we apply the abstract to the concrete. It is also properly called mathematical physics, that is, the application of mathematics to physics or natural science. This in turn can have two branches, theoretical mathematical physics, and practical mathematical physics which has many branches (all types of engineering of accounting, and of statistic-making). We may illustrate this by means of the diagram below. Notice that the word "applied" is used in three different senses, indicated by the three arrows in the diagram.
Thus "application" can mean: 1) the application of algebra to geometry; 2) the application of any type of mathematics to natural science; 3) the application of theoretical mathematical science to practical problems. It is the second sense which particularly concerns us in the rest of this chapter .
Applied mathematics as a theoretical study was developed by the same group of scholars in Plato's Academy who first developed pure mathematics. Plato pointed out to his pupils that the movements of those heavenly bodies we call the "planets" is very mysterious. The other stars move across the sky with a regular motion every 24 hours Without ever changing their relative positions. The Big Dipper, for example (the Greeks called it the Great Bear), turns about the pole star every 24 hours, but never alters its pattern. The planets, on the other hand, seem to wander across the sky from week to week in a strange fashion, gradually working their way east-ward but not without frequently reversing their movements. Hence the very name "planet" means a "wanderer."
Plato said that a celestial being could not be moving aimlessly. There must be a pattern to these movements that could be reduced to a definite law. Therefore he suggested to his followers that they seek to find an order in these seemingly confused movements and express it by a mathematical diagram. just as an artist looking at nature disentangles its beautiful patterns from apparent confusion, so the astronomer should detect the mathematical order of the heavens.
Hence it was that the first important application of the geometry of the Greeks was a mathematical theory of the motion of the planets which would agree with careful observations which had been kept for a long time by the priests of Babylon and to which the Greeks added still better measurements. This theory was worked out by Eudoxus and gradually improved by a long series of astronomers. They suggested many possible theories, some of them placing the sun at the center with the earth and planets moving about it, others placing the earth at the center with the sun and planets moving about it, sonic supposing that the center is hidden from us.
According to the facts which they were able to gather without the aid of a telescope, the theory that the planets and sun move about the earth was found to be the most satisfying for two reasons: 1) It best accounted for the known faces; 2) it could be given the most simple and clear mathematical diagram. It was Ptolemy (page 359) who gave the most perfect mathematical theory based on this arrangement. He knew that it might not really be true (as it is not, since the earth moves about the sun), but he showed that according to the known facts it was more probable than the other theories.
This study of astronomy also led to the use of mathematics in the study of light and its reflection from mirrors and its refraction by water and glass. Euclid himself wrote an Optics, showing some of the mathematical laws which determine the path of a ray of light. This science was also sometimes called perspective, and it made it possible for painters to represent three dimensions on a flat surface, so that some Greek and Roman paintings had the appearance of great depth and roundness.
We have already seen that Pythagoras applied numbers to explain the musical scale, and the later Greeks and Romans made further progress in this branch of science called acoustics.
Still more interesting was the application of mathematics to building machines, some using water power (hydraulics), some using steam power (pneumatics). This study in all its branches is called mechanics. It was originally a practical application of mathematics, but it soon led to interesting theoretical applications in natural science. The great name in this field was Archimedes who lived about a generation after Euclid (about 287-212 B.C.). It is said that he claimed, "I could move the earth if I had a fulcrum on which to put the lever." It was he who showed how mathematics could be made to apply to all sorts of physical problems if it were only used with ingenuity and imagination, so that all who read his work could clearly see that mathematics is an indispensable tool for natural science.
Why is this the case? Students beginning their study of science often wonder why it is necessary to use so much mathematics. It appears to make the subject very complicated, difficult, and abstract, while a more popular, descriptive approach to science would be much more easy and pleasant. It is because of the use of mathematics that so many people fear to study science, and find its reasoning meaningless and remote from daily experience.
There is some real sense in this objection. The physical world in reality is not a set of mathematical formulae or geometrical diagrams. It is a colorful, moving, tangible collection of things, each of which has its own inner nature and way of behaving. The richness and the unity of the real world can never be reduced to a mere formula. Natural science is not all mathematics, nor is it even basically mathematics. Mathematics deals with something abstract and idealized, figures and numbers which we construct in our imagination, while natural science deals with something very concrete which we do not imagine but bump into all around us. Aristotle at the very beginning of natural science protested against the inclination of his fellow pupils in the Academy of Plato to treat the world as if it were only a mathematician's diagram. He insisted that natural science must keep close to sense observation of the material world, and explain that world in its own terms.
Yet Plato, Aristotle, and Archimedes all agreed that we cannot understand the material world if we neglect its quantities. Quantity, according to Aristotle, is the first of all the accidents (see page 344), so that every other accident has a quantitative aspect. Furthermore, the quantity of a substance is one of its properties; hence if we know the quantity that naturally belongs to a thing, we can define it in terms of quantity, and we can come to an understanding of its essential nature. Figure, the kind of quality intimately connected with quantity (see page 249), is one of the best indications we have of the nature of a thing since it shows us the proper quantity in its connection with proper quality, and from this all the other properties of the thing are explicable.
Hence natural science must give special attention to the proper quantities of things if it is to discover their inner natures. We may take as an example the experiment of Pythagoras with the musical string. Tones are qualities, not quantities. Yet when we discover that a certain quality of sound or tone is produced by a certain length or quantity of a string, and that another tone is produced by a different quantity, and that the relations between one quality and another resemble the relations between one quantity and another, we are beginning to understand the nature of sound. When we understand the nature of sound we also come to understand the nature of the vibrating body (a substance) in which this accident exists.
It was an obvious step, therefore, but a very important one, which the Greeks took when they developed mathematical physics, that is, an application of mathematics to natural science. Unfortunately, not all of them kept clearly in mind the warning of Aristotle that we must not confuse mathematical quantity (which is abstract) with physical quantity (which is concrete). The number 2, and 2 apples, are very different things. When we say that a pair of apples are 2, we say something very true and very important, but we are far from telling the whole truth about the apples. It is an equivocation (see page 59) to use the word "two" of apples and of the number 2, unless we carefully distinguish the different senses of the word.
This means that mathematics and natural science are distinct and separate sciences, although both treat of quantity. Natural science treats of concrete quantity as it actually exists as a property of changing things. Mathematics treats of abstract quantity considered apart from the nature and other accidents of the thing which has quantity. The triangle which we study in mathematics is considered without deciding whether it is of paper, or wood, or metal, or what produced it, or what it is to be used for.
Mathematical physics is a combination of these two distinct sciences. How can we combine two sciences which are distinct without running into confusion? We can do this in two ways:
- 1. We know that if we prove something is impossible in abstract quantity, then it will also be impossible in concrete, quantity.
For example, if we know that it is impossible to construct a regular polygon with 11 sides in geometry, then we will never find a crystal in nature that has this shape. Thus mathematics will eliminate for us many guesses about the natural world, because it will prove their impossibility.
- 2. If we know that a natural object has a certain definite figure or number, then by mathematics we can show that it must have certain other quantitative properties that belong to such a figure.
For example, if we know that a drop of water is round, we know at once that it is exposing the least possible area per unit of volume to the air, since a sphere has the least possible surface per unit of volume.
It may be objected, however, that natural figures are never perfect, and that even in counting natural units it is often very difficult to be accurate. Hence it would follow that in applying the conclusions of mathematics drawn from perfect figures and numbers we cannot be sure that our application is anything more than approximate, and hence our conclusion will be probable and not certain.
This objection is a very good one. A very slight change in a measurement in science may lead to an entirely different mathematical figure. After all, the difference between a polygon of many sides and a circle is very small in measurement, and yet these two figures have completely different definitions in mathematics. The result is that when we apply mathematics in science we become more exact in one sense, and less exact in another.
- 1. We are more exact in that mathematical relations are perfectly definite and clear (6
is exactly twice 3).
- 2. We are less exact in that we are substituting a mathematical quantity for the physical quantity which actually exists, although they are only approximately equivalent.
It is not surprising, therefore, that a very large part of mathematical physics is only probable and not certain, since an improvement of measurement may greatly change the theory. Thus if we consider what parts of Greek science still remain valid today, we see that:
- 1. Greek pure mathematics still remains valid, with only minor corrections.
- 2. Greek natural science of the non-mathematical sort (e.g., biology) still remains essentially sound, although many corrections (some quite important) are needed.
- 3. Greek astronomy, which was mathematical physics, has had to be completely revised, although some parts of it are still valid (theory of eclipses, method of measuring the earth).
- 2. Greek natural science of the non-mathematical sort (e.g., biology) still remains essentially sound, although many corrections (some quite important) are needed.
Mathematical physics is thus the most shifting part of science and is constantly undergoing revisions. Does this mean that mathematical physics never gives certain Conclusions? It can give certitude if we can be certain of the range of the physical measurements to which it is applied. For example, if we know that the earth is approximately a sphere and that its flattening at the poles is not greater than a certain maximum nor less than a certain minimum, then we can treat it mathematically as if it were a sphere, and be sure that our conclusions as to its properties are valid within that range. If we extend our argumentation beyond this range, then, of course, it becomes merely probable. If, moreover, we do not know the range of validity with certitude, then our whole theory will be only probable.
Consequently, the astronomy of Ptolemy which put the earth at the center of the universe was never more than a probable theory, because there was no way to be certain of the range of accuracy of the measurements used. Ptolemy assumed that the fixed stars were close enough to the earth that, if the earth moved, at least a small displacement of these stars should be observed. Since such a displacement was not observed, he concluded that the earth did not move. As a matter of fact, the stars are very remote, and consequently their displacement is very small, too small to be measured without very fine instruments. Since Ptolemy could not estimate the distance of the stars, he had no way of being sure that his argument was conclusive. Hence his whole theory was probable. On the other hand, the theory much later devised by Newton was known to be certain within a given range, although, as we now know, it is not accurate beyond this.
Hence it is a mistake to believe that in science new theories are constantly overthrowing all previous belief. Some theories which were always regarded as being merely probable are being replaced by more probable theories, or finally by ones which are genuinely certain within it given range of accuracy.
Since the application of mathematics depends on measurement, we must perform these measurements carefully:
- 1. There must be a standard of measurement which is as invariable as possible, and
whose range of variation is known within limits. The best standards are those
which are natural (for example, the rotation of the earth as a standard for the day),
because their variation is held within a definite range by natural causes.
- 2. We must try to isolate the thing to be measured from as many disturbing factors as possible. For example, if we wish to measure a metal rod we must keep its temperature constant.
- 3. We must perform the measurement repeatedly. This shows us the range or limit of error within which our measurement is accurate.
- 2. We must try to isolate the thing to be measured from as many disturbing factors as possible. For example, if we wish to measure a metal rod we must keep its temperature constant.
In every case we are not merely looking for the measurement of an individual quantity but of a natural regularity. For example, the height of John Jones or the temperature of this sick man is not itself a scientific fact. Rather the range of height for any human being, or the normal temperature of a man, or the maximum temperature of a sick man who will survive are scientific facts, because they regularly recur in nature and are properties of some natural thing.
Often our measurements are statistical in character, that is, they are expressed in terms of a ratio between quantities. For example, we may discover that the height of 3 men out of 4 falls within a certain range, or that 7 out of 10 times a certain chemical reaction occurs. We speak then of the statistical probability. Actually there is true certitude in these measurements, since the existence of the ratio is certain. The probability lie s in the individual cases, since if we are asked whether in a given case the chemical reaction will occur, we may say, "It is more probable that it will, since it occurs in 7 out of every 10 cases."
Indeed, we may admit that all measurement has a statistical character; for whatever quantity we find to be a property of natural things admits of some variation, and the operation of measurement which we perform on it is also of varying accuracy. This does not destroy the certitude of our knowledge, because science is not concerned with individual cases but with what is true in most cases.
Natural laws are not infallible; they are subject to interference from the carrying out of other natural laws or of the action of free wills not compelled to obey or to limit themselves to these laws (the wills of men, angels, and of God.) The interference of one natural force with another we call chance. Chance is possible because the universe is made up of a multitude of different substances, each following the end of its own nature, and because in the material world things by reason of their matter are constantly subject to outside influences. In his wisdom God rules over this whole multitude of natural things and through his angels keeps this conflict of natural forces from destroying the general order of the universe. Yet since he rules things according to their natures, he permits chance events, as part of the pattern of the world, although he keeps them within limits.
Furthermore, God permits free creatures, the angels and men, to share in guiding their own lives. Because of the sin of the angels and of men there is more chance and disorder in the world than God intended in the beginning, and he wills to remove this undesirable disorder through his Son and the Church and man's intelligent and right use of his own power over lower things.
In studying the world of nature, therefore, and also the world of human society, we need to lay bare the order often obscured by chance and by moral evil. Because quantity is the basic accident, natural order is very clearly manifested in quantitative order. Hence mathematics can play a great role in helping us to sort out regularities.
The branch of applied mathematics called statistics (from late Latin for "a statesman," because statistics were first used by government officials) deals with this attempt to discover regularities in quantitative data (measurements).
Statistics is concerned (1) with describing in a clear fashion the actual results of measurement, and (2) with induction of some generalization from this data. In carrying out the first of these tasks, the statistician arranges his data in an array from the smallest to the largest item. He then attempts to fit an equation or curve (using analytic geometry) to this array so as to show a frequency distribution. It will commonly be found that the items fit a normal curve in which must of the items have a central tendency. For example, if he has repeatedly measured the weight of a quantity of material, the actual measurements will spread out over a considerable range, but most of the results will cluster around one value, and this value, therefore, is probably the true one.
Various types of averages (mode, median, arithmetical mean, geometrical mean, etc.) can be calculated which serve as a convenient approximation to this true value, It is necessary also to indicate how the deviant values are "scattered" around this center; hence the statistician attempts to measure this variability, to estimate the lack of balance or skewness in the curve, and to estimate the bunching of cases around the average (the flatness of the curve). Mathematical formulae and indices have been developed which express this central tendency and variability in different ways, each of which gives a better approximation for different types of data.
Besides a simple curve showing such a frequency distribution, descriptive statistics also deals with complex curves that show, for example, the variation of a quantity through a series of time periods, as in the familiar business chart.
The problem of arriving at a generalization or induction is principally that of discovering a correlation between two arrays of measurement, especially when these arrays indicate an actual change of the quantity in time. If quantities bear a constant relation to each other, and especially if this relation continues throughout a series of changing conditions, then we have a sign that there is a cause and effect connection between them. It will be noticed that such a correlation does not tell us which is cause and which is effect, for these can only be known by a study of the natures of the two things involved; but it does indicate to us which things may be causally connected.
In actual fact, however, correlations are not perfect, so that again it is necessary to resort to a statistical procedure to determine whether the deviation between the expected values and the actual values is so great as to render the correlation meaningless.
Basic to all these statistical processes is the theory of probability. A mathematician knowing, for example, that in throwing dice only a certain number of results are possible, and that they are independent of each other and due to so many different factors that the individual result is pure chance, can predict an ideal pattern to which the successive throws will approximate as a limit. If this pattern is not actually realized, then we must suppose that, besides chance, some special cause (loaded dice) is operating. Hence the knowledge or probability makes it possible both to perceive order in data, and to eliminate what is pure chance.
Finally we may mention that mathematics can help to sort out the operation of several independent causes (factor analysis), For example, in testing human intelligence by a variety of tests we may get very complex data. The mathematician determines whether this complexity could be explained by the fact that human beings have several relatively independent abilities, some being superior in memory, others in the use of words, etc. Once these factors have been hypothetically isolated, special tests may be designed to test each.
All use of statistics is only instrumental to a genuine physical or sociological understanding of the realities which are being measured. If we have some understanding of the natures of things, we can devise measurements which will help us detect regularities in their behavior, and these in turn will lead us to a better insight into their natures. But without this understanding the collection of data is aimless and meaningless. Hence statistical methods are only a preparation for a process of analysis and reasoning.
Once we have established by measurement that a certain quantity regularly occurs in a natural thing and hence is its property, we can draw further conclusions by syllogistic reasoning. In a mathematical-physical argument the middle term must be a measurement which is known to be a physical fact, but which can be treated as an abstract mathematical quantity. Take, for example, the demonstration given by Aristotle from biology:
A wound whose area
compared to its circum-
ference is large is slow to heal
And: a round wound is a wound whose area compared
to its circumference is large.
Therefore: a round wound is slow to heal.
Major: Is proved biologically from the fact that a wound
heals from its edges.
Minor: If we consider this abstractly we get:
A circle is the figure whose area compared to its
circumference is greatest -- a fact
which can be proved to be true from
mathematical principles.
Thus in the demonstration the middle term can be considered concretely and physically (a wound whose area, etc.), or abstractly and mathematically (a figure whose area compared to its circumference is greatest). Considered abstractly, it can be treated as a part of mathematics, and its various properties and relations can be demonstrated mathematically. The conclusion of the syllogism, however, is not a mathematical truth, but a physical truth: Round wounds heal slowly.
The Greeks had a very clear picture of these three types of science: 1) mathematics; 2) natural science; 3) mathematical natural science (physics); and of their mutual relationships. During the five hundred years after Aristotle had proposed the program of research in these fields, the Greeks made a remarkable advancement in them all. Then a strange decline set in, and scientific progress became very slow.
Many factors played their part in this decline, but the most obvious one is that the Greeks had already exhausted the possibilities of advance without the introduction of better instruments for gathering facts. The Greeks had no telescope or microscope. They did not know much about the various methods of purifying and combining chemicals. They made only a meager use of dissection in studying living things. It was not that they were totally ignorant of the advantages of such methods, but that they never carried any of them very far.
Why was it that such a brilliant people did not develop the technology which would have helped them extend their science? Most historians believe it was because Greek and Roman civilization was built on the institution of slave-labor. A slave is not much interested in inventing new methods of doing things, because he does not own what he produces. On the other hand, those who own slaves are likely to feel that such problems are beneath them. At any rate, Greek and Roman civilization were weakest in just this technological aspect of culture, so that when Roman civilization was at its height, the interest in science was in decline.
Then came the terrible period of the civil wars in the Roman Empire, the invasions by barbarians, and the disruption of transportation and city life. There were few places that men had the leisure or the opportunity to study, although men like the Roman senator Boethius (who died a martyr at the hands of a barbarian king in 525) labored earnestly to try to restore education. Yet during these Dark Ages the Christian Church was able to check the downward course of civilization which had resulted from the corruption of paganism, and to begin to build a firmer Christian civilization. The basis of this new civilization was an education which combined the study of the Sacred Scriptures with the study of the liberal arts which bad come down from the schools of Plato and Aristotle.
By the thirteenth century the Church was able to revive these schools in Christian form in her great universities in western and southern Europe. Fortunately, in eastern Europe (the Byzantine Empire) the learning of Greece had been preserved, if not greatly advanced, while the Mohammedans who had overrun the rest of the Byzantine Empire in the East and in Africa had taken over this learning from the Christians. Hence the West was able to obtain the works of the great Greeks -- either in Greek manuscripts from the
Byzantines or in Arabian translations from the Mohammedans -- and have them translated into Latin. In this way Aristotle, Euclid, Ptolemy, and finally, Archimedes were studied diligently; mathematics, natural science, and mathematical Physics began to revive and to develop once more. At the new University of Oxford the great Bishop of Lincoln, Robert Grosseteste, especially awakened interest in science, and he was soon followed in other places by St. Albert the Great, Roger Bacon, and St. Thomas Aquinas, who all promoted these studies.
What was even more significant was that this time real advances in the technological side of the sciences were being made. The Mohammedans had already made advances in astronomical measurement and in chemistry (alchemy), and now the Christians added to this the mechanical clock, the use of lenses, new metallurgical processes, etc. The time was ripe for an immense flowering of science. During the 14th and 15th centuries, however, this progress was slowed down by the social upheavals of the period and the decline of the universities under the influence of the type of philosophy called Nominalism.
In the 16th century, however, and especially in the 17th, very rapid scientific progress was made, the great University of Padua being especially influential. This University was strongly Aristotelian, and it fostered the great interest in natural science and the direct observation of nature which is to be found in Aristotle's works. Among the men whom it produced was the great Galileo (1564-1642), who took the final steps in establishing the method of mathematical physics. Galileo was the first to make effective use of the telescope, and at the same time, through his study of Archimedes, he was able to apply mathematics in a brilliant fashion to the new facts he was discovering. Thus it became clear to all that the way for the advance in mathematical physics would be through an improvement of methods of observation and measurement, combined with the analytic geometry of Descartes and the subsequent development of higher mathematics. The method of Galileo and Descartes remains dominant in science today, and Newton and Einstein were happy to follow in their path.
The career of Galileo, however, well illustrates a danger to which mathematical physics is subject. Aristotle had made clear that physics of this kind is not the whole study of nature. Not all aspects of nature can be known by a quantitative procedure, and even those which are known in this way have to be interpreted in the, light of the nature of a thing which underlies its quantity. This was well understood by the great biologist, William Harvey (1578-1657), who had also studied at Padua and who discovered the circulation of the blood.
This discovery was as great an advance over the knowledge possessed by Aristotle as were Galileo's discoveries about falling bodies, but it was not arrived at by mathematical methods (see the proof on page 586). Galileo, however, was a genius of a very impetuous sort, and he wished to forget the whole past and to rush forward with his mathematical methods without attempting to control these methods by other types of investigation. He claimed to have proved things that actually he had not proved (for example, his claim to have shown that the tides were due to rotation of the earth). He lacked the critical and conservative thinking which are required for a perfect scientist.
It is little wonder that such a man eventually went too far. Although a Catholic, be was led by his enthusiasm for his own scientific guesswork to reinterpret the meaning of the Bible. The interpretation of the Bible belongs only to the Church to whom God has given that right, and a scientist is stepping outside the limits of his science when he attempts to do so. As a result, Galileo was required by the authorities of the Church to retract his teachings, if he was to remain a Catholic, and as a penance he was required to remain the rest of his life in house-confinement, although he was allowed to continue his studies. This penalty today seems to us severe, but the Church well understood how important it is for scientists to remain within the field of science, and within that field to proceed by the scientific method of carefully testing their discoveries. If later scientists had submitted to this guidance as Galileo finally did, the world would be further advanced along the road of sound science than it is today. The theologians who condemned Galileo were wrong in some of their own scientific ideas, where he (genius that he was) was right in some guesses; but they were right about the strict standards of the scientific method, and he was wrong.
The period from Galileo to our own has seen a marvelous development of mathematical physics, but it has also seen the weaknesses of Galileo multiplied a thousandfold. The philosophical bases of science which Galileo treated so lightly have been ignored by many scientists, until today science has broken up into a thousand specialties which cannot be put back together again. What is worse, it is often at odds with religion, philosophy, art, and morality, because the common foundation which united them all has been shattered.
Before us lies a new age of science in which the advance will probably include a return once more to a sounder foundation for modern science. On the part of the students, this requires that they learn mathematics as a pure science, and also mathematical physics as an independent science and a tool for natural science. At the collegiate level they should begin their study of natural science itself with a consideration of its fundamental principles, through which all the different branches of science can be integrated.