BENEDICT M. ASHLEY, O.P.: THE ARTS OF LEARNING AND COMMUNICATION
Wider Vistas in Mathematics
THE RICHES OF MODERN MATHEMATICS
The Elements of Euclid did not exhaust the mathematics known in his day, but they gave a good survey of the principal topics which had then been studied in pure mathematics. A look at a modern library of mathematical works gives at once a bewildering impression of the immense growth in the number and the complexity of topics in mathematics. One of the sources of this complexity is that each time some new idea in mathematics is discovered, mathematicians then make haste to apply it to all the topics previously known, so as to give to each of them a new treatment. It is like walking into a hall of mirrors in which we see in one mirror reflections of other mirrors and in each of them also reflections of others and so on. Thus in every field and branch of mathematics all the other fields are reflected.
This makes it very difficult to give the beginner a map of mathematics in the way that we might outline, for example, the branches of natural science. Yet it is important that a student who has become acquainted with elementary geometry and algebra should have seen that these fundamental mathematical studies are the door to a whole world of mathematics.
ADVANCES IN GEOMETRY
In ancient times many interesting theorems were added to those contained in Euclid's Elements. A student can get an idea of this achievement by examining the Conic Sections of Appolonius of Perga (260-170 B.C., a generation after Archimedes), in which a beautiful theory is given of the figures and curves produced by a plane intersecting a cone at various angles.
The middle ages (lid very little to advance geometrical studies because the mathematicians of that period were more concerned with algebra, and this tendency continued even in modern times because of the invention of analytic geometry, which is algebraic in character. Nevertheless, the extension of algebra to new theorems continued.
More significant, however, is the effort to give the Euclidean geometry a more rigorous logical structure. The Greeks themselves made many criticisms of Euclid's definitions, axioms, postulates, and proofs, and showed that many of them could be made more exact. In particular, they attempted to show that the complicated parallel line postulate (see page 331) was not a postulate, but a theorem, although they never succeeded in this attempt.
Modern geometers also have been chiefly interested in this problem of making geometry rigorously exact. To do this it is necessary to admit nothing in geometry that is not entirely justified by the explicit postulates. Furthermore, these postulates must be exactly stated, and be independent of each other. By "independent" is meant that one cannot be proved from another. The clearest proof of such independence, of course, is had when it can be shown that one postulate can be contradicted without contradicting the others.
In applying these strict standards to Euclid, numerous defects were found. In listing only five postulates, Euclid had implicitly included others which needed to be stated. Once this point had been cleared up it was also necessary to show the mutual independence of the postulates. Once again suspicion centered on the parallel line postulate. Was the statement that through a point outside a given line one and only one parallel can be drawn* [*This is equivalent to Euclid's own statement of the postulate, given on page 331.] a theorem or a postulate? The only way to settle this question was either to prove it as a theorem, using the other postulates, or at least to show that it was dependent on the other postulates by showing that to assume its contradictory would lead to it contradiction of the other postulates. If this last proved impossible, then Euclid would have been vindicated, and it would be evident that this statement is not a theorem but a true postulate independent of the others.
Since after hundreds of years of effort no positive proof had been found that it was a theorem, Father Geronimo Saccheri, S.J. (1667-1733), attempted the negative approach of assuming the contradictory of the postulate and seeking to show its dependence on the other postulates in this way.
The importance of this new approach was not recognized until the 19th century. The Russian, Nicholas Lobachevski, and the Hungarian, John Bolyai, assumed that in fact there are many lines parallel to a given line passing through a point outside the line. The German, Bernhard Riemann, took the opposite assumption, that there are no parallel lines. In both cases the geometers were able to show that geometries of this sort are self-consistent. Hence Euclid was quite right in holding that his postulate was not a theorem, but a statement independent of the others.
This proof of the independence of the parallel line postulate did not, of course, prove that Euclid's postulate was false, nor that it was a mere assumption. Nor did it prove that either the assumption of Lobachevski-Bolyai, or that of Riemann, was true. What it showed was that if this postulate is not taken as a first principle, then the complete Euclidean system with its use of similar figures cannot be demonstrated.
On the other hand, besides the geometry of Euclid there now existed a group of Non-Euclidean systems (other sub-varieties besides the two we have mentioned were soon added), which as self-consistent systems of propositions were quite marvelous and as rich
in arguments as the Euclidean geometry. How were these to be understood? It was soon seen that all these geometries, including that of Euclid, could be considered as forming a more general type of geometry. This more general geometry shows the analogy between various different figures.
Thus in Euclidean geometry distances are measured by straight lines. On the surface of a sphere, however, distance is measured by a curved line (called a geodesic, "earth-line"). Hence there is an analogy between a straight and a curved line in this respect. Not every curved line on a sphere, however, is the shortest distance between two points, but only a certain type of line depending on the curvature of the sphere itself. Even on a sphere, therefore, it is possible to distinguish between two classes of lines, and we may consider the geodesic as straight, and the other as curved. Since this is only an analogy, however, the properties of a geodesic are not altogether the same as the properties of a straight line-for example, all geodesics on a sphere intersect.
Hence "straight lines" on a sphere cannot be parallel, and in this sense the assumption of Riemann is valid for the surface of a sphere. Similarly, concave solids can be constructed on whose surfaces the geodesics are such that many never meet (at least on the limited surface of such solids). For such surfaces, then, the assumption of Lobacbevski and Bolyai holds.
So the Euclidean and Non-Euclidean geometries are not contradictory, but apply to surfaces or spaces in which the measurement of distance is made by lines which have zero curvature (Euclidean geometry), positive curvature (Lobachevski-Bolyai), or negative curvature (Riemann). Between figures of these three sorts there is thus revealed a beautiful analogy, since in each of the geometries there are similar theorems which can be translated into each other by an appropriate change in the definition of the terms. For example, the figures on the surface of a sphere do not have the same definition as figures drawn on a flat surface, and yet for every figure on a flat surface there is a corresponding figure on the sphere and corresponding theorems can be proved of it.
In this way a greater unity is given to the whole of geometry. For it becomes apparent that once we have proved a theorem for a figure on a flat surface, we will be able to prove many analogous theorems which apply to a great variety of curved surfaces.
The Euclidean and Non-Euclidean geometries are often called metric (Greek metron means "measure") geometries since in them the measurement of distance is of fundamental importance, and they differ as to the way in which this measurement is made by a straight or curved line. It was discovered even by the Greeks that it is possible to compare one figure with another, abstracting from considerations of measurement.
This discovery was made in the applied mathematics called perspective, in which a painter tries to solve the problem of representing a three-dimensional object on a two-dimensional surface. As we all know, in such a representation parallel lines (a railroad track, for example) must be shown as converging to a distant point, and all objects appear as foreshortened. Again, the circular top of a vase or cup appears as an ellipse in a painting, although in reality the edge of the cup is at all points equidistant from the center. Thus in the painting the distances of the parts are greatly altered, and yet a very exact analogy between the real figures and their representation is maintained; otherwise they would not look alike.
We have no practical difficulty in interpreting a picture, but the history of art shows us that it was no mean feat for painters to learn this trick of perspective. Geometers attempting to develop an exact theory of such perspective soon realized that it had important implications for pure geometry. This development began in modern times with the Frenchmen, Girard Desargues (1593-1662), and the great Blaise Pascal (1623-1662), who was noted alike as a physicist, mathematician, and Christian apologist.
Projective geometry considers the relations between figures which correspond to each other point for point, although their shape, size, and proportions may be quite different. The name is derived from the fact that one figure is then thought of as being altered into the other by a process of projection on a different surface or a different part of surface. We can, for example, cause a movie camera to cast its picture on a flat surface facing the camera, or on the ceiling or walls which are perpendicular to the camera, or on the curved surface of a pillar. The picture undergoes all sorts of stretching and shrinking, and still every point of the picture as it appears on the flat surface appears also in its other projections.
Obviously the properties which are common to classes of figures related to each other in this way are very fundamental ones, and projective geometry is therefore a very penetrating and profound study. The postulates which are required to prove its theorems are of it very general type relating to the order of points in space. As Non-Euclidean geometry omits the similarity of figures, so projective geometry abstracts from the congruence of figures.
It is possible, however, to abstract still further, leaving out not only the similarity and congruence of figures, but even the point-to-point correspondence which still remains in projective geometry. Just as we imagined a figure projected on another surface so that the distances between its points is stretched or shrunken, so now we may imagine it as twisted, bent, knotted, or deformed in any fashion, provided it is not torn or divided. Even when treated in this fashion a physical figure retains certain properties. A circle of wire, for example, can be twisted and crushed into a ball, yet this does not alter the fact that it is a single line which is endless. A rubber balloon could be forced into the shape of an egg, of a dumbbell, or made to fill the interior of a square box; it remains a closed surface.
Topology (Greek topos, a place) is the study, only recently developed to any extent, of the properties of figures which are of this fundamental character and which imply only the connectedness or continuity of the parts. With it we reach the limits of pure geometry, since if we go a step further and remove continuity we cannot properly be said to be in the science of geometry. Actually, this further abstraction is made, as we will see below.
THE SCIENCE OF NUMBERS
We have seen how the science of magnitudes has been pushed to deeper and deeper depths of analysis. What of the development of the science of number? The problems of number theory raised by Euclid and Nicomachus (see page 353) have continued to occupy mathematicians, and are still of very great interest. For example, the following two problems dealing with prime numbers have never been solved:
- 1. Is it true that any even number (except 2, which is itself a prime) can be represented as the sum of two primes? If you try it with 4, 6, 8, 48, 100, etc., you will see that it is true in these cases. Is it always true? No one knows. This is called Goldbach's conjecture after an obscure mathematician in the 18th century who first suggested it.
- 2. Are there infinitely many pairs of primes which differ by 2? There are many such pairs -- for example, 3 and 5, 11 and 13, 29 and 31. Is this a regular recurrence throughout the series of natural numbers?
Thus it is apparent that the study of the properties of natural numbers is far from exhausted. Indeed, as we shall see below, all modern mathematics ultimately rests on the theory of numbers.
We have seen in Chapter I of this part (see page 311) that algebra arose from the emphasis on the construction of whole classes of numbers bearing certain relations or functions with respect to one another. Thus for algebra the fundamental concept is that of function.
We have also seen that, in order to construct such classes of numbers, it was convenient to enlarge the concept of number so as to include various types of operational or artificial numbers which signify not a quantity (as do the natural numbers) but the construction of a quantity by a certain operation. Each of these artificial numbers can be defined as a certain process involving the fundamental operations of addition, subtraction, multiplication, and division, performed on specified natural numbers or classes of natural numbers. In this way, zero, negative and positive numbers, fractions, irrational numbers, and complex numbers were all devised and proved to form a number field which is closed under the four fundamental operations. This means that any process of addition, subtraction, multiplication, or division (with the sole exception of division by zero) on numbers in this field will construct another number pertaining to this same field (see page 321).
This does not mean, however, that "numbers" cannot be found which cannot be constructed in this fashion. Transcendental numbers have been discovered, such as (3.1415...), which are not algebraic numbers and which cannot be constructed by algebraic processes (that is, they cannot be roots of all algebraic equation with rational coefficients). Furthermore, George Cantor (1845-1918) introduced transfinite numbers, concerning which there has always been much mathematical controversy, but which certainly have some mathematical meaning. Cantor pointed out that it is possible to distinguish different infinite sets of numbers, and to give arguments to show that one infinity is greater than another, while some infinites are equal, It then becomes possible to assign a cardinal number to these infinities by ordering them according to smaller and greater, and then perform some of the ordinary mathematical operations upon them. The results are very curious. For example, we must admit that the set of odd and the set of even numbers are both infinite, as well as the set of all natural numbers. Yet we must also admit all three sets are equal to each other, since we can find an odd and even number for every natural number as follows:natural number: 1 2 3 4 5 6 7 8 9 10 odd number: 3 5 7 9 11 13 15 17 19 21 even number: 2 4 6 8 10 12 14 16 18 20
From this it would appear that axioms such as "the whole is greater than the part" do not apply to such sets.
In order to introduce numbers other than the integers, the concepts of a series of numbers and of a limit were required. Thus an irrational number can be defined as the limit of a series of fractions, a limit which is approached more and more closely the further this series is constructed, but which is never reached.
Given these numbers all constructed by means of fundamental operations which are justified by postulates, the work of algebra is to study functions, or the relations between classes of numbers. Such a relation is expressed by an equation which indicates that certain operations on certain classes of numbers (variables) will yield a number which can also be constructed in a different (or sometimes identical) fashion. So 3² + 4² = 5² means that the same number, 25, can be constructed by two different operations. a² + b² = c² means that there are classes of numbers for which this same relation holds.
In elementary algebra the student is taught to solve many such equations of the simplest types, generally simultaneous equations (equations of first degree) and quadratic equations (equations of second degree), named from the coefficients which occur in the equation. The Greeks by the time of Diophantus (who died about 330 A.D.) had advanced this far in algebra. The general equations of the third and fourth degree were solved by the Italians, Nicholas Tartaglia and Ludovici Ferrari, and first published by Girolamo Cardano in 1545.
For a long time mathematicians attempted to find general solutions for equations of higher degree; such efforts were unsuccessful, although they advanced rapidly in solving particular types of equations, Finally, a Norwegian, Niles Abel (1802-1829), showed that this search was hopeless, since equations of the fifth degree and higher have no general solution. Henceforth it became necessary to deal rather with the immense array of particular types of equations, and the brilliant Evariste Galois (who died in the year 1832 when lie was only 21) laid a general foundation for this higher algebra by showing a way to classify the various types of equations, so that each group could be dealt with as a whole.
In the meantime, the development of algebra was being strongly influenced by the great effort to reduce geometry to algebra. We have already seen (page 359) how the basis of such a reduction had been provided by Descartes (1596-1650) in his system of analytic geometry. This completed the work of trigonometry (see page 359), which had already provided a method of expressing the ratios of lines and the size of angles in numerical terms. By an extension of Descartes' method of co-ordinates, any geometrical figure -- whether plane or solid, even those in the Non-Euclidean and projective geometries -- could be expressed agebraically. Each point in space could be specified by three numbers, each line and plane by an equation, and thus geometry seemed to be immensely simplified.
Even more surprising was the fact that by pursuing this method it became possible to speak of geometries of more than three dimensions. Since in algebra a point in space is defined by 3 numbers giving the distance of that point from certain fixed lines (co-ordinates), there is no algebraic reason why it should not be possible to speak of points which require 4 such numbers, or any set n of numbers to determine its location. Each such number required to determine a position may be called a "dimension," and any number of such dimensions can be dealt with in algebra. The fourth dimension could even be pictured graphically by projecting it upon a solid or a plane, just as it is possible in perspective to project a solid figure of three dimensions on a plane of one less dimension.
When an equation can be graphically represented as a straight line (a simultaneous equation), it is quite easy to see how it behaves for every value of the independent variable. The slope of the line shows how, as we increase the independent variable, the dependent variable increases. If the line slopes upward sharply, this increase is rapid; if it slopes slowly, the increase of the dependent variable for each unit of increase in the independent variable is small. But equations of more than the first degree are found to be curves. Since a curve varies its direction continuously, the slope of the curve is also constantly changing. How, then, can we reduce this to a convenient geometrical expression? Or to put it algebraically, how can we determine the value of a function for any value whatsoever of its independent variable?
Many mathematicians contributed to the development of a method of answering this question-which obviously is fundamental to algebra, since on it depends the solution of many types of equations which cannot be solved by the more elementary methods. The ultimate solution, however, was proposed by Leibnitz and Newton in the 17th century. Newton was concerned with the problem principally because without such a mathematical method it becomes well nigh impossible to deal with natural motions in which there is acceleration of change (for example, a falling body whose speed is constantly increasing).
The method found is called the calculus (method of calculating), which is spoken of as differential when it is used to determine the slope of a line for a given difference of the variable, and integral when it is used to determine the area bounded by the line (i.e., to determine the rate of change in a function, or the accumulated change). It consists essentially in the use of the concept of limit.
We can convert a continuous line into a series of numbers by dividing the line into units as small as we please, so that the series approximates to a description of the line as closely as we wish to make it. So also we can convert a curved line into a series of straight lines (tangents to successive points in it) of varying slopes, with as perfect an approximation as we wish, by considering as small a segment of the curve as we wish. For each such small segment the slope of the curve is roughly the same as a certain tangent straight line. The smaller we make this segment the more accurate our description, although the exact description is a limit which cannot be attained.
Such a study of the approach of a series to a limit can also be applied to many types of problems other than those of geometrical curves. For example, when we speak of the velocity of a moving body we are considering a relation between a change in time and a change in distance, so that if we speak of the velocity of a body at an instant of time, we are speaking of a limit, namely the change in distance during a smaller and smaller interval of time.
It is possible, therefore, to generalize the calculus so that it becomes the branch of algebra which considers any type of problem dealing with the ratio between the difference of a dependent variable required when the independent variable is made to differ by a smaller and smaller amount (differential calculus), or problems dealing with the sum of a series of such difference (integral calculus).
Of particular interest in studying functions in this way is the determination of whether they are continuous or not, and also the determination of their maximum and minimum values.
GROUP AND SET THEORY
This reduction of geometry to more and more general systems such as topology (which we have described), then its reduction to algebra, and the advance of algebra to methods of studying every type of function by a general method of calculating every possible value of a variable-this could not but lead on to a still more general study of mathematics.
We have already mentioned Galois, who showed that an approach could be made to the solution of equations of higher degree by classifying them. This idea was soon extended to all of geometry and of algebra, so that mathematicians became interested, not merely in particular equations, but in whole classes of transformations. By this is meant that we can find very general classes of mathematical objects which have some properties in common, so that they could be transformed into each other by a finite number of specified mathematical operations. Thus in our survey of geometry we saw, as we advanced from Euclid's geometry to topology, that we were dealing with more and more basic properties which applied to wider and wider groups of figures. In topology a circle could be classed with a vast number of other closed figures, into which it could be transformed by a series of operations of twisting and bending and rearrangement. Similarly, as we advanced in algebra we saw that mathematicians were ever seeking to group equations into great classes, each of which could be solved by a series of defined operations.
In order to reach this generality, it is necessary first to find analogies between unlike things. Thus projective geometry finds an analogy between a solid figure and its projection on a plane. In somewhat the same way analytic geometry finds an analogy between a line and an equation, and algebra finds an analogy between a rational and irrational Dumber, or a number and an infinite collection. Such analogies, however, can lead to all sorts of confused and paradoxical thinking. Hence the tendency of mathematics has been to try and reduce these to univocal definitions, which are extremely generalized. When this is carried to a limit, the result is called set theory.
In set theory the mathematician begins with definitions and postulates which abstract from all geometric notions and which reduce numbers to sets of collections of unspecified objects. These objects might be interpreted as points, as lines, as spheres, as integers, as real numbers, as irrational numbers, or in any other fashion, just so long as they form a collection or set which can be defined by some common property. It then remains possible to perform operations on these abstract sets according to operational laws which are taken as postulates. In this way we may add sets, divide them, consider the part they have or do not have in common, etc.
Such a theory seems to satisfy the famous definition of mathematics by Bertrand Russell: "Mathematics is a science in which we do not know what we are talking about, or if what we are saying about it is true." Yet it is extremely powerful. In set theory it is possible to prove theorems so general that they will apply to any part of geometry or algebra. These theorems make it apparent that most of the theorems previously proved in these sciences are only special cases of more general theorems, in which the basic relations involved are much more evident. In consequence, set theory casts light on the whole of mathematics and gives it an orderliness never before suspected.
MATHEMATICAL OR SYMBOLIC LOGIC
We have still not reached the end of our survey. As soon as the ideal of rigor in mathematics began to be prominent in modern mathematics, it was evident that a mathematician in making a demonstration must not only list his mathematical postulates, but also must state his logical rules of reasoning as well. We saw in Chapter II that Euclid's proofs are not convincing unless we can show that he also observed the rules of syllogistic reasoning. Hence arose the attention to mathematical logic, or the methodology required in mathematics, and attempts were made to make very explicit the logical rules which were being followed.
As soon as this was done, mathematicians were rather horrified to discover that "traditional logic" -- that is, the logic current in the schools of the 19th century -- was itself not a very rigorous science. They attempted, therefore, to improve logic by the same methods of explicit postulation and careful definition as they were using in mathematics, and to facilitate this they introduced a mathematical type of notation. This new rigorous system of logic using a mathematical type of notation is what we call symbolic logic. Again, this was only reviving an idea which Aristotle had glimpsed, and which was later considered by the medieval thinker, Blessed Ramon Lull, in the 13th century, and again by Leibnitz in the 17th. But in our times this new logic has been developed with amazing perfection.
We will not describe this system of logic here in detail, but only mention the following features:
1. It has a notation, similar to that used in mathematics, by which all logical relations found in formal logic can be expressed. This notation includes
- a. Letters which stand for variables, just as in mathematics, except that in this case the variables are classes of statements or terms. For instance, the letters p and q may each stand for any statement belonging to a particular set of statements.
- b. Symbols which indicate logical relations. For example, the symbol ~ placed before a propositional variable (one standing for a set of statements) negates it, so that ~ p is the denial of the statement p. Again, the symbol connecting two variables indicates their conditional relation, so that p q means "If the statement p is true, then the statement q is true."
2. It begins with a small number of axioms expressing such basic relations. If it introduces symbols other than the primitive ones, it defines these new symbols in terms of the primitive symbols. Furthermore, it has rules of inference which are similar to the rules of operation in mathematics, and which permit theorems to be proved by reduction to the axioms. Finally, it may have formation rules which regulate the formation of statements from terms, and combinations of statements from simple statements. All these axioms, definitions, and rules are assumed without proof. It has been shown that, just as a number of geometries or algebras are possible depending on the axioms assumed, so many logistic systems are possible in the same way.
3. From these axioms theorems are deduced according to the assumed rules of inference. These theorems are of two sorts:
- a. Theorems which are the various laws of logic; for example, the moods of the syllogism, etc.
- b. Meta-theorems which deal with the completeness and order of logic as a science. For example, if we prove our list of the moods of the syllogism to be complete, this conclusion is a meta-theorem. These meta-theorems are, so to speak, the logic of logic itself. (See page 568 ff.).
Once this logistic system is complete we have a full set of logical rules which can be used to regulate the development of some particular science. The particular sciences themselves can then be formulated in the same fashion, by adding new symbols which stand for their special terms, and axioms containing these terms. It may also be necessary to provide some special logical rules for a given science, but these can be either postulated or proved in the logistic system.
To date it has been found too complicated to formalize in this rigorous way any of the sciences except mathematics, and here a curious result has been achieved. It is the contention of Alfred North Whitehead and Bertrand Russell, who at the beginning of this century were leaders in the revival of logic, that when mathematics is formalized it becomes possible to reduce it to logic, because the postulates required for mathematics all relate to numbers, and numbers can be defined in purely logical terms. Russell defines number as "the set of all those classes whose members have a one-to-one correspondence." Thus the set of all couples is the number 2, the set of all trios the number 3, etc. Since in this definition no terms except logical terms ("set," " class ... .. correspondence") are used, it is argued that number can be defined as a purely logical entity.
If this attempt had succeeded, then the final outcome would be that mathematics would turn out to be pure logic. However this result has not been universally accepted, so that at present there are four well-known opinions about the ultimate nature of mathematics:
- The Logistic Theory of Whitehead and Russell, which holds that pure mathematics is a branch of logic, expressed in their famous book Principia Mathematica (1913).
- The Formalistic Theory, of which David Hilbert (1862-1943) was the leader, according to which mathematics is the science of the formal structure of symbols. This school admits that mathematics is not logic since it concerns the structure of real objects, but they contend that it is possible to replace this structure by symbols which represent this structure. Once a correct symbolization has been achieved, then mathematics consists in the study of the structure of these symbols and only indirectly in the study of the real things.
- The Theory of Intuitionism, of which L. E. J. Brouwer is the leader, holds that mathematics must be based on the natural numbers which we are intuitively sure we can construct.
- The Aristotelian Theory, which prevailed in mathematics before Descartes, according to which geometry and algebra are essentially distinct sciences of real quantity.
THE PROBLEM OF MODERN MATHEMATICS
Some Difficulties about Modern Mathematics
This survey makes clear that in mathematics as in every other science there is progress, but at the same time there is often an increase in confusion and error. As every field of learning advances, brilliant ideas are proposed which often rapidly develop and become the fashion. Some of these eventually prove to be false, but their exposure usually brings to light some new truth or at least makes us understand an old truth better. In the history of mathematics, the many "proofs" of the parallel line postulate are an example of how errors may be made even by great thinkers, errors, however, which eventually are profitable.
Other new ideas prove to be true, but even in this case they are often proposed at first in an inaccurate and paradoxical fashion. Expressed in this incorrect and exaggerated fashion they often seem to overthrow the truths which have already been achieved, and there are always enthusiasts who claim that a new discovery has destroyed all that men previously thought to be true. Thus the success of the calculus led some (including one of its discoverers, the great Leibnitz) to suppose that at last it bad been shown that a line is made up of points. But this conclusion was due to inaccurate definitions of terms; it has been removed by a better understanding of the notion of a limit.
The lesson which this must teach us, and which the history of mathematics so well illustrates, is that we must constantly analyze and criticize each new theory until we are sure that it is correctly expressed and basically sound. In the elementary study of a subject (and this is specially true of mathematics) textbooks and teachers often find it convenient to define terms in a loose and metaphorical way. If the student takes this illustrative language literally lie will fall into many errors.
For example, many terms used in mathematics seem to imply motion or change. We define a circle as "a line described by a point moving, at a constant distance from another point." We speak of "constructing" a geometrical figure or a number, of a "variable" and
"function," of "approaching a limit," of a series "converging," etc. Quantities as they are abstractly considered in mathematics have no motion or change whatsoever. We do not really "project" a figure in projective geometry, nor do we move and superimpose a figure in proving congruence. All such expressions are only figures of speech derived from physical quantity. Hence in giving accurate definitions in mathematics we should seek to eliminate all implication of motion. Even the term "operation" must be understood very accurately, since we do not really make anything in mathematics (see pages 309 f.).
Another sort of confusion which we have pointed out several times is to be found in the analogical use of terms. The term number, for example, strictly applies only to the natural numbers, beginning with 2. Zero, the unit, negative numbers, fractions, irrational numbers, imaginary numbers, complex numbers, transfinite numbers, all are numbers only in an analogical sense. Similarly, the term "dimension" and the term "space" have many meanings. Nothing could be more absurd than the idea of some people that modern mathematics has proved the existence of a fourth dimension in the sense that length, breadth, and depth are dimensions.
Fortunately the last fifty years has seen mathematicians become very careful about distinguishing these different senses of terms, but the student will find them still often misused in mathematical writing and reasoning.
The Chief Difficulty about Modern, Mathematics
Yet even when these confusions in the use of terms is cleared up, a basic weakness in modern mathematics remains. It is the same weakness which Aristotle criticized in the mathematics which he learned in Plato's Academy, namely, the failure to distinguish between science and dialectics. Aristotle showed that no matter how beautifully constructed and how logical a system of thought might appear, it remained only dialectical unless it could be shown to rest on first principles (axioms and postulates) which are immediately evident. A building is only as firm as its foundations and a house built on sand will fall. The problem about modern mathematics, therefore, must be whether it is built on immediately evident first principles.
Modern mathematicians generally concede that it is not. Indeed many deny that any such principles are possible. The modern view is that mathematics begins with "axioms" which are pure assumptions and which do not have to be known to be true or false. They concede then that mathematics is not a science in the Aristotelian sense, but only a dialectical system, and they deny that it can ever become a science.
What is the cause of this strangely pessimistic attitude? Mathematicians generally give two reasons:
- The use of this method of basing mathematics on assumptions has proved wonderfully fruitful. If we compare the mathematics of Euclid with modern mathematics, we see that modern mathematics is incomparably richer and more powerful.
- 2. The classical example of immediately evident first principles -- namely, the axioms and postulates of Euclid -- has been shown to be a set of assumptions whose contradictories can just as well be assumed as a base for mathematics.
The consequences of this position are serious. If all of mathematics is only probable, and mathematics is the most certain of all sciences, then all knowledge is only probable. Since it is contradictory to hold that all knowledge is probable (see page 79), this leads to the admission of contradiction or absurdity in the real world, and hence to complete scepticism and to atheism. The Catholic Faith tells us very plainly that the human reason is capable by its own power of arriving at certain truth, and this is also evident from our own daily experience in which everyone is aware of many truths which cannot be reasonably doubted.
How This Difficulty Can Be Solved
Pessimism and scepticism in this matter, as in many others, is the result of failure to analyze conflicting views in a critical way. Confusion leads to a despair of finding the truth. Let us consider the two main difficulties:
- 1. It is not surprising that the use of assumptions is fruitful in mathematics. We have seen (page 187) that in every science dialectics is required in opening up new problems. In mathematics also new discoveries have been made by beginning with an assumption and drawing out of it a series of conclusions. If paradoxes were then encountered the assumptions had to be abandoned. If the results seemed self-consistent, then the investigation was continued until it became possible to connect the system with the evident axioms and postulates of mathematics. In this manner the calculus was developed quite extensively before it was possible to show in a precise way how it could be given a secure foundation by the concept of the limit. Hence it is not necessary that a particular mathematical system be abandoned merely because it is dialectical. We need only acknowledge that it is tentative and imperfect, and so open to further research.
- 2. It is not true to say that the development of the Non-Euclidean geometries proved that the parallel line postulate is not immediately evident. They only proved that it is a true postulate independent of the other Euclidean postulates. The NonEuclidean geometries as originally posed were based on an arbitrary assumption of a substitute for this postulate. As such they were merely dialectical. Eventually it was shown that these geometries could be verified in Euclidean space if geodesics of positive or negative curvature were taken as analogous to a straight line. Thus in Euclidean space it is possible to have Non-Euclidean figures. That Euclidean space is the true space of geometry is. by no means disproved. In it alone are there similar figures and zero curvature. Other spaces have to be constructed by us as modifications of this space. Furthermore, it is a three-dimensional space, since it cannot be proved that any other space can be constructed, except in an analytical geometry in which "dimension" is only used in an analogical sense.
The same thing is true of arguments drawn from Cantor's transfinite numbers (see page 388), for which the whole is not greater than the part, etc. These are supposed by some to disprove the very axioms of Euclid. If we consider such arguments carefully we will find them riddled with equivocation. Can we see that an infinite multitude is a whole with parts? The term "whole" implies some kind of unity, but we cannot conceive how an infinite multitude could have any kind of unity of itself, and it certainly cannot have unity from our counting its elements in a series, since we cannot count to infinity. In mathematics we have the potentially infinite; for example, a line or any magnitude can be divided and subdivided as often as we please. Similarly, the series of natural numbers or the length of a line can be made as great as we please. But we can never construct an actually infinite magnitude or number. As the criticisms made by mathematicians of the Intuitionist school (see page 396) have shown, the assumption of actually infinite quantities is only dialectical. We cannot prove that such quantities are not contradictory, although conversely (as St. Thomas Aquinas pointed out in the 13th century) it seems difficult to prove them impossible.
The Test of Mathematical Truth
Some have tried to argue that the only test of mathematical truth is the self -consistency of a system based on a set of assumptions -- a defense given by many for the Non-Euclidean geometries. If only this test is met, however, a system is dialectical and not scientific. A poem can be self -consistent, to all appearances, and still be a mere fiction. But as a matter of fact even this test is not possible in a mathematics based on assumptions.
It has been proved by Goedel and other modern mathematicians that it is not possible to prove the self -consistency of a system from within the system. The mere fact that a contradiction has not yet been discovered in a system does not prove it consistent, since in a science it is usually possible to draw an unlimited number of conclusions, and therefore such a contradiction may appear in the next theorem. Furthermore, since it is possible for contradictory conclusions to be deduced from a false premise, two systems contradictory to each other but each consistent with itself might both be deduced from the same false assumptions.
Hence in actual practice mathematicians have never been content with resting a theory on self-consistency. Ordinarily they justify it by an existence theorem in which it is shown that the theory is valid in a particular case, or that it is strictly analogous to something which is known to be valid. This is the way in which the Non-Euclidean geometries, and in general any system of this type, is validated, by showing that it is analogous in a strict way with Euclidean geometry, or with the series of natural numbers. Indeed, as we have seen, all modern mathematics has been developed in such a way that it is ultimately tested by reference to the natural numbers.
Since this is the case, it also appears why it is not true to say that modern mathematics is no longer about quantity, but merely about relations. Relations (see page 60) cannot exist of themselves. They must be relations between things, and they must have a foundation in things. The relations with which modern mathematics deals are those studied in set theory. What is a set? It is defined as "a collection of my kind of objects having some common property." If we examine the word "collection" we see that it means "a multitude having some type of unity." Thus none of the relations studied in modern mathematics of the most advanced type can be considered or defined except in terms of "unified multitudes."
A "unified multitude" is either made up of substances or accidents. Of the accidents, quantity is the first, and it is only through it that other accidents can be multiplied. Substance is either material or immaterial, and material substances are multiplied only by means of quantity. From this it follows that, since set theory treats of a multitude, it must either treat of quantity, or it must treat of immaterial objects. These immaterial objects, however, (such as the angels) can be understood by us only by analogy to material objects, so that when we speak, of a "multitude" of angels we make use of transcendental quantity conceived analogously to the quantity which is in the categories (categorical or predicamental quantity). There must be, therefore, a science of quantity before there can be any set theory. If set theory is considered as embracing spiritual beings, it properly belongs to metaphysics and it presupposes another distinct science of mathematics which treats of the quantity of material objects. This objection holds also for Russell's definition of number, which is supposed to eliminate quantity but which still makes use of the notion of a "collection," or "class" in the sense of a collection, and which is therefore a circular definition.
The only alternative to this is to say, with Russell, that mathematics is not about real things, but about logical beings, and that the relations in question are purely logical. However, if this is the case, then the things studied by mathematics cannot exist in physical reality, since a mental being is one which cannot exist outside the mind. Since mathematicians know very well that they deal with many things which do have an extra-mental existence, and which are not merely creations of the mind, this view of mathematics would destroy almost the entire existing body of mathematical doctrine.
We may conclude that modern mathematics is still about quantity when it is accurately developed, although it often deals with extremely general quantitative relations.
GEOMETRY AND ARITHMETIC
In surveying mathematics we cannot help but be struck by the way in which the two branches of geometry and arithmetic have survived in spite of all the attempts to reduce them to a single science. The fact is that modern mathematicians in general grant that there is a genuine and irreducible distinction between continuous and discrete quantity, between lines and numbers.
Of course numbers are more abstract and clear than are magnitudes. Hence it will always be highly useful to use analytic geometry and other methods to approximate geometry by means of algebra; but this approximation of numbers to magnitudes is an approach to a limit, it can never be realized. The reason has already been given by Aristotle. Geometrical objects are infinitely divisible, while arithmetical objects are made tip of units which are indivisible. Thus the two sciences remain essentially distinct, with different sets of definitions and postulates which are only analogous to each other.
Our knowledge of immediately evident principles comes from the power of our intelligence, which moderns call "intuition" or "insight." We must not be scandalized at the many attacks made on "intuition" by formalists in mathematics and logic. They usually disparage intuition by pointing out the fact that very frequently in mathematics and logic we discover surprising conclusions which "at first sight" we would have thought unlikely, for example, the possibility of Non-Euclidean Geometries, the irrationality of relations like, etc.
If by "intuition" is meant our rough, uncritical impressions of reality, then of course " intuition" has to be eliminated from scientific knowledge -- and replaced by precise and accurate knowledge. This is just what Euclid was trying to do when he produced his Elements; in this work the rough "intuitive" guesses and imperfect demonstrations of earlier mathematicians were given critical and scientific form. His efforts were imperfect, but in recent years the efforts of every logician and mathematician to construct perfect systems have also been shown by critics to contain many weak spots. The great Principia Mathematica of Russell and Whitehead has been found to be riddled with difficulties more fundamental than any in Euclid's Elements.
But if by "intuition" we mean critical intelligence, then all mathematics must rest on the power of the human mind to abstract from experience certain things which it sees to be certain and evident. Those selected by Euclid have never been shown to be false, and must be used today as the basis of all mathematics, although we can formulate them more precisely than he did. They are true because they are not assumed arbitrarily, but abstracted from experience by a clear intelligence.
The efforts of modern thinkers to turn mathematics into a strictly formal system in which the conclusions can be deduced by purely mechanical methods and without thinking is not to be rejected. Mathematics at all times needs a system of calculation. As we have seen, arithmetic in our ordinary modern sense, and much of algebra and calculus, is a method of calculating; it is only an instrument of the science of mathematics, whose business is not the working of problems but the demonstration of truths. Such calculations can be very well carried out by electric-brains, but an electric-brain cannot see why its results are true. Only the human intelligence can see truth.
In learning mathematics, therefore, the student must constantly seek to understand why an answer is true. This means that he must trace it back to fundamental axioms and postulates, which he is certain are true from his own experience. Only then does he have scientific knowledge. If he cannot base a conclusion on these truths, but only on assumptions, then he must regard it as tentative or dialectical reasoning, which may or may not turn out to be true on further examination. The fact that such dialectical reasoning appears at the same time both highly plausible and yet in contradiction to the axioms or postulates should not surprise him in the least; on the contrary, it should be a stimulant for him to reconcile it with these axioms and postulates, or to discover its inherent fallacy.
This is the straight road which Euclid gave to mathematics, and it is only by traveling it that mathematics has advanced.
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