MacTutor
Marston Morse: Twentieth Century Mathematics
Marston Morse wrote the article Twentieth Century Mathematics which was published in The American Scholar 9 (4) (1940), 499-504. We give a version below.
Twentieth Century Mathematics
An understanding of mathematics is latent in every intelligent person. Unfortunately this understanding is for the most part undeveloped and at times it is even denied. I have a friend who can argue almost flawlessly that he understands no mathematics, not realising that the logic of his discourse proves the contrary.
Mathematics is both an art and a science, and the lack of appreciation of this fact is responsible for much misunderstanding. There is very little in the experience of the student sated with problems in arithmetic that makes him believe that mathematics is an art. Courses in geometry and algebra which degenerate into exercises in memory, add to the conviction that mathematics is dead and mechanical. Misconceptions with regard to mathematics are common even among those otherwise well in formed. There is a literary critic of rank for whom mathematics is typified by the binomial theorem. To confound mathematics with its applications is general.
This confusion is as great in America as anywhere in the world. In France and in Italy, where appreciation of the arts is high, the nature of mathematics seems to be better understood. Henri Poincaré has written that mathematics, the art of pure thought, is the highest form of art. In France and in Italy, as in ancient Greece, mathematics reached a high estate before the day of engineering applications. The uses of mathematics that were made by Leonardo de Vinci, Lagrange, Leibniz and Newton did not conceal its inherent beauty.
To insist on the aesthetic aspects of mathematical thought is not to detract from the merit of its applications. The development of mathematics for its own sake, and the practical applications of mathematics go hand in hand. Objective advances must be revised in form to make them aesthetically acceptable and logically comprehensible, while advances of a more subjective nature, if complete and harmonious, will not long remain unapplied. Mathematicians are by no means agreed among themselves as to what is important either in content or in form. In the days when the relativity theory awaited its tests, its acceptance or rejection was a matter of taste. Otherwise convictions could not have been so violent. Mathematics grows best when tended by both artist and engineer.
These remarks concerning the nature of mathematics are necessary if one is to understand recent progress. I shall refer to advances which have been made in the present century and for the most part in the last decade. I shall try to give some slight answer to the ever-recurring question, "But what can one do in research in mathematics?" The idea that all essentials were discovered long ago is all too prevalent.
The truth is that higher mathematics has been revolutionised in the last fifty years and that the body of discoveries during that period is comparable to the advances made in all the preceding years. To clear away another source of misunderstanding one should explain in what sense research mathematicians use the word "problem." It is the sense in which Newton refers to the problem of gravitation, a philosopher to the problem of being, or an economist to the problem of equilibrium. It is more than likely that such problems admit no final solution. Contributions consist of new modes of attack, new theorems, or a new theory which relates the problem to other branches of mathematics. What is the nature of space, time, and matter? What is a system of logic? How are prime numbers distributed among all numbers? What is the mathematical meaning of probability? How is equilibrium defined mathematically, and under what conditions does it exist? What are the different possible kinds of algebra and geometry? Do electron phenomena admit a unified mathematical theory? These problems are typical of mathematical research today.
Questions such as these have an immediate intuitive meaning and have been selected in part for that reason. There are other problems which are equally important but for which the difficulties of popular exposition are nearly prohibitive. An attempt to convey to the non-mathematical a real approximation to recent mathematical ideas is an attempt at the impossible. My only hope is that my failure may be instructive and that thereby the reader may be encouraged to search more deeply.
Modern mathematics is particularly characterised by abstraction. This is partly a reflection of the taste of mathematicians and partly a matter of necessity. It is associated by some with the political and economic troubles of the day but I am more inclined to find its cause in the nature of mathematics itself. Popular opinion to the contrary, abstract studies are often simplest. The process of abstraction rules out the irrelevant and permits greater generality. Mathematicians abstract in order to unify, simplify, comprehend and extend. The great complexity of modern mathematics, and its vast scope, make the process a necessity.
Mathematicians differ as to the value of abstraction as a mode of discovery. A large group believes that the most significant discoveries begin with concrete cases, and that the final abstract forms of mathematical theories are relatively obvious generalisations of these typical cases. There is one brilliant young mathematician who maintains that the process of abstract generalisation may be come an affliction for science and that the right to make abstractions should be rationed, the quota for each mathematician to be proportioned to his earlier contributions of a concrete nature. It is true that the tendency to start with abstraction is often an escape from the necessity of mastering the difficult and more concrete problems of classical mathematics. But one must distinguish between the tendency to abstract as an end and the use of abstraction as a means to an end. In any case it is too early to pass a final judgment.
Another characteristic of modern mathematics is its tendency to ignore the barriers between the different branches of mathematics, between geometry and algebra or algebra and analysis. This tendency is now so marked that the older distinctions between the fields of mathematics frequently become meaningless. A problem in geometry, for example, may be found to have a structure that makes it appear as a problem in algebra and from this new point of view the road to the solution may be clear. Such developments mark the maturity of mathematics and give it a new unity.
To approximate a universal knowledge is no longer possible, but it is still possible to think profoundly. In seeking to rebuild the foundations of logic, mathematical philosophers must indeed think deeply. Their efforts have been crowned with success, and the last few years have witnessed advances in logic unequalled since the time of Leibniz.
Over a hundred years ago Lobachevsky showed that the so-called parallel axiom of Euclid was not necessarily true; it could not be proved from the other axioms of Euclid. Through a point not on a line , there can be drawn, in the plane of and , one and only one line that never meets . So runs the axiom and for two thousand years mathematicians tried to prove it. Lobachevsky showed that there are geometries in which it fails, and that these geometries are as simple and complete as Euclid's.
Emancipated from the rule of the "obvious," mathematicians celebrated their freedom by the invention of numerous types of geometry, including the Riemannian geometry so important for the relativity theory. It was inevitable that this realism should extend beyond the bounds of geometry into logic. The attempt to put logic on a completely formal basis was begun some time ago. The Principia Mathematica of Russell and Whitehead was one of the first great efforts. To understand the need for such a work one has only to note the looseness with which classical metaphysicians use terms. Consider for example the dictum, "I think, therefore I am." In what sense shall one understand "I am"? The origins of such a phrase lie in the dim past. Mathematical logicians would begin again with symbols whose meanings, fixed at birth and preserved in logical alcohol, would be immune from all change or contamination. This procedure may seem cold, but it is necessary if science is to be firmly and permanently based.
I have space for the statement of one theorem in logic. It is the epoch-making discovery of a young Viennese by the name of Gödel. In any ordinary system of formal logic a proof of freedom from contradiction is formally impossible in the system itself. It is not that ordinary mathematics is affirmed to be inconsistent but only that its consistency is incapable of internal proof. From Gödel's theorem it can be inferred that no one system of formal logic can embrace all forms of correct reasoning.
The positive integers form the raison d'être of many a mathematician. Their pleasing, if deceptive, simplicity is in striking contrast with the perplexing metaphysics which threatens logic. Among the many problems which are associated with the integers is that of the distribution of the primes 2, 3, 5, 7, 11, etc. These are the numbers greater than 1 with no factors other than themselves and 1. Euclid showed that there exists an infinite number of primes. A little experimentation will indicate that the density of the primes decreases rapidly as the numbers increase. It can be shown that on the average there is less than one prime in every million consecutive integers.
To give a more precise result let denote the number of primes less than the integer . For example, there are four primes, 2, 3, 5, 7, less than 8 so that . As we have seen tends to infinity with . But is much smaller than , in fact so small that divided by f tends to 0 as tends to infinity. A result such as this shows that the primes are rare but it does not show how rare. To give a precise statement we shall consider the ratio . As f tends to infinity does likewise, but f is so much smaller than that f tends to infinity with . It was conjectured that ) and become infinite in about the same way, in fact that their ratio tends to 1 as tends to infinity. This conjecture is called the "prime number theorem" and although brought to the attention of mathematicians over a hundred years ago it was not proved until 1896. Since then the theorem has been proved in simpler ways and various extensions have been made. There are, however, many related problems of great interest which are still unsolved.
We come to topology, a subject which belongs almost exclusively to our day. To define topology it will be helpful to begin with a definition of ordinary Euclidean geometry. Euclidean geometry is a theory of rigid bodies and rigid motions that is, motions in which a body is neither stretched nor distorted. An edge of a rigid body may be straight, or two edges may be parallel. If the body is subjected to a rigid motion this straightness or parallelism is unchanged. More generally Euclidean geometry may be defined as the study of those properties of rigid bodies which are unchanged by a rigid motion.
To define topology one must broaden the class of motions so as to include motions which distort or stretch. These more general motions are called space transformations. They are termed continuous because they do not tear space at any point. Topology can now be defined as the study of those properties of configurations which are unchanged by a space transformation.
A closed curve tied in a knot is an object of study in topology because this property of being knotted is unchanged under any space transformation. From the point of view of topology, surfaces are regarded as in the same class if they can be transformed one into the other under some space transformation. From this point of view a spherical surface and an ellipsoid are in the same class, but not in the class of the surface of a doughnut (with a hole). We thus have a basis for the classification of surfaces and this has been fully exploited. When however one goes to higher dimensional configurations the field is wide open.
One of the favourite devices of modern mathematics is to consider curves, surfaces etc., as points. This is done to bring out the fact that certain relations between these configurations are abstractly similar to those between points. For example, the objects considered may be closed curves in ordinary space. Given two such curves one can move either curve so as to make it coincide with the other (assuming a physical body capable of moving through a part of another like a ghost). This movement can be made so that the maximum distance moved by the points is the least possible. When so moved will be termed the distance between the two curves. This distance is not the ordinary distance between the two curves but it has some of the essential properties of ordinary distance.
In ordinary geometry three points define a triangle, and in this triangle the length of each side is at most the sum of the lengths of the other two sides. This is called the "triangle axiom." The statement of this axiom does not involve the existence of the sides of the triangle as line segments, but depends only on the definition of distance between the points involved. If one chooses to regard closed curves as points and defines a distance between them as in the preceding paragraph, it can be shown that any three such curves satisfy the triangle axiom that is, the three distances involved are related as are the three distances defined by an ordinary triangle. Working abstractly with this triangle axiom one can derive many properties of closed curves.
A modern approach to the mysterious problem of the existence of closed planetary orbits is to regard the closed curves involved as points . Certain elementary physical conceptions then enable one to discover a function of these pseudo points with the property that the points (maximum, minimum, saddle, etc.), of this function occur only when is a planetary orbit. This type of reduction of a problem in celestial mechanics to one in function theory belongs to a modern branch of mathematics - "the calculus of variations in the large." This theory shows that each topological kink in the space of the points p implies a planetary orbit. The extent and nature of the topological kinks is at present unknown but it is conjectured that these kinks are sufficiently numerous to explain the existence of the planets. This theory illustrates the possibilities of a suitable combination of topology and analysis.
If space permitted, one might go further and show how the classical notions of volume and area are replaced in modern analysis by a much more refined notion of measure, and how this measure theory contains a large part of probability theory and of dynamics. We should refer to the great technical advances of modern algebra and the ramifications of differential geometry. The examples which we have given are typical of modern mathematics in spirit rather than in extent. Unsolved problems are many and deep. Courage and imagination are needed to solve them. But mathematics belongs essentially to youth, and strong minds are ready to go ahead. We may explain and evaluate mathematics in many ways, yet the impulse to understand always remains and cannot be denied. Mathematics will advance regardless of its difficulties and, however great the advance, mathematicians will find still greater fields to conquer.
Last Updated July 2026