Science in Christian Perspective
The Impact of Three Mathematical Discoveries on Human Knowledge
ROBERT L. BRABENEC
Department of Mathematics
Wheaton College Wheaton, Illinois 60187
From: JASA 30 (March
This paper was supported by a grant from the Wheaton College Alumni Association during the summer of 1975.
The period from 1826 to 1931 was a century filled with significant developments in every area of mathematics. The year 1826 is noteworthy, marking the discovery of non-Euclidean geometry. This discovery changed the way in which man thought about mathematics and led to a rapid growth of many new kinds of mathematical structures as well as the adoption of the axiomatic method as the format for these developments.
In the middle of this period, Georg Cantor's work with the concept of infinity in his theory of sets and the subsequent paradoxes that were discovered in his work led to the development of different philosophies of mathematics, each with sharply drawn lines of distinction regarding the nature and extent of mathematics.
The close of this period in 1931 refers to the publication of the remarkable paper by Kurt Godel on the foundations of mathematics. His results ended the dreams of some mathematicians, notably David Hubert, of the potential of the axiomatic method and initiated a fresh investigation into the nature of mathematics that has continued until today.
If one holds the common misconception that mathematics is little more than a collection of clever devices for working with numbers, then it should come as a surprise that modern mathematics contributes significant insights toward the answers of such basic questions as What is truth? What is the nature of the infinite? What are the limits to the extent and kind of man's knowledge?
This paper discusses three discoveries that are important not only within the field of mathematics, but which also have broader significance. These results are probably not widely known because, although they are understandable by one who does not possess extensive mathematical facility and training, they are not the material commonly taught in the high schools and colleges. These three results are the discovery of non-Euclidean geometry in the 1820's, the development of infinite set theory by Georg Cantor in the 1870's and the foundational results of Kurt G–del in the 1930's.
Most people are familiar with Euclidean geometry to some extent because they studied it in high school. But few know about non-Euclidean geometries or their significance for present world views. Let us consider now this interesting and far-reaching development.
In the 4th century B.C., the Greek mathematician Euclid compiled many of his results and those of his contemporaries into a remarkable work, the Elements. Beginning with 23 definitions and five postulates, Euclid used deductive reasoning to obtain 465 theorems, dealing mainly with geometry, but also including number theory and algebra. We now know he omitted some necessary postulates, but this has no effect on our discussion. The fifth postulate is popularly known today as the parallel postulate. It asserts that through any point P not on a given line L, exactly one line can be drawn that is parallel to L, in the sense that the lines will never meet, no matter how far they are extended.
For some reason still not clear today, this parallel postulate provoked great concern among many who felt it could be proved as a theorem using only the first four postulates, rather than being independent of them. Although this question in no way affected the content of the Elements, it remained an absorbing and unsolved problem to a large number of mathematicians for more than 2000 years. Then around the year 1825, three men (the great German mathematician and astronomer Gauss, the Hungarian army officer Bolyai, the the Russian professor Lobatchevsky) working independently of each other, came to a surprising discovery: it had been so difficult to prove that the parallel postulate depended on the first four postulates because it was not true. Not only was the parallel postulate independent of the first four postulates, it was also independent of the physical world.
At this point, it is pertinent to review the mindset of the early nineteenth century. The main theorems and formulas of the calculus had been discovered near the end of the seventeenth century by men such as Newton and Leibniz. In the eighteenth century, men such as Euler and Laplace applied the calculus with remarkable success to describe and solve an abundance of physical problems such as the motion of heavenly bodies. In all of this, Euclidean geometry was believed to furnish the unique mathematical description of the physical world. It consisted of results that were universally accepted as the absolute truth about our world; e.g. in every triangle, the sum of the angles is 180 degrees. Thus, man was confident of his ability to reach certainties in mathematics. In fact, mathematics was the last area of knowledge in which man thought he could obtain absolute certainty; now this idea was shattered forever. For the essence of the discovery of non-Euclidean geometry is that Euclidean geometry becomes but one of many possible consistent geometries, any of which might be a valid model for the physical world. Instead of one unique parallel line to L through P, there might just as easily be none, two, three, or even an infinite number.
Several decades passed before more than a few scattered individuals realized the implications of nonEuclidean geometries. Then in the early 1900's, Albert Einstein used one of the nonEuclidean geometries as the model for the physical world in his famous theory of relativity. The popular form of this theory quickly captured the attention of the public, giving dramatic evidence that non-Euclidean geometry could apply to a physical situation. Many believe the introduction of the concept of relativity into all areas of human knowledge and activity to be a consequence of non-Euclidean geometry. Morris Kline expresses this very well.
All people, prior to non-Euclidean geometry, had shared the fundamental belief that man can obtain certainties. The solid basis for this belief had been that man had already obtained some truths-witness, mathematics. No system of thought has ever been so widely and completely accepted as Euclidean geometry . . . Men such as Plato and Descartes were convinced that mathematical truths were innate in human beings. Kant based his entire philosophy on the existence of mathematical truths. But now philosophy is haunted by the specter that the search for truths may be a search for phantoms.
The implication of non-Euclidean geometry, namely, that man may not be able to acquire truths, affects all thought. Past ages have sought absolute standards in law, ethics, government, economics, and other fields. They believed that by reasoning one could determine the perfect state, the perfect economic system, the ideals
Mathematics possesses a vitality that seems to guarantee a long future of new ideas and significance for all areas of human knowledge.
of human behavior, and the like. The standards sought were not
just the most effective ones, but the unique, the correct ones.
Our own century is the first to feel the impact of nonEuclidean geometry because the theory of relativity brought it into prominence. It is very likely that the abandonment of absolutes has seeped into the minds of all intellectuals. We no longer search for the ideal political system or ideal code of ethics but rather for the most workable.1
It is essential to realize that the discovery of nonEuclidean geometries did not prove absolute truth did not exist. The mathematicians were asserting only that absolute truth could not be proved within their discipline. This assertion does not abolish truth, but rather indicates it may transcend the highest level of human thought and effort. Unfortunately, most disciplines reacted by denying the existence of absolute truth, asserting all things to be relative, whether they are in politics, ethics, or even theology. While this reaction may have been in response to a mathematical discovery, it must he emphasized it was not a necessary reaction. How often we see non-scientists alter their beliefs because they misunderstand the meaning of a scientific or mathematical discovery.
On the other hand, mathematicians, rather than despairing at this apparent deficiency in man's ability to know truth, accepted it as an open invitation to expand their searches for new results in all directions. They would now emphasize the ideas of consistency and validity, rather than that of truth. Endless variety thus became another consequence of the discovery of nonEuclidean geometry for the mathematical world. Instead of one geometry (i.e., Euclidean), there were now many. Instead of one number system (i.e., our usual one), many others were discovered in the years after 1845, leading to the modern structures in abstract algebra. Instead of one infinity, we will see how Cantor showed in the lS7O's that there were many different infinities. And instead of one logic (i.e., Aristotelian two-valued logic), the years after 1920 saw the development of new logics, each leading to different mathematical systems.
Infinite Set Theory
One problem area had consistently surfaced throughout the history of mathematics, that of the infinite. In the ease of Euclidean geometry, it was the parallel postulate that dealt with the infinite. The problems that arose when an attempt was made to decide whether two lines would meet if extended indefinitely has already been discussed. Also, one who has studied the calculus will recall the centrality there of the limit concept, which is the mathematician's way of dealing with the infinite. However, in spite of the frequent occurrence of the concept of the infinite before the 1870's, people had not thought with much clarity about it. It remained a fuzzy concept, somewhere beyond the reach of man. Infinity was viewed as an absolute and unique entity, a vast pool into which everything non-finite blended.
Then a German mathematician by the name of Georg Cantor, motivated by his researches in an advanced area of the calculus (convergence of Fourier series), began a careful investigation of the theory of sets, and especially the properties of infinite sets, those possessing a non-finite number of members. The results that he discovered revolutionized several areas. His concept of a set has proven fruitful as the basic language to use in describing most areas of modem mathematics. In fact, the twentieth century has been the scene for use of the language of set theory and the axiomatic method to reformulate much of the known mathematical results. What we refer to as the "new mathematics" is really in large part the old mathematics expressed in the new language of set theory.
But it was when Cantor dealt with infinite sets that some truly significant results arose. Upon sending some of these results to a friend, Cantor remarked, "I see it, but I don't believe it." Another mathematician, on reading Cantor's results exclaimed, "This is not mathematics, this is theology." Let us examine the nature of results that would evoke such responses.
The set of counting numbers 1, 2, 3, . . . is almost universally recognized as a familiar infinite set. No matter how large a number is chosen from this set, there is always a larger one. This common view of infinity-when there is to end but always another element-has been called the concept of "potential infinity" by David Hilbert. The infinite is never reached, but it is potentially there in the sense that there is always another element beyond any chosen one.
Cantor extended his work to the realm of the "actual infinite." That is, he considered the counting numbers as a completed set and began to form and to work with subsets of this set. This approach led to several remarkable results, a few of which are discussed here.
Cantor began by defining when two infinite sets were of the same level of infinity, or equivalent, in the sense that their elements could be matched up in a one-to-one manner. Thus, even though the set of all counting numbers N== [1,2,3,4,. . .] is obviously different from the set of even counting numbers E= [2,4,6,8 . . .] they are seen to he at the same level of infinity under Cantor's definition by observing the one-to-one matching of is from the set N with 2n from the set E. Even though the set R of all rational numbers seems to be infinitely larger than the set of natural numbers N (for between any two natural numbers, there are an infinite number of rational numbers), Cantor showed R to be of the same level of infinity as N. Then he was able to demonstrate two sets (the natural numbers and the real numbers) which could not be matched up according to his definition of equivalence, and thus were of different levels of infinity. In fact, Cantor showed there must be an unending string of larger and larger infinities.
Before Cantor, whether one accepted the position of a potential or actual infinity, whether one was a mathematician or not, whether one was thinking from a Christian perspective or not, whenever one spoke of infinity or the infinite, he thought there was only one infinity: whatever was beyond the finite. For a theologian such as Strong, this was the basis of arguing for one God and for a finite universe.2 For if there is but one infinite, he argued, it would be contradictory to speak of two different infinite beings, or of both an infinite God and an infinite universe. The two could not subsist together. It would make an interesting question to try to relate the one infinity of the theologian to the endless number of infinities of the mathematician.
The consideration of eternity as endless time is not generally accepted by theologians, who prefer to consider eternity as an entity distinct from time. We can think of eternity, however, only in the framework of time, and thus often carelessly speak of eternity as endless time. We should rather think of eternity as qualitatively different from time, not as quantitatively different. For example, in John 17:3, eternal life is not viewed as an endless life, but as an experience of knowing God, "And this is eternal life that they may know Thee the only true God and Jesus Christ whom Thou hast sent." TI Corinthians 4:18 suggests that eternity consists of entities beyond the realm of man's insight-"The things which are not seen are eternal."
The Philosophies of Mathematics
Widespread disbelief of the validity of Cantor's findings developed quickly, so much so that personal attacks were mounted against Cantor, especially by Kronecker, a mathematician at the University of Berlin. These led to mental breakdowns and denial of a university position at the University of Berlin that Cantor desired. A number of men began to find paradoxes within set theory which seemed to augur evil for all of mathematics. The attention of many turned to attempts to vindicate mathematics by resolving the paradoxes and developing philosophies of mathematics that would describe the true nature of mathematics; the paradox of the barber who shaves all those men and only those men in his village who do not shave themselves, is probably the best known.
Though the sharp distinctions have mellowed considerably over the years, the original situation was the development of three distinct philosophies of mathematics, each with vigorous proponents and well-defined battle lines. This inauguration of a serious investigation into the foundations of mathematics and the development of philosophies of mathematics was another significant consequence of Cantor's work with the infinite.
The first of these three philosophies was intuitionism, advocated mainly by Kronecker in the last decades of the nineteenth century, followed by Brouwer in the opening years of the twentieth century. Their rallying cry was the famous statement of Kronecker, "God made the integers, all else is the work of man." It was especially the work of the man Cantor in infinite set theory which infuriated Kronecker. He insisted that only those results that could be proved in a finite number of constructive steps were acceptable as mathematics. While this viewpoint had the advantage of making mathematicians more cautious of the reasons given in their proofs, it also severely limited the scope of mathematics. Many basic results could not he accepted under the strict limitations of the intuitionists.
Another objector to the infinite set theory was Bertrand Russell; he was among the first to discover some of the paradoxes it contained. Since so much of mathematics had been restated in terms of this new universal language of set theory, the paradoxes necessitated a drastic reassessment. Russell's answer was to begin the second philosophical school, logicism. His thesis was that mathematics was but a branch of logic, so if one could place logic on a firm axiomatic basis, then the results of mathematics would he safe in this new universal language. The Principia Mat hematica was written by Bertrand Russell and Alfred Whitehead to demonstrate the validity of this thesis. Although the Principia provided the tool of symbolic logic, their commitment to logicism made long, plodding proofs of the most elementary mathematical results. As an example, they most go more than 200 pages into the second volume before proving that 1 + 1=2.
Then, the German mathematician, David Hilbert, began the third philosophical school, formalism, in strong reaction to the strict curtailment of his beloved mathematics that was imposed by the intuitinnists and logicists. The name "formalism" refers to Hilbert's strong dependence on an axiom system in which the symbols are manipulated according to the formal rules of the system without an attempt to attribute any meaning or interpretation to the symbols themselves. It was his goal to demonstrate both completeness and absolute consistency for an axiom system which had the natural numbers as a model. Since the set of real numbers, as well as the calculus and its applications, are ultimately based on the natural numbers, and since Euclidean geometry had been proven consistent if the natural numbers were consistent, such an achievement would vindicate mathematics in large measure.
Completeness would be established if every question that could be posed in the language of the axiom system could be answered within the system. Absolute consistency would be established if it could be proven as a theorem within the system that there were no contradictions in the system. It should be stated that Hilbert was the outstanding mathematician of his day and so his efforts were widely followed by the mathematical community. Throughout the 1920's, Hilbert and his followers made slow but continual progress in their assigned task. Some areas were proven to be consistent within themselves and complete; one example was the predicate calculus of Russell and Whitehead.
Then in 1931, a 25-year-old mathematician at the University of Vienna, Kurt Godel, published what is probably the most startling result in the foundations of mathematics, showing that Hilbert's goal was unattainable. Godel's paper is deeply involved with formal logic, but in simple terms he proved that no axiom system significant enough to contain our usual number system among its possible models could be proved consistent except by going outside the system. Furthermore, even if such a system could be proved consistent, it would necessarily be incomplete in that one could always state propositions that could not be proved true or false within the system. This inability to simultaneously obtain both consistency and completeness in an axiom system reminds one of the indeterminacy principle of Heisenberg with the similar inability to simultaneously know both position and velocity. It was very disappointing to realize that the axiomatic approach which seems to furnish man his best means for knowing, and which had served so admirably in the 'SOO's for generating and expressing new mathematics was now shown to be unavoidably defective in this way.
The believer, upon reflection, will realize the extensive use of axiomatics in Scripture. For one example, the parables of Christ are axiomatic in nature, with a presentation of the primitive terms and the axioms, leaving it to the hearer to assign an interpretation to the system. In the parable of the prodigal son, we have as the undefined primitives such concepts as the older son, the younger son, the father, the far country and the fatted calf. The axioms would include such statements as, "The younger son went to the far country" and "The father had the fatted calf killed." While no theorems or conclusions are presented in the parable, we find ourselves compelled to conclude that the father loved the younger son dearly. The numerous models or interpretations of this parable that have been presented from the pulpit indicate the attempt of men to derive the theorems implicit in this axiomatic system.
As a second example, we consider the reasoning of Paul in his presentations of the gospel along the route of his missionary journeys. In axiomatic language, we might say that he was trying to demonstrate that Jesus of Nazareth was the only one who satisfied all the "axioms" for the Messiah that were presented in the Old Testament. Those who were open to follow his logical arguments accepted Jesus as Christ. Those who already "knew" what they thought was the truth, furnished the opposition to Paul's ministry.
The widespread usefulness of the axiomatic method both within mathematics and outside it, encouraged man to try to find some way to justify his usage of axiomatics, even though the method had been shown to have inherent weaknesses. And indeed, the initial reaction of Hilbert and most mathematicians to these limitations was one of despair, but the realization gradually came that there was also a positive side to Godel's results. For they opened whole new areas in the foundations of mathematics, as well as new vistas to our understanding of the power of the human mind.
Toward the end of the eighteenth century, some mathematicians feared that mathematics was almost a closed subject, with all its questions answered and no new fields to investigate. However, we have seen that the discovery of non-Euclidean geometry opened the gates for the proliferation of new structures and new topics using the axiomatic method. Then the remarkable achievements of Cantor in the field of infinite set theory led to a new language for mathematics as well as furnishing the impetus for an investigation into the nature of mathematics. The resultant philosophies arid the reactions to the results of Godel have shown us that mathematics possesses a vitality that seems to guarantee a long future of new ideas and significance for all areas of human knowledge.
1M. Kline, Mathematics for Liberal Arts (Addison-Wesley, 1967), pp. 475-6.
2A. H. Strong, Systematic Theology, Vol. 1, (Judson Press, 1907), pp. 254-255.