Science in Christian Perspective

The Paradoxes of Mathematics*
R. P. DILWORTH
California Institute of Technology, Pasadena, Calif.

In popular usage a paradox is a true statement which apparently has false consequences. The explanation of the paradox consists in showing that the false consequences do not, in fact, follow from the statement. Now, from the point of view of mathematics, which treats the strictly logical consequences of propositions, there is no difficulty at all with such statements. Hence, for mathematics, the term paradox has a sharper meaning; namely, a self-contradictory proposition. At first glance it would appear unlikely that self-contradictory statements could occur in a rigorous, deductive, mathematical system. Unfortunately, they do indeed occur and this paper will be devoted to a description of some of the more important paradoxes which have arisen to plague the mathematician.

It will be instructive to, mention first a non-mathematical paradox which illustrates the basic principle underlying most of the mathematical paradoxes. This is the so called "Barber's paradox."

The barber in a certain military unit is ordered by his commanding officer to shave those and only those members of the company who do not shave themselves. Now apply this order to the barber. If he does not shave himself then he fails to obey the order since he should then shave himself but if he shaves himself he is likewise failing to obey the order.

The difficulty with the Barber's paradox is simply that the officer's order does not determine unambiguously the class of men who will be shaved by the barber. It fails, in fact, for the barber and whether or not the barber shaves himself must then be specified by the officer.

A quite similar non-mathematical paradox can be formulated as follows: Let us agree to call a word 11 autological" if it modifies itself. For example, the word "short" is autological. On the other hand we will call a word "heterological" if it does not modify itself. Thus the word "long" is heterological. According to a basic principle of logic every word should be either autological or heterological. Let us try to determine whether "heterological" is heterological or not. If "heterological" is heterological then it does not modify itself and since this is the meaning of heterological it must be autological. But if it is autological it must modify itself and hence be heterological. The explanation of this paradox is similar to that of the Barber's paradox.

Now it may be argued that the class of all classes which are not members of themselves is indeed not a well-defined class just as is the case of the Barber's paradox where the officer's order was ambiguous with regard to the barber himself. But if we adopt this point of view then we have a property, namely, that of not being a member of itself which does not determine a class. This immediately raises a question concerning ,he validity of other classes. For example, can we be .Sure that the class of all integers is a well-defined class which will not lead us to contradictions? Clearly, a wide variety of classes are needed for the purposes of mathematics. On the other hand, too wide a flexibility in the definition of classes leads to a contradiction. Thus Russell's paradox emphasizes the need for a formulation of the language underlying mathematics which is sufficient to express the propositions of mathematics and yet which is consistent, that is contains no self-contradictory propositions.

The next paradox to be considered is due to Richard. It arises from considering the names of the integers in the English language. Now, since there are only a finite number of words in the English language and since the integers form an infinite set it is clear that not every integer can be nameable in English in less than 13 words. Hence, "the least integer not nameable in English in less than 13 words" is a definite integer and is a name consisting of only 12 words. But then this integer is indeed nameable in English in less than 13 words and we have a self -contradictory statement. As in the case of Russell's paradox any consistent formulation of the language of mathematics must be such that sentences like that of Richard cannot be formulated in the system. It is interesting that a number of systems which have been proposed for the foundations of mathematics have later been shown to have such a flexibility of expression that they were susceptible to paradoxes analogous to Richard's.

What then is the present situation in regard to the consistency of mathematics? Systems of language have been proposed which are adequate for all of mathematics and in which no contradictions have been detected. Furthermore at least one of these systems has been proved to be consistent. The proof, however, involves methods which cannot be expressed in the system itself and, indeed, the validity of these methods are questioned by some mathematicians. On the other hand, this, result seems to be about the best that can be hoped for since Godel 131 has proved that any system which is sufficient f or all of mathematics cannot be proved consistent by methods expressible within the system. This rather paradoxical result appears to close the door as far as a completely satisfactory logical foundation of mathematics is concerned. Nevertheless, the gap between adequacy and consistency is very narrow since systems have been developed which are adequate for a large part of mathematics and which can be proved consistent by methods expressible in the system itself (Church [21). This curious situation with regard to the foundations of mathematics has prompted Andre Weil to remark, "God exists since mathematics is consistent and the Devil exists since we cannot prove it".

It has been mentioned in a preceding paragraph that there are principles frequently used in mathematics concerning which there is strong disagreement among mathematicians concerning their validity. One of these principles which has played an important role in the development of mathematics and which has been used in the construction of consistency proofs is the "axiom of choice" first formulated by the mathematician, Zermelo. In order to describe this axiom, let us consider a class of mutually disjoint, non-empty classes. The axiom of choice postulates the existence off a class which has the property that it contains exactly one element from each of the classes in the original class. Or putting it in another way, the axiom of choice asserts that it is possible to pick one element out of each of the classes in the collection and put them together to form a single class. Intuitively, this principle seems quite harmless. Nevertheless, the principle has far-reaching consequences, some of which even contradict our basic intuitions. One such consequence is a theorem due to Banach and Tarski [11] which is certainly paradoxical in the usual sense of the word. This theorem asserts that a sphere of radius one can be decomposed into five parts which can then be put together again in such a way as to form two spheres of radius one. Of course, the parts into which the sphere is decomposed have an exceedingly complicated and complex structure. As a matter of fact the parts cannot be constructed in a finite number of operations. And it is here that the axiom of choice comes into play. Nevertheless, the conclusion of the theorem seems to be contrary to our intuitions of three dimensional bodies. In spite of these consequences Godel [4] has proved it is possible to adjoin the axiom of choice to one of the standard systems which is sufficient for mathematics and if the original system is consistent then the new system will also be consistent. Many mathematicians feel that this theorem justifies the use of the Zermelo principle as a standard part of mathematical methodology. On the other hand, there are some mathematicians who feel that a proof using this principle is, in fact, no proof at all. In view of the nature of the problem it seems unlikely that this controversy will be resolved in the near future.

Finally, we turn to the question of the implications of these considerations concerning the foundations of mathematics for philosophy in general and, in particular, for Christian philosophy.

Now if there are serious difficulties associated with the logical foundations of mathematics where very precise and rigorous methods are available f or exploring the consequences of propositions, it would be presumptous to suppose that basic difficulties of a similar nature are not present in other areas of knowledge. In fact, it is because of the high precision associated with the concepts and deductive procedures of mathematics that the detection of the subtle contradictions becomes possible. In a field where the basic ideas are not so carefully formulated, fundamental logical difficulties may be obscured by ambiguities in the definition of terms. Furthermore, since the language required for mathematics is, in many respects, similar to the language of philosophy, these considerations indicate points at which trouble is likely to occur. For example, use of the word "all" in philosophical or theological arguments should be carefully examined to insure that there are no hidden inconsistencies. In point of fact, many classical theological controversies have centered about words with a similar inclusive connotation.

Next, it should be noted that while the reasoning of mathematics is formally deductive, much of the reasoning of philosophy and theology is intuitive in character. The formalization of the reasoning would, in many cases, be very difficult indeed. Now we have already pointed out the unreliability of intuition even in the domain of the foundations of mathematics where it would be expected to be accurate. Again, it is the existence of a rigorous deductive method which enables the mathematician to detect the errors in an intuitive argument. It seems reasonable, therefore, to suppose that errors in intuitive reasoning are just as likely to occur in areas where a rigorous method of checking the argument is not available. If this is the case, it emphasizes the need for a critical and tentative attitude toward intuitive thinking. This applies both to the professional philosopher in his ivory tower and to man in his daily conversations. In particular, Christian folk have a special obligation in this regard. For if-their words betray a foolish and careless habit of mind, serious damage may be. done to the Christian cause. By way of example, consider the very common practice among evangelical Christians of interpreting as the working of God the occurrence of an unexpectedly pleasant or, perhaps, longed for event. This is clearly an intuitive conclusion. If it were formalized it would probably run as follows: God is good-This event is good-Hence God is responsible for this event. When it is, presented in this form, the weakness of the argument is obvious even though, in some instances, the conclusion itself may be true. However, in many cases, a little careful reflection shows that what at the moment appeared to be good would, from a long range point of view, indeed be evil. Thus in place of having been honored, God has been dishonored.

Clearly there are only a few who have the time and ability to acquire the intellectual sophistication of the professional logician. On the other hand, there is available to everyone the opportunity to acquire the modest amount of critical judgment and logical habit of mind which distinguishes the wise man from the foolish.

REFERENCES

1. St. Banach and A. Tarski, Sur la decomposition des encembles de points en parties respectivement congruent, Fundamenti Mathematicae, vol. 6 (1924) pp. 244-277.
2. A. Church, A proof of freedom from contradiction, Proc. Nat, Acad. Sci., vol. 21 (1935) pp. 275-281.
3. K. Godel, Uber formal unentchiedbare Satze der Principia Mathematica und verwandler Systeme, Monatschefte ffir Mathe matic und Physic, vol. 38 (1931) pp. 173-198.
4. --, The consistencv of the axiom of choice and the generalized continuum-hypp thesis, Princeton University Press, Princeton (1940).

GENERAL REFERENCES

5. B. Russell, Introduction to mathematical philosophy, London (1920).
6. P. Rosenbloom, The elements of mathematical logic, Dover publications, New York (1950).