C. RALPH VERNO
Associate Professor of Mathematics
West Chester State College,
West Chester, PA
Many people have misconceptions about the nature of mathematics and mathematical thinking. Such thinking, called postulational reasoning, involves reasoning deductively from assumptions containing undefined terms. A sound Christian apologetic must make use of this type of thinking; we cannot logically prove the Christian faith true and we should not try. We acknowledge our presuppositions which we have by reason of God's work on our souls.
This thinking is also important for doctrinal discussion and exposition. We should state undefined terms and postulates, and precisely define other terms. Christian doctrines, even "mysterious" ones, do not involve contradiction but are rejected due to unscriptural assumptions.
To many people the thought of mathematics suggests some mysterious collection of devices and tricks used to calculate with numbers and to solve certain kinds of problems. Those who have such a view of mathematics fail to understand its nature. At its core mathematics involves a way of thinking. It is not until a person begins to understand and appreciate this way of thinking, which we call postulational thinking, that he can begin to see the structure, the system, the significance, indeed the beauty of mathematics. Only then can one begin to appreciate the marvellous creativity involved in pure mathematics. It is failure to appreciate the nature of mathematics and mathematical thinking which causes many people to react negatively to the ideas of contemporary mathematics. Too often uninformed people, not knowing what is involved in mathematical thinking, prejudicially reject so-called modem mathematics. Such people (and there are sadly too many), content with what they learned in school, not only fail to understand what mathematics is but often are not interested in finding out.
Postulational thinking is not only at the heart of mathematics; it can also be of great value in other areas of life and thought. We as individuals and members of society would be able to act more rationally and live together more congenially if we could and would apply the mathematical way of thinking to matters involving various human relations. Moreover, postulational thinking is vital to a sound Christian apologetic and to intelligent and rational theological discussion. It is the purpose of this paper to explain the nature of postulational reasoning and to indicate its significance for Christian apologetics and theological discussion involving matters of doctrine.
The first fundamental element of an abstract mathematical system is a set of undefined terms. Certain terms must necessarily be left undefined simply because we cannot define everything. Any good definition must use terms previously understood. If we have no undefined terms, then the terms in our definition must be previously defined using terms which are previously defined. Since we cannot continue this indefinitely, we will ultimately be guilty of circular definition, i.e. defining something in terms of itself. To define lines in terms of points and points in terms of lines is to say nothing about either. Evidently the dictionary in trying to define everything is attempting what is impossible. So we must have some undefined terms.
Secondly a mathematical system contains good definitions which, of course, make use of undefined terms (and ordinary connectives of our language). A good definition is far more than a partial description. The essentials of a good definition are as follows:1. It names the term being defined and places it in a set or category to which it belongs.
Thus "A square is a geometric figure having four sides" is not a good definition. Nor is "God is love" notwithstanding what some preachers may say. (Is "God" a term which we as Christians can adequately define, or is it a term which we should technically leave undefined and then only give partial descriptions given in Scripture?) We should note that definitions are not essential in a mathematical system but are used for the sake of efficiency and conciseness. For every time we wanted to refer to a circle we could instead refer to the set of all points in a plane, and only those points, which are equidistant from a given point; this is in effect to repeat a definition of circle. When mathematicians introduce a term that is not left undefined, they clearly and precisely define it so that there is no reason for doubt as to what they mean.
Another essential of a mathematical system is a set of axioms or postulates. We use these terms synonymously to mean assumptions or presuppositions; they are statements or propositions which we accept without proof, which we assume. Now these are absolutely essential. Everyone has axioms in every subject for thought even though he may not be aware of nor openly acknowledge the assumptions. When we endeavor to logically prove some conclusion to someone, we do so by citing previously agreed upon statements as the reasons for the necessary conclusion. If these reasons were in turn previously proven, then they were based on other statements which implied them. Again we cannot continue indefinitely and prove everything. We must have certain statements we assume without proof or we will surely be guilty of circular reasoning; and when we engage in the all-too-common practice of circular reasoning we are in effect saying that something is so because it is so. An example of a mathematical axiom would be what we call the commutative property of addition of counting numbers (called natural numbers), namely, that when we add two such numbers the order of the terms does not matter.
From this set of axioms (which contain undefined terms) we then draw necessary conclusions using deductive logic, i.e. we prove theorems. In proving theorems we may use previously proven theorems, but since these previously proven theorems are derived from our set of axioms it should be clear that every theorem in our system ultimately rests upon the set of postulates. Deductive logic is the means by which we prove theorems. In deductive logic the conclusion is inescapable, i.e., it would be impossible for the hypotheses to be true and the conclusion false. Deductive reasoning is "If . . ., then . . ." kind of reasoning. We must very clearly distinguish this from what is called inductive reasoning. When using inductive reasoning one comes to a conclusion about things not yet observed on the basis of a common characteristic already observed in many cases. Inductive reasoning is indeed important in providing conjectures or hypotheses in science or mathematics, but it is not proof. The conclusion in inductive reasoning is never more than probable at best. Although it is sometimes called inductive logic, it seems preferable not to call it logic; by logic we refer to deductive reasoning, or what is also called necessary inference.
Postulational thinking, then, is simply reasoning deductively from a set of axioms which use undefined terms, drawing necessary conclusions from postulates. If the axioms are accepted, then the conclusions which are deduced must be accepted. Postulational thinking is drawing valid conclusions from assumptions; a valid conclusion is one that necessarily follows. We must be careful to distinguish between validity which is a property of an argument (although we may refer to the conclusion of a valid argument as a valid conclusion) and truth which is the property of a proposition.
For example, if we accept the hypotheses "All Pennsylvanians are Californians" and "All Californians are Russians", then we must necessarily conclude that all Pennsylvanians are Russians. This conclusion is not as matter of fact true, but it is a necessary conclusion of valid argument. Proof always requires hypotheses whose acceptance forces the acceptance of the conclusion. When we realize the tremendous role of this reasoning in mathematics, it is not presumptuous to say that postulational thinking is mathematical thinking.
Now how is this mathematical way of thinking, this postulational reasoning, related to Christian apologetics? The answer of many theologians would be that it is not. Failing to understand postulational thinking or its relation to the Christian faith, they endeavor to do what cannot be done, namely, to logically prove to the unbeliever that the Christian faith is true. They try to present a presumably logical argument to prove that the Bible is the Word of God or that the Christian view of God and the universe is the true one. The reason why this cannot be done is simple enough. In order to logically prove to the unbeliever that the Christian faith is true, the Christian must agree with the unbeliever upon the propositions which are the hypotheses requiring the conclusion. Nobody can be forced to accept the conclusion of a valid argument unless he agrees to the hypotheses. Now if these hypotheses were conclusions to previous arguments, then they in turn depend upon previously agreed upon statements. To avoid circular reasoning we must ultimately derive the conclusion to the first argument from axioms or presuppositions. As mentioned above, we cannot prove everything; something must be assumed at the start. Now what ultimate philosophical or theological presuppositions are there about the universe, God, life, man, etc. upon which we agree with the unbeliever? The simple answer to that question is that there are none. To prove conclusions we need axioms and we do not have any axioms in common with unbelief. Thus any effort to prove the Christian faith true is hopeless and impossible.
Our presentation of, or apologetic for the Christian faith must therefore be axiomatic or presuppositional. We must openly acknowledge that we have postulates or presuppositions to begin with. (It is not the purpose of this paper to deal with the most desirable list of propositions which we should presuppose; in short, we might speak of presupposing the Triune God of the Bible who has infallibly revealed himself therein.) The main point is that our axioms are not arbitrary nor invented by us, but come from our experience of God's work on our souls; we presuppose what God has made known to us, not that which we have created. The Christian apologetic must be postulational or presuppositional. Any other so-called apologetic is not worth the effort. Incidentally, we should point out that even if we agreed with the unbeliever on certain primitive propositions by means of which we could logically prove that Christianity is true, our system would still be axiomatic. We cannot avoid having assumptions as starting points. The simple fact is that we do not, as mentioned above, have philosophical axioms in common with unbelief. We presuppose the truth of the Christian faith.
This apologetic approach must make clear that these presuppositions are the only ones which meaningfully explain the universe, its "disorderliness" as well as its coherence and the uniformity of the natural world which provide the basis for scientific pursuit. These alone are the axioms which explain man as be is, his ability to think, to reflect, to relate, to know, and the world as he finds it. (Something analogous to this occurs in certain areas of mathematical creativity. The mathematician is free to assume whatever axioms he wishes, but he uses his freedom wisely and responsibly. He is free to choose any axioms for the set of counting numbers; why, then, does he assume the commutative property mentioned above, as well as the other properties of these natural numbers? He does so because this is the way the natural numbers "behave" in the world in which he lives.)
Since we cannot prove the Christian faith to be true by a logical argument, we should not even try. The unbeliever can come to accept its truth as a result of God's use of our witness to open his eyes, to renew him and enable him to believe and thus accept the axioms with which he had not agreed. The "conclusions" we want the unbeliever to accept are really our presuppositions. In saying that we must acknowledge our presuppositions as well as the presuppositions of those who disagree with us, we must emphasize that we are not engaging in circular reasoning. In logic or mathematics circular reasoning is the worst of "sins"; when one commits the mistake of circular reasoning, he makes use of or assumes the very thing he is trying to prove by a logical argument. This is not what we do when we have a presuppositional Christian apologetic, because we are not trying to logically prove the Christian faith to be true. The effort to lead the unbeliever to accept the Christian faith is not one of logical deduction. Rather, we presuppose the Triune God of the Bible in his relation to the created universe, we point out the unbeliever's own presuppositions, and we also point out the meaninglessness of his axioms ' meanwhile praying that the sovereign God will open his eyes and bring him to faith.
Now what relevance does the mathematical way of thinking have for theological discussion, for matters pertaining to doctrines of the Bible? For one thing, as indicated above, we should in our discussion, whether with unbeliever or believer, acknowledge our assumptions at the start. Discussions would be less heated and more intelligent, less emotional and more rational, if we were to state clearly not only our own assumptions but also those of the people with whom we deal. In mathematics we state our axioms clearly and precisely before we use them. If we were to do this in theological discussion, our dialogue would be more fruitful and amicable, even if it were not to end in agreement.
Another desirable aspect of theological discussion is to acknowledge the use of undefined terms, realizing that we cannot define everything. It is not our purpose here to determine a precise list of terms to be left undefined; perhaps we should not define such words as "God", "reality", "existence", "universe". Moreover, when we define words, we should do so clearly and precisely before we use them as does the mathematician. What point is there in discussing religion, theology or doctrine (or anything else, for that matter), and perhaps getting angry, when we use the same words (such as "revelation", "creation", "inspiration", "sin", "liberty", "foreknowledge", "faith", etc.) but mean different things by them? Intelligent theological discussion will openly state the meanings of terms and phrases used.
In much discussion of Christian doctrine certain "mysterious" doctrines (such as the Trinity, the person of Christ and predestination) are often said to be contrary to human logic. On the one hand this is stated by unbelieving philosophers and theologians whose position is based on the assumption of the autonomy of the human mind and who consequently reject these doctrines. On the other hand such statements are often made by believing theologians and preachers while setting forth the mysteriousness of these doctrines which should be believed. They in effect say that these doctrines involve contradictory propositions but we nevertheless take them by faith. Now this is simply not so. These doctrines are not contrary to logic. They do not involve contradictory propositions. God does not reason deductively in a manner different than we do. God deduces as we do; rather, we deduce as God does, for in thinking logically we are using our God-given minds properly. Suppose we give God the premises "All M is N" and "All N is P"; will God deduce something different than "All M is P"? Of course not. To think logically is to think in an orderly fashion with minds made by the God of order.
When a person who is liberal, neo-liberal, cultist or otherwise, rejects the doctrine of the Trinity or the doctrine of the person of Christ as historically held by the church to be scriptural, or when an Arminian reJ . ects the central doctrine of Calvinism, such rejection is not based on logic but on a different set of axioms. A person who rejects the doctrine of the Trinity really reasons as follows on the basis of his assumed hypotheses:
Hypothesis 1: If God is one, he cannot have a plurality of persons.
Hypothesis 2: God is one (clearly taught in the Bible). Conclusion: Therefore God cannot have a plurality of persons.
The person who rejects the orthodox doctrine of the person of Christ also has assumptions and reasons as follows:
Hypothesis 1: If a person is human he cannot be divine. Hypothesis 2: Jesus is human (clearly taught in the Bible).Conclusion: Therefore Jesus is not divine.
Hypothesis 1: If a human act is foreordained by God, then the human cannot be responsible for it.
Hypothesis 2: We are responsible for all our acts (clearly taught in the Bible).
Conclusion: Therefore our acts are not foreordained by God.
The first hypothesis in each of the above arguments is assumed, contrary to the teaching of Scripture. Such a person does not come to the Bible to see what it teaches, but comes already begging the question by assuming that the Bible cannot teach such a doctrine. These doctrines do not involve logical contradiction; they are rejected because the persons doing so have faulty, i.e. unscriptural, axioms.
The Bible does not say that God is Triune and also that God is not Triune. The Bible does not say that Jesus Christ is one person with two natures and also that he is not one person with two natures. The Bible does not say that God foreordains whatsoever comes to pass and also that God does not foreordain whatsoever comes to pass. Those who deny the Trinity assume that the assertion of the unity of God is a denial of threeness of personality. Those who deny the doctrine of the person of Christ assume that humanity precludes deity. Those who deny God's absolute sovereignty assume that human responsibility precludes foreordination. The Bible does not teach contradictory propositions. The assertion that God has three persons does Dot contradict, i.e. is not the negation of, the assertion that God is one. The contradiction of the assertion that God is one is that God is not one. The assertion that Christ is God is not the negation of the assertion that Christ is man. The contradiction of the proposition that Christ is man is that Christ is not man. The assertion of God's foreordination of whatsoever comes to pass does not contradict the assertion of human responsibility. The contradiction of the assertion that man is responsible for his acts is that man is not responsible for his acts. The Bible does not make contradictory assertions simply because the Bible itself tells us that God, who speaks therein, cannot lie. God cannot assert proposition "I"' and also proposition "not P" for God does not contradict himself. This is at the very heart of the precious doctrine of God's unchanging covenant faithfulness.
It should be noted that we did not speak of the doctrine of the sovereignty of God involving foreordination and human freedom, but rather of foreordination with human responsibility. Some people try to "resolve" the debate between the Calvinist and the Arminian by saying that both are extremists, and that the Bible teaches both God's sovereignty and man's freedom. Some Calvinists in stating the doctrine say that God foreordains all things but that man is free, that we believe in two seemingly contradictory truths. Now perhaps the Bible teaches that man, even sinful man, has some kind of freedom which even Reformed theologians refer to as moral freedom or free agency or even freedom of the will; by this they mean rational self-determination through which man acts in a manner consistent with his nature. (The explanation of even the very capable Reformed theologian Louis Berkoff is not completely satisfying to this writer.' Why must what is called free agency by theologians be so called? This is not the language of Scripture for man's rational self-determination in accord with his disposition. Use of such language confuses the issue.) This, however, is not what the ordinary believer (or unbeliever) thinks of when he thinks of freedom or free will. The Arminian view of freedom is that of man with an autonomous will which is the sole cause of what he does, which could resist what God might wish to come to pass. The Bible does not teach this kind of freedom, and so it does not assert both foreordination and freedom in this sense. If it did, it would be asserting both a statement and its overt contradiction or negation which, as we noted above, God cannot do. Because of widespread misconceptions about freedom, it is better to speak of the Bible as teaching both God's foreordination of whatsoever comes to pass and man's responsibility. If we do use the word "freedom", we should, as stated above, clearly define it in distinction from the Arminian view.
The trouble people have with these doctrines, then, is not a matter of logic but rather a matter of assumptions. Those who disbelieve need better hypotheses which are based on Scripture, rather than their own assumptions. Scripture teaches that God does have unity and also exists in three persons, that Christ is both God and man, and that God foreordains acts for which man is responsible. These mysterious doctrines of the Bible do not run counter to human logic, but to human assumptions that there is a contradiction when there is not. We must reject such assumptions and base our hypotheses on the teaching of Scripture. These hypotheses based on the Bible may still be mysterious to us, but they do not involve, and we do not believe in, contradictory propositions.
proper understanding of the mathematical way of thinking and its application can
contribute to a better presentation and understanding of the Christian faith. To
many people the subjects of mathematics and theology are far removed from each
other. It is not so and need not be so in the minds of thoughtful people.
(Perhaps it is not insignificant that a number of mathematicians of history have
been theologians or clergymen). Strange as it may sound, we suggest that some
effective study of mathematics and mathematical thinking would better prepare
ministers and theologians for the understanding, defense and exposition of the
Christian faith. The study of mathematics is the best training one can get in
logical reasoning or postulational thinking, better than the study of logic per
se. Whether we acknowledge it or not, a sound approach to Christian
apologetics and biblical exposition must use mathematical or postulational
1. Berkhof, L., Systematic Theology, Grand Rapids, 1941, pp. 106,248.