Korean Article Bank

On the Motivations of Goedel's Ontological Proof

Woosuk Park (KAIST, Taejon)

1. Introduction

In recent years there has been a surge of interest in Goedel's ontological proof of the existence of God. Goedel showed his proof (Goedel *1970) to Scott, and Scott made a note of the proof and presented it in his seminar at Princeton University in the fall of 1970. From then on, Goedel's proof has become widely circulated. It was finally published in Sobel 1987 as an appendix and later included in volume three of Goedel's Collected Works. Recent discussions of Goedel's proof mostly start from Sobel's crticisms. As is well known, the most influential crticism of Sobel is that Goedel's proof leads to a consequence unacceptable to most philosophers, i.e. that all truths are necessary truths. Anderson 1990 viewed this as the modal collapse of Goedel's assumptions, and tried to save Goedel's proof by some plausible modifications. Anderson's emendation secured many interesting responses including Oppy 1996, where a parody of the Goedelian proof reminicent of Gaunilo's objection to Anselm's proof is presented. As one might expect, such a parody has invited friends of ontological proofs to follow in the footsteps of Anselm.

In spite of all this extensive concern, it is not certain whether there is any improvement in understanding the motivations of Goedel's ontological proof. Why was Goedel so preoccupied with completing his own ontological proof? To the best of my knowledge, no one has dealt with this basic question seriously enough to answer it.

In this article, I propose to examine Goedel's ideas against a somewhat larger background in order to understand his motivation for establishing the ontological proof. I shall point out that the value of Goedel's proof is to be found in the possible role of his proof of the existence of God in his philosophy as a whole as well as in its relative merit as an ontological proof. Hopefully, my guiding question as to Goedel's motivation will turn out to be extremely fruitful by enabling us to fathom his mind regarding God and mathematics

2. Back to Goedel's Original Proof

It should be noted that what Sobel discussed was not the proof given in Goedel's own writing but the proof in Scott's note. Sobel did so on the assumption that the ideas presented in Scott's note are those of Goedel's substantially agreeing with "ideas conveyed in two pages of notes in Goedel's own hand dated 10 February 1970 and entitled "Ontologischer Beweis", which appears in Collected Works Vol. III as *1970. To say the least, it is not certain whether such a treatment of Goedel's proof is legitimate. In the appendix, Sobel juxtaposed both Goedel's version and that of Scott's. At a glance, they are significantly different. For example, Goedel's axiom 1 is missing or transformed into axiom 2 in Scott's version. It is left as homework for the reader to check their comprehensional, intensional, extensional (or what not) equivalence.

Anderson heartily accepted Sobel's announcement of the modal collapse of Scott's version of Goedel's ontological proof. But with slight modification, he thought, the collapse can be avoided. So, we have Anderson's emendation of Scott's version of Goedel's ontological proof. Now Oppy freely identified Anderson's emendation of Scott's version of Goedel's ontological proof as Goedel's proof and parodied it to hail Gaunilist victory. What is going on?

Presumably Anderson is one of the best friends of ontological proofs, and in view of the damaging effect of the charge of modal collapse, his alleged emendation must be an example of heroic benevolence to save non-existent Goedel. But was emendation needed at all? In his introductory note to Goedel *1970, Adams expressed exactly that kind of suspicion:

"It is characteristic of Leibnizian philosophical theology to be in some danger of leaving no truths contingent (See Adams 1977). And it is not altogether clear that Goedel was determined to avoid such a necessitarian conclusion". (See also Dawson 1997, p. 266)

It is widely known that Leibniz was one of the intellectual heroes of Goedel. Further, as Adams noted, "The study of Leibniz is known to have been a major intellectual preoccupation for Goedel during the 1930's(Menger 1981, §§ 8, 12) and especially during 1943-46 (Wang 1987, pages 19, 21, 27)". We can easily pile up more testimonies to advocate Goedel's Leibniz scholarship. Also, it has been duly noted that Goedel's ontological proof is basically Leibnizian. Adams seems to be primarily interested in the Leibnizian character of Goedel's ontological proof in such a way that he organized his introductory note to Goedel's ontological proof on the premise that at least Goedel knew pretty well both that "Leibniz held that Descartes's ontological proof is incomplete" and that "Leibniz also held that the ontological proof can be completed by proving the possibility of God's existence".

There seems to be, then, reasonable doubt as to whether Goedel would have been impaled by the threat of modal collapse. If so, Anderson's kindness could have been either misplaced or presumptuous on Goedel's part. Needless to say, the original must be consulted if possible. In this case, Goedel's original proof in perfect shape. There seems to be no reason why one should discuss allegedly Goedelian proofs rather than Goedel's original proof.

One more reason to go back to Goedel's original proof is this. Though cryptic, ample clues are found for fathoming his mind in the philosophical annotations in *1970. Goedel's Collected Works also contain "Texts relating to the ontological proof" as Appendix B, where more clues are found. Curiously enough, commentators of Goedel's ontological proof have not in general fully utilized these clues. Wang 1996 seems the only case where Goedel's annotations are discussed in detail. Unfortunately, Wang was not sympathetic at all to Goedel's interest in the ontological proof. (See Wang 1988, p. 195 and Wang 1996, p. 121) Be that as it may, he failed to treat Goedel's ontological proof in close conncetion with Goedel's views on other philosophical issues, in particular on the philosophy of mathematics. Apparently, commentators of Goedel's ontological proof did not treat Goedel seriously as a professional philosopher let alone as a historian of ontological proofs or a Leibniz expert. As a consequence, they have failed to discuss some of the most enlightenling clues found in Goedel's writing. At least, so I shall argue.

3. Gathering Clues

I would like to gather as many clues as possible for understanding Goedel's motivation for drafting his ontological proof from his philosophical annotations in *1970 and from the "Texts relating to the ontological proof".

(1) axiomatic method

The most prominent aspect of Goedel's ontological proof must be that it utilizes the axiomatic method. There may have been predecessors in the history of ontological arguments, but let us not forget the fact that we are dealing with Goedel's axiomatic method in the post-Hilbertian era. Goedel's proof has axioms and definitions, and if Goedel has his own view of the roles of axioms and definitions as a leading mathematical logician, that must be relevant to understanding his ontological proof.

In Excerpt from "Phil XIV", we read:

"Philosophy: The ontological proof must be grounded on the concept of value (p better than ~p) and on the axioms:X"

In the footnoteX Goedel elaborated the point: "It can be grounded only on axioms and not on a definition (=construction) of "positive," for a construction is compatible with an arbitrary relationship."

These remarks provide us with at least two hints for further query. First, if we remember the question as to whether "that than which nothing greater can be thought" in Anselm's proof was meant to be a definition of God, then here we might have some data about Goedel's view on Anselm's ontological proof. But do we have any axioms in Anselm's proof in proslogion? Even if it has some, do they play the role as Goedel would assign to axioms? If there is none, then Goedel already has refuted Anselm's proof. We may generalize the point to all previous ontological proofs to check whether all of them are vulnerable to Goedel's (implicit) criticism.

Secondly, though inseparable from the first point, we have to examine Goedel's remark here to see whether he was criticizing Hilbert's view regarding implicit definitions and other issues of axiomatic method. No one has shed light on Goedel's use of the axiomatic method in *1970 by fully exploiting Goedel's views on the foundations of logic and mathematics.

(2) Why ontological proof?

However, probably the most relevant to our guiding question is found in Excerpts from "Max XI". In the item [From page 97] we read:

"Remark(Philosophy): If the ontological proof is correct, then one can obtain insight a priori into the existence (actuality) of a non-conceptual object".

But what is securing a priori the existence (actuality) of a non-conceptual object for? Except as a possible proof of platonism in mathematics and mathematical intuition, I cannot figure out any other use. But let us postpone this issue for a while.

Another possible reading of "Why ontological proof?" is to interpret the question as meaning "Why not cosmological proof but ontological proof?" Though some would view this kind of approach as reading too much history of ontological and other traditional arguments in favor of God's existence into Goedel, the mathematician and amateur philosopher, such a qualm would have no ground. For in the item [From page 149] we read the following remarkable passage:

"Remark (Theology): The reflection: according to the Principle of Sufficient Reason the world must have a cause. This must be necessary in itself (otherwise it would require a further cause). Proof of the existence of an a priori proof of the existence of God (the proof it contains fails to be one)."

This passage is remarkable since it is a rare example of trying to understand the relationship between cosmological and ontological proofs. Why did Aquinas adopt cosmological proofs rather than ontological proofs in his five ways? Why did Scotus or Descartes attempt ontological proof again with full understanding of Aquinas's attack? Regardless of the intrinsic value of questions like these, Goedel must have been struggling with the similar questions. Intuitively, I tend to interpret his reflection as saying that cosmological proof is merely a quia demonstration (demonstration of fact) not a propter quid demonstration (demonstration of reasoned fact). (See Mancosu 1996, pp. 11-13) What is nice in this particular reading is that it reminds one of the plausible purpose of seeking ontological proofs. Anselm is a perfect example. Immediately before he presented his celebrated ontological proof, Anselm wanted to make sure that the sole purpose lies in the attitude of "faith seeking understanding". As a believer, he already had faith in the existence of God. But he wanted to understand God by ontological proof. Goedel seems to be on a par with Anselm in this respect.

Of course, it is difficult to make sense of the final part of the passage. If I am right, Goedel is saying that cosmological proof itself cannot be the proof of the existence of an a priori proof of the existence of God. Probably, the only possible way of providing such a proof of the existence of an a priori proof of the existence of God is to deliver a priori proof of the existence of God. So, Goedel tried to give the ontological proof.

(3) positive, privation, etc.

Another important clue for in depth understanding of Goedel's proof must be found in what Goedel meant by "positive". In fact, Goedel was anxious to comment on it in *1970 itself:

"Positive means positive in the moral aesthetic sense (independent of the accidental structure of the world). Only then [are] the axioms true. It may also mean pure "attribution" as opposed to "privation" (or containing privation). This interpretation [supports a ] simpler proof."

Commentators of Goedel's ontological proof have been prudent enough not to indulge in far-fetched speculation about what Goedel meant by "positive". At best, they alluded to simple or atomic propositions. But if that is all Goedel meant, why did he invoke "moral" and "aesthetic"? Again Wang 1996 seems to be the only case that contains some useful, though hardly satisfactory, discussion of the possible meaning of "positive". Speculation is needed here! My hunch is that Goedel is deeply indebted to the medieval theory of transcendentals, which was rooted in neoplatonism. In the context of discussing ontological proof, being and truth were already at issue, and Goedel seems to remind us of the convertibility of transcendentals by introducing the moral aesthetic sense of "positive". Wang 1996 (p. 120) has interesting comments possibly supporting my hunch: "Goedel seems to identify the true with the good (and the beautiful). The affirmation of being is both the cause and the purpose of the world." Mentioning "privation" seems perfectly consistent with interpreting Goedel as a neoplatonist. The item entitled "Philosophy: Ontological Proof" in Excerpt from "Phil XIV" contains three possible interpretaions of "positive" and other more pregnant information for our subject. It is simply impossible for me to discuss them now. Let it suffice to note that these remarks further demonstrate how seriously Goedel was involved in uncovering the meaning of "positive", and that there is no counter evidence against viewing Goedel as a neoplatonist.

(4) Cause, Analyticity, Etc.

Still another important clue we cannot ignore in "Texts relating to the ontological proof" is related to the concept of cause, analyticity, the relationship between mathematics and physics, and other cognate issues. In the items entitled "Philosophy" in Excerpt from "Phil XIV", Goedel declares:

"The fundamental philosophical concept is cause. It involves: will,x force, enjoyment,x God, time, space.* (*Being near = possibility of influence. xHence life and affirmation and negation.)" (p. 433-4) (See Wang 1996, p. 120)

Later in the same item he wrote:

"Perhaps the other Kantian categories (that is, the logical [categories], including necessity) can be defined in terms of causality, and the logical (set-theoretical) axioms can be derived from the axioms for causality. [Property = cause of the difference of things]. Moreover, it should be expected that analytical mechanics would follow from such an axiom." (p. 433-4)

Finally, in the item entitled "Philosophy:?", we read:

"The only synthetic propositions are those of the form (a) (for example: I have this property), for these have no objective meaning, or: They depend not on God, but on the thing a."(p. 437)

What is revealing in lumping together all these remarks is that we can get a glimpse of Goedel's architectonic of sciences. The idea that analytical mechanics would follow from logical(set-theoretical) axioms, and thereby ultimately from axioms of causality alone deserves extensive philosophical discussion. It is indeed a radical vision encompassing all human knowledge on a hierarchical system. Wang reports that Goedel once told him that "there is a sense of cause according to which axioms cause theorems". (Wang 1996, p. 120) This can be a link with the clue we secured in connection with the axiomatic method.

4. The Role of the Proof of the Existence of God in Mathematics and Science

In the previous section we have gathered many suggestive clues from *1979 and "Texts relating to the ontological proof". With all these clues we can more efficiently look into Goedel's other writings in Collected Works or from biographies of Goedel. In order not to be led astray by the formidable data, I would like to exploit one working hypothesis: Goedel's preoccupation with the ontological proof must be found somewhere between God and mathematics. We should ask what possible role the ontological proof was supposed to play in the Goedelian edifice of human knowledge including mathematics and science.

After all, Goedel is one of the greatest mathematicians. This obvious but all too easily forgotten point leads one to realize why Goedel's ontological proof does matter. It does matter simply because it is Goedel's proof. Goedel is above all a mathematician. Probably no further justification is necessay for my choice of the working hypothesis. I would also say that all the clues are pointing to that hypothesis. They are not unfamiliar to us simply because they are relevant to the same old issues extensively discussed by commentators of Goedel's philosophy of mathematics.

Now, with all the clues and the working hypothesis, the most pertinent texts we have to turn to seem to be *1951, *1953/9, and *1961/? in vol. III of Collected Works. In particular, *1951 seems the best for my purpose. The key issues are centering around Hilbert Program, Carnap, platonism in mathematics, and the relationship between mathematics and physics. I will focus on how much light the clues from *1970 and "Texts relating to the ontological proof" shed on Goedel's philosophy of mathematics by raising the following two questions: (1) Does Goedel's ontological proof improve our understanding of the philosophical implications of his incompleteness result?; (2) Does it have a role in his criticism of Carnap's conventionalism in logic and mathematics, and thereby in his defense(or proof) of platonism in mathematics?

The answer to the first question seems to be "Yes"! Let us take a look at a pertinent text from *1951 where Goedel presents an argument to the effect that his second incompleteness theorem makes the incompletability(or inexhaustibility) of mathematics evident:

"It is this theorem which makes the incompletability of mathematics particularly evident. For, it makes it impossible that someone should set up a certain well-defined system of axioms and rules and consistently make the following assertion about it: All of these axioms and rules I perceive (with mathematical certitude) to be correct, and moreover I believe that they contain all of mathematics. If someone makes such a statement he contradicts himself. For if he perceives the axioms under consideration to be correct, he also perceives (with the same certainty) that they are consistent. Hence he has a mathematical insight not derivable from his axioms."(10-11; p. 309)

How should we understand what Goedel meant by the incompletability of mathematics? He himself is here cautious enough to guard against possible misunderstanding. He wrote:

"Does it mean that no well-defined system of correct axioms can contain all of mathematics proper? It does, if by mathematics proper is understood the system of all true mathematical propositions; it does not, however, if one understands by the system all demonstrable mathematical propositions."(11)

He calls the two different senses "mathematics in the objective sense" and "in the subjective sense" respectively. If the enterprise of ontological proof is successful, we may freely talk about mathematics from God's point of view. That must be nothing but mathematics in the objective sense.

If so, it is hard not to remember Aquinas's distinction between "in itself" and "to us". In launching his Five Ways, Aquinas dealt with three questions consecutively: (i) Is it self-evident that there is God?; (ii) Can it be made evident?; and (iii) Is there God?. Bluntly speaking, his idea is that even if God's existence is self-evident, still there is need for the proof. For it is not evident to us. But it is possible to make it evident to us by his Five Ways. I think, we may safely assume that Goedel was at least well versed with Aquinas's Five Ways to the extent of adopting a useful distinction. Just as Aquinas had to prove God's existence, which in itself is self-evident but not to us, Goedel had to prove the second incompleteness theorem, which is true in mathematics in the objective sense but not in the subjective sense. If this analogy is well taken, then we can understand how important the ontological proof was to Goedel. Without the solid distinction between mathematics in the objective sense and in the subjective sense, his project of proving his theorem would never get off the ground.

If the first question gets the positive answer, so does the second question. For we are already well informed of the fact that Goedel extensively used his incompleteness results in his attack on conventionalism. In his introductory note to *1953/9 Goldfarb even claims that "[t]he heart of Goedel's criticism is an argument based on his Second Incompleteness Theorem." (p. 327) So, if ontological proof is instrumental for understanding his incompleteness theorem, it must be valuable for understanding his defense of platonism and his criticism of conventionalism. Let us again look into *1951 in order to confirm this point. As is well-known, Goedel was anxious to refute the view that mathematics is our own creation. For that very reason, in refuting that view, he had to reveal his meaning of creatio and creator. For example, Goedel wrote:

"On the other hand, the second alternative, where there exist absolutely undecidable mathematical propositions, seems to disprove the view that mathematics is only our own creation; for the creator necessarily knows all properties of his creatures, because they can't have any others except those he has given to them." (15-16; p. 311)

Goedel's proof of God's omniscience from creature's total dependence on God is in itself interesting. And it becomes more interesting if we remember the total dependence of painting on the painter in Anselm's proof. In Anselm's proof, the painter already had the complete idea of the painting. The painting cannot have any other properties except those the painter has given to them. Whether Goedel himself had some such a picture in mind, the parallel seems striking. It could be confusing whether the concept of God as Artist does help the philosophy of art or the concept of creation in art can help us sharpen our view of creation. An analogous situation seems to be involved in Goedel's two great projects: incompleteness proof and the ontological proof of the existence of God. Be that as it may, it is tempting to speculate whether Goedel was already helped by his idea of ontological proof in achieving his incompleteness result. If it is indeed the case, then we may speculate even further that that was the reason for Goedel's persistent preoccupation with the ontological proof. Didn't Goedel want to solve some extremely difficult problem by sharpening his idea of God? As was pointed out, for Goedel, improving the ontological proof by searching for the appropriate axioms was nothing but the way to understand God.

Since we are dealing with Goedel's platonism, it would not be inappropriate to reinvoke our clues, especially the possibility that Goedel was a neoplatonist. In *1951, there is an interesting passage which reminds us of Augustine's criticism of Academician skepticism. Goedel wanted to differentiate his own concept of analyticity from the usual logical positivist concept of "truth owing to our definitions". "Analytic" means to him "rather "true owing to the nature of the concepts occurring [therein]", in contradistinction to "true owing to the properties and the behaviour of things." (34; p. 321) In that context, Goedel wants to protest against the frequent allegation of the paradoxes of set theory as a disproof of platonism. He counts that as unjust on the ground of an analogy with visual perception:

"Our visual perceptions sometimes contradict our tactile perceptions, for example, in the case of a rod immersed in water, but nobody in his right mind will conclude from this fact that the outer world does not exist."(34)

In Augustine's Against the Academicians, 3.11.26 we read:

"Surely it's the truth! There is a cause intervening so that the oar should seem bent. If it were to appear straight while dipped in the water, then with good reason I would blame my eyes for giving a false report."

Conclusion

Despite the recent surge of interest in Goedel's ontological proof, the level of understanding concerning it is incredibly low in that we are still ignorant of why Goedel was preoccupied with it. In this article, I tried to show that by probing its role in Goedel's philosophy as a whole or his philosophy of mathematics in particular we can improve our understanding of both. One of the lessons is the realization that Goedel had ample historical knowledge of philosophy in general and the ontological proofs in particular.

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