1The remark that is going to be interpreted in the following has already been subject of another article but some important aspects had to be left open at the time and could not be examined and discussed further on.1 Nevertheless, it is chosen again for interpretation, because it is an exemplary case for the point that is to be shown, namely that Gödels’s Philosophical Remarks can be seen as a philosophical monadology, and a new interpretation will give the opportunity to find answers for the unsolved questions at issue.

2We will see that the subjects that are referred to by Gödel in the remark in question are central to some of his philosophical convictions, which he has stated himself as follows:

My philosophical viewpoint2
1. The world is rational.
2. Human reason can, in principle, be developed more highly (through certain techniques).
3. There are systematic methods for the solutions of all problems (also art, etc).
4. There are other worlds and rational beings, who are of a different and higher kind.
5. The world in which we live is not the only one in which we shall live or have lived.
6. Incomparably more is knowable a priori than is currently known.
7. The development of human thought since the Renaissance is thoroughly one-sided.
8. Reason in mankind will be developed on every side.
9. The formally correct is part of a science of reality.
10. Materialism is false.3
11. The higher beings are connected to the others by analogy, not by composition.
12. Concepts have an objective existence (likewise mathematical theorems).
13. There is a scientific (exact) philosophy and theology (this is also more highly fruitful for science), which deals with the concepts of the highest abstractness.4
14. Religions are, for the most part, bad. But Religion is not.
Meine ph<ilosophischen> Ansicht<en>:5
1. Die Welt ist vernünftig.
2. Die Vernunft im Menschen kann prinzipiell höher entwickelt werden (durch gewisse Techn<iken>).
3. Es gibt syst<ematische> Methoden zur Lösung aller Probl<eme> (auch Kunst etc.).
4. Es gibt andere Welten und vernünftige Wesen der anderen {und höheren} Art.
5. Die Welt, in der wir jetzt leben, ist nicht die einzige, in der wir leben oder gelebt haben.
6. Es ist unvergleichlich mehr a priori erkennbar als jetzt bekannt ist.
7. Die Entwicklung des menschlichen Denkens seit der Ren<aissance> ist eine durchaus einseitige.
8. Die Vernunft wird in der Menschheit allseitig entwickelt werden.
9. Das formal Richtige6 ist<?> eine Wirklichkeit<s>wissenschaft.
10. Der Materialismus ist falsch.
11. Die höheren Wesen sind durch Analogie nicht durch Komposition mit den anderen <Wesen> verbunden.
12. Die Begriffe haben eine objektive Existenz (ebenso die mat<hematischen> Th<eoreme>).
13. Es gibt eine wissenschaftliche (exakte) Ph<ilosophie> {und Th<eologie>7}, welche die Begriffe der höchsten Abstraktheit behandeln.
14. Die Religionen sind zum größten Teil schlecht, aber nicht die Religion.

3The two philosophical viewpoints out of fourteen that are relevant for this article are emphasized. One is the very well known philosophical position of Gödel that materialism is false and the other one concerns his view on an exact philosophy and theology. An exact philosophy and an exact theology deal with basic principles, basic ideas and are therefore insofar most fruitful for science as they have been a kind of paradigmatic model for this kind of thinking.

4Both of these topics, among other things, are alluded to in the following remark from the notebooks:

Remark Philosophy: Aristotle’s proof that the intellect is not corporeal and has no bodily organ at all (and that it has after all no “nature” at all but is a mere possibility, that everybody possesses) is after all the antinomical character of the “all”.^* (But he says that if it had a nature this would prevent it from perceiving something different from this nature.⁺)

But could one not also assume types of the intellect? The absence of an organ is also proven by the fact that the intellect and only the intellect is the “place” of the forms [species], because an organ would then also be a place of this <form>.

*Or better, the possibility to make “all” in turn an object and to transcend beyond that (the unboundedness8)Est Deus sed potentia tantum.⁺Or at least one would see everything in this light. [Gödel, Max Phil VI, 404]9
Bem<erkung> (Phil<osophie >): Beweis des Arist<oteles>, dass der Intellekt nicht körperlich ist und überhaupt kein körperliches Organ hat (und dass er letzten Endes überhaupt keine„ Natur “hat, sondern eine bloße Möglichkeit ist, die alle haben), ist letzten Endes der antinomische Charakter des„ Alle “.^* (Er sagt aber, wenn er eine bestimmte Natur hätte, würde diese ihn hindern, etwas von dieser Natur Verschiedenes wahrzunehmen.⁺)
Könnte man aber nicht auch Typ<en> des Intellektes annehmen? Der Mangel des Organs wird auch daraus bewiesen, dass der Intellekt und nur dieser der„ Ort “der Formen [species] ist, denn ein Organ wäre dann auch ein Ort dieser <Form>.
*Oder besser, die Möglichkeit„ alle “wieder zum Objekt zu machen und darüber hinauszugehen (die Uferlosigkeit)Est Deus sed potentia tantum.⁺Oder zumindest würde man alles in diesem Lichte sehen. “[Gödel, Max Phil VI, 404]10

5The remark is to be found in notebook VI of Gödel’s philosophical remarks, written in 1942. Before discussing the details of this remark, I will give an overview of its content. It starts with the observation that no organ belongs to the intellect according to Aristotle. This statement has to be understood in the larger context of Aristotle’s theory of perception in which each of the five senses has an organ that is directed to specific kinds of phenomena in the world. The eye for example perceives darkness, light and colours, the ear sounds, and the nose odours. Gödel’s remark is in this context referring to Aristotle’s consideration whether the intellect (nous) has an organ and is therefore something corporeal or not. Aristotle’s conclusions can insofar be called anti-materialistic as the intellect (nous) does not have an organ and is independent from the body. It is this ground for anti-materialism that Gödel will associate with set theory.

6When Aristotle compares sense perception and intellect ( “nous”) he presents a number of arguments for the claim that sense perception and intellect are not alike. Gödel alludes to some of them in his remark. Aristotle states for example that thinking and thoughts can be right and wrong whereas sense perception is always veridical. It is on the other hand also always true that each of the senses has its specific blind spots. Whereas temperature can only be perceived when it differs from the temperature of the sense organ (e. g. the temperature of the skin), the intellect can think everything and about everything (even about itself as one may already point out). When Gödel alludes to the incorporeity of the intellect he is implicitly alluding to Aristotle’s consideration that the intellect is not compound with the body and that it has no nature other than the capacity to think everything (“πάντα νοεῖ”, De anima III, 429a, 18). Insofar as the intellect can think everything, it must be able to become everything, according to Aristotle, and cannot have a nature.

7Gödel is alluding to the passage at III, 429a 18-29 in Aristotle’s De anima with his remark, but he is adding a phrase and an anti-materialistic argument for the intellect that can only be understood in the context of Gödel’s own thinking about mathematics and his own results in that discipline. He is quoting Aristotle’s comment that the organs of the senses are corresponding to something in the world, e. g. the nose to odours. Now if there were something like an organ of the intellect it would have to correspond to something in the world but there is no set that contains everything. Nevertheless, the intellect is very well capable of thinking about such a set. Therefore, according to an Aristotelian reasoning, the intellect cannot be corporeal respectively material.

1. Russell’s antinomy and the iterative conception of ‘set’

8The possibility to make “everything” respectively “all” again an object of thought and to go on with this application is linked by Gödel with his considerations in set theory and at the end also with his considerations as to what his incompleteness theorems mean for a theory of mind. The key hints at set theory are given by the phrase “the antinomical character of the ‘all’”, and even more so by the notion “Uferlosigkeit” (unboundedness). The notion “uferlos’’, “Uferlosigkeit’’ is for example used by Felix Hausdorff in his Grundzüge der Mengenlehre from 1914 on page 2:

If there is for each set of objects another and yet another object that is different from them, then the totality of all objects is obviously not a set.… E. Zermelo has made the accordingly necessary attempt to restrict the process of unbounded formation of sets trough adequate requirements.

Wenn es zu jeder Menge von Dingen noch ein weiteres, von ihnen allen verschiedenes Ding gibt, so ist die Gesamtheit aller Dinge offenbar keine Menge.... Den hiernach notwendigen Versuch, den Prozeß der uferlosen Mengenbildung11 durch geeignete Forderungen einzuschränken, hat E. Zermelo unternommen.

9And on page 46:

G. Cantor defines the potency of a set as the general concept which comes into being through abstraction of the individual nature of its elements. B. Russell defines it essentially as the totality or class of “all” sets that are equivalent with this set. Given the unbounded and antinomical nature of these classes we think that this is questionable.

G. Cantor definiert die Mächtigkeit einer Menge als den Allgemeinbegriff der durch Abstraktion von der individuellen Beschaffenheit ihrer Elemente entsteht. B. Russell definiert sie geradezu als die Gesamtheit oder Klasse„ aller “mit jener Menge äquivalenten Mengen; dies halten wir bei der uferlosen und antinomischen Beschaffenheit dieser Klassen12 für bedenklich.

10The phrase and the notion indicate that Gödel is having set theoretical considerations in mind that accompanied its development from Gottlob Frege’s attempt on to derive the set theoretic axioms from an intuitively evident conception of set. This effort was ruined by Russell’s antinomy. Russell’s antinomy tells us that there exist sets in Frege’s and Richard Dedekind’s approaches to sets that do not contain themselves and also sets that do contain themselves as for example the set of all sets that is itself a set.

11Now, Russell’s set is defined as the set of all sets that do not contain themselves. Therefore Russell’s set contains itself when it does not contain itself, and it does not contain itself when it is containing itself, however this is a contradiction. In other words, there is no set of all sets, because it would belong to itself.13 This is where the phrase “the possibility to make “all” again an object and to transcend beyond that” in the remark in question is hinting at.

12The problem was unravelled in set theory when John von Neumann introduced the axiom of foundation (as Ernst Zermelo did in a different formulation in ZFC) in order to avoid Russell’s antinomy. This axiom is a generalization of the characteristic of sets that no set can belong to itself. In addition the von Neumann-Bernays-Gödel set theory (NBG) classifies the particular classes that, fatally, must be taken to be sets in naïve set theories (like Frege’s which rendered the antinomy possible) as proper classes that cannot be members of other classes because they are “too big”.14 Now, in NBG one can have a class of sets that do not contain themselves but not a set of sets that do not contain themselves because this would result in Russell’s antinomy again. And we cannot construct a class of all classes in this theory because classes contain only sets by definition. Classes that are too big to be sets are called “proper classes”, in German “echte Klassen”, but also in an out-dated mode of speaking “uferlos große Klassen”. Therefore Gödel is alluding to this topic with his use of the notion “Uferlosigkeit”. It indicates that when a set is “too big” —and this would lead to an antinomy— a set cannot be an element of a class but is a proper (uferlose) class itself whereas in naïve set theory any definable collection is a set. The iterative concept of sets understands sets as determined by their extensions15 and the question of an antinomy does not arise in this case until one considers the whole or totality of all V_\alpha. And therefore Hao Wang reports, that “Gödel repeatedly relates objectivism in set theory to the axiom that a collection of sets is a set if and only if it is not as large as the universe V of all sets”.16

13The concept of ‘set’ is the intuitive basis for the axioms of set theory and is thus at the bottom of formalisations of set theory. The iterative concept of set aims to provide an intuitive justification for the axiom system ZFC (of which NBG is an extension) by appealing to a transfinite iterated process of formation of sets of sets. The phrase “the possibility to make “all” again an object and to transcend beyond that” alludes to the transfinitely iterated process of formation of sets of sets. It is a process that cannot be carried out by a mortal being and not even by an immortal being because it would still be a being living in space-time. Since the iteration of the operation is transfinite it would need a non-temporal being, a God to conduct these operations.17 We will come back to this when going into greater detail of interpreting Gödel’s remark in question. Among those who proposed the idea of transfinite iteration of the operation “set of” as the basic conception behind axiomatic set theory was in fact Gödel in the early 1930s. But the iterative picture is only spelled out in full clarity in Gödel’s famous paper on Cantor’s continuum problem that was published in 1947.18

14One might take the iterative conception of set as a starting point when taking a closer look at Gödel’s analogy between perception and mathematical intuition–an analogy that is vexing19 scholars in the philosophy of mathematics until today. The iterative conception of set and our grasp of the axioms of set theory are one of the best-known examples of how we make use of mathematical intuition. It is widely assumed that the iterative conception of set constitutes thereby an intuitive basis for the axioms.20

2. Mathematical intuition and sensory intuition

15Moreover, mathematical (or rational) intuition is quite often compared with sensory intuition by Gödel or by Gödel scholars. As Michael Potter puts it:

What we need to do, then, is to explain how our grasp of the meaning of terms such as ‘number’ or ‘set’ is possible, given that this grasp cannot be described in any finite formalism. Gödel thought it was necessary to appeal to something he called ‘mathematical intuition’ to explain this grasp, and he has, [...] become famous for thinking of it as a ‘mysterious faculty’ in some ways analogous to sense perception.21

16That Gödel himself considered mathematical intuition to be merely analogous to sense perception is documented by many quotations from his published articles. The major source for these quotations is the two versions plus the supplement of his famous article “What is Cantor’s continuum problem”. There he writes for example:

Despite their remoteness from sense experience we do have something like a perception of the objects of set theory. [...] Mathematical intuition need not be conceived of as a faculty giving an immediate knowledge of the objects concerned. Rather it seems that, as in the case of physical experience, we form our ideas also of those objects on the basis of something else which is not immediately given. Only this something else here is not, or not primarily, the sensations.22

17Gödel then goes on to explain that thinking is a mere combinatorial rearrangement and that we cannot create any new elements by it but only reproduce or recombine those that are given. Therefore the analogy between mathematical intuition and sense perception is, as Potter explains, grounded in the circumstance that both of them provide the thinking process with an input that thinking itself cannot originate.23 But, as many commentators have complained, Gödel does unfortunately not explain what he has in mind when he hints to a “kind of relationship between us and reality” in the case of mathematical intuition.24

18One reason why Gödel considers an analogy between perception of things (i.e. physical objects) and perception of mathematical objects to hold especially in set theory is that sets are objects for him but classes are not.25 For him sets are thus somehow perceived by the mind or the intellect.26 Now we can read in Wang (1996) the following statement and should try to integrate it with what has been said so far:

At least in his discussion with me, Gödel did not identify mathematical intuition with something like the perception of sets. That may be why he told me specifically that it was a mistake in the original text to say “in this perception, i.e., in mathematical intuition”, pointing out that such perception is but a special type of mathematical intuition. In fact, my impression is that mathematical intuition for him is primarily our intuition that certain propositions are true–such as modus ponens [...], some of the axioms of set theory, and so on. Only derivatively may we also speak of the perception of sets and concepts as mathematical intuition.27

19If we take this statement by Wang seriously it will not suffice to say that the notion of set is a paradigmatic example for mathematical intuition in order to understand what Gödel might have meant when he spoke about “perceiving sets”. So far we have seen that a set is an object for Gödel but as we do not see sets like we see for example trees we have to know more about how we have epistemic access to sets.

20The iterative conception of set offers some insight into how we manage to do it. According to it a set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set28 and the important point is that the elements have already to be there before they are gathered together. Therefore a set cannot belong to itself because it cannot be there before it is gathered together.

21But the gathering together is a process that has to be performed. This raises the questions, how this process can possibly be called to be objective and how Gödel can say that sets are objects that are perceived and hold to an iterative (and therefore process-related) conception of set at the same time.29

22A comparison between perception of an object and gathering together elements for a set might help to get an idea. When we see a tree for example, we see it as an object despite its being a compound of leaves, branches and a trunk—they are integrated in our perception. There is an equivalent when we perceive sets. The perceiving of a set is a gathering together of the elements. This is done according to objective criteria that determine in advance (that is to say before we have gathered the elements together) whether there is such a set or not.30 One constraint for this conception is that a set must be understood as the result of a gathering together of its elements and we cannot “perceive” a set before we have done the gathering together.

Reason and the senses

23When Gödel is bringing together Aristotle’s reflections on the incorporeality of the intellect in contrast to the corporeality of the sense organs with his own reflections on set theory, he must have had his own contemplations on mathematical intuition and perception in mind. Let us therefore see what Gödel remarks about the intellect, perception and sense organs in the very beginning of the 1940s. His statements about these topics are quite indistinct in those days. The reason for this might be that he was not so certain how to put it and that he read Aristotle (and Thomas of Aquinas) in order to clear things up.

24It is not a secret that “many philosophers are put off by Gödel’s language of mathematical (or conceptual) perception and intuition”, as Wang has put it.31 Therefore we might try at first to understand Gödel’s philosophical motives to speak in the way he did. And here we might take into account that what he wanted to grasp was a kind of relationship between ourselves and reality, ourselves and mathematical objects, ourselves and mathematical concepts, and so on. He conceives similarities respectively analogies between reason and the senses, perception and understanding, sense organs and the faculty to grasp abstract concepts or objects in the early forties and still in the early fifties.32 And there are good reasons to regard their similarity only and not to put them on the same level because the intellect and reason can for example refer to themselves whereas “sight”, “hearing” etc. cannot refer to themselves. (I can of course conceive that I see or hear but this is a question of consciousness and not of hearing the hearing or seeing the sight. Whereas one can think of one’s own thinking or reasoning.)

25Now, let us turn to Gödel’s remarks on sense organs in his notebook VI that has been written at the beginning of the forties.

26There he used the word “organ” in two different meanings. One use is to name a physical organ, the other one is to use it for the function that an organ has. This is a bit strange because one would for example call an eye an organ but not what one does with it namely seeing. An explanation for this might be that Gödel uses “organ” for anything that is a faculty to establish a connection between an organism and something else (be it things, concepts, sets or ideas). The organ is an instrument for establishing such a link, but Gödel also calls the faculty to establish such a connection an organ. One can see this exemplarily in a remark on page 381 in notebook VI,33 where Gödel says that the soul is an organ of the I because its essence and aim is to relate us to different parts of the world, namely sensibility with the material world and reason with the world of ideas.34 This indicates that Gödel use of “organ” is not always a literal one in those days. We might just make a note that an organ of a human being is for Gödel something that is relating the human being to different parts of the world and also to “things” like sets and ideas.

27Gödel is specifying how he conceives the relation between sense (Gefühl) and reason (Verstand). He says that “sense” might be the organ of perception for concepts and states of affairs, and that reason is the perception of words and abstract thinking. One could suggest that what he means to say is that reason is an organ of perception for words, but one can just as well remark, again, that Gödel does not care too much about the difference between an organ and the activity or faculty that go along with an organ in those days.35

28Concerning another question he is however quite clear. Reason is not mechanistic36 whereas he states elsewhere that the brain is mechanistic.37 One of his reasons to study Leibniz so intensely as he did, and later on Husserl, is probably that he wanted to be able to give an answer to the problem how the non-mechanistic spirit is related to a mechanistic computing machine.

29But we have not come to an end yet with the difficulties to interpret Gödel’s use of the term “organ” in his philosophical notebooks. Gödel challenges us with the following list of human organs for perception: 1. senses for perceiving bodies, 2. inner perception (and empathy), 3. sense, 4. reason. There seems to be no problem to understand what is meant with the first. The sense organs for a human being are: eye, ear, nose, tongue and skin. But Gödel does not say so. What he is speaking of are the senses not the sense organs and the former are: sense of sight, sense of hearing, sense of smell, sense of taste, sense of touch. He goes then on to call the inner sense an inner perception. What is meant here is what has been called the common sense in Aristotle’s De anima. The common sense makes us aware that we have sensations at all and it links and categorizes different sensations in order to perceive things. Again one might add that one calls this a sense and not a sense organ, although one might ask the question what the sense organ for the inner sense is. The same holds true for sense (Gefühl) and reason. To complicate things further on, one might conclude that sensory perception (the faculty for perception or the activity of perceiving) and its accompanying sense organ seem to be the same for Gödel.

30A peculiarity of the German word “Sinn” (sense) might provide an explanation for this. According to the authorative German dictionary Duden, “Sinn” (sense) has two meanings among others: 1. the sensitive faculty for perceiving and feeling (that is located in the sense organs), 2. feeling for, understanding of something; an inner relation towards something.38 Gödel confounds the faculty, its organ and the function of it with each other. But although all three of them might be called by the same word in German that does not mean that they are not separable and indeed separated virtually always.

[303’] Remark Psychology: It may be the case that the sense is the organ of perception for concepts and states of affairs in general [nor with regard to myself, e.g. in the case of “I have the feeling, that this and this will happen”], but reason the perception of the word [this is abstract thinking]. Since it cannot be a mechanical thinking, it is either guided by the sense [but only very rough perceptions of the sense are required and then they will improve automatically] or it is the perception of the principle of concept formation. In any case humans have four organs of perception: 1. senses for perceiving bodies, 2. inner perception (and empathy), 3. sense, 4. reason.39

31This rests puzzling not only in this unpublished remark but also in published articles by Gödel. John P. Burgess discusses also the already cited passage in Gödel’s paper from 1947 “What is Cantor’s continuum problem?”:

But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception [...]40

32Burgess observes rightly that Gödel seems to leap from an intuition of set-theoretic objects to an intuition that set-theoretic axioms are true. Whereas it is understandable why perceiving “set-theoretic concepts (set and elementhood) would involve [...] perceiving or grasping that set-theoretic axioms are supposed to hold”; it is not clear why one should “think the same about perceiving set-theoretic objects (sets and classes)”. Burgess’s objection rests on an epistemological observation, namely that “we have not just “something like” a perception but an outright perception of the objects of astronomy, but when we look up at the starry heavens above, no astronomical axioms force themselves upon us as true”.41

33However, Gödel gave an implicit answer to this epistemological objection because according to his position we do neither have outright perceptions of mathematical objects nor of sense objects.42 When we perceive an object as an object the notion of “object” is for example not given by the senses. What Burgess points out with his objection is nevertheless something more than this. It covers also two different aspects. One is the question of causality in sense perception that is not involved in the perception of mathematical objects and the other one are the roles of objects for the axioms of a theory. We can unfortunately not continue this discussion but want nevertheless to indicate a route that one would have to pursue. On the one hand one has to consider uninhibitedly that knowledge does not only come about with objects towards which we are causally related.43 As for the other question one would have to track Wang’s suggestion that Gödel was willing to endorse other approaches to axioms than direct intuition although the last one was the most favourable method for him in mathematics.44 And in addition to this one should take Wang’s observation seriously that Gödel made a difference between perception and intuition.45

4. Rational intuition and the non-mechanistic human mind

The inexhaustibility of mathematics appears not only through [...] the incompleteness theorems, but also in [...] the unlimited series of axioms of infinity in set theory, which are analytic (and evident) in the sense that they only explicate the content of the general concept of set. That such a series may involve a great (perhaps even infinite) number of actually realizable independent rational perceptions is seen in the fact that the axioms are not evident from the beginning [...].46

34Statements by Gödel like this (that the unlimited series of axioms of infinity in set theory are in need of “perceptions”) are at the basis for Potter’s conclusion that the process of forming stronger and stronger axioms of infinity in set theory involves more than mere mechanical checking and that his conception of the mind as non-mechanistic was hence “intimately related in Gödel’s view to the incompletable nature of set theory”.47

35Potter explains this further on by saying that in the process of forming stronger and stronger axioms of infinity in set theory we assent to them because we recognise them as axioms of infinity and therefore true of the concept “set”.48 But how do we do this? It seems hardly possible to give a formalized account for it and hence in Gödel’s view a sign of the human mind in contrast to a machine.

36This is well known with respect to Gödel’s considerations about the reach of his incompleteness theorems, where he says:

Either mathematics is incomplete in this sense, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified. [...] The philosophical implications prima facie will be disjunctive too; however, under either alternative they are very decidedly opposed to materialistic philosophy. Namely, if the first alternative holds, this seems to imply that the working of the human mind cannot be reduced to the working of the brain, which to all appearances is a finite machine [...] On the other hand, the second alternative, where there exist absolutely undecideable 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.49

37With respect to the remark in question we are now able to make an educated guess: The fact that the human mind is able to make “all” again an object and to transcend beyond that (the unboundedness) might be an indication for Gödel that the human mind infinitely surpasses the powers of any finite machine.50 But as Gödel declared that only the solution of the intensional paradoxes would give a clear proof that mind is not a machine51 it cannot be the solution of the extensional paradoxes alone that indicates a superiority of the human mind over machines. However the fact that the human mind is incapable of formulating (or mechanizing) all of its mathematical intuitions52 but is nevertheless capable of detecting for example new axioms of set theory is a paradigm of the “incompletability” of mathematics and therefore for the superiority of the human mind.

5. Set Theory and God

38The last part of this essay will be dedicated to the attempt to disclose what set theoretical considerations might have to do with the notion of God. The last sentence of the quotation can be a starting point to interpret the Latin phrase to which Gödel alludes right after having mentioned the “unboundedness” of set theory.53 He inserts an arrow pointing to a Latin sentence that can be traced in parts to Thomas of Aquinas: “Est Deus sed potentia tantum”. Gödel has used Thomas of Aquinas Commentary on Aristotle’s De anima54 and “sed potentia tantum” is a phrase that is to be found frequently in Thomas’ Commentary on Aristotle’s De anima and, even more significantly, one can find it in contexts that are related to Gödel’s remark in question. Here are three examples:

On the other hand Anaxagoras thought it [i.e., the intellect] was simple and pure and detached from all bodily things. And therefore, precisely because the intellect, as he has just said, is not in act of understanding, but in potency only, and in potency to know everything, Aristotle argues that it cannot be compounded of bodily things as Empedocles thought, but must be separated from such things, as Anaxagoras thought. (Translation by Foster/Humphries, p. 206)
For it is not sight that receives visible forms, but the eye. It follows that the soul as a whole is not the place of forms, but only that part of it which lacks a bodily organ, i. e., the intellect; and even this part does not, as such, possess them actually, but potentially only”. (Translation by Foster/Humphries, pp. 207-208)
The mind, then, is called passive just in so far as it is in potency, somehow to intelligible objects, which are not actual in it until understood by it. It is like a sheet of paper on which no word is yet written. Such is the condition of the intellect as a potency, so long as it lacks actual knowledge of intelligible objects. (Translation by Foster/ Humphries, p. 217).55

39But this does not explain what Gödel means by referring from the “unboundedness” to “Est Deus sed potentia tantum”.

40The difficulty starts with translating the last sentence. If we translate the phrase as if it were written in classical Latin, one might translate it as: “God exists but only as <mental> capability”. But in medieval Latin “potentia” is quite often used as “possibility”, and on other occasions Gödel uses it in that sense (as well as Thomas of Aquinas does). In the case that “potentia” is used as it is in medieval Latin the verbatim translation would then be: “There is56 God but only as possibility”.57

41We have already hinted at one possible interpretation of this while discussing the phrase “the possibility to make “all” again an object and to transcend beyond that” that alludes to the transfinite iterated process of formation of sets of sets. As it is a process that cannot be carried out by a mortal being and not even by an immortal being (that is, although immortal, still temporal) it would have to be a God to conduct these operations, and Gödel might allude to this endless and timeless procedure.

42Another plausible interpretation can be found in Mark van Atten’s observation that a corollary of a Leibnizian proof of God’s existence (as Gödel’s proof is) “would be that a single, fixed universe of all sets V indeed exists, and hence that there is a privileged model for the axioms of set theory”.58

43It is very well known that Gödel proposed an ontological proof of God’s existence in the tradition of Leibniz’s proof.59 This proof includes intermediate steps that draw on the concept of possibility. Therefore it is worthwhile to have a look at it. In colloquial language the theorems that are relevant are the following:

44Theorem: If something is a deity then it holds necessarily that some deity existsThere: If some deity exists then it holds necessarily that some deity existsthere: if it is possible that some deity exists then it is also possible that it holds necessarily that some deity exists If it is possible that some deity exists, then it holds necessarily that some deity exists.60

45Following this sequence of the proof it is clear that it is essential for Gödel’s proof that it is possible that God exists but this requires that the set of God’s properties is a consistent set of positive properties. The properties must not be contradictory to each other. If they are contradictory it is not a consistent set. Gödel avoids this Leibnizian problem by presupposing its solution but it is not solved and it would have to be if Gödel’s proof is to be a proof for God’s existence.61

46The possibility to refer again to “everything” or “all” as an object and to transcend beyond that (the unboundedness) does not prove that God is possible because the consistency of the notion of God is not shown by it and therefore it is not shown that the notion does not contain contradictions (here antinomies) from which anything or nothing can be derived. In order to complete the proof Gödel would have had to demonstrate that the set of God’s positive properties is consistent but this he has not done.62

47The possibility to refer again to “everything” or “all” as an object might have been a sign for Gödel that God exists but it is not a proof for the consistency of the notion of God. Yet another aspect that comes to one’s mind in relation with the phrase in question is formulated in one of Richard Tieszen’s papers:

For an act to be directed in a particular way means that it is not directed in other ways. Cognitive acts are perspectival and we cannot take all possible perspectives on an object or a domain. We do not experience everything all at once. Perhaps only a god could do that, but we cannot. Our beliefs and other cognitive acts at any given time are, in other words, always about certain categories of objects. The mind always categorizes in this way.63

48At first one would not think, that this has anything to do with antinomies, proper classes and the inexhaustibility of mathematics, but as Tieszen points out in several of his publications, Gödel himself stressed the similarity between reason and the senses, because it shows that there exists a practically unlimited number of independent “perceptions” also of reason.64 Gödel writes about “rational perceptions” in the context of inexhaustibility in his paper “Is mathematics a syntax of language?” from 1953:

The “inexhaustibility” of mathematics makes the similarity between reason and the senses still closer, because it shows that there exists a practically unlimited number of independent perceptions also of this “sense”. Note that the inexhaustibility of mathematics appears, not only through foundational investigations, but also in the actual development of mathematics, e. g., in the unlimited series of axioms of infinity in set theory, [...]. That such series may involve a very great [...] number of actually realizable independent rational perceptions is seen from the fact that the axioms concerned are not evident from the beginning [...].65

49Therefore one might interpret the phrase in question also in the following way: When our cognitive acts are related to the domain of “all” we cannot take all possible perspectives of it all at once, only a God could do this.

50The subsequent phrase of the remark in question ( “But he says that if it had a nature this would prevent it from perceiving something different from this nature”) is a modification of Aristotle’s argument according to which a sense organ and its matching perceptive faculty only perceive what is specific for them. It holds that the intellect cannot have a nature or be corporeal because if this were the case it could only perceive what corresponds to its own specific nature.

51The last passage of the remark is also dedicated to the question of the incorporeal intellect. It presents another argument by Aristotle for the intellect being incorporeal in saying that “the absence of an organ is also proven by the circumstance that the intellect and only the intellect is the ‘place’ of the forms [species], because an organ would then also be a place of this <form>”. It says: If the intellect had an organ, the organ would also be a place of ideas or forms. As a place of ideas and forms it would also be the place of the idea of the organ as a place for ideas or forms itself. Therefore the intellect and its organ would have the same function and this function is not to perceive something in the world.

52The feature of self-referentiality that is implied in this argument might have attracted Gödel’s attention because it is a figure of thought that he uses in his incompleteness theorems and a few other places.