ON KURT GÖDEL'S PHILOSOPHY OF MATHEMATICS MARTIN K. SOLOMON

Martin K. Solomon Assistant Chair And Professor Dept. of Computer Science and Engineering Florida Atlantic University 777 W. Glades Rd. Boca Raton, FL 33431-6498 www. cse. fau. edu/~marty

Kurt Gödel (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic - January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher. One of the most significant logicians of all time, Gödel's work has had immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were attempting to use logic and set theory to understand the foundations of mathematics. Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age.

ABSTRACT

We characterize Gödel's philosophy of mathematics, as presented in his published works, with possible clarification and support provided by his posthumously published drafts, as being formulated by Gödel as an optimistic neo-Kantian epistemology superimposed on a Platonic metaphysics. We compare Gödel's philosophy of mathematics to Steiner's "epistemological structuralism." We present a conjecture that Gödel may have considered himself a kind of optimistic neo-Kantian Platonist even with respect to the physical world.

§1. Introduction

We show that Gödel's philosophy of mathematics, as presented in his published works, with possible clarification and support provided by his posthumously published drafts, can be considered as being formulated by Gödel as an optimistic neo-Kantian epistemology (obtained from Kant's epistemology regarding the physical world in terms of sensory appearances as distinct from things in themselves, not obtained from Kant's epistemology of mathematics as being synthetic a priori knowledge) superimposed on a platonic metaphysics. By Platonic metaphysics, we of course mean that abstract objects have an objective existence. By neo-Kantian we mean obtained from the Kantian epistemology with one important modification, namely, removing the doctrine of the unknowability of things in themselves.

Indeed, we will see in section 2.2 that Gödel thought that abstract things in themselves may be progressively knowable. Furthermore, it is pointed out in section 2.4 that he explicitly indicated that the knowability of physical things in themselves is possible through the progressive advancement of modern science. It is also pointed out in section 2.4 that Gödel didn't think that Kant would consider such a modification to be as significant as might some of Kant's followers.

In other words, in Gödel's well-known analogy of mathematical intuition to sense perception (see the passage from [16, p. 268] in section 2.1), he is clearly using (what he views as) a Kantian model of the sensory world of experience, optimistically modified. However, apparently missing from this Kantian model when Gödel applies it to abstract intuitions is the subjective a priori component (i. e., Gödel never mentions a mathematical intuition analog of anything akin to the a priori intuitions of space and time that Kant held for physical perception). Based on Gödel's letter to Greenberg and the passage from [12, p. 241] given is section  2.1, it appears that Gödel agreed with Kant in the existence of such a subjective a priori component for physical perception. Another bit of interesting information supplied by Gödel's letter to Greenberg is that Gödel in [16, p. 268] was specifically discussing set theoretical intuition as not necessarily providing "immediate Knowledge of the object concerned," whereas Gödel feels that geometric intuition ("in its purely mathematical aspect") does provide such immediate knowledge.

In section 2.4 we point out that Gödel apparently believed that mathematical intuitions are more "direct" than sense perceptions, presumably because the mathematical intuitions are abstract impressions of abstract objects, whereas we are advancing toward knowledge of physical things in themselves only by viewing the sensory world through the abstract lenses of modern physics. Also, in section 2.4, we conjecture that the seeming absence of the above mentioned a priori subjective component from Gödel's view of mathematical perception, as contrasted with his apparent agreement that such an a priori component exists for sense perception, could have contributed to his considering sense perception less direct than mathematical perception.

In section 3, we compare Gödel's philosophy of mathematics to (what we may call) the epistemological structuralist philosophy of mathematics that is briefly presented by Mark Steiner in his book "Mathematical Knowledge" [24]. We observe that, although there are some clear differences between the approaches of Gödel and Steiner, there are also some surprising similarities between their approaches. Specifically, both approaches center around the distinction between mathematical things in themselves and our intuitions regarding these things, both approaches consider intuitions to synthesize unities out of manifolds, both approaches distinguish between different kinds of mathematical intuition, and both approaches consider the content of mathematical statements to regard the relationship between abstract objects. Thus, we will see that Gödel's philosophy of mathematics has some elements in common with a certain kind of structuralist philosophy of mathematics.

One important difference between the two approaches is that Steiner is pessimistic (as is Kant with regard to physical things in themselves) in that he arguably considers abstract things in themselves to be unknowable. (Steiner states that "the only things of value to know about abstract objects are such relationships" [24, p. 134]; we argue in section 3.1 that "the only things of value to know" in his statement can be equivalently replaced with "the only things that can be known".) On the other hand, Gödel, as mentioned previously, is optimistic in that he allows for the convergence on knowledge of things in themselves. Therefore, Steiner's philosophy of mathematics can also be considered to be a variety of neo-Kantian Platonism "that is more Kantian" than Gödel's variety of neo-Kantian Platonism.

In our conclusion, we conjecture that Gödel may have considered himself to be a kind of optimistic neo-Kantian Platonist, not only for mathematics, but regarding the physical world as well.

§2. Gödel's philosophy of mathematics

2.1. Abstract reality and appearance.

Even in Gödel's 1944 "Russell's Mathematical Logic," along with postulating the existence of an abstract reality, there is a hint of Gödel's distinction between abstract reality and what our intuition may provide us with concerning that reality.

Classes and concepts may, however, also be conceived as real objects, namely classes as "pluralities of things" or as structures consisting of a plurality of things [Gödel is referring to the membership trees of Mirimanoff; see [1] for a good discussion of Mirimanoff's ideas] and concepts as the properties and relations of things existing independently of our definitions and constructions. [10, p. 128]

... the objects to be analyzed (e. g., the classes or proposition) soon for the most part turned into "logical fictions". Though perhaps this need not necessarily mean (according to the sense in which Russell uses this term) that these things do not exist, but only that we have no direct perception of them. [10 p. 121]

From these two passages it is clear that:

Gödel considers classes and concepts to be abstract objects that are real.

Gödel is suggesting (using Russell's name) an epistemology for abstract objects that is analogous to an epistemology of the physical world, in which we have a distinction between abstract things in themselves and our indirect intuitions concerning these abstract things in themselves.

In the 1964 version of "What is Cantor's Continuum Problem?" [16] Gödel elaborates in more detail his ideas concerning the above distinction, and he further clarifies his point of view in his letter to Marvin Jay Greenberg, which was sent to provide material for Greenberg's book [19]. To start, in [16, p. 259 n14], Gödel states that "set of x's" exists as a thing in itself, even though at the present time we do not have a clear grasp of the general concept of set (or "random sets," as Gödel puts it):

(footnote 14) The operation "set of x's" (where the variable x ranges over some given kind of objects) cannot be defined satisfactorily (at least not in the present state of knowledge), but can only be paraphrased by other expressions involving again the concept of set, such as: "multitude" ("combination", "part") is conceived of as something which exists in itself no matter whether we can define it in a finite number of words (so that random sets are not excluded).

Observe the hint of optimism in this footnote, in which Gödel implies that the gap between the set concept as a thing in itself and our intuitions concerning that concept, may be narrowed in the future. We can see from other remarks of Gödel in section 2.2, that this may be more than a cautious parenthetical, but actually may reflect an important optimistic component of Gödel's philosophy of mathematics.

Then, in the following intriguing (and much-cited) passage from [16], Gödel gives his most direct presentation of his epistemological ideas:

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 being 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. It should be noted that 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 immediately given. Only this something else here is not, or not primarily, the sensations. That something besides the sensations actually is immediately given follows (independently of mathematics) from the fact that even our ideas referring to physical objects contain constituents qualitatively different from sensations or mere combinations of sensations, e. g., the idea of object itself, whereas, on the other hand, by our thinking we cannot create only qualitatively new elements, but only reproduce and combine those that are given. Evidently the "given" underlying mathematics is closely related to the abstract elements contained in our empirical ideas. 40 It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective, as Kant asserted. Rather they, too, may represent an aspect of objective reality, but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality. [16, p. 268]

40 Note that there is a close relationship between the concept of set explained in footnote 14 [16, p. 259 n14] and the categories of pure understanding in Kant's sense. Namely, the function of both is "synthesis", i. e., the generating of unities out of manifolds (e. g., in Kant, of the idea of one object out of its various aspects). [16, p. 268 n40]

Here Gödel identifies what mathematical intuition provides to us (actually, according to his letter to Greenberg, Gödel means specifically set theoretic intuition) as being something that synthesizes a unity out of a manifold (data of the second kind). Gödel refers to such "data of the second kind" as "abstract impressions" in [25], as we shall see in section 2.2. Data of the second kind in mathematics provides abstract impressions of abstract objects (objects which themselves, by the above footnote 14, also synthesize unities out of manifolds).

Also, rather surprisingly, as part of his argument that data of the second kind is also involved in our cognition of the physical world, Gödel characterizes thinking in a manner that Potter calls "trivial" [20, p. 9] (is it also mechanical?). Given Gödel's well-known view that mind is more powerful than machine, if thinking is mechanical then the intuition "input facility" is what gives the mind its power. We will reexamine Gödel's "trivial" concept of mind in section 2.3, when we consider the implications of Gödel's conjecture concerning the existence of an abstract sense organ.

Gödel's letter to Marvin Jay Greenberg further clarifies the above passages from [16]. Greenberg mailed Gödel, asking Gödel's permission to quote as follows from that article:

I don't see any reason why we should have less confidence in this kind of perception, i. e., in mathematical intuition, than is sense perception, which induces us to build up physical theories and to expect that future sense perceptions will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in the future. The set theoretical paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics. ... Evidently the "given" underlying mathematics is closely related to the abstract elements contained in our empirical ideas. It by no means follows, however, that the data of this second kind [mathematical intuitions], because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective, as Kant asserted. Rather, they, too, may represent an aspect of objective reality. But as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality. [19, p. 305]

Gödel responded to Greenberg as follows:

1. I also have to request that you give this reference in full because you omit important parts of my exposition, and, moreover, the passage you quote does not occur in my original paper, but only in the supplement to the second edition. [18, pp. 453-454]

From Gödel's letter we can see that:

Gödel distinguishes between different kinds of mathematical intuition, specifically geometric intuition (which is mathematical intuition in a restricted sense), and set theoretical intuition (which is presumably mathematical intuition in the inclusive sense, as Wang [26, p. 184] indicates that Gödel considers mathematics to be the study of pure sets, and Gödel indicates in [14, p. 305] that he feels that all of mathematics is reducible to abstract set theory).

Unlike our set theoretical intuition in its current state, Gödel considers our Euclidean space intuition (in its mathematical aspect) to penetrate the realm of abstract objects in themselves (presumably because Euclidean geometry is complete, indeed categorical) so as to be "perfectly correct" as it "represents correctly" Euclidean space structure. Thus, certain abstract objects (i. e., certain classes and concepts in themselves) are knowable to us, whereas others, such as, presumably, (the isomorphism type of) the standard model of set theory, are currently only indirectly and vaguely known to us. We will discuss Gödel's optimistic view of mathematical intuition further in section 2.2.

Gödel asserts that physical space is an a priori intuition (presumably in the Kantian sense of being subjective because it's an aspect of the structure of our cognitive apparatus). This as contrasted with "the purely mathematical aspect of our Euclidean space intuition" which is "perfectly correct" because "it represents correctly a certain structure existing in the realm of mathematical objects." Thus, mathematical truths are not true a priori because of the structure of our cognitive apparatus, but are objectively true contingent on the way the abstract world actually is. The following passage from [12, p. 241] also suggests agreement between Gödel and Kant on the a priori nature of spatial intuition:

In the case of geometry, e. g., the fact that the physical bodies surrounding us move by the laws of a non-Euclidean geometry does not exclude in the least that we should have a Euclidean "form of sense perception", i. e., that we should possess an a priori representation of Euclidean space and be able to form images of outer objects only by projecting our sensations on this representation of space, so that, even if we were born in some strongly non-Euclidean world, we would nevertheless invariably imagine space to be Euclidean, but material objects to change their size and shape in a certain regular manner, when they move with respect to us or we with respect to them.

2.2 Gödel's optimistic epistemology for abstract objects.

We have already noted in section 2.1 the optimistic tone in Gödel's letter to Greenberg [18] and a footnote in the 1964 version of "What is Cantor's Continuum Problem?" [16, p. 259 n14].

In his 1946 "Remarks Before the Princeton Bicentennial Conference," [11] Gödel expressed optimism concerning the possibility of discovering, in the future, a concept of demonstrability (with a nonmechanical, but humanly generated axiom set) that is complete for mathematics (i. e., set theory), and hence absolute, just as Turing discovered the absolute concept of computability [11, p. 151].

Also, Gödel expresses in [25, pp. 324-325] (where he is summarizing from his Gibbs lecture) the view that there do not exist number theoretical propositions that are undecidable for the human mind (i. e., that are absolutely undecidable). From the argument he gives there, Gödel also seems to be rejecting the existence of any mathematical truths that are absolutely undecidable. In "The Modern Development of the Foundations of Mathematics in the Light of Philosophy," [15], which apparently is a draft of a lecture that Gödel planned to deliver before the American Philosophical Society but never delivered, he states:

It is not at all excluded by the negative results mentioned earlier [his incompleteness theorems] that nevertheless every clearly posed mathematical yes-or-no question is solvable in this way [by the "intuitive grasping of even newer axioms"]. [15, p. 385]

In [25, pp. 84-85] Gödel further elaborates on his optimistic epistemology for abstract objects. In particular, he describes how we can begin with an abstract impression (called data of the second kind in [16]) of an abstract concept (in itself) that is vague, and we can end up with the sharp concept that faithfully represents the abstract concept in itself:

Gödel points out that the precise notion of mechanical procedures is brought out clearly by Turing machines ... The resulting definition of the concept of mechanical by the sharp concept of 'performable by a Turing machine' is both correct and unique. ... Gödel emphasizes that there is at least one highly interesting concept which is made precise by the unqualified notion of a Turing machine. Namely a formal system is nothing but a mechanical procedure for producing theorems. ... In fact, the concept of formal systems was not clear at all in 1931. Otherwise Gödel would have then proved his incompleteness results in a more general form. ... 'If we begin with a vague intuitive concept, how can we find a sharp concept to correspond to it faithfully?' The answer Gödel gives is that the sharp concept is there all along, only we did not perceive it clearly at first. This is similar to our perception of an animal first far away and then nearby. We had not perceived the sharp concept of mechanical procedures sharply before Turing, who brought us the right perspective. And then we do perceive clearly the sharp concept. There are more similarities than differences between sense perceptions and the perceptions of concepts. In fact, physical objects are perceived more indirectly than concepts. The analog of perceiving sense objects from different angles is the perception of different logically equivalent concepts. If there is nothing sharp to begin with, it is hard to understand how, in many cases, a vague concept can uniquely determine a sharp one without even the slightest freedom of choice. ... Gödel conjectures that some physical organ is necessary to make the handling of abstract impressions (as opposed to sense impressions) possible, because we have some weakness in the handling of abstract impressions which is remedied by viewing them in comparison with or on the occasion of sense impressions. Such a sensory organ must be closely related to the neural center for language.

2.3 Digression on Gödel's abstract sense organ

Please observe that Gödel's thoughts in the above passage concerning a "necessary" abstract impression physical organ can be viewed as being motivated by a desire to strongly counter the "Kantian assertion" stated in [16, p. 268]:

It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective, as Kant asserted.

Similarly, the existence of such a physical organ could be used to counter Benacerraf's criticism of Platonism that is based on the causal account of knowledge [2].

Finally, we might comment on the apparent inconsistency of Gödel's belief in the existence of such an abstract sense organ with his view that the mind is not mechanical (this view is expressed, in particular, in [25, pp.
324-326] and [17, p. 306]). We express this apparent inconsistency by synthesizing Gödel's views in the following argument:

Gödel appears to have discussed only two functions of the mind with regard to mathematical activity: thinking and intuition (see [20, p. 340] for a discussion of these two functions from the Gödelian point of view).

Gödel feels that "thinking cannot create any qualitatively new elements but only reproduce and combine those given" [16, p. 268]. In fact, as we indicated previously, Potter [20, p. 340] refers to thinking from this point of view as being "trivial." Whether or not it is trivial, thinking from Gödel's point of view is clearly mechanical.

Intuition is implemented by the abstract sense organ, which Gödel says is "closely related to the neural center for language."

Since the neural center of language is part of the brain, and Gödel stated in [25, p. 326] that "very likely" the brain is mechanical, the abstract sense organ, according to Gödel's view, is also very likely to be mechanical. Note also that Gödel states more strongly in [14, p. 311] that the brain "is a finite machine with a finite number of parts, namely, the neurons and their connections."

Therefore, by 1, 2, and 4 above, it follows from Gödel's beliefs that the mind is (or is very likely to be) mechanical.

However, there are several possible ways out of the apparent inconsistency of the above argument with Gödel's non-mechanical view of mind:

Gödel may have felt that the mind involves functions with regard to mathematical activity other than thinking and intuition, although he apparently never communicated this feeling.

As "trivial" as Gödel's concept of thinking may appear, he still may have felt that thinking is not mechanical.

Gödel may have intended his remark about thinking only to pertain to thinking regarding the processing of sensory input (which is the context in which he presents it), and not to thinking in general. In fact, in his "Some Remarks on the Undecidability Results" as presented in [17, p. 306], Gödel appears to divide the credit between thinking and intuition for the mind developing toward an infinite number of states (and therefore, in his view, being non-mechanical):

What Turing disregards completely is the fact that mind, in its use, is not static, but constantly developing, i. e., that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. ... Therefore, although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing's number of distinguishable states of mind) may converge toward infinity ...

However, the version of this passage that appears in [25, pp. 325-326], a somewhat later version according to Wang [27, pp. 123-124], is not explicit about there being two ways for producing an infinite number of mental states.

Gödel may have believed that the abstract sense organ is necessary but not sufficient for the implementation of intuition, and that the implementation must also require some nonmechanical feature of the mind. For example, Wang reports that Gödel told him in 1971 that:

Even if the finite brain cannot store an infinite amount of information, the spirit [mind] may be able to. The brain is a computing machine [situated in the special manner of being] connected with a spirit. If the brain is taken as physical and as a digital computer, from quantum mechanics there are then only a finite number of states. Only by connecting it to a spirit might it work some other way. [27, p. 127]

This infinite amount of memory would be needed to store the infinite number of intuitions that "a mind of an unlimited life span" would generate. And Gödel told Wang that "a mind of an unlimited life span" is what Gödel meant by mind, and is "close to the real situation" where "people constantly introduce new axioms" [27, p. 121].

However, although such infinite memory might distinguish the mind from the brain, it would not distinguish the mind from a Turing machine (which also has potentially infinite memory), unless the mind could store an infinite amount of information, such as a burst of an infinite number of axioms, at once.

We close this section with passages from two letters provided in Gödel's Collected Works in which Gödel succinctly presents his view that the intuition of abstract objects provides the mind the ability to surpass the machine.

Here is one formulation of the philosophical meaning of my result, which I have given once in answer to an inquiry: The few immediately evident axioms from which all of contemporary mathematics can be derived do not suffice for answering all Diophantine yes or no questions of a certain well-defined simple kind. Rather, for answering all these questions, infinitely many new axioms are necessary, whose truth can (if at all) be apprehended only by constantly renewed appeals to a mathematical intuition, which is actualized in the course of the development of mathematics. Such an intuition appears, e. g., in the axioms of infinity of set theory. [8, p. 330, letter to George A. Brutian]

2.4 Comparison with Gödel's optimistic epistemology for physical reality.

In Gödel's drafts of [13] he apparently feels that Kant's "pessimistic" view that physical things in themselves are unknowable should be modified so as to bring Kant's epistemology into agreement with modern science:

A real contradiction between relativity theory and Kantian philosophy seems to me to exist only in one point, namely, as to Kant's opinion that natural science in the description it gives of the world must necessarily retain the forms of our sense perception and can do nothing else but set up relations between appearances within this frame.

This view of Kant has doubtless its source in his conviction of the unknowability (at least by theoretical reason) of the things in themselves, and at this point, it seems to me, Kant should be modified, if one wants to establish agreement between his doctrines and modern physics; i. e., it should be assumed that it is possible for scientific knowledge, at least partially and step by step, to go beyond the appearances and approach the world of things. The abandoning of that "natural" picture of the world which Kant calls the world of "appearance" is exactly the main characteristic distinguishing modern physics from Newtonian physics. Newtonian physics ... is only a refinement, but not a correction, of this picture of the world; modern physics however has an entirely different character. This is seen most clearly from the distinction which has developed between "laboratory language" and the theory, whereas Newtonian physics can be completely expressed in a refined laboratory. [12, p. 244]

... one may find a description in more detail of these steps or "levels of objectivation", each of which is obtained from the preceding one by the elimination of certain subjective elements. The "natural" world picture, i. e., Kant's world of appearances itself, also must of course be considered as one such level, in which a great many subjective elements of the "world of sensations" are already eliminated. Unfortunately whenever this fruitful viewpoint of a distinction between subjective and objective elements in our knowledge (which is so impressively suggested by Kant's comparison with the Copernican system) appears in epistemology, there is at once a tendency to exaggerate it into a boundless subjectivism, whereby its effect is annulled. Kant's thesis of the unknowability of the things in themselves is one example; and another one is the prejudice that the positivistic interpretation of quantum mechanics, the only one known at present, must necessarily be the final stage of the theory. [12, p. 240 n24]

Note how the last passage also nicely characterizes Gödel's epistemology of mathematics where we distinguish intuitions of abstract objects from the intuitions of those objects in themselves, but consider our intuitions to converge on those objects in themselves. The applicability of this passage to Gödel's epistemology of mathematics is not surprising, since, as we have seen in the preceding sections, that in Gödel's well-known analogy of mathematical intuition to sensory perception is clearly based on (what he views as) the Kantian model for the sensory world of experience that is based on the contrast between that world of experience with the world of things in themselves, after having optimistically modified this model.

In the following passage Gödel expresses his belief that the unknowability of things in themselves is more a tenet of Kant's followers than Kant himself:

We conclude this section by observing that in several places Gödel expressed the view that abstract objects are perceived more directly than physical objects. This is expressed in the [25, p. 85] passage, which was given in section 2.2, as "In fact physical objects are perceived more indirectly than concepts." It is also expressed, somewhat more demurely, in the passage from [21, p. 217], which is given in section 2.5, as "we perceive mathematical objects and facts just as immediately as physical objects, or perhaps more so." This is presumably the case because mathematical (or conceptual) intuitions are abstract impressions that converge on the abstract objects in themselves, whereas the advance toward knowledge of physical things in themselves only by viewing the sensory world through the abstract lenses of modern physics. Also, we conjecture that the seeming absence of an a priori subjective component from Gödel's view of mathematical intuition (at least he never mentions such a component), as contrasted with his apparent agreement with Kant that such an a priori component exists for sense perception (see section 2.1), could have contributed to Gödel's judgement that sense perception is less direct than mathematical intuition.

2.5 Intuition of abstract objects and mathematical facts.

Throughout Gödel's published philosophical writings, it appears to us that he considers mathematical intuition as providing both objects as well as facts (truths) to the mind, but he is not very explicit about this. However, in [21, p. 217], Gödel explicitly states this to be the case:

There exist experiences, namely those of mathematical intuition, in which we perceive mathematical objects and facts just as immediately as physical objects, or perhaps more so. It is arbitrary to consider "this is red" an immediate datum, but not so to consider modus ponens or complete induction (or perhaps some simpler propositions from which the latter follows). For the difference, as far as it is relevant here, consists solely in the fact that in the first case a relationship between a concept and a particular object is perceived, while in the second case it is a relationship between concepts.

Note also this passage suggests that the content of a mathematical truth is the relationship between concepts that it expresses. This is also indicated in a different passage from the same draft: ... the reasoning which leads to the conclusion that no mathematical facts exist is nothing but a petitio principii, i. e., "fact" from the beginning is identified with "empirical fact", i. e. "fact in the world of sense perception." Platonists should agree that mathematics has no content of this kind. For its content, according to Platonism, does not consist in facts perceptible with the senses, but in relations between concepts or other ideal objects. [21, p. 184]

and is indicated again in the following passage from [14, p. 320]:

Therefore a mathematical proposition, although it does not say anything about space-time reality, still may have a very sound objective content, insofar as it says something about relations of concepts.

We will summarize the preceding observations on Gödel's philosophy of mathematics in the following diagram. First we remark that in the diagram, we place the "mathematical truth" node on a branch separate from the "data of the second kind" node because for simplicity of presentation we consider these to be separate kinds of intuitions, as opposed to treating the intuition of a mathematical truth as being a special case of the intuition of data of the second kind. Although this separation makes sense to us, it may or may not coincide with the way Gödel felt. (In fact, we will see in section 3.2 that Wang [28, pp. 226-227] appears to have a contrary impression.)

2.6 Gödel diagram

We will summarize the preceding observations on Gödel's philosophy of mathematics in the following diagram. First we remark that in the diagram, we place the "mathematical truth" node on a branch separate from the "data of the second kind" node because for simplicity of presentation we consider these to be separate kinds of intuitions, as opposed to treating the intuition of a mathematical truth as being a special case of the intuition of data of the second kind. Although this separation makes sense to us, it may or may not coincide with the way Gödel felt. (In fact, we will see in section 3.2 that Wang [28, pp. 226-227] appears to have a contrary impression.)

2.7 A few remarks on the Gödelian view of concepts

What Gödel means by "concept" is not completely clear. In [10, p. 128], Gödel defines concepts as the properties and relations of things. Charles Parsons in his note to [10] given in [6, p. 108] observes that:

Gödel evidently means objects signified in some way by predicates.

Wang remarks in [26, p. 189] that:

Even though G seems to speak of mathematical and conceptual realism interchangeably, the obvious connotations are different. He takes mathematics as the study of (pure) sets and logic as a more inclusive domain that studies (pure) concepts. This already suggests that conceptual realism is a stronger position than mathematical realism, except perhaps in the metaphorical sense of having been suggested by his mathematical experience but covering much more. At any rate, G's conceptual realism goes far beyond mathematics.

In fact, Wang links Gödel's "optimistic" view of philosophy as an exact science to Gödel's conceptual realism.

Gödel believes that the development of philosophy into an exact science is not only possible, but will take place within the next one hundred years or even sooner [25, p. 85]. In [26, p. 192],

Wang states:

Philosophy as an exact theory may be viewed as a special application of G's conceptual realism. It is to bring about the right perspective so as to see clearly the basic metaphysical concepts.

According to this view, the diagram given in section 2.5 can be taken to represent not only Gödel's philosophy of mathematics, but also a more general aspect of his approach to philosophy.

§3. Epistemological structuralism and neo-Kantian Platonism

We now discuss a structuralist philosophy of mathematics that is briefly presented by Mark Steiner in his book Mathematical Knowledge [24]. In particular, we will argue that, although there are clear differences between Gödel and Steiner's views, there are also some surprising similarities. We shall phrase the comparison between Gödel's and Steiner's view by calling them different kinds of epistemological structuralism. The comparison with Steiner's views will reveal that Gödel's philosophy of mathematics bears some elements in common with certain (at least Steiner's) structuralist philosophies of mathematics that postulate the existence of abstract objects. The comparison will also indicate such a structuralist philosophy of mathematics can be considered a variety of neo-Kantian Platonism that is "more Kantian" than Gödel's variety of neo-Kantian Platonism.

3.1 Steiner's view

Steiner's view is given by the following statement:

We might view with Benacerraf (1965) the point of mathematics as the study of "structures," rather than individual mathematical objects. The point, however, is epistemological rather than ontological - we accept mathematical objects, contra Benacerraf, but we agree that the only things to know about these objects of any value are their relationships with other things. (This is the mark of abstract objects.) Intuition becomes then the intuition of [abstract] structures rather than the intuition either of truths or of individual objects. This new point of view has the virtue of conforming to the way mathematicians speak of intuition. One speaks of set-theoretic intuition, analytic intuition, geometric intuition, topological intuition, and so forth. Intuition in one branch of mathematics, furthermore, is alleged not to go with intuition in another. [24, p. 134]

To clarify Steiner's passage, let us first consider the view of Benacerraf that Steiner references. Benacerraf argues in [3] that "there are no such things as numbers" [3, p. 294]. However, in that paper he doesn't make the same claim about sets or any other mathematical object. (However, Steiner makes his "epistemological" point about all mathematical objects.) Benacerraf makes his "ontological" argument as follows:

The pointlessness of trying to determine which objects the numbers are thus derives directly from the pointlessness of asking the question of any individual member. For arithmetical purposes the properties of numbers which do not stem from the relations they bear to one another in virtue of being arranged in a progression are of no consequence whatsoever. But it would be only these properties that would single out a number as this object or that. Therefore, numbers are not objects at all, because in giving the properties (that is, necessary and sufficient) of numbers you merely characterize an abstract structure - and the distinction lies in the fact that the "elements" of the structure have no properties other than those relating them to other "elements" of the same structure. If we identify an abstract structure with a system of relations (in intension, of course, or else with the set of all relations in extension isomorphic to a given system of relations), we get arithmetic elaborating the properties of the "less-than" relation, or of all systems of objects (that is, concrete structures) exhibiting that abstract structure. [3, p. 291]

Now back to Steiner.

We see that Steiner is making a sharp distinction between the mathematical appearances, namely, relationships, which mathematical intuition provides us with, and the mathematical objects which participate in those relationships. We have intuition of the abstract structures that encapsulate those relationships, but not of the objects themselves. In fact, we claim that Steiner's assertion that "the only things to know about these objects of any value are their relationships with other things (i. e., their appearances) is equivalent to "the only things it is possible to know about these objects are their relationships with other things." For the things that are of value about these objects are the things about these objects that are reflected in our world of experience, which are the only things that it is possible for us to know. Thus, Steiner's inclusion "of any value" can be viewed as epistemological sour grapes. Recall that Benacerraf identifies an abstract structure with either the set of all relational systems in extension that are isomorphic to a given system of relations, or with a system of relations in intension. Since Steiner considers sets as being mathematical objects, and hence beyond intuition, he would presumably opt for the latter choice. But with either choice, the abstract structure synthesizes a unity out of a manifold.

We present the following diagram as representing Steiner's view:

3.2 Comparison with Gödel's view

Although, both Gödel and Steiner have their intuitions synthesizing unities out of manifolds, Steiner's manifolds appear to be specifically isomorphism types of relational systems (with the unity being the underlying intensional form of such an isomorphism type), whereas Gödel presumably has these, but also many more kinds of manifolds in mind. Steiner has us intuiting structures that encapsulate relationships between abstract objects, but not abstract objects or mathematical truths, whereas Gödel seems to have us intuiting mathematical truths, the contents of which consists of relationships between mathematical objects (Steiner would obviously agree with Gödel that mathematical truths consist of such content), as well as having us intuit abstract impressions that converge on abstract objects. When Gödel discusses the intuiting of structure in his letter to Greenberg (that was given in section 2.1), he means structure to be a mathematical object (such as, presumably, the extension notion of structure that Benacerraf mentioned).

We remark that Wang, in his posthumously published [28] expressed an impression that is contrary to the evidence (given in section 2.4):

In fact, my impression is that mathematical intuition for him is primarily our intuition that certain propositions are true - such as modus ponens, mathematical induction, 4 is an even number, 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. [28, pp. 226-227]

If this is indeed the case, then given Gödel's view that the content of mathematical propositions consists of relationships between abstract objects, Gödel's view would be a lot closer to Steiner's than we think it is: Gödel would have us intuiting only mathematical truths, whose content consists of relationships between abstract objects, whereas Steiner has us intuiting only abstract structures, which encapsulate relationships between abstract objects.

Although both Gödel and Steiner distinguish between abstract appearance and abstract thing in itself, with Steiner the gap is unbridgeable, there is a complete separation, whereas with Gödel the appearance can converge on the object. In this way, Steiner is more Kantian than Gödel (see the passage from [7, pp. 257-258 n27] that was given in section 2.4, and our comment on it).

Observe that both Gödel and Steiner are in agreement that there are different kinds of mathematical intuition, e. g., set theoretical intuition and geometric intuition (see Gödel's letter to Greenberg that was given in section 2.1).

We conclude this section by formulating the following definitions of epistemological structuralism, optimistic epistemological structuralism, and pessimistic epistemological structuralism as constituting a vehicle of comparison between Gödel's and Steiner's views. We consider both Gödel and Steiner to be epistemological structuralists, as defined here, with Gödel being optimistic, and Steiner being pessimistic.

Epistemological structuralism distinguishes between abstract things in themselves, which it postulates to exist, and our intuitions concerning these things in themselves. An important property of our intuitions is that they synthesize unities out of manifolds.

Optimistic epistemological structuralism is a form of epistemological structuralism where our intuitions of abstract things in themselves vary as to their degree of sharpness, and sharp intuitions can provide direct knowledge of these abstract things in themselves.

Pessimistic epistemological structuralism is a form of epistemological structuralism where it is postulated that the gap between abstract things in themselves and our intuitions concerning these things in themselves
(we do not have direct intuitions of these things in themselves) is unbridgeable and consequently the abstract things in themselves are unknowable to us (they are completely outside our world of experience). These intuitions constitute the underlying form that synthesizes a unity out of a manifold of concrete relational systems.

§4. Conclusion

In this paper we observed that Gödel appeared to have formulated his philosophy of mathematics as a neo-Kantian epistemology superimposed on a Platonic metaphysics, i. e., by basing an epistemology for abstract objects, which are postulated to exist, on (what Gödel considers to be) Kant's epistemology for the physical world. Gödel modified Kant's epistemology for the physical world in an "optimistic" way, so as to allow for knowledge of physical things in themselves, and Gödel appeared to have his epistemology for abstract objects inherit this optimistic feature.

Gödel appears to have taken perception of abstract objects through intuition to be more direct than the data that sense perception provides of the physical world. This may reflect a greater trust that Gödel may have had in abstractions than in time - a time that he argued in [13] to be illusory. Gödel seems to have felt that Kant was misled into holding the unknowability of physical things in themselves, because the science of Kant's day was insufficiently abstract to progress beyond the subjective world of appearances (see section 2.4). Thus, the science of Kant's day couldn't progress beyond the shadows in the cave.

In fact, it can be conjectured that Gödel's non-materialism was much stronger than he explicitly expressed in [14] and [15] (as strong as the non-materialism of Parmenides and McTaggert, whom Gödel sympathetically cites in [13]). Possibly, one can take the sum of Gödel's published philosophical writings as providing a Platonic metaphysics (penetrable by abstract intuition), not just for mathematics, but for the physical world as well: the physical world is an imperfect reflection of a more perfect abstract world, which science enables us to penetrate that underlying abstract world as it becomes increasingly abstract. Perhaps the sum of Gödel's published philosophy presents him as viewing himself as an optimistic neo-Kantian Platonist, not just for mathematics, but with regard to his more general world view as well.

REFERENCES

[1] Aczel, Peter, Non-Well-Founded Sets, CLSI Lecture Notes, No. 14, CLSI, Stanford, CA, 1988.

[2] Benacerraf, Paul, "Mathematical Truth," Journal of Philosophy, Vol. 70 (1973), reprinted in [4], pp. 403-420.

[3] Benacerraf, Paul, "What Numbers Could Not Be," Philosophical Review, Vol. 74 (1965), reprinted in [4], pp. 272-294.

[4] Benacerraf, Paul and Putnam, Hilary, Philosophy of Mathematics: Selected Readings, Second Edition, Cambridge University Press, New York, NY, 1983.

[5] Davis, Martin (ed.), The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Raven Press, Hewlett, NY, 1965.

[6] Feferman, Solomon, et al (eds.), Kurt Gödel Collected Works, Vol. II: Publications 1938-1974, Oxford University Press, NY, 1990.

[7] Feferman, Solomon, et al (eds.), Kurt Gödel Collected Works, Vol. III: Unpublished Essays and Lectures, Oxford University Press, NY, 1995.

[8] Feferman, Solomon, et al (eds.), Kurt Gödel Collected Works, Vol. IV: Correspondence A-G, Oxford University Press, Oxford, 2003.

[9] Feferman, Solomon, et al (eds.), Kurt Gödel Collected Works, Vol. V: Correspondence H-Z, Oxford University Press, Oxford, 2003.

[10] Gödel, Kurt, "Russell's Mathematical Logic," 1944, in [22, pp. 123-153], reprinted in [6, pp. 119-141].

[11] Gödel, Kurt, "Remarks Before the Princeton Bicentennial Conference," 1946, in [5, pp. 84-88], reprinted in [6, pp. 150-153].

[12] Gödel, Kurt, "Some Observations About the Relationship Between Theory of Relativity and Kantian Philosophy," draft B2 in [7, pp. 230-259], 1946/9.

[13] Gödel, Kurt, "A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy," in [23, pp. 555-562], 1949, reprinted in [6, pp. 202-207].

[14] Gödel, Kurt, "Some Basic Theorems on the Foundations of Mathematics and their Implications," 25th Josiah Willard Gibbs Lecture, American Mathematics Society, Brown University, 1951, in [7, pp. 304-323].

[15] Gödel, Kurt, "The Modern Development of the Foundations of Mathematics in the Light of Philosophy," in [7, pp. 374-387], 1961.

[16] Gödel, Kurt, "What is Cantor's Continuum Problem?", in [4, pp. 258-273], reprinted in [6, pp. 254-270], 1964.

[17] Gödel, Kurt, "Some Remarks on the Undecidability Results" in [6, pp. 305-306], 1972.

[18] Gödel, Kurt, Letter to Greenberg, in [8, pp. 453-454], 1973.

[19] Greenberg, Marvin Jay, Euclidean and Non-Euclidean Geometries: Development and History, 3rd Edition, W. H. Freeman and Company, New York, 1993.

[20] Potter, Michael, "Was Gödel a Gödelian Platonist?" Philosophia Mathematica (3) Vol. 9(2001), pp. 331-346.

[21] Rodriguez-Consuegra, Francisco A. (ed.), Kurt Gödel: Unpublished Philosophical Essays, Birkhäuser Verlag, Boston, MA, 1995.

[22] Schilpp, Paul A. (ed.), The Philosophy of Bertrand Russell, Library of Living Philosophers, Vol. 5, Northwestern University Press, Evanston, IL, 1944.

[23] Schilpp, Paul A. (ed.), Albert Einstein, Philosopher-Scientist, Library of Living Philosophers, Vol. 7, Northwestern University, Evanston, IL, 1949.

[24] Steiner, Mark, Mathematical Knowledge, Cornell University Press, Ithaca, NY, 1975.

[25] Wang, Hao, From Mathematics to Philosophy, Humanities Press, New York, NY, 1974.

[26] Wang, Hao, Reflections on Kurt Gödel, MIT Press, Cambridge, MA, 1987.

[27] Wang, Hao, "Physicalism and Algorithmism: Can Machines Think," Philosophia Mathematica (3) Vol. 1 (1993), pp. 97-138.

[28] Wang, Hao, MIT Press, Cambridge, MA, 1996.

EDITOR'S NOTE

MORE OF PROFESSOR SOLOMON'S EXCELLENT PAPERS CAN BE FOUND ON THE INTERNET IN PDF FORMAT
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1. Abstract Complexity Theory and the Mind-Machine Problem
2. A Connection between Blum Speedable Sets and Gödel's Speed-Up Theorem
3. Some Results on Measure Independent Gödel Speed-Ups
4. Measure Independent Gödel Speed-Ups & the Relative Difficulty of Recognizing Sets
5. Relativized Gödel Speed-Ups & the Degree of Succinctness of Representations