Gödel''''s Philosophical Views

Kurt Gödel 

Stanford Encyclopedia of Philosophy)

https://plato./entries/goedel/

3. Gödel's Philosophical Views

Gödel's philosophical views can be broadly characterized by two points of focus, or, in modern parlance, commitments. These are: realism, namely the belief that mathematics is a descriptive science in the way that the empirical sciences are. The second commitment is to a form of Leibnizian rationalism in philosophy; and in fact Gödel's principal philosophical influences, in this regard particularly but also many others, were Leibniz, Kant and Husserl. (For further discussion of how these philosophers influenced Gödel, see van Atten and Kennedy 2003.)

The terms “Gödel's realism” and “Gödel's rationalism” must be prefaced with a disclaimer: there is no single view one could associate with each of these terms. Gödel's realism underwent a complex development over time, in both the nature of its ontological claims as well as in Gödel's level of commitment to those claims. Similarly Gödel's rationalism underwent a complex development over time, from a tentative version of it at the beginning, to what was adjudged to be a fairly strong version of it in the 1950's. Around 1959 and for some time afterward Gödel fused his rationalistic program of developing exact philosophy with the phenomenological method as developed by Husserl.

We examine these two strains of Gödel's thinking below:

3.1 Gödel's Rationalism

Gödel's rationalism has its roots in the Leibnizian thought that the world, not that which we immanently experience but that which itself gives rise to immanent experience, is perfect and beautiful, and therefore rational and ordered. Gödel's justification of this belief rests partly on an inductive generalization from the perfection and beauty of mathematics:

Rationalism is connected with Platonism because it is directed to the conceptual aspect rather than toward the (real) world. One uses inductive evidence…Mathematics has a form of perfection…We may expect that the conceptual world is perfect, and, furthermore, that objective reality is beautiful, good, and perfect. (Wang 1996, 9.4.18)

Our total reality and total experience are beautiful and meaningful—this is also a Leibnizian thought. We should judge reality by the little which we truly know of it. Since that part which conceptually we know fully turns out to be so beautiful, the real world of which we know so little should also be beautiful. (9.4.20)

Although the roots of Gödel's belief in rationalism are metaphysical in nature, his long-standing aspirations in that domain had always been practical ones. Namely, to develop exact methods in philosophy; to transform it into an exact science, or strenge Wissenschaft, to use Husserl's term.

What this means in practice is taking the strictest view possible of what constitutes the dialectical grounds for the acceptance of an assertion; put another way, a level of rigor is aspired to in philosophical arguments approaching that which is found in mathematical proofs. A formulation of the view—one which is somewhat phenomenologically colored (see below)—can be found in a document in the Gödel Nachlass. This is a fourteen item list Gödel drew up in about 1960, entitled “My Philosophical Viewpoint.” Two items on the list are relevant here:

3. There are systematic methods for the solution of all problems (also art, etc.).

13. There is a scientific (exact) philosophy and theology, which deals with concepts of the highest abstractness; and this is also most highly fruitful for science.

(The list was transcribed by Cheryl Dawson and was published in Wang 1996, p. 316.)

Gödel's earlier conception of rationalism refers to mathematical rigor and includes the concept of having a genuine proof, and is therefore in some sense a more radical one than that to which he would later subscribe. One can see it at work at the end of the Gibbs lecture, after a sequence of arguments in favor of realism are given:

Of course I do not claim that the foregoing considerations amount to a real proof of this view about the nature of mathematics. The most I could assert would be to have disproved the nominalistic view, which considers mathematics to consist solely in syntactical conventions and their consequences. Moreover, I have adduced some strong arguments against the more general view that mathematics is our own creation. There are, however, other alternatives to Platonism, in particular psychologism and Aristotelian realism. In order to establish Platonic realism, these theories would have to be disproved one after the other, and then it would have to be shown that they exhaust all possibilities. I am not in a position to do this now; however I would like to give some indications along these lines. (Gödel 1995, p. 321–2).

(For a penetrating analysis of this passage see Tait 2001.) Such an analysis must be based on conceptual analysis:

I am under the impression that after sufficient clarification of the concepts in question it will be possible to conduct these discussions with mathematical rigour and that the result will then be…that the Platonistic view is the only one tenable. (Gödel 1995, p. 322).

Along with the methodological component, as can be seen from the items on Gödel's list, there was also an “optimistic” component to Gödel's rationalism: once the appropriate methods have been developed, philosophical problems such as, for example, those in ethics (e.g., item 9 on the list is: “Formal rights comprise a real science.”) can be decisively solved. As for mathematical assertions, such as the Continuum Hypothesis in set theory, once conceptual analysis has been carried out in the right way, that is, once the basic concepts, such as that of “set,” have been completely clarified, the Continuum Hypothesis should be able to be decided.

Although at the time of the Gibbs lecture the analogy in Gödel's mind between philosophical and mathematical reasoning may have been a very close one, Gödel's view at other periods was that the envisaged methods will not be mathematical in nature. What was wanted was a general, informal science of conceptual analysis.

Philosophy is more general than science. Already the theory of concepts is more general than mathematics…True philosophy is precise but not specialized.

Perhaps the reason why no progress is made in mathematics (and there are so many unsolved problems), is that one confines oneself to the ext[ensional]—thence also the feeling of disappointment in the case of many theories, e.g., propositional logic and formalisation altogether. (Wang 1996, 9.3.20, 9.3.21)[22]

(See notebook Max IV, p. 198 (Gödel Nachlaß, Firestone Library, Princeton, item 030090). Transcription Cheryl Dawson; translation from the German ours; amendment ours. Gödel's dating of Max IV indicates that it is from May 1941 to April 1942. See also Gödel's letter to Bernays, Gödel 2003a, p. 283.)

An important source for understanding Gödel's advance toward a general theory of concepts are Gödel's remarks on conceptual analysis published by Hao Wang in Logical Journey. In remark 8.6.10 for example, Gödel expresses the belief that extensionality fails for concepts, contrary to what he said in his 1944 “Russell's Mathematical Logic,” a remark which he now wishes to retract:

I do not (no longer) believe that generally sameness of range is sufficient to exclude the distinctness of two concepts.

In some of Gödel's later discussions another component of conceptual analysis emerges, namely the project of finding the so-called primitive terms or concepts, and their relations. These are roughly terms or concepts which comprise a theoretical “starting point,” on the basis of their meaning being completely definite and clear. For example, the concept of “the application of a concept to another concept” is a primitive term, along with “force”. (Wang 1996, 9.1.29).

He spoke to Wang about the general project in 1972:

Phenomenology is not the only approach. Another approach is to find a list of the main categories (e.g., causation, substance, action) and their interrelations, which, however, are to be arrived at phenomenologically. The task must be done in the right manner. (Wang 1996, 5.3.7).

Gödel spoke with Sue Toledo between 1972 and 1975 about the project of finding primitive terms, as well as other aspects of phenomenology. See Toledo 2011. We discuss Gödel's involvement with phenomenology further in the supplementary document Gödel's Turn to Phenomenology.

The judgement levied upon Gödel's rationalism by contemporary philosophers was a harsh one. (See for example Gödel 1995, pp. 303–4). Nevertheless Gödel himself remained optimistic. As he commented to Wang:

It is not appropriate to say that philosophy as rigorous science is not realizable in the foreseeable future. Time is not the main factor; it can happen anytime when the right idea appears. (Wang 1996, 4.3.14).

Gödel concluded his 1944 on a similarly optimistic note.

3.2 Gödel's Realism

Gödel's realist views were formulated mostly in the context of the foundations of mathematics and set theory.

We referred above the list “What I believe,” thought to have been written in 1960 or thereabouts. Out of 14 items, only two refer to realism, remarks 10 and 12:

10. Materialism is false.

12. Concepts have an objective existence.

Gödel published his views on realism for the first time in his 1944. The following is one of his most quoted passages on the subject:

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 and concepts as the properties and relations of things existing independently of our definitions and constructions.

It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions and in both cases it is impossible to interpret the propositions one wants to assert about these entities as propositions about the “data,” i.e., in the latter case the actually occurring sense perceptions.

Gödel's reference to the impossibility of interpreting empirical laws, or more precisely, instantiations of them—the statements “one wants to assert,”—as statements about sense perceptions, is likely an endorsement of the (then) contemporary critique of phenomenalism. The critique was based on the observation that sense data are so inextricably bound up with the conditions under which they are experienced, that no correspondence between statements about those and the statements “we want to assert” can be given (see Chisholm 1948 for example). More generally Gödel was against verificationism, namely the idea that the meaning of a statement is its mode of verification.

The analogical point in the first part of the passage was amplified by Gödel in the draft manuscript “Is Mathematics a Syntax of Language?”:

It is arbitrary to consider “This is red” an immediate datum, but not so to consider the proposition expressing modus ponens or complete induction (or perhaps some simpler propositions from which the latter follows). (Gödel 1995, p. 359)

Some writers have interpreted Gödel in this and similar passages pragmatically, attributing to him the view that because empirical statements are paradigmatic of successful reference, reference in the case of abstract concepts should be modelled causally. (See Maddy 1990.) Interpreting reference to abstract objects this way, it is argued, addresses the main difficulty associated with realism, the problem how we can come to have knowledge of abstract objects. Others have argued that Gödel had no paradigm case in mind; that for him both the empirical and the abstract case are either equally problematic, or equally unproblematic. (See Tait 1986.) The latter view is referred to as epistemological parity in van Atten and Kennedy 2003. (See also Kennedy and van Atten 2004.)

In his 1947 “What is Cantor's Continuum Problem?”, Gödel expounds the view that in the case of meaningful propositions of mathematics, there is always a fact of the matter to be decided in a yes or no fashion. This is a direct consequence of realism, for if there exists a domain of mathematical objects or concepts, then any meaningful proposition concerning them must be either true or false.[23] The Continuum Hypothesis is Gödel's example of a meaningful question. The concept “how many” leads “unambiguously” to a definite meaning of the hypothesis, and therefore it should be decidable—at least in principle. Most strikingly Gödel does not leave the matter there but goes on to offer a practical strategy for determining the value of the continuum, as well as the truth value of other axioms extending ZFC. Specifically, he offers two criteria for their decidability: the first involves conceptual analysis and is associated with Gödel's rationalistic program. (See the above section on Gödel's rationalism.) Secondly one must keep an eye on the so-called success of the axiom, as a check or indicator of which direction to look to for the solution of its truth. For example, Gödel notes in the paper that none of the consequences of the Axiom of Constructibility are very plausible. It is, then, likely false. See Maddy 2011 and Koellner 2014 for discussion of intrinsic vs extrinsic justifications for new axioms of set theory.