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$bV>>>X>>>>>>>> ~2*i&How Carnap Could Have Replied to Gdel
S. Awodey and A.W. Carus
In 1995, with the publication of volume 3 of his collected papers, Gdels sustained effort during the 1950s to refute the Vienna Circles conception of mathematics (which he called a combination of nominalism and conventionalism) was made widely available; already it has stimulated extensive philosophical commentary. Attention has focussed, above all, on several drafts of what was to have been Gdels contribution to the Carnap volume in the Library of Living Philosophers (Gdel *1953/9), a series to which Gdel had contributed twice already. In the end, Gdel decided against publication, and the volume appeared without his paper. Gdel himself, though convinced that very weighty and striking arguments in favor of my views could be alleged against the positions of Carnap he was attacking, seems to have had doubts about the effectiveness of his argument. (Goldfarb 1995, p. 324)
The recent commentators have tended, on the whole, to dismiss these doubts and support Gdels view that Carnaps overall framework, based on the principle of tolerance, is self-undermining. Even those who share Gdels doubts, like Goldfarb and Ricketts (1992), have held that Carnaps view can only be upheld in a weakened or diluted (and rather empty) form. But Carnaps own later views on these issues have hardly been given a hearing. If Gdels confidence hadnt wavered, and something like one of his extant drafts had been published, Carnap would certainly not have replied in the style of Goldfarb and Ricketts. Our purpose in this paper is not so much to reconstruct what the historical Carnap might actually have said in such a response, though, as to suggest what he could have said within the framework of the overall position he had sketched out in his later period. This not only results in some interesting clarifications of his later position, casting doubt on the still widespread tendency to dismiss it unheard, but also leads us to the diagnosis, in Gdels argument, of a rather subtle fallacy that has so far escaped the commentators.
I.
Gdel summarizes the syntactic program he attributes to Carnap in two assertions, that mathematics can be interpreted to be syntax of language and that mathematical sentences have no content. The investigations in Carnaps Logical Syntax of Language, he claimed, as well as those of the Hilbert school, had shown the following about these two points:
(1) Mathematics can be interpreted to be syntax of language only if the terms language or syntax or interpreting are taken in a very generalized or attenuated sense, or if only a small part of what is commonly regarded as mathematics is acknowledged as such. . . . (2) Mathematical sentences have no content only if the term content is taken from the beginning in a sense acceptable only to empiricists and not well founded even from the empirical standpoint. Thereby these results become unfit to. . . support . . . the philosophical views in question (such as nominalism or conventionalism). (Gdel *1953/9-III, p. 337)
Regarding assertion (2), Gdel maintains that the examination of the syntactical viewpoint leads to the conclusion that there do exist mathematical objects and facts which are exactly as objective (i.e., independent of our conventions or constructions) as physical or psychological objects and facts. (ibid.) Carnap, of course, would have regarded such questions (asked outside the context of a particular linguistic framework) about the existence or non-existence of any objects, whether physical or mathematical, as empty of cognitive significance. To that extent, he would have been happy to agree with Gdel that there is no difference between the two kinds of objects.
He would even have gone along with Gdels insistence that they are equally objective in the sense that they are independent of our conventions or constructions, and would of course have agreed that analytic and synthetic are nonetheless to be sharply distinguished. Their agreement on these matters is superficial, however; it conceals a basic difference. For Gdel, analytic and synthetic sentences represent or describe two different aspects of reality, to which we have access by means of different processing systems: sense perception for synthetic statements, intellectual intuition for analytic ones (Gdel 1964, pp. 260-261, pp. 267-269). For Carnap, in contrast, the distinction between analytic and synthetic is a pragmatic constraint on the construction of an acceptable language of science: the language must allow us to distinguish genuine empirical information from mere artifacts of the language.
The point of this distinction for Carnap is that languages permitting it are more useful and economical for science, not that some ultimate reality is thereby accounted for or truly represented. It is a misunderstanding to say, then, as Michael Friedman does (1999, pp. 219-221), that Carnaps principle of tolerance depends on the presupposition prior to any language choice that analytic and synthetic sentences are distinguishable. For Carnap, Friedman says, . . . what shows us that external questions are purely pragmatic is precisely the circumstance that they concern, in the end, only the question of which primitive analytic sentences to adopt. It is precisely for this reason, that is, that such questions involve us with no matters of fact. (ibid., p. 220)
A question is external to a given language framework, in Carnaps sense (Carnap 1950a), if it is not answerable by following the specified rules governing that framework. The question which framework to adopt is therefore external, at least to the frameworks among which we are choosing. Among the considerations bearing on this question is the availability, within the language, of a distinction between sentences that convey information and those that only convey artifacts of the language itself. We can discuss, from an external viewpoint, how to make this distinction (even whether to make it). But the distinction is only vague and intuitive until we have made it precise in a rule-specified framework; we are not able to classify sentences outside a framework as in any sense inherently analytic or synthetic. A sentence can only be so classified when it is identified (or designated) as belonging to a language framework whose rules are fully explicit. (In section III below we discuss how Carnap conceived of the relation between vague, pre-formal notions and precisely specified ones.) It is therefore strictly speaking impossible to decide, from an external standpoint, which analytic sentences to adopt, as Friedman suggests. The analytic-synthetic distinction itself must be internal to be applicable, and the external question whether there is such a distinction schlechthin has exactly the same status, for Carnap, as the external question whether numbers or other abstract entities really exist.
II.
But it is the first of the two assertions Gdel makes in the above quotation that has excited all the recent commentary: his assertion that the syntactic interpretation of mathematics, as maintained by Carnap and other members of the Vienna Circle, can be proved false. Carnap would have found Gdels argument very interesting, but would not have accepted it. To understand why, it will be best to treat this rather subtle point with Carnap-like, perhaps almost pedantic, thoroughness. Gdels argument can be paraphrased in the following four steps:
(i) For mathematics to be interpreted as syntax of language and thus empty of empirical content it must be proved that no syntactic (i.e. purely linguistic) stipulation can possibly have empirical consequences; otherwise mathematics is in danger of making claims about the empirical world on purely arbitrary, definitional (however convenient or practical) grounds.
(ii) But even the choice of a very weak language framework (as restricted as primitive recursive arithmetic) has the consequence, by Gdels own second incompleteness theorem, that the consistency of our chosen language cannot be proven with its own resources.
(iii) Any proof that our chosen language is consistent, then, presupposes the consistency of the stronger metalanguage required for the proof, so the attempt to prove consistency at any level incurs an infinite regress, and we cannot completely exclude the possibility that the chosen language is inconsistent, and thus has empirical consequences (as it would imply not only every mathematical sentence, but every empirical sentence).
(iv) Conclusion: The requirement of step (i) cannot be met, so mathematics cannot be syntax of language.
Carnap would have pointed out that Gdels assertion in step (i), though, that the consistency of any stipulated language must be provable, is not implied by what Gdel calls the syntactic interpretation of mathematics (SIM). He is right to point out that the SIM entails the consistency of any language stipulated, but this is not the same as provable consistency. Carnap could have agreed with Gdel on if P then Q, in other words, but Gdel uses if P then provably Q as the basis for his argument. This stronger assertion rests on an apparent non-sequitur in step (i); let us examine the argument more carefully. Gdel begins with the correct statement that
(A) SIM implies that mathematics is empirically vacuous.
He also reminds us, correctly, that
(B) If a stipulated language for mathematics is inconsistent, then it may have empirical consequences.
From this it follows that
(B) If a stipulated language for mathematics is to be empirically vacuous, it must be consistent.
But from A and B it follows only that
(C) SIM requires the consistency of any stipulated language for mathematics.
It does not follow, as Gdel suggests in (i), that
(D) SIM requires the provable consistency of any stipulated language for mathematics.
In short, where Gdel says the rules of syntax must be demonstrably consistent, since from an inconsistency every proposition follows (ibid.), he should correctly say the rules of syntax must be consistent, since from an inconsistency every proposition follows.
This small difference is an important one, as can perhaps be seen more clearly by considering Gdels argument (shown on the left-hand side below) in juxtaposition with an analogous one which might easily have occurred to Carnap if he had written his own reply to Gdel shown on the right-hand side:
(i) For mathematics to be syntax of language, it must be proved that no stipulation can have empirical consequences.
(ii) But even a very weak language cannot be proved consistent without further assumptions. Any proof that our chosen language is consistent, then, presupposes the consistency of the stronger meta-language required for the proof.
(iii) We can never be certain that the chosen language is consistent, and has no empirical consequences.
(iv) Conclusion: The requirement of step (i) cannot be met, so mathematics cannot be syntax of language.
(i) For spacetime to be flat, it must be shown that such and such conditions (indicating curvature) do not obtain in any region of the universe.
(ii) But the required observations may be affected by the presence of curvature; e.g. measurements may be distorted or instru-ments become unreliable. Moreover, the uni-verse may be infinite, or there may be regions that are in principle inaccessible to us.
(iii) We can therefore never be certain that our observations are conclusive, and that the required conditions obtain.
(iv) Conclusion: The requirement of step (i) cannot be met, so spacetime cannot be flat.
The argument on the right differs from that on the left in that it concerns an empirical question. But it has the same logical form as Gdels argument (on the left), and it is no less sound. In both cases, the conclusion is unwarranted.
The erroneous claim in the right-hand argument is it must be shown that . . . in (i). It would be correct to say it must be the case that . . ., for spacetime to be flat, such and such conditions do not obtain. It would also be correct to say for spacetime to be shown to be flat, it must be shown that such and such conditions do not obtain. In Gdels argument, likewise, though if P then Q is correct, and if provably P then provably Q is correct, neither of these is equivalent to Gdels if P then provably Q.
Gdels argument, therefore, does not refute the possibility that mathematics can be interpreted as syntax of language, as he claims. However, a slight modification of this argument does show something else of equal interest. Though we saw that (D) above is unwarranted, it would be correct to say
(E) A proof of SIM requires the provable consistency of any stipulated language for mathematics.
As Gdel correctly argues, the consistency of any stipulated language cannot be proved. So by (E) there can be no proof of SIM. We cannot prove that mathematics is syntax of language by mathematical reasoning. But this result, far from undermining SIM, is in complete harmony with it. For the syntactic view implies the vacuity of mathematics, which would surely be violated if that viewpoint could itself be proved mathematically, as such a result would itself be a non-trivial mathematical proposition.
III.
But what Gdel refers to as the syntactic interpretation of mathematics, and we have accordingly been calling SIM, would not have been accepted by Carnap as an appropriate description of his position either in the 1930s or, certainly, in the 1950s. The idea of an interpretation of mathematics Carnap would have thought vague and misleadingly suggestive of a foundational theory, while his earlier syntactic view had given way by the mid-1930s to a broader framework in which the meta-language of science was semantic (incorporated designation, reference, meaning, and truth). By the end of the 1930s, Carnap had added to this account the perspective of the language user, so that the meta-theory now had three parts: syntax, semantics, and pragmatics. The significance of these moves would only gradually become apparent, however, and has still not been widely understood. To clarify what we have been referring to as SIM, the most important aspect to stress in the present context is the framework Carnap arrived at, in his later years, for what he called explication. An explication, for him, was a local replacement for a vague or informal concept of everyday language, not with the goal of capturing its intuitive meaning more precisely, but of finding a precise concept, defined within a framework of specified rules, that could function, in at least some respects of interest, analogously to the vague concept it explicates. Implicit here is a distinction between the vague (ordinary or natural) language of the explicandum and the precise (rule-governed) language of the explicatum. But the replacement of explicanda by explicata was not a one-way street, as it had been regarded by the 1920s Vienna Circle, since no criterion internal to the explicatum-language can tell us whether a given proposed explicatum is correct or true. (Truth, in Tarskis definition, has become language-relative, and there is nothing external to the explicatum-language, no regulative reality of true meanings, to which it could conform so as to establish its correctness.) Nor can adequacy to the explicandum be an absolute criterion of adequacy for an explicatum, since that can be determined only within the (ordinary) explicandum-language, which is not fully intertranslatable with the explicatum-language (it lacks its precision, for instance), and is unable to tell us whether the proposal is useful (or fruitful, or whatever) within the scientific language (the explicatum-language) in which it is couched. Such usefulness can only be gauged by reference to the purposes for which the language as a whole is to be employed which need not coincide with closeness of correspondence in usage between a particular explicatum and its explicandum within their respective languages.
The user community on the one hand pursues practical goals (such as the construction of theoretical languages and other tools) within an evolved language suited to such practical activity a language it continually upgrades to reflect its cognitive progress. This progress itself, on the other hand, takes place within precisely specified rule-frameworks, in which knowledge is codified, written down. The relation between these two kinds of communication or representation systems is characterized by mutual feedback; in Carnaps mature view the principle of tolerance comes to mean not only that the user community has choices (ultimately practical ones) among precise-rule systems, but also that the precise rule-systems do not globally replace the informal practical systems, but rather inform and upgrade them. The relation between theoretical and practical becomes one of continual interaction and mutual adjustment. The language users live within the Lebenswelt of their ordinary objects and ordinary intuitions, but at the same time they try to use precise frameworks as tools to improve those intuitions (as well as their lives more generally).
Carnaps view in the 1950s, then, was not a syntactic interpretation of mathematics; it was neither syntactic nor was it an interpretation (it was not something that interprets something else truly or falsely, correctly or incorrectly). Carnaps accounts of mathematics, in the Syntax as well as in later works, were not assertions about some state of affairs, but proposals for clarifying the nature and uses of mathematics. The criterion of adequacy for such proposals does not and cannot reside in their correctness with respect to something they represent, truly or falsely; it resides in the helpfulness of the proposal as a whole in clarifying the nature of human mathematical activity.
IV.
But does such an approach nonetheless make certain presuppositions? Michael Potter has argued, in a recent book, that it does in fact presuppose provable consistency of the mathematical part of the language which Gdel was therefore right to require of it; Carnaps position is therefore impossible. Potters argument invokes the constraints on the principle of tolerance resulting from the addition of semantics to the study of scientific language. As Carnap himself put it, the syntactical rules
. . . can be chosen arbitrarily and hence are conventional if they are taken as the basis of the construction of the language system and if the interpretation of the system is later superimposed. On the other hand, a system of logic not a matter of choice, but either right or wrong, if an interpretation of the logical signs is given in advance. (Carnap 1939, p. 48, quoted by Potter 2000, p. 272)
In other words, Potter says, the introduction of a theory of semantics is to be seen as a constraint on the principle of tolerance. This is not quite right. The introduction of a theory of semantics is not yet a constraint; a particular interpretation must be given first. And even then, this interpretation is not a constraint on the principle of tolerance as such, but on the freedom to choose a syntax for that particular interpretation. These distinctions might seem trivial, but weighty consequences follow. For Potter takes Carnap, by the introduction of this constraint of interpretability (Potter 2000, p. 272), to be committed to a particular fixed semantics in which at least some descriptive sentences mirror the world indefeasibly (ibid., p. 277).
Potter sees Carnaps position as resting on a kind of transcendental argument by which consistency is a necessary condition for the possibility of the descriptiveness of language. On Carnaps account, Potter argues, we can never be sure that the language we use actually describes anything unless we prove that it is consistent; for the discovery of an inconsistency would deprive the entire language of its descriptive capability (since every sentence is derivable, every sentence is determinate, and the language has no descriptive vocabulary at all; ibid., p. 271). So to refute this account, Potter says, there need be only one sentence in my language which I am indefeasibly sure I can use to express an empirical claim (ibid., p. 277), since Carnaps account, he claims, forces such a sentence to imply certain knowledge of consistency (which we cannot have). And the constraint of interpretability, which concedes the existence of such indefeasible sentences (ibid.), Potter says, thereby commits Carnap to the claim that we know mathematics to be consistent. Since we know by Gdels second incompleteness theorem that we cannot know this, Carnap contradicts himself; in fact, says Potter, this is as close to a straightforward contradiction as one is likely to encounter in philosophy. (ibid.)
But Potter is quite mistaken in attributing to Carnap a fixed constraint of interpretation, or the acceptance of even a single indefeasibly descriptive sentence. It is fundamental to the later Carnaps view, as we saw above (section I), that there is no fixed partition, antecedent to any language, of sentences into analytic ones and synthetic ones. Any sentence whatever, including Potters favorite This table is black, could, if it were for some reason convenient, be made into a constitutive language rule, and thereby deprived of its descriptive capacity within that language. No sentence, regarded in isolation, is inherently descriptive.
Potter sees a deeper problem, though, in what he regards as Carnaps attempt to eliminate any role for subjective intuition (what Potter calls the self) in our grasp of arithmetic, or to reduce such understanding to the bare manipulation of rules. The result is that, unmoored to a human user, a Carnapian language floats at too high a level, so to speak, above its application instances and is therefore able to express concrete sentences only if perfectly aligned as a whole with the facts they are intended to represent, i.e. only if it happens to be consistent. (It is connected to the ground the user only at one point, by the consequence relation.) This would appear to make the entire language vulnerable to an inconsistency within its logical part; Carnaps account does not permit a plausible picture of our grasp of language as meaningful, Potter says, and the reason is ultimately that its holism fails to leave room for the transparency of the grasp of meaning to the language user himself. (ibid., p. 276) By trying to account for language comprehension entirely by rules, and leaving no room for subjective intuition, Carnap seems to be making the absurd claim that the practical viability of our ordinary linguistic competence stands or falls by stipulated rules of which we may be ignorant and which cant even be finitely enumerated.
But as we have seen in section III, this is not Carnaps view of language. Though his conception of the scientific, theoretical language was holistic (ibid., p. 246), he specifically saw the practical, intuitive part of language as distinct from it and as serving a different purpose. The ordinary linguistic competence of human language user communities is the practical setting (not a precise framework) within which theoretical languages are precisely specified; whether particular concepts or rules specified in the theoretical language adequately reflect intuitions that may originally have motivated them is not a sufficient criterion for their acceptance or rejection. So the imputed transcendental argument misrepresents Carnaps view. The descriptive capacity of a language is not destroyed by the discovery of an inconsistency since we can change the rules or postulates of the theoretical language without also changing the vocabulary of ordinary objects in our everyday language. The purely descriptive sentences in the latter are unaffected by an inconsistency in the theoretical language. So both of the premises Potter attributes to Carnap rest on misinterpretations, and the contradiction vanishes.
V.
However, there is reason to believe that Gdel systematically misunderstood Carnaps project and his terms, and attributed a meaning to the notion syntax of language quite different from that intended by Carnap; one clue is the remark already quoted, to the effect that mathematics could only be considered syntax of language if only a small part of what is commonly recognized as mathematics is acknowledge as such. It seems Gdel may have taken syntactic to mean something like inherently syntactic, in a sense that can be exactly specified and then investigated mathematically, like Hilberts contentful [inhaltliche] meta-mathematics, or the property of definability or constructability that occupied an important place in Gdels investigations regarding the consistency of the continuum hypothesis. For instance, the notion of provability in a formal language could be regarded as strictly syntactical, in that it refers only to the formal, combinatorial features of the expressions in the language, and not to their intended interpretation (if any). Gdel does actually seem, in his published Remarks before the Princeton Bicentennial Conference, to have countenanced such an idea. He cites the example of Turing computability, which succeeds for the first time, he says, in giving an absolute definition of an interesting epistemological notion, i.e. one not depending on the formalism chosen. (Gdel 1946, p. 150) In all other cases treated previously, he says, particularly citing demonstrability as an example, one has been able to define them only relative to a given language, and for each individual language it is clear that the one thus obtained is not the one looked for. But the example of Turing computability should encourage one to expect the same thing [i.e. an absolute definition] to be possible also in other cases, in particular, demonstrability. (ibid., pp. 150-151) Propositions involving the concept of demonstrability would thus become part of what Gdel elsewhere called mathematics proper, i.e. the body of those mathematical propositions which hold in an absolute sense, without any further hypothesis, in contrast to axiomatic or hypothetico-deductive systems such as geometry (Gdel 1951*, p. 305).
Gdel seems to have attributed to Carnap the substantive claim that all of mathematics can be constructed hypothetically, as Hilbert attempted to do, on the basis of some absolute mathematics (only a small part of what is commonly recognized as mathematics). But Carnap was obviously doing something rather different from Hilbert, and it isnt clear what basis in absolute mathematics Gdel took Carnap to be using. Based on the critique in his Gibbs Lecture (Gdel *1951), it seems he may have thought Carnap to be claiming that hypothetical mathematics can be formulated using just one single absolute concept, that of demonstrability or syntactic derivability, to derive consequences from arbitrary axioms, man-made. . . conventions (ibid., p. 320).
That something like this was the object of his critique becomes clear when he challenges the very idea of the distinction he has himself formulated, earlier in the Gibbs Lecture, between absolute and hypothetical mathematics, arguing that the latter partake of the same objective reality of their own, which we cannot create or change, but only perceive and describe as the former, since the syntactic rules are based on the idea of a finite manifold (namely, of a finite sequence of symbols), and this idea and its properties are entirely independent of our free choice. In fact, its theory is equivalent to the theory of [the] integers. (ibid., p. 320) The possibility of so constructing a language that this theory is incorporated into it in the form of syntactic rules, he concludes, proves nothing. Hypothetical mathematics, in other words, is effectively part of absolute mathematics. More generally, he argues that although the truth of the mathematical axioms is derivable from suitably chosen rules, these same concepts and axioms must be used in the derivation (in a special application, namely, as referring to symbols, combinations of symbols, sets of such combinations, etc.), which is therefore circular: So while the original idea of this viewpoint was to make the truth of the mathematical axioms understandable by showing that they are tautologies, it ends up with the just the opposite, that is, the truth of the axioms must first be assumed and then it can be shown that, in a suitably chosen language, they are tautologies. (ibid., p. 317) Again, the conclusion is that there is no genuinely hypothetical or purely formal part of mathematics; there is really only absolute mathematics, so the idea of attempting to make demonstrability or purely formal derivability the only absolute basis for mathematics fails.
VI.
Carnap would have regarded Gdels conception of absolute mathematics unclear, and might have tried to clarify it by using Gdels own investigation of the continuum hypothesis as an example. Gdel insisted, of course, that the axioms of set theory were part of absolute mathematics. Despite their remoteness from sense experience, he said, 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. (Gdel 1964, pp. 267-268) But this idea of forcing themselves upon us is given an important qualification:
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. . . (ibid., p. 268)
Though the truth of the axioms forces itself on us, it does not do so immediately; it does so in the context, rather, of the larger picture, the practice, we might say, of set theory as a whole, and in this context Gdel sees an open-ended development of new mathematical intuitions on the basis of previous ones, and holds that new mathematical intuitions leading to a decision of such problems as Cantors continuum hypothesis are perfectly possible. (ibid.) In a manuscript from this same period, Gdel specifically says that such intuitions go beyond common sense; again and again, new axioms become obvious, and despite his incompleteness results he thinks it conceivable that every clearly formulated mathematical problem can be solved, precisely because the human mind has this power, which machines lack, of becoming intuitively acquainted with more and more new axioms on the basis of the sense of the basic concepts. (Gdel *1961, p. 384) He explicitly sees an analogy, in this, to natural science, in which the theoretical creativity of the scientist nonetheless corresponds to an external reality conveyed to us by observations mediated by the senses. The creativity by which we generate and become acquainted with new axioms (and they become obvious to us) also corresponds to an external, abstract reality.
It is in this sense, then, that Gdel claims that the question of the truth of the continuum hypothesis would not lose its meaning if it were proved to be undecidable from the accepted axioms of set theory" (Gdel 1964, p. 267), since the mere psychological fact of the existence of an intuition which is sufficiently clear to produce the axioms of set theory and an open series of extensions of them suffices to give meaning to the question of the truth or falsity of propositions like Cantors continuum hypothesis. (ibid., p. 268)
Carnap would have responded that this conception presumes a development of set theory (and the larger world of mathematics) in the future that we have no way of knowing in advance or even making plausible guesses about. Though it is entirely possible that our current axioms will remain firmly in place and obvious to us, and that future axioms will enjoy the same acceptance (in a smooth cumulative development to the end of time), it is also possible that a contradiction will arise that forces us to revise or replace an existing axiom, or that it will turn out that somewhat different sets of axioms are suitable for different purposes or different uses, like set theory without the axiom of foundation, or even its negation. Why, Carnap would ask, should we foreclose these possibilities? We have not the slightest justification for predicting that one of these futures is any more likely than the others.
Since the continuum hypothesis does not have a truth value within the language of existing axiomatic set theory, he would have said, the question whether it is true lacks a reference language, is incorrectly posed. To give it an unambiguous meaning we must change or expand the language, or switch to a different one, and this, Carnap said, presents us with a new question that is not mathematical, but requires explication. Gdels criterion of meaning in this case was, as we have seen, the mere psychological fact of the existence of an intuition which is sufficiently clear to produce the axioms of set theory and an open series of extensions of them. This psychological fact, moreover, is not merely a necessary criterion but suffices to give meaning to the question of the truth or falsity of propositions like Cantors continuum hypothesis (our emphasis). But obviously Gdel did not think the truth (as opposed to the meaning) of the continuum hypothesis forces itself on us; or at least the form in which it did so had to be a proof of some kind. He did for a time, though, think that the axiom of constructibility, from which he was able to derive the continuum hypothesis, might be true. Did he think, then, that the axiom of constructibility was a candidate for membership in this open series of extensions of the previous axioms? Perhaps not, as he also had another, entirely different criterion for accepting new axioms, which, as he says, applies even disregarding the intrinsic necessity of the new axiom, and even in case it has no intrinsic necessity at all! (1964, p. 261) This criterion can justify only a probable decision about the truth of the new axiom, however, not a certain one; still, the truth of a new axiom can be justified, he says,
. . . inductively, by studying its success. Success here means fruitfulness in consequences, in particular in verifiable consequences, i.e. consequences demonstrable without the new axiom, whose proofs with the help of the new axiom, however, are considerably simpler and easier to discover, and make it possible to condense into one proof many different proofs. (ibid.)
The example he gives are the axioms for the system of real numbers, rejected by the intuitionists, which have in this way been verified to some extent, owing to the fact that analytical number theory frequently allows one to prove number-theoretical theorems which, in a more cumbersome way, can subsequently be verified by elementary methods. But he specifically does not take this case to be a very strong example, or to be exhaustive of this criterion for accepting new axioms, and adds a scenario that for this criterion also invokes the analogy of physical science:
A much higher degree of verification than that, however, is conceivable. There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole field, and yielding such powerful methods for solving problems (and even solving them constructively, as far as that is possible) that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any well-established physical theory. (ibid.)
Gdel makes no effort, though, to reconcile these two criteria; though he obviously gives a kind of priority to intrinsic necessity, there is no indication how inductive success relates to it, if at all. Indeed, it has been pointed out that from the viewpoint of their justification, the two criteria seem difficult to reconcile with each other. Extensionally, on the other hand, they might be expected more or less to coincide; it would be an odd sort of intrinsic necessity that did not carry some degree of inductive success with it (though even an overwhelming inductive success of the kind Gdel describes might not bring about a feeling that the axiom in question forced itself on our intuition). In the present case, there would appear to be prima facie little to choose, on the basis solely of intrinsic necessity, between a minimal axiom like constructability and maximal ones of the kind Gdel later favored (1964, pp. 262-263). But the considerations Gdel actually advances for the falsity of the axiom of constructability and the continuum hypothesis focus not on their lack of intrinsic necessity, but on the highly implausible consequences of the continuum hypothesis (ibid., p. 263, our emphasis).
Carnap might well have asked, therefore, whether Gdels own actual procedure did not correspond more closely to Carnaps pragmatic criterion (as described in section III above) which seems essentially to be the same as Gdels second, inductive success, criterion than to Gdels official primary criterion of intrinsic necessity. Gdel and I, Carnap might have said, both admit that new axioms are required; where we differ is not about a matter of fact, or about anything with cognitive content, but in our attitude toward this search for new axioms. Gdel is impressed with the tendency to convergence he observes in the history of mathematics, the tendency toward unification, even as mathematics develops outward in breadth and scope. While I, though admitting this tendency, believe that much, perhaps most, actual progress in mathematics has come about through the free creation of conceptual possibilities on the one hand and concerted efforts to give precise meaning to vague, intuitive concepts on the other. The vague concepts on which mathematicians have focussed have often been ones with immediate physical application, like the area under a curve, or the idea of a uniform motion or rate of change, but even where this wasnt the primary motivation, such clarifications are often fruitful in applications (like Riemann extending the notion of curvature to spaces of more than two dimensions). And the path to the convergence and unification Gdel points to has not always been smooth; there have been enough changes of course and enough discarded ideas in the history of mathematics to justify the conjecture that the convergence we observe is at least partly the result of our desire to unify, our practical need for an integrated and unified body of mathematical concepts. It seems to me that only harm can result from the assumption that all future mathematics must remain within the fundamental axioms we have been working within for the past century; the assumption of a mathematical or conceptual reality described by these axioms can only discourage new attempts to articulate conceptual possibilities in new and different ways.
Of course the ultimate test of the two approaches can only, in Carnaps view, be a practical one. As in his confrontation with Quine, he might have said, In my view, both programs. . . ought to be pursued; and I think that if [Gdel] and I could live, say, for two hundred years, it would be possible at the end of that time for us to agree on which of the two programs had proved more successful. (as reported by Stein 1992, p. 279)
VII.
Unlike Quine, though, Gdel would probably not have acquiesced to such an adjudication procedure; it would have seemed to him to miss the point. But we need not leave Carnap and Gdel at quite the stand-off Goldfarb and Ricketts abandon them to. (Goldfarb and Ricketts 1995, p. 71) Carnap had available, and Gdel might have been willing to consider, an alternative description of their difference as amounting, in the end, to a question of values.
This might appear to put their disagreement beyond discussion, especially as Carnap thought our language for dealing with values extremely vague and unsatisfactory. Nonetheless, he was willing to attempt improvements (Carnap 1963, 32); in the spirit of clarifying the explicandum, he might have said something like this: To maximize the practical value of our scientific knowledge (both in the sense of technological applications and in the sense of understanding), it is essential that we distinguish clearly between practical or normative questions and those that have clear, unambiguous answers, at least in principle, within the terms we have defined and the rules we have laid down. Gdel, on the other hand, believes that the value of our knowledge (again in both senses) is maximized if we assume that certain essential features of the world are not accessible to our senses even in principle (and thus not accessible to empirical science), and that there may be other routes toward their cognition. Such an assumption conflicts with Carnaps proposed sharp distinction between practical and cognitive questions, as the cognitive category would then be much more difficult to characterize precisely. Though on the surface it might appear that Carnaps and Gdels goals were the same to maximize the practical value of knowledge it seems that the difference between them arises from the different value they attached to certain subjective feelings of insight. Gdel valued such feelings very highly, and was inclined to believe that casting doubt on the real existence of the apparent objects of these feelings would restrain or dampen our ability to feel them. Carnap also valued these feelings, but saw their value as residing mainly in a heuristic function; he saw them as explicanda, vague starting points as raw material that we can, with effort, make something from. Pressed further about why he believed that the practical value of science would be maximized if we distinguish sharply between practical and cognitive, Carnap would undoubtedly have pointed to history to the practical result of the distinction. But he might also have said that clarity of understanding was itself a value for him, and one that Gdel certainly shared (though he might have subordinated it to other values).
The vague term clarity Carnap might in turn have explicated in terms of formal (or semantic) explicitness, in a context where alternatives are available for comparison: clarity would be regarded as maximized in a framework of discussion that allowed proponents of various views or proposals to state their assumptions and their terms (including categorial terms) as explicitly and unambiguously as possible, in a way that both (a) their own users regarded as adequate, and (b) their consequences for all questions of interest to both sides are directly comparable but excluding all normative or optative components of the alternatives in question. (Not that these normative considerations should not be discussed at all; on the contrary, Carnap regarded them as very important. But he did not think they could be adequately addressed until the contending proposals themselves are clearly explicitly identified so that their consequences can be worked out and assessed.)
This formulation has the advantage that it leaves Carnap and Gdel (potentially) agreeing about what they disagree about. It also has the advantage, from Carnaps point of view, that it enables him to express the disagreement within the terms of his own meta-framework, which does not, therefore, require modifications of the kind Goldfarb and Ricketts suggest, in response to Gdels critique. That meta-framework itself (based on the principle of tolerance) cannot, then, be disproved, as Gdel tried to do. It is not the sort of thing that could be proved or disproved. It is not a claim at all, but a proposal; it can be found attractive or unattractive, useful or useless, elegant or clumsy, by those who elect to use it.
Literature Cited
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Awodey, S. and A.W. Carus 2002 Carnap vs. Gdel on Syntax and Analyticity, in M. Salmon, ed. Proceedings of the Florence Conference (Pittsburgh: University of Pittsburgh Press).
Bird, G.H. 1993 Carnap and Quine: Internal and External Questions, Erkenntnis 42, pp. 41-64.
Carnap, R. 1932 Erwiderung auf die vorstehenden Aufstze von E. Zilsel und K. Duncker, Erkenntnis 2, pp. 177-188.
Carnap, R. 1934 Logische Syntax der Sprache (Vienna: Springer).
Carnap, R. 1939 Foundations of Logic and Mathematics (Chicago: University of Chicago Press).
Carnap, R. 1950a Empiricism, Semantics, and Ontology Revue Internationale de Philosophie 4, pp. 20-40, repr. in Meaning and Necessity, 2nd ed. 1956 (Chicago: University of Chicago Press), pp. 205-221.
Carnap, R. 1950b Logical Foundations of Probability (Chicago: University of Chicago Press).
Carnap, R. 1956 The Methodological Character of Theoretical Concepts, in H. Feigl and M. Scriven, eds., The Foundations of Science and the Concepts of Psychology and Psychoanalysis (Minneapolis: University of Minnesota Press), pp. 38-76.
Carnap, R. 1963 Replies and Systematic Expositions, in P. Schilpp, ed. The Philosophy of Rudolf Carnap (LaSalle, IL: Open Court), pp. 859-1016.
Carnap, R. 1966 Philosophical Foundations of Physics: An Introduction to the Philosophy of Science, edited by Martin Gardner (New York: Basic Books).
Carnap, R. and W.V.O. Quine 1990 Dear Carnap, Dear Van: The Quine-Carnap Correspondence and Related Work, edited by R. Creath (Berkeley: University of California Press).
Creath, R. (ed.) 1990, Introduction to Carnap and Quine 1990, pp. 1-43.
Einstein, A. 1921 Geometrie und Erfahrung (Berlin: Springer).
Friedman, M. 1999 Tolerance and Analyticity in Carnaps Philosophy of Mathematics, in his Reconsidering Logical Positivism (Cambridge: Cambridge University Press), pp. 198-234.
Gdel, K. *1939 Vortrag Gttingen, in his Collected Works, vol. 3, pp. 126-155.
Gdel, K. 1946 Remarks before the Princeton Bicentennial Conference on Problems in Mathematics, in his Collected Works, vol. 2, pp. 150-152.
Gdel, K. *1951 Some Basic Theorems on the Foundations of Mathematics and their Implications, in his Collected Works, vol. 3, pp. 304-323.
Gdel, K. *1953/9, version III Is Mathematics Syntax of Language?, in his Collected Works, vol. 3, pp. 334-355.
Gdel, K. *1961 The Modern Development of the Foundations of Mathematics in the Light of Philosophy, in his Collected Works, vol. III, pp. 374-387.
Gdel, K. 1964 What Is Cantors Continuum Problem? (revised and expanded version), in his Collected Works, vol. 2, pp. 254-270.
Goldfarb, W. 1995 Introductory Note to *1953/9, in K. Gdel Collected Works, vol. 3, pp. 324-333.
Goldfarb, W. and T. Ricketts 1992 Carnap and the Philosophy of Mathematics, in D. Bell and W. Vossenkuhl, eds. Science and Subjectivity; The Vienna Circle and Twentieth Century Philosophy (Berlin: Akademie-Verlag), pp. 61-78.
Jeffrey, R. 1994 Carnaps Voluntarism, in D. Prawitz, B. Skyrms, and D. Westersthl, eds. Logic, Methodology, and Philosophy of Science IX (Amsterdam: Elsevier), pp. 847-866.
Parsons, C. 1995a Quine and Gdel on Analyticity, in On Quine; New Essays (CUP), ed. by Paolo Leonardi and Marco Santambrogio, pp. 297-313.
Parsons, C. 1995b Platonism and Mathematical Intuition in Kurt Gdels Thought Bulletin of Symbolic Logic 1, pp. 44-74.
Potter, M. 2000 Reasons Nearest Kin: Philosophies of Arithmetic from Kant to Carnap (Oxford: Oxford University Press).
Richardson, A. 1994 The Limits of Tolerance: Carnaps Logico-Philosophical Project in Logical Syntax of Language Proceedings of the Aristotelian Society, supplementary volume, pp. 67-82.
Ricketts, T. 1994 Carnaps Principle of Tolerance, Empiricism, and Conventionalism, in P. Clark and B. Hale, eds., Reading Putnam (Oxford: Blackwell), pp. 176-200
Ricketts, T. 1996 Carnap: From Logical Syntax to Semantics, in R. Giere and A. Richardson, eds., Origins of Logical Empiricism (Minneapolis: University of Minnesota Press), pp. 231-250.
Stein, H. 1992 Was Carnap Entirely Wrong, After All? Synthese 93, pp. 275-295.
Stein, H. 1994 On the Structure of our Knowledge in Physics, in D. Prawitz, B. Skyrms, and D. Westersthl, eds. Logic, Methodology, and Philosophy of Science IX (Amsterdam: Elsevier), pp. 633-655.
Tait, W.W. 2001 Gdels Unpublished Papers on Foundations of Mathematics, Philosophia Mathematica 9, pp. 87-126.
From, among others, Goldfarb (1995), Goldfarb & Ricketts (1992), Richardson (1994), Ricketts (1994, 1996), Parsons (1995a, 1995b), Friedman (1999), Potter (2000), and Tait (2001). We thank Michael Friedman and Michael Potter for discussion of their respective positions with us, and Charles Parsons for comments on the final version of this paper.
This view is most evident in Richardson (1994), Friedman (1999), and Potter (2000).
An actual response of Carnaps to Gdel would have been colored by Carnaps enormous respect for Gdel and feeling of indebtedness to him (though there was also a debt in the other direction; see Awodey and Carus 2001). And Carnap was hardly ever an outspoken controversialist; one of the frustrations of his debate with Quine about analyticity, for example, is his reluctance to spell out his full position, at least in published form (Creath 1990); he tended in all of his replies (Carnap 1963) to construct his critics points as narrowly as possible. In telling Carnap what he could have said to Gdel, then, we make the unrealistic assumption that he would have felt free to spell out his fundamental differences with Gdel quite explicitly and place them in a wider context.
This framework was very much a sketch, and has only recently been the subject of tentative discussion and interpretation, above all by Stein (1992), Bird (1993), and Jeffrey (1994). The general understanding of Carnaps later position portrayed in this paper is greatly indebted to Howard Stein, whose comments also have much improved it.
Though he acknowledges that Carnaps view has changed since the Logical Syntax (Carnap 1934), what Gdel attributes to Carnap is not the doctrine of that book but a misunderstanding of it, as will become clear especially in section V below. In section II we bracket Carnaps actual view (of either the 1930s or the 1950s), and discuss the one Gdel attributes to him and seeks to refute. Beginning in section III, Carnaps actual view during the 1950s is added to the picture. In a separate paper (Awodey and Carus 2002), we offer a defense of Carnaps actual position in the Syntax against Gdels argument.
This appears to be the conception of analyticity in the passages cited here, at least, as well as in the drafts of *1953/9; for discussion of variant conceptions Gdel may have entertained, see Parsons 1995a, pp. 299-306.
In this connection Carnap often referred to Einsteins paper Geometrie und Erfahrung which contains the famous passage: Insofar as the sentences of mathematics refer to reality, they are not certain, and insofar as they are certain, they do not refer to reality . . . I place such a high value on this conception of geometry because without it, the discovery of the theory of relativity would have been impossible for me. (Einstein 1921, pp. 3-6)
See Carnap 1966, pp. 257ff. as well as the discussions in Stein 1992 and Bird 1993.
It may perhaps be thought of as internal to some larger, possibly imaginable, normative super-framework (Bird 1993, pp. 44-46). Of course such a framework would itself, in turn, be subject to external choice at the next level up.
This straightforward picture is complicated by the loose but customary usage (even Carnap employed it on occasion) by which any rule specifying a framework is informally called analytic (in the sense of trivially provable from itself, within a system it partly defines). Ordinarily this causes no confusion. But Carnap also wanted to keep open the possibility that among the transformation rules governing a language we might specify not just logical laws but also physical ones. In such a language, those physical laws would have the status, not of contingent sentences, but of rules of inference. Relative to such a language, these P-rules, as he called them (Carnap 1934, pp. 133-135), could not be falsified (if we had some reason to doubt them, we would have to change the language). They would be empirical, and thus synthetic, but only by fiat. Still, they would be empirical, and this is what causes confusion when all rules specifying a language are loosely called analytic; the P-rules would then be called analytic from the informal, external standpoint, and synthetic within the framework they contribute toward specifying. This confusion, we suspect, lies at the basis of Friedmans statement quoted above.
Following the passage quoted at the beginning of section I above, Gdel says, . . . if the terms occurring are taken in their ordinary sense, then assertion 1 [mathematics can be interpreted to be syntax of language] is disprovable. (Gdel *1953/9, p. 337)
In draft III, published as the first part of Gdel *1953/9, he formulates the argument as follows: . . . a rule about the truth of sentences can be called syntactical only if it is clear from its formulation, or if it somehow can be known beforehand, that it does not imply the truth or falsehood of any factual sentences (i.e. one whose truth, owing to the semantical rules of the language, depends on extra-linguistic facts). This requirement not only follows from the concept of a convention about the use of symbols, but also from the fact that it is the lack of content of mathematics upon which its apriori admissibility despite strict empiricism is to be based. The requirement under discussion implies that the rules of syntax must be demonstrably consistent, since from an inconsistency every proposition follows, all factual propositions included. (p. 339) Similarly, in the same version: To eliminate intuition or empirical induction by positing the mathematical axioms to be true by convention is not possible. For, before any convention can be made, mathematical axioms of the same power or empirical findings with a similar content are necessary already in order to prove the consistency of the envisaged convention. A consistency proof, however, is indispensable because it belongs to the concept of a convention that one knows it does not imply any propositions which can be falsified by observation (which, in the case of mathematical conventions, is equivalent with consistency . . .). (ibid., p. 347)
It would not be surprising if Gdel had not considered the implications of these changes, as there is no evidence that he studied closely anything Carnap wrote after the first draft of the Logical Syntax in 1932. (We discuss Gdels contribution to the composition of the Syntax in a forthcoming paper.) The very use of the word syntax in the title of his piece, and in the discussion of Carnaps doctrine, indicates that he was focussing on a phase of Carnaps development that Carnap himself no longer referred to in such terms.
We can, for instance, imagine explicata that have no equivalent at all in ordinary language, newly-created concepts that are very useful in the theoretical language but have no antecedent in our common-sense intuition, and of course Carnap was always ready to point out that certain intuitive or ordinary-language concepts have no equivalent in a more precise language, i.e. are cognitively meaningless, as he put it.
For Carnap himself, the precise frameworks entailed sentences in an observation sublanguage that he regarded as part of the intuitive language of the ordinary life-world, to which the sentences of the precise framework could thus be regarded as applying directly. Carnap was never, of course, able to make this account work, possibly because it cant work (Stein 1994, p. 638). However, a modification of Carnaps view avoids this difficulty; the precise language of science need not be regarded as, strictly speaking, directly applicable to the language of the ordinary life-world, or to ordinary objects (or qualities), at all; it can be regarded as applying to a schematic representation of that world, which the users can fill in for their purposes when they use it. (ibid., pp. 649-650)
Note that while Gdel speaks of the empirical consequences derivable from an inconsistency, Potter deprives the language of any possibility of descriptive (or empirical) force in that case. This is only a terminological difference; Gdel uses empirical to mean non-logical (as we saw in section I, he accepts an analytic-synthetic distinction prior to any language, reflected in different human faculties for perceiving facts in the logical and empirical realms), while Potter uses descriptive (in Carnaps sense, putatively) to mean contingent or not derivable.
This possibility is specifically mentioned by Carnap (1934, p. 133); see also footnote 9 above.
Except the fragment of it which he called the observation language; but it is not an essential part of Carnaps larger meta-framework (as opposed to his successive theories of empirical verification; see footnote 14 above) that this observation language be either a part of the theoretical language or that sentences containing its terms be logically derivable from the theoretical language.
Carnaps own view was that the observation language is given a complete interpretation by the fact that a user community uses it as its means of communication and all its sentences are understood by all members of the group in the same sense (Carnap 1956, p. 40). But the purely descriptive sentences remain unaffected by an inconsistency in the theoretical language whether they are in such a Carnapian observation language or simply taken as being in ordinary language, since the interpretation of this language is given by practical agreement in its use, not by the rules that constitute the theoretical language. Note also that maintaining an observational vocabulary across changes in rules or postulates is not tantamount to the assertion of a language-independent empirical world; it means only that the two choices (of observational language and theoretical language) are, in principle, independent.
A more straightforward way of understanding what Gdel meant by the idea that the axioms of set theory, though given by intuition, are not immediately so given, is suggested in a 1948 conversation with Carnap, who notes, He sees a strong analogy between theoretical physics and set theory. Physics is confirmed by sense perceptions; set theory is confirmed by its consequences in elementary arithmetic. The fundamental insights in arithmetic, that are not reducible to anything still simpler, are analogous to the sense perceptions. (ASP/RC 088-30-03, Princeton, 26 March 1948) Parsons (1995b, pp. 62-64) argues, on the basis of the published and unpublished texts, for a similar understanding of Gdels notion that the axioms of set theory, though intuited in a way analogous to perception, are not immediately intuited, though he cites no passage that is quite as unambiguous as the one we quote here from Carnaps notes.
Interestingly, Gdel himself had once been willing to consider such possibilities: it is very plausible that with [V = L] one is dealing with an absolutely undecidable proposition, on which set theory bifurcates into two different systems, similar to Euclidean and non-Euclidean geometry (Gdel *1939, p. 155, quoted by Parsons 1995a, p. 67)
By W.W. Tait, for instance, who points out that Gdels suggested criterion of inductive success is difficult to reconcile with the iterative conception of the universe of sets; To introduce a new axiom as true on this conception because of its success, would have no more justification than introducing, in the study of Euclidean space, points and lines at infinity because of their success. One may obtain an interesting theory in this way and one worthy of study; but it wont be Euclidean geometry. (Tait 2001, p. 96)
That Gdel was not unconcerned with scientific and practical applications is illustrated in a 1940 conversation with Carnap, in which (according to Carnaps notes), Gdel suggests that an axiom system with concepts that are ordinarily considered metaphysical, like God, soul, ideas could be scientifically very fruitful. Though agreeing that such a system would not be nonsense, as it would be relatable to empirical consequences in the same way as Maxwells electromagnetic field and other theories, Carnap is skeptical. But Gdel says one should try it and see: Decisive progress in science, including physics, is often possible only through change of direction. Carnap still demurs, but Gdel insists the attempt should be made, and we should judge by results. (ASP/RC 102-43-06, 13 November 1940)
We assume, though neither Carnap nor anyone else, as far as we know, has ever attempted to show this rigorously, that (a) and (b) are to some degree incompatible, and must be traded off against each other in any given framework; more of one means less of the other. This is presumably why some philosophers (e.g. Richard McKeon; see Stein 1992, p. 279) have claimed that differences of philosophical principle are fundamentally irremediable. But this is simply to insist on a framework that optimizes (a) alone, while omitting (b); there is no a priori reason for choosing such a framework over one that sacrifices some (a) for some gain in (b).
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