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Revue internationale de philosophie

2005/4 (n° 234)



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1

Godel is routinely called a Platonist in his philosophy of logic and mathematics. What does that mean ? From the point of view of a working mathematician or logician such labels make little difference. For instance, take the idea of antirealism. A typical manifestation of antirealism in the foundations of mathematics is the claim that in mathematical contexts quantifiers do not really express existence and universality with respect to some subject matter, as they do in everyday applications. (Cf. here Benacerraf 1973.) But what is really meant by "not really" here ? What difference does it make for one's actual work in logic ? It is almost certain that when a soi-disant antirealist logician practices model theory, he or she will treat quantifiers as ranging over a class of values, which is tantamount to taking them to express existence and universality in that class of values. And this is likely to be the case no matter whether the logician in question pays homage in his or her philosophical Sunday prayers to Plato or to Michael Dummett.

2

I am of course keenly aware that there are approaches to semantics that are not referential and in which the meaning of quantifiers is supposed to be explained in terms other than existence and universality. I am equally painfully aware that I cannot refute them in one single paper. All I can say is therefore that I find such treatments of semantics to be on a par with attempts to stage Hamlet without the Prince of Denmark. In any case it is eminently clear that Godel wanted to interpret quantifiers in a mathematical context realistically, no matter whether this is thought of as a reason for calling him a Platonist or not.

3

To take another example of this vacuity of current philosophical terminology, almost all philosophers of mathematics call David Hilbert a "formalist", even though on closer examination he turns out to be an axiomatist and even a proto-model-theorist. (Cf. here Hintikka, forfhcoming(a).)

4

Needless to say, even though such no-brainer terms as "realist", "formalist" and "Platonist" do not have any explanatory force or much descriptive content either, they usually hint at something relevant and often interesting. However, this real force of such labels is much subtler than philosophers seem to realize, and has to be uncovered case by case.

5

In this paper I am concerned with what could be meant by Godel's "Platonism". It is far from clear what the import of such a label is apart from historians' purely descriptive taxonomies. Does his Platonism show up in his actual logical and mathematical work and if so how ? If it does not, what is the cash value of calling him a Platonist ? At first sight, it is not obvious what the specific consequences of his Platonism might have been. Does one have to be a Platonist in order to make a distinction between truth and provability ? In order to consider different axioms for set theory and their model-theoretical consequences ? Are set theorists committed to Platonism as soon as they use quantification over sets ? I do not expect to receive interesting answers to such questions. Yet I do believe that something that can be taken as Godel's Platonism throws sharp light on his positions on the map of different ideas about the foundations of mathematics.

6

There is in fact in the light of hindsight a major puzzle about Godel's insights and about the way he put them to use. One of his great achievements, arguably the greatest one, was to show the deductive incompleteness of elementary arithmetic. We know from his own testimony that the crucial step in reaching this result was the realization of the undefinability of arithmetical truth by means of arithmetic itself. However, the two insights, though not unrelated, are essentially different in nature. The undefinability of truth is a semantical result. As a consequence this undefinability can be overcome by enriching the logical language that we are using and modifying its semantics while staying on the first-order level. (See e.g. Hintikka 1996, ch. 6.) This undefinability can essentially be thought of as being due to the poverty of the Frege-Russell logic that Godel was presupposing. And because the undefinability result is semantical it applies to many different kinds of logical and mathematical systems.

7

In contrast, the deductive incompleteness of elementary arithmetic is at bottom a combinatorial result in a wide sense of the term. It says that the set of true propositions of elementary arithmetic is not recursively enumerable. This is an intrinsic property of the set of all arithmetical truths. As such this kind of incompleteness this kind of impossibility cannot be overcome by "completing" the language in question or its logic by means of further concepts and assumptions.

8

The puzzle I find here can perhaps be seen from Tarski's pointed remarks to the effect that Godel applied his idea of the impossibility of defining truth only to arithmetic, and even there only implicitly. (See e.g. Tarski 1944, note 18.) I do not know what Tarski might have had in the back of his mind when he made such remarks, but in the light of hindsight they can be given an interesting twist. This twist can be seen by asking : To what other systems could Godel have applied his insight concerning the undefinability of truth ? One obvious answer is : axiomatic set theory. Later Godel was going to devote decades of his working life to thinking about axiomatic set theory. His thoughts were addressed largely to the search of new axioms and the independence of certain important propositions of the current axioms of set theory.

9

The sixty-four thousand dollar question here is : What would have happened if Godel had instead applied to axiomatic set theory the same lines of thought as he applied to elementary arithmetic ? Admittedly, the kind of counterfactual history that is exemplified by this question is an iffy project, notwithstanding the current fashion of "virtual history" practiced by Niall Ferguson and his peers (Cf. Ferguson 1999 (a),(b).) But counterfactual history can clarify what happened by placing it on the map of different possibilities, just as a chess master's genius can sometimes seen only by considering the moves he could have made if the game had taken a different turn. (Cf. here Pandolfini 1986.)

10

The main features of Godel's situation are in any case clear. The syntax of axiomatic set theory can be represented in set theory itself, just as the syntax of arithmetic can be "arithmetizised" by means of Godel numbering. This is clearly so because arithmetic should naturally be possible to reconstruct within any respectable set theory. A liar-type argument will then show that there cannot exist any syntactical predicate that would capture set-theoretical truth. All this is perfectly straightforward, and was accessible to Godel in 1931.

11

What is not equally obvious is the consequences of these results for the different truth definitions that one might be tempted to propose. What are they ? One of them is connected with the idea of what is called the T-schema. If we had a formula F[x] which for any Godel number x = g(S) of a sentence S is logically equivalent with S, then we would have a truth predicate T(x), for then it would be the case that

12

(1) (Vx)(T(x) F[x])

13

But such a truth definition is impossible by Tarski's result. Why so ? The explanation lies in the fact that the quantifiers in F[x] are in (1) unavoidably dependent on (Vx), which can be seen to make nonsense of (1) as a truth definition. This reason is in a sense syntactical, not purely semantical. Hence it can be understood how Godel, who was led from the undefinability of arithmetical truth to the formal (deductive) incompleteness of elementary arithmetic, could miss the true reason for the impossibility of a first-order truth definition — and missed the fact that this impossibility is consequently due to the expressive limitations of the particular first-order languages he was using. Needless to say, Tarski did not acknowledge this diagnosis, either, even though it is the underlying reason why his T-schema could not be converted into a truth definition

14

It is nevertheless instructive to realize that while Godel's first incompleteness theorem cannot be overcome by using a more flexible logic, his second incompleteness theorem can. The consistency of elementary arithmetic can be proved in elementary arithmetic if instead of the received Frege-Russell logic we use independence-friendly logic. (See Hintikka, forthcoming (a).)

15

All this applies to first-order axiomatic set theory quite as much as to elementary arithmetic — not that anyone expected such an axiom system to be complete. But there is a crucial difference between the two cases which is so simple that it is a little surprising that it has not been discussed by logicians and philosophers — and more that a little surprising to my mind that it was not pointed out by Godel. The difference is that in an interesting sense truth should be definable in axiomatic set theory AX for the same theory. This sense can be explained by reference to a suitable representation of the syntax of AX in AX itself. As was pointed out, there are no obstacles in principle to doing so. One way of doing so — perhaps a little cheap one — is to assume that in such a Godelization — or should one rather say, Tarskification ? — of axiomatic set theory there is a function n(x) which to every set x assigns a suitable set to it as its name. This is not an outrageous demand, however. For instance, if we assume a global axiom of choice, we can take n(x) to be the ordinal of x in some fixed global well-ordering.

16

What we can then do is to take a one-place predicate variable X and say that when applied to the Godel number g(S) of a set-theoretical sentence S it satisfied the requirement of a truth predicate. For instance, it will be required that if it applies to g(S, & S2), it applies to g(S,) and to g(S2). The crucial clauses for quantifiers will be the following :

(2) X(g((3x)F[x])) D (3x)X(g(F[n(x)]))

X(g(Vx)F[x])) D (Vx)X(g(F[n(x)]))

17

If the conjunction of all these requirement is T(X), then the truth predicate is

(3) (3X)(T(X) & X(y))

18

But by Tarski's impossibility result, this cannot be a genuine truth predicate. Hence there must exist a set-theoretical sentence G such that G is true but the following sentence false :

(4) (3X)(T(X) & X(g(G))

19

But what (4) says is that the structure of G is such that it would by our ordinary pretheoretical standards be called true. Hence, intuitively speaking, the following instance of the T-schema will be false

(5) rGn is true ** G

20

In this sense, if the logic of our natural language (or our mathematical language) is on a par with language of axiomatic set theory, there will be violations of the T-schema. This poses a painful dilemma to the adherents of the so-called minimalist approach to truth who rely on the T-schema as their sole guidepost. They have to give up either their minimalist stance or first-order axiomatic set theory.

21

I have pointed out earlier that if minimalists really tried to give a serious structural account of the notion of truth and its logic, they would run into the very same problems that independence-friendly logic is calculated to solve. The dilemma which minimalists have to face here is an example of those problems.

22

More generally, Tarski notwithstanding, we cannot use the T-schema as an adequacy criterion for truth definitions (or conceptions of truth) without giving up axiomatic set theory. Here one is easily moved to ask : Was Tarski somehow aware of this conundrum, and was it what led him to conceive the idea of doing set theory without variables ? (See Tarski and Givant 1987.) If so, Tarski and Godel were worlds apart (or should I say, models apart ?) from each other when it comes to one's view of the prospects of set theory.

23

But T-schema is not the only way in which to think about truth conditions and truth predicates. It is not the most interesting one philosophically, either, for it does not offer any kind of analysis of what it means for a proposition to be true, not even the case of first-order (quantificational) propositions. (This failure has been noted in criticisms of Tarski-type truth definitions. (For an early example, see e.g. Black 1948.) More natural and more instructive truth conditions are nevertheless easily available. (Cf. here Hintikka, forthcoming (b).) A first-order proposition S is true if and only if there exist suitable "witness individuals" vouchsafing its truth. For instance, if S is (3x)F[x], it is true if and only if there exists an individual b satisfying F[x]. A sentence (Vx)(3y)F[x, y] is true if and only if for each individual a there exists a "witness individual" b satisfying F[a, y] and so on. As the latter example shows, witness individuals can depend on other witness individuals. The functions that produce them as their values are known as Skolem functions. The existence of the requisite witness individuals guaranteeing the truth of S thus means the existence of a full array of Skolem functions for S. (Unconditional constant witness individuals can be considered as extreme cases of Skolem functions .)

24

This is indeed our natural pretheoretic conception of the truth conditions of first-order propositions. It is hard to think that Godel was not congnizant of it, especially as Skolem had used the idea of Skolem functions already in the twenties in arguments not entirely different from Godel's completeness proof for first-order logic. Somewhat later, Skolem functions were discussed in the second volume of Hilbert and Bernays (1934-1936).

25

What happens when Godelian ideas are applied to this kind of truth condition in the context of a self-applied axiomatic set theory ? In the same way as in elementary arithmetic we can formulate a quasi-truth-predicate T by means of Skolem functions. This T can be stipulated to apply to g(S) only if there are Skolem functions that satisfy it. For instance, we have

26

(7) T(g((VxX3y)F[x, y]))

27

only if

28

(8) (3f)(Vx) T(g(F[n(x), f(n(x))])

29

Again, Tarski's impossibility result shows that this approach cannot yield a genuine truth predicate. Hence there must be in any model of axiomatic set theory false sentence of the form

30

(9) (S D S*)

31

where S* is the sentence that asserts that there exists a full array of Skolem functions for S. Since this existence is nothing but an expression of our pretheoretical notion of truth, there will be in any model of axiomatic set theory materially false sentences. This presupposes of course that our axiomatic set theory is considered as a set theory and not a theory of some unspecified membership relation whose interpretation need not be set-theoretical.

32

Moreover, suppose that we try to force our quasi-truth-predicate to do its job and adjoin to the axioms of set theory all the sentences of the form (9). What happens then is of course that by Tarski's impossibility result the resulting axiom system is inconsistent. By compactness, there will then be a finite disjunction of negations of sentences of the form (9) provable from the previous axioms of set theory. In other words, there will be materially false sentences provable in our set theory.

33

This is no mean result — or perhaps in a different sense it is mean indeed, for it implies among other things that unprovability of a theorem in axiomatic set theory does not per se tell us anything about the truth or falsity of set theoretical propositions. Ergo, Godel's own results about the consistency of the continuum hypothesis (CH) with the axioms of set theory are not automatically relevant to the question of the truth or falsity of CH, and ditto for Paul Cohen's results.

34

These observations open new perspectives systematically and historically. They mean that first-order axiomatization is not a good way of approaching set-theoretical truth. This in truth suggests that the entire research project of first-order axiomatic set theory is little more than a wild goose chase, if considered as a foundational enterprise and not merely as one particular mathematical theory among many others. Here you can perhaps see why I have called — tongue in cheek — axiomatic set theory "Fraenkelstein's Monster". Arguments from silence are dangerous, but even knowing the danger I cannot help wondering whether it is entirely a historical accident that all the incredibly ingenious studies of axiomatic set theory have not even produced a solution to the continuum problem.

35

Even apart from Godel, what has been pointed out here has important repercussions on our overall research strategies in the foundations of mathematics. Even though logicians and mathematicians have not always acknowledged their motives, one important reason for resorting to set theory in the foundations of mathematics has been the failure of first-order languages of expressing their own metatheory. Truth is impossible to define for a theory using the received first-order logic according to Tarski's result, and logical necessity cannot be expressed by any syntactical predicate, either. (See here e.g. Montague (1963) who develops further ideas codified in Godel's second incompleteness theorem.) It has apparently been thought that in set theory we can speak at the same time of our language and of the set-theoretical structures it can serve to represent. This idea of set theory as "poor man's model theory" now turns out to be an instance of wishful thinking. The semantics of set theory cannot be theorized about in axiomatic set theory. Here the old developments sometimes called semantical paradoxes suddenly receive new interest. (They are exemplified e.g. by Richard's paradox.) As Frank Ramsey (1926) in effect forcefully and correctly pointed out such quasi-paradoxes do not matter for the usual axiomatic treatments of the foundations of mathematics. However, they can be taken to bring out limitations of such foundational themes when one tries to express their semantics in the theory itself. Speaking personally, when I came to realize this flaw of all first-order axiomatic set theories, I was not sure whether I was more surprised by the insight itself or by the fact that people had not reached it long ago. And the test case of those people is undoubtedly Godel. We have seen that this result was easily accessible to Godel in the thirties. But if he had reached it, he could scarcely have spent years' and years' efforts on enterprises that are not obviously relevant to the most important basic questions in set theory. One's name does not have to be Niall Ferguson in order to wonder most seriously what would have happened if Godel had realized to the development of set theory and more generally to the foundations of mathematics. Would there exist any mathematicians in that possible world who would think of set theory as the natural medium of mathematical reasoning ad systematization ? I doubt it.

36

But even if you have no taste for virtual history, you are facing an intriguing question here. Why did Godel not realize the consequences of his own insights for set theory ? It is not a full answer to surmise that he did not know, or did not pay enough attention to, the notion of Skolem function. No reader of Hilbert & Bernays (1934-36) could not have been unaware of the notion, and in any case Godel was eminently capable of re-inventing the notion by himself at the drop of a quantifier.

37

One partial explanation of Godel's failure — or refusal — to follow the line of thought presented here lies in the nature of the truth predicates explained above. They are in terms of languages with an infinite number of individual constants. Godel's idea of a formal system might have been too puritanical to allow for infinite languages even in this minimal sense.

38

This cannot be a complete explanation, however. For some of the consequences of the existence of a truth predicate are independent of the use of an infinity of individual constants. In particular, this applies to the existence of false theorems.

39

Moreover, as I have pointed out elsewhere (Hintikka forfhcoming(c)), a somewhat less paradoxical result can be obtained without considering truth-definitions at all, by a simple modification of the usual proofs of Godel's first incompleteness theorem. We can reconstruct the syntax of axiomatic set theory in itself, as we can reconstruct the syntax of elementary arithmetic in itself. Then we can define a numerical predicate Sc(x) which is the Godel number of (S D S*) where x is the Godel number of S and S* is the sentence asserting the existence of all Skolem functions for S. If Prov(y) is the provability predicate (of Godel numbers of sentences), then by the diagonal argument there is a number n represented by the numeral n in such that n is the Godel number of

40

(10) -. Prov(Sc(n).

41

It is easily seen that (10) is true but unprovable. If our basic logic is the received first-order logic, which is complete, this implies that in some models of our set theory there is a true sentence whose Skolem functions do not exist. This is already a clear violation of our pretheoretical notion of truth, and enough to show that first-order axiomatic set theory is a poor guide to set-theoretical truth.

42

If Godel had realized these consequences, he would scarcely have spent as much time and energy as he did thinking about possible new axioms of set theory. For if false theorems are derivable already from the present axioms, they cannot be eliminated by introducing new ones.

43

Someone might attempt to answer the question psychohistorically. For all of his revolutionary accomplishments, Godel was in different walks of life a conservative thinker who needed some safe accepted field where to operate. Within that field, Godel discovered all its hidden problems and limitations, but he was not temperamentally prepared to venture beyond it, much less to replace it in its entirety. In social and political matters, he shocked Einstein by voting for General Eisenhower rather than for Adlai Stevenson, but he could find a clever series of steps that could theoretically turn the United States into a dictatorship completely constitutionally, of course without any thought of anybody's actually doing so. Likewise, as Solomon Feferman (1984) has pointed out, he put forward the ideas that prompted the next major developments in logic, including model theory and recursive function theory, but never contributed to the systematic construction of these new disciplines. Thus it might be suggested that in set theory, too, Godel considered the axiomatic theory created by Zermelo, Fraenkel and others as the safe home field which he was loath to abandon.

44

Purely historically, this may perhaps be what happened. But we are doing Godel serious injustice, I believe, if we do not also heed his intellectual reasons to do what he did. Indeed, I have been impressed more than once by how even Godel's most idiosyncratic ideas are grounded in his general theoretical ideas. For instance, I have argued that Godel's qualified acceptance of the ontological argument for God's existence is closely related to his general actualist ("one-world") stance. (See here Hintikka 1998.)

45

So why did Godel overlook the subversive lines of thought concerning axiomatic set theory that I have sketched, even though they scarcely went beyond his own ideas ? One can generalize this question and ask why any working mathematician ever thought that first-order axiomatic set theory is the medium of all mathematical reasoning. (I have seen a professional mathematician considering the entire body of mathematics as so many deductions from the axioms of the Zermelo-Fraenkel set theory.) But axiomatic set theory is not a logic, even though in practice the things it is supposed to do could in some sense be done by means of second-order logic. Set theory is not just a system of rules of inference. It is formulated as a theory about a given universe discourse, that is, some part of reality in principle on a par with Euclidean geometry as a theory of a certain kind of space, not to speak of axiomatizations of thermodynamics or quantum theory. There is something more than a little awkward in proposing to formulate principles of logical inference as truths about some particular domain of entities. What kind of reality is set theory supposed to be a theory of ? (It does not matter if we acknowledge that it is only a theory of the structure of a certain kind of actual or possible reality.)

46

It is here that Godel's "Platonism" becomes relevant. Godel acknowledged in so many words that in logic and in set theory we are dealing with a universe of entities of a certain kind, viz. structures and that we are dealing with them in a manner that is analogous with a natural scientist's study of natural objects. (Godel 1944) In this analogy, a scientist's sense perception has according to Godel a counterpart in mathematical intuition. Thus here we can finally see the real manifestations of what might be called Godel's Platonism. He in some sense thought of the set-theoretical universe as a realm of actually existing entities, albeit abstract ones. This was for him an important idea which guided his thinking about set theory. If the term "Platonism" is calculated to call attention to the fact that Godel dealt with set theory on a par with such aspects of the actual reality, then I have no objections to using it.

47

But, even after having admitted that, one cannot help asking further questions. For one thing, we clearly cannot simply identify Godel's Platonism with his realistic attitude to the objects of set theory. Even more importantly, we have to ask : What is more basic here, a certain philosophical — virtually metaphysical — doctrine or a certain methodological preference ? Should we see Godel's Platonism as a rationalization of his methodological approach to this area ? I do not think that we can, or should, opt for either answer. The important thing is to be aware of both perspectives. For one thing, Godel's Platonistic doctrine has important systematic connections with his other views, especially with his actualism. Like Frege and Russell (at least the sufficiently early Russell) he believed that logical and mathematical truths are truths about the actual world, albeit about its more abstract regions. He quoted with approval Russell's (1919, p.169) saying that logical truths are quite as much about the actual world as the truths of zoology, although about its more abstract aspects. Hence we certainly have to take Godel's Platonism seriously as a theoretical position. For one thing, it certainly could serve as a rationale of an axiomatic treatment of set theory.

48

It may nevertheless be suggested that what characterizes Godel's views is not so much his so-called Platonism per se as a combination of a reliance of abstract entities (whether you call this reliance Platonism or realism) and his actualism. A philosopher or a logician can for instance rely on his or her reasoning on the idea of possible worlds. They are abstract entities if anything is, and one can adopt as realistic an attitude toward them as Meinong adopted toward his possible objects. But that is compatible with thinking of each possible world completely nominalistically, as a structure of particular objects. Is a logician who does so ipso facto a Platonist ? I for one am not interested in an answer. What may have been most characteristic of Godel is not just his taking all and sundry abstract entities realistically, but doing so only to entities that can be assigned a home in this actual world of ours, albeit in a special higher region of it. This makes a huge difference conceptually, for the job description of possible worlds is to embody the different alternative possibilities that there may be concerning some part or aspect of reality. In contrast, only one configuration of abstract entities can exist in the upstairs part of our one and only real world. All this illustrates further the ambiguities and subtleties of notions like Platonism or realism. And some of those subtleties affect also the idea of a "structure of all structures". Is that structure to be thought of as a part of our real world, or as a set of alternatives to it ? Godel apparently opted for the former alternative, even though his reasons are not clear. For instance, he never put the notion of possible world to any systematic use, even though it was the central concept of his admired Leibniz. Apparently a Platonist need not accept all the different kinds of Platonic entities.

49

Yet one can still ask what the precise consequences of Godel's philosophical position are. With a practicing logician like Godel, no matter how keenly aware he is of philosophical issues, it is hard not to suspect that his commitment to a philosophical position like Platonism is more an indication of his preferences than a guideline for actual thinking and argumentation. A test case if offered by the axiom of choice. If there ever was an issue to which Platonism dictates an unequivocal answer, it is the question of the existence of the choice functions that this axiom postulates. To borrow Russell's proverbial illustration (Russell 1919, p.126), we may be blissfully or banefully ignorant of any particular function which for each one of our infinity of pairs of socks picks out one of them, but surely in a Platonist's combinatorial heaven there exists such a function. Hence a Platonist can be expected to accept the axiom of choice unconditionally. Yet in his work on the axiomatics of set theory, Godel considered models of axiomatic set theory in which the axiom of choice is not true. This is a telling example of the irrelevance of a logician's philosophical beliefs to his or her actual research.

50

Or do we have here an instance of Godel's conservatism ? Did he perhaps once again think a move or two further in the game of logical theory than other players ? When truth conditions are formulated as requiring the existence of full sets of Skolem functions for all true sentences, they are nothing but instances of a suitably generalized and strengthened axiom of choice. If your intuitions lead you to accept the usual axiom of choice, you must by the same toke accept the kind of truth definition described above. But, as was explained above, in a set theoretical context the codified acceptance of all such truth conditions gives rise to a contradiction. (By codified acceptance, I mean of course the acceptance of all sentences of the form (S D S*) as axioms, where S* asserts the existence of a full array of Skolem functions for S.) Perhaps Godel saw this, but did not want to give up in one fell swoop the entire structure of first-order axiomatic set theories. But if so, a critic is entitled to ask whether he really trusted his own Platonistic intuitions about the acceptability of the axiom of choice.

51

It does not contradict the attribution of Platonism to Godel, either, if I surmise that we might as a matter of philosophical research strategy gain better insights into Godel's ideas by considering them in their historical context rather than by thinking of them as timeless theses carved in the stone of a structure of eternal philosophical ideas. In particular, it seems to me interesting to ask where Godel's so-called Platonism came from. Was it Godel's own pet idea, derived from his own intuitions, or did it have a history ? If it did, who before him had thought of the set-theoretical universe as a structure of all possible structures ? I have a tentative — very tentative — answer to this question that is unfortunately hard to document. The answer is : Almost everybody in the foundations of mathematics in Godel's immediate background.

52

This answer may seem surprising, but perhaps it should not really be so. Why should there have been a general interest in the totality of all possible structures ? One of the initial reasons came from non-Euclidean geometries. Their apparent deductive consistency did not persuade everybody of their reality. Such a persuasion was reached only by actually constructing models for them. Some of them were somewhat ad hoc, as Felix Klein's relative interpretation of a certain non-Euclidian geometry within the Euclidean one. However an overview over such models could be reached only through some more general theory of different structures. An example from the foundations geometry is Riemann's theory of what he called Marmigfaltigkeiten (manifolds). But the need of such structures was not restricted to geometry. Systems of axioms in other parts of mathematics needed models, too. Where do we find them ? Somehow an only partly articulated idea seems to have gained currency according to which we need a general study of all different models, somewhat in the same way as different kinds of geometry could be thought of as particular cases of the manifolds Riemann studied. The subject matter of such a theory would be the totality of all different models which in mathematical practice means all different structures. This "structure of all structures" was sometimes assimilated to the universe of set theory, which was initially also called Mannigfaltigkeitslehre.

53

There are indications that this semi-identification persisted longer than philosophers in our days realize. To give but one example, for what kind of universe of discourse did Tarski (1936) think he was defining truth in his famous monograph ? Present-day philosophers and logicians often think that Tarski was showing how to define truth in a model (structure). Tarski distances himself from such interpretations (Tarski 1935, p.200 of the English translation). At one point (p. 199) he considers the structures Hilbert and his school had dealt with as a subset of his universe of discourse. Thus it is hard to avoid the impression that Tarski is in his own mind defining truth in some "structure of all structures", possibly in some kind of set-theoretical universe. In fact, Tarski was ostensibly dealing with a theory of classes. Perhaps — just perhaps — those classes should be thought of, not as subsets of some given superset, but as extensions of all the different possible models (structures) there are.

54

It must be pointed out, however, that later — and not that much later — Tarski strongly opposed any idea of a single universal language of science. I have not seen any analysis of the development of his ideas and I do not know what happened but there is some evidence in Tarski (1935) itself suggesting that his views were changing in the early thirties. (Cf. the appendix added in the 1935 German version of his monograph.)

55

Hilbert, too, seems to have held some version of the same idea of a universe of all possible structures. At the very least, this would enable us to make sense of his strange claim that consistency is a sufficient condition of existence. This claim is not only strange in itself, it is especially strange in Hilbert's mouth. Hilbert was an axiomatist. He envisaged each axiom system as delineating a class of structures as its models. We can then study these structures by deducing logical consequences from the axioms. As such an axiomatist, Hilbert was keenly aware that for many such models there exist individuals that are consistent with the axioms and with the model but which do not exist in it. Otherwise Hilbert would not have needed his nortorious "axiom of completeness" in his systematization of geometry. (See Hilbert 1903.) Hilbert's proposed inference from consistency to existence becomes obvious, however, if we assume that he is speaking of existence of structures in a super-universe of all possible structures. Indeed, this structure comprises by definition all structures specified by consistent propositions or consistent sets of propositions including consistent axiom systems.

56

Hilbert's position differs from Godel's in that the structures Godel is thinking of are structures that involve essentially general concepts roughly in the same sense in which Frege's Begriffsschrift dealt with concepts (intensions) rather than mere extensional structures. In Frege (1895), p. 455 of the original, he says that "in the conflict between extensional and intensional logicians" he holds that "the concept is logically prior to its extension." Admittedly, for Godel set theory deals with extensions, but this is so only because it is the extensional aspect of a general theory of concepts. (See Hao Wang 1996, ch.8.) And the best known model Godel actually invented, the constructive model of axiomatic set theory, was essentially a structure of sets, not of individuals.

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Thus what is properly meant by Godel's Platonism is not sufficiently characterized by his use of the "structure of all structures" idea. It also has to be the case that those structures involve higher-order entities, presumably in the sense of their being quantified over. Perhaps it may also be required that concepts rather than their extensions have to be involved. Ironically, it seems to me unmistakable that the most important manifestation of the ways of thinking by Godel that have been labeled "Platonistic" is the "structure of all structures" idea, reflected in the idea of treating set theory as a realistic axiomatic theory. This illustrates vividly the relative unimportance of the philosophical labels like "Platonism". As far as the actual working problems in the foundations of mathematics are concerned, a consequence of Godel's "Platonism" turns out to be much more interesting than the label itself, a consequence that is even shared by some non-Platonists.

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A belief in a "structure of all structures" on the part of Godel and others might also help to understand why they thought that all the different modes of mathematical reasoning could be discovered by studying one particular set-theoretical universe. Such a study would have to include a study of the interrelations of different structures, which could then serve as a basis of a theory of mathematical inferences. These thus might be here a link between Godel's Platonism and his methodological approach. Why did he cultivate set theory and not second-order logic ? Perhaps we have here the beginning of an explanation.

59

There are other indications of the currency of the "structure of all structures" idea in the literature, although even the best of them are only faintly smoking guns. In any case, the study of manifolds in the generalized geometrical sense grew apart from set theory. The outlines of Godel's background are nevertheless clear. In a historical perspective, his "Platonistic" idea of set theory as the science of all possible structures is unmistakably a latter-day version of the "structure of all structures" idea that has just been described. Awareness of this ancestry of Godel "s ideas is a far better guide to understanding them than pidgeonholing them in rigid philosophical categories like "Platonism".

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Among other things, the historical story suggests that we have to make further distinctions in speaking of Godel's Platonism. In order to see this, let us assume that the interpretation sketched above of Hilbert's position is correct and that he, too, believed in some kind of structure of all structures. Would that make him a Platonist ? I would hesitate to say so, for Hilbert makes it clear that the structures he wants mathematicians to consider are structures of particular objects. (See Hilbert 1922, p. 262, and cf Hintikka, forthcoming (a).) In other words, for Hilbert the "structure of all structures" idea is apparently compatible with a form of nominalism. Accordingly, the label "Platonist" does not apply very happily to Hilbert.

61

Somewhat surprisingly — it was certainly a surprise to me — the thinker that articulated most explicitly the "structure of all structures" idea turns out to have been Edmund Husserl. (See the materials published in Husserl 1970.) His views changed over time, but during the crucial period in the early years of the twentieth century he definitely held the kind of vision that has been referred to here as the idea of structure of all structured. As might be expected, he related this idea both to Cantor's set theory, aka Cantor's Mannigfaltigkeitslehre, and to Riemann's namesake theory of different kinds of manifolds. Husserl was also interested in how an axiomatic theory determines its models, including the unique determination of a model by a complete (or, rather in our present-day terminology, categorical) theory. Hence the structure of all structures came to mean for him a study of a theory of all theories. It is therefore not surprising that Godel was fascinated by Husserl's ideas and studied them seriously.

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Husserl's ideas in this direction are lent some additional interest by his association with Hilbert while his stay in Gottingen. In particular Husserl claimed that his ideas about completeness were closely related to Hilbert's.

63

Moreover, Husserl's proposed phenomenological study of the structure of all structures was apparently calculated to involve general concepts and did not pertain only to structures of individuals. In this respect, Husserl's preferences seem to have been somewhat different from Hilbert's. However, they agreed with Godel's in this respect. For Husserl, too, there existed an a priori study of formal essences. In this sense, they were both Platonists.

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However, for Husserl, essences were not given by what he called intuition (Anschauung), which for Husserly meant simply immediate givenness of any sort. The essences had to be extracted from the given by the process Husserl called Wesensschau. Neither the Husserlian notion of Anschauung nor the notion of Wesensschau seem to have precise analogues in Godel's thinking. In particular, Godel's notion of intuition as providing us with putative new truths is different from both these Husserlian notions. It differs from Husserl's Anschauung which only give us our most directly known truths, while Godelian intuition can pertain to complicated general truths and also can be suggestive rather than conclusive. It differs from Wesensschau which give us essences rather than truths.

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Hence interpreters face here some intriguing questions. What was it that fascinated Godel in Husserl's ideas ? Was it the axiomatic approach to set theory ? The idea of a structure of all structures ? The idea that we can access general essences by means of Wesensschau ? Or was it the archetypally Platonic idea of an abstract reason of the actual world populated by abstract entities ?

66

Future research will be needed to answer all these questions. Yet some observations can already be made. For instance, even though Godel's idea of intuition does not match its Husserlian namesake, it could be compared with Husserl's Wesensschau which reveals at one and the same time to us general concepts and truths about them.


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Pour citer cet article

Hintikka Jaakko, « What Platonism ? Reflections on the thought of Kurt Godel », Revue internationale de philosophie 4/ 2005 (n° 234), p. 535-552
URL : www.cairn.info/revue-internationale-de-philosophie-2005-4-page-535.htm.


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