Ramsey and the Vienna Circle 07

E CKEHART K ÖHLER

RAMSEY AND THE VIENNA CIRCLE ON LOGICISM

1. S CANDAL OF P HILOSOPHY AND M ATHEMATICS I think it is a scandal of philosophy that Logicism – the reducibility of mathematics to logic – receives so little attention anymore, as if it were dead. It is not dead. Instead it has been shamelessly abandoned. Issues for which some of the greatest thinkers such as Frege, Russell and Ramsey fought passionately now barely elicit yawns from “sophisticated” philosophical logicians, who seem to assume that the issue was long ago “settled” in the twenties. Oddly enough, the issue of what relation logic has to mathematics has in the meantime been settled by default in another way: mathematics departments have simply requisitioned logic from philosophy as one of the specialties of mathematics, on the one hand – philosophy having given up the mathematized beast they never had much patience with anyway. But as such a specialty, on the other hand, the mathematicians have, by way of compensation, at least given set theory back to logic, where it has always belonged.1 I first got acquainted with Logicism in Stegmüller’s philosophical institute at the University of Munich in 1960, when Frege was still virtually unknown. Once I read his analysis of the natural number concept, especially the concept of the ancestrals using his new theory of relations, and his sophisticated treatment of measurement,2 I became completely convinced that mathematics was reducible to logic and that the main thesis of Logicism was proven. When I later read more literature on foundations of mathematics, I was quite distressed to discover that, in the meantime, Logicism was widely regarded as a failure.3 What had happened? Well, the antinomies had happened, stopping Frege, and Russell had taken over the torch. Logicism had become a complete hostage of special, newly discovered thorny problems in the logic of Russell which I never thought were so immediately relevant to the reduction of mathematics. Russell created Type Theory, with all its special complexities and artificialities, getting caught in thickets of controversies and blind alleys. What Russell did or did not achieve is still a central question concerning Logicism even now – all the more so because Carnap’s logic (in particular the Logical Syntax of Language) uses a variant of Russell’s Type Theory. Gödel (1944) wrote the severest and most exacting critique of Russell’s contributions, including comments on and praise for Ramsey’s 91 M. C. Galavotti (ed.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, 91–121. © 2006 Springer. Printed in the Netherlands.

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refinements. Nevertheless, the complexities of Type Theory unfairly draw attention away from the central claim of Logicism, which should be judged in partial isolation – a standpoint shared by Boolos (1998) shortly before his death, and by Clark (2004). Part of the problem lies simply in Frege’s having raised expectations of rigor to a very high degree in his effort to demonstrate that no intuitions extraneous to logic, no Anschauungen were surreptitiously slipping in to spoil the reduction of mathematics to pure logic. If he had merely kept to the lower level of rigor which Cantor, Dedekind and Zermelo applied, he would perhaps have escaped much of the onus he suffered, as Cantor did with his comparatively informal approach. Instead, Frege (1879) developed his famous Begriffschrift, the world’s first “logistic system” (Church), comparable in rigor to a programming language and only surpassed by Gödel when proving the Incompleteness Theorems.4 Russell in his Type Theory backslid considerably from Frege’s level of rigor back to that of Peano, who had been the “discovery of his life” (not Frege!). But even that level of rigor was sufficient to quickly entangle him in many distracting thorny issues. As everybody knows, Ramsey played a major role in rescuing Type Theory from some of these thorns, but unfortunately his yeoman work was insufficient to persuade either the philosophical community or mathematicians of the virtues of Logicism. One of the worst aspects of Russell’s original Type Theory involved “Ramified Types” or orders, used to solve semantic or intensional antinomies such as Richard’s paradox. Ramsey simply pointed out that the reduction of mathematics to logic doesn’t require any intensional principles, so ramification could be eliminated. In retrospect, we may say that Russell was simply trying to do too many things at once: in his zeal to make logic do all things for all seasons, attempting to realize Leibniz’s dream of a characteristica universalis by creating a universal language, Russell crammed machinery into his formalism to deal with a few semantic issues which we nowadays treat much better in semantic metatheories.5 Instead, he should have ignored some of his ambitions and concentrated on the most important issue in the grand British tradition of dividing and conquering: that of reducing mathematics to a more streamlined logic. And in the same years that Russell was developing his baroque Ramified Type Theory, more practical-minded mathematicians such as Zermelo were “streamlining” Cantor’s jungle of set theory, coming up with a axiom systems (Z, ZF, NB) paradigmatic for the majority of mathematicians today. Bingo! Classical mathematics is easily reducible to set theory! It is peculiar that practically every mathematician is roughly familiar with the fact that, in both major variants of set theory, Zermelo–Fraenkel (ZF) and von Neumann–Bernays–Gödel (NBG),6 as well as in the systems of Quine (NF and ML), natural number theory and analysis are straightforwardly derivable, once suitable models for progressions and continuous point sets are set up. And yet, although set theory has meanwhile been conventionally classified as a branch of logic, one would think the issue of Logicism ought to be regarded as settled positively – open-and-shut case! Even if one still regarded set theory as contro-

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versial (which “good” logic is not supposed to be!), it is so primarily with respect to exotic questions of large cardinality axioms of uncertain relevance for classical mathematics, and with respect to the fact that no axioms have been found which imply the Continuum Hypothesis (CH), whose solution will always remain the ultimate goal of set theory and all of mathematics since Cantor and Hilbert. But we are farther away from this goal than we ever seemed to be before. There is a patent injustice in the inconsistent ways in which mathematicians judge the success of the Bolzano–Weierstrass reduction program for analysis and the closely related Frege–Russell reduction program for arithmetic. Since the old Bolzano–Weierstrass program of “arithmetizing” analysis has met with very widespread approval (except perhaps among the lovers of the recently revived infinitesimal theory, and among the minority of constructivists – and even Brouwer was a radical arithmetizer), and since this program uses the same set theory sometimes regarded as too dubious for the reduction of arithmetic, it would seem patently unjust to withhold approval of the “remaining” reduction of cardinal-number arithmetic to set theory alone. It seems two significantly different criteria are being applied here, for if Frege–Russell Logicism is rejected because of doubts about Type Theory (a variety of set theory), then the Bolzano– Weierstrass arithmetization of analysis should also be rejected for the same doubts. Conversely, if set theory is sound enough for Bolzano–Weierstrass, it is sound enough for Frege–Russell. So there is a strange reluctance to concede victory to Logicists – as if mathematics were to lose its freedom and integrity if hordes of impertinent logicians and proof theoreticians were suddenly to invade analysis and topology seminars and begin inquisitions on proofs and axioms! (Come to think of it, maybe that would be good at times!) At the same time that mathematicians were being unfair to logic, logicians were being eccentric, too. Even the great disciple of Russell and Carnap, Van Orman Quine, ultimately put set theory outside of logic!7 It is not supposed to be logic’s business to make such contentual, ontologically committing claims; it should rather remain just “structural”; but set theory is famous particularly for claiming existence of infinite sets – most notoriously for power sets. Kneale & Kneale (1962) take the same position as Quine in restricting logic to first-order predicate theory (only second-order and higher-order theories explicitly assume concepts or sets). This seems absurd to me. Logic always dealt with concepts and concept formation, with their intensions and extensions, in addition to propositions and inference; and sets are extensions of concepts (Frege).8 Indeed, as the early Russell saw, doing set theory doesn’t even require sets, all it needs is concepts (his intensional “propositional functions”). But of course, the pioneer of set theory, Cantor, was officially a mathematician, not a philosopher, and his formulations obscured the connection with the logic of concepts. His ally, the mathematician Dedekind, was fairer: he proclaimed that set theory definitely belongs to logic and fell under the “laws of reasoning”. The way Frege separated his symbols into those designating apparently intensional concepts9 and

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extensional objects (Gegenstände), including sets (Wertverläufe), made it seem as if they were fundamentally different, and Russell inherited this view. But Ramsey saw the light: there is no fundamental difference between concepts and objects, both depend equally on their role in propositions. This is close to the position Carnap (1947) came to, in which all signs in the language have both extensions and intensions assigned to them, where individual objects have “individual concepts” assigned to them. The membership relationship of set theory ∈ is really identical to the satisfaction or fulfilment relationship between properties and the objects falling under them, and in this sense, it belongs to logic if ever anything does. Since all existence claims of set theory may be regarded as statements about ∈ , (hence about satisfaction), they are statements about logic. Furthermore, existence claims for sets can all be formulated syncategorematically, i.e. without using empirical concepts. This makes them all candidates – acceptable or not – for logical postulates. We see the main problem was unclarity on the nature of logic in general, and in particular whether certain axioms, principles or rules were logically valid, such as the Axiom of Infinity; and a secondary problem was finding an acceptable boundary line between logic and mathematics. I shall say more about these issues in the course of this paper. It should of course never be forgotten how blindingly new modern logic really is, considering the considerable age of logic; how long it took philosophy to recover from the cultural devastation of the Dark Ages; how late the great publications of Frege came; and how long the great predecessors of Frege and Russell, Leibniz and Bolzano, remained largely unpublished or unread in old journals, virtually unknown among academic philosophers. People’s judgments concerning mathematics and logic are dominated by intuitions which stubbornly resist change and remain anchored in the half-knowledge and prejudices of their age. 2. T HE U NCERTAIN B OUNDARY BETWEEN L OGIC AND M ATHEMATICS At the beginning of the twentieth century, when the Logicist claims of Dedekind, Peano, and especially Frege and Russell first attracted attention, Henri Poincaré, always somewhat conservative, doubted the new doctrine. To be sure, there is a certain justice to one of Poincaré’s objections to Logicism: that it illicitly hijacks mathematical motifs into its new logic, and only by doing this can it get the Principle of Complete Induction, for example. Frege’s theory of relations was “brand new”, not accepted in the classical cannon of logic, because this was monomaniacally fixated on the (monadic) subject-predicate forms of Syllogistics, due to Aristotle’s domination of Scholasticism. Hence Poincaré, with seeming plausibility, could argue that Frege’s treatment of concepts as functions (enthusiastically taken over by Russell as “propositional functions”) simply incorporates mathematical functions right into the basis of the new logic. Thus, Frege’s

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(1879) reduction of complete mathematical induction to ancestrals of relations could be seen as a reduction of mathematics to mathematics, so far as relation theory is regarded as a theory of (mathematical) functions in disguise. Despite Poincaré, most people nevertheless agreed that relation theory naturally did belong to logic, simply because relations treated as two-placed (dyadic) predicates so obviously seem to be extended from the notion of one-placed predicates, used in Syllogistics, as the grammar of ordinary language persuades us. In addition, despite the “second-rank” status of the logic of relations, various principles belonging to it are repeatedly mentioned or used in traditional logical literature, where it seemed to take a natural place, prominently even in Aristotle. In retrospect it seems amazing to us now how hard a struggle Leibniz, perhaps the most brilliant philosopher in history, had with relations and orderings.10 Much more deeply perplexing was the status of Cantor’s set theory, to which Whitehead & Russell devoted major attention in the Principia Mathematica. Set theory was in a long tradition of generalizations in algebra, especially applications of function theory using the notion of series. Sets obviously fit into the mathematical zoo together with groups, fields and series. Russell saw no problem reinterpreting set theory as logic, since sets were easily identifiable with classes interpreted as extensions of one-place attributes or predicates.11 Here Russell was influenced by Frege’s notion of “range of values” (Wertverläufe); Russell already had clearly enough spelled out his view independently of Frege in his Principles of Mathematics (1903), following traditional lines established by Venn, Boole and Dodgson (Lewis Carroll) on the way in which extensional classes are related to intensional properties: Just as Frege had characterized his “Sinn” as “die Art des Gegebenseins eines Gegenstands” (sense is the manner in which objects are presented), Russell originally thought of classes as being generated by intensional procedures.12 That is, every (intensional) property has a procedure which discriminates objects satisfying the property into a class obtained by running through the procedure. Russell naturally supposed that intensions were mental, in the tradition of Idealism, whereas classes (or sets) were thought to be extra-mental, quasi-physical (Gödel).13 Russell’s approach seemed to crash into a dead end when Zermelo (1903, 1908) published his famous proof of the Well-Ordering Theorem using his newly discovered Axiom of Choice (AC). The Axiom of Choice immediately became a great bone of contention between Constructivists and Platonists (or Realists).14 Russell had by this time, after his pioneering study on Leibniz (1900), become a Platonist and was perfectly willing to grant the validity of the AC (which Whitehead had discovered independently of Zermelo and had called the Multiplicative Axiom), especially since the AC was crucial for several famous mathematical theorems, as Zermelo had made clear. But what did worry Russell quite a bit was that the AC seemed to guarantee the existence of sets independently of generation by any intensional classification procedure. If the AC somehow was a characteristic set theoretical axiom, this fact seemed to imply that sets were sui generis and existed independently of being generated by any (mental or logical)

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classifying procedure; or similarly, that they were “quasi-physical” in Gödel’s term. Consequently set theory now seemed to Russell to be independent of logic! Frege also came to just this conclusion after Russell sent him news of his famous Antinomy in 1902, and Frege abandoned the disgraced Grundgesetz V which converted equivalences of intensional properties into identities of extensional sets. Apparently as a result of breaking the previously assumed bond with the mental, most mathematicians began regarding sets as quasi-physical entities sui generis, typically beginning with the famous “purely mathematical” notion of the null set and iterating all other sets from that narrow starting point. I don’t see what the fuss is about. Russell should have interpreted the AC not as divorcing choice sets from intensional classifications (i.e., concepts), but rather as simply making the admittedly rather powerful claim that, even in the infinite, some selection procedure can always be devised by a sufficiently strong mind to generate any choice class. (The “strong mind” simply runs through all subsets, randomly selects a member of each and assigns it to the choice set.) This is perfectly compatible with Idealism; the problem is merely that it accentuates the theological, perhaps Hegelian, side of Idealism; in particular, the strong mind generating the choice class would be outside the realm of the elements. Mathematics routinely makes many extreme and transfinite rationality assumptions like this, e.g. that ʌ can be uniquely determined by the completion of an infinite calculation. Mathematicians accept transcendental numbers which can’t even be calculated like ʌ . So then what’s so special about the AC? I say: not much! Ramsey (1926a, §V) himself was sure that under his interpretation the AC was “the most evident tautology” (albeit, under Russell’s original interpretation, the AC was “really doubtful”). Moreover, all sets, just like the classes in von Neumann–Bernays set theory, can be regarded as extensions of properties; and indeed all sets, whether Zermelo-Fraenkel’s or von Neumann–Bernays’s, can be reduced to intensions, as Carnap, Church and others later showed from the forties onward: Set theory can always be embedded within concept theory. I will return to this later when I discuss Quine’s Set Theory and Its Logic. 3. T ROUBLES WITH T YPE T HEORY , R AMSEY ’ S R ESCUE , AND V IENNA By the twenties, opinions on the success or failure of Logicism focused on Russell’s Type Theory, since Russell was the most prominent Logicist.15 It was a serious mistake to restrict such opinions to Type Theory without also considering Zermelo’s axiomatized set theory of 1908, later extended by Fraenkel, since set theory competes with Type Theory as a logic. Frege could or even should have taken the path of using Zermelo’s comprehension axioms instead of his Grundgesetz V. Frege’s Logicist program thus became unfortunately and unnecessarily tied to the misfortunes of Russell’s procrustean Type Theory. A balanced assessment of Logicism’s fate taking set theory into consideration would have resulted in a much more favorable reception by the ’30s. To be sure, Type

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Theory is to an extent completely natural the way it forms hierarchies; but the restrictions it poses on theory formation makes it at the same time distressingly unnatural, forcing us into contortions when approaching even elementary mathematical problems. For example, in analysis, it leads to rather artificial difficulties with the concepts of limits and accumulation points. Such problems provided Quine (1940) with his principal motivation for using set theory rather than Type Theory. Thus attention was fixated on the special problems which Type Theory faced, which the main alternative, Zermelo’s set theory, did not. Briefly, those problems included 1) the Byzantine labyrinths of ramified types and its horridly ad hoc Reducibility Axiom, and 2) the Axiom of Infinity, needed in order to obtain number theory on the third type level. In addition, 3) the Axiom of Choice (AC) was regarded as somehow more dubious in Type Theory than in set theory, because it seemed less plausible to consider it to be logically valid than as mathematically useful. At this point, the young and promising Ramsey entered the picture, amazingly gifted and solid both in his technical mathematical ability as well as in his philosophical judgment. And he was a glowing adherent of the idea of Logicism, so he was very set on revising Type Theory sufficiently to persuade philosophers and mathematicians of the virtues of Logicism. This effort early on caught everyone’s attention in the Vienna Circle, from Hahn’s to Carnap’s and the younger students like Gödel’s. Most attractive was Ramsey’s elimination of ramified types and the concomitant Reducibility Axiom,16 immediately making Type Theory much more presentable. This was Ramsey’s main influence on Carnap’s Logical Syntax of Language (1934), duly honored by Carnap in §60a and elsewhere in that work. Ramsey’s approach was guided by his famous classification of antinomies into two classes, which we may call intensional and extensional, or perhaps better semantical and object-theoretical; Ramsey found that Ramified Type Theory was needed to deal with semantical antinomies like Richard’s Paradox, but not needed to deal with object-theoretical antinomies like Russell’s Paradox. Ramsey proposed ridding Type Theory of intensions (whose semantic nature was only implicit before Tarski) and moving to a purely extensional theory. This move ironically contravened a prior move of Russell’s for which Russell had become famous, viz. the so-called “no-class” approach, whereby an entire ontological category was subjected to Occam’s razor, leaving only intensional propositions and propositional functions. (Tarski furthermore pointed out that only one category is needed here, since propositions could be construed as zero-argument propositional functions.) Ramsey preferred wielding the razor in the other direction, and the extensionalists generally approved. That included Carnap at the time. In fact, Carnap’s extensionalist formulation of Type Theory in his Abriss der Logistik (1929, §30), the first textbook of mathematical logic for philosophers, has long been regarded as the standard in the field. Carnap was personally directly indebted to Russell, who had written out in longhand for him the main definitions and theorems of

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Principia Mathematica in the early twenties when it was impossible for Carnap to purchase any books. Church (1940) finally provided the definitive formalization, later (1951) extending this to a sophisticated formalization of intensions and extensions, close to Russell’s original motive, although directly aimed at explicating Frege. But as we all know, Carnap (1947) himself became a famous intensional logician, placing meaning at the center of logic again – this time using Tarski’s quasi-Hilbertian metatheoretical approach. It is appropriate at this place to insert an anachronistic remark concerning Quine’s much later, didactically brilliant treatment of Type Theory and set theory within one homologous setting in his Set Theory and Its Logic (1963). Usually this book of Quine’s is taken to imply that Type Theory is just a variety of set theory, in a way pushing Ramsey’s approach to its natural conclusion. This then suggests, however, that Type Theory is really as “mathematical” as set theory is usually held to be, whereas the older, intensionally-oriented view was that Type Theory was “logical” because it explicitly dealt with concepts (“propositional functions”), whose intensionality implied semantics – and semantics had been naïvely but perceptively (and correctly!) called “psychological” by both Ramsey and Carnap in the ’20s. But au contraire , Quine could have drawn quite the opposite conclusion from his homologization, namely that set theory is at least as logical as Type Theory – and as a follower of Russell he really ought to have done so. However Quine was perhaps constrained by his famous prejudice against intensional theories, which he didn’t like and preferred to avoid, no matter how inherently logical and traditional they were. Therefore, although Quine’s main program originally was to improve on Type Theory in deriving arithmetic, he was curiously poorly motivated to rescue Logicism. 4. R AMSEY AND C ARNAP ON THE A XIOM OF I NFINITY Once the main problem of Russell’s original Type Theory was removed by Ramsey, the Axiom of Infinity (AI) took center stage as the remaining serious bone of contention. Ordinary Type Theory cannot be revised to avoid this axiom, however, without sacrificing much of mathematics, which requires the existence of infinite sets. The problem is that, looking at type level 0, we aren’t sure if the universe has infinitely many particles in it.17 But we need that many to obtain infinite sets at any level above 0. Thus is AI usually interpreted to be an empirical claim. Ramsey (1925), at the end of his “Foundations of Mathematics”, gave a nice argument showing that, however many individuals n there may exist at type level 0, the proposition that it has exactly n individuals will be tautological, and that it has > n individuals is contradictory; from which he concludes … the Axiom of Infinity in the logic of the whole world, if it is a tautology, cannot be proved, but must be taken as a primitive proposition. And this is the course which we must adopt, unless we prefer the view that all analysis is self-contradictory and

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meaningless. We do not have to assume that any particular set of things, e.g. atoms, is infinite, but merely that there is some infinite type which we can take to be the type of individuals.

By “some infinite type” Ramsey means “some type level with infinitely many classes”. But we might ask immediately: must all types be < Ȧ ? What about a type levels beyond the infinite? We will always have all the mathematics we want at levels • Ȧ ! And, without AI, we don’t care that number theory and analysis are invalid at levels < Ȧ . Hilbert (1926, p. 184) already proposed transfinite types, and Gödel (1931) suggested them concerning attempts at completeness. Wolfgang Degen (1993, 1999, 2000) has worked with great success on exactly this idea, “vindicating” Logicism. Of course, conservative theoreticians may protest that transfinite types are egregious monsters. But set theory also is exceedingly generous – no one blinks at ℘(Ȧ) or at 2ℵ3 . It just has betterstreamlined axioms. Those prejudiced against any reduction of arithmetic to logic such as Poincaré would no doubt find something to complain about, come what may.18 Ramsey considered the Axiom of Infinity an extra-logical proposition which could be tautological, depending on whether the universe contained infinitely many bodies. Russell before him was more straightforward: he conceded that it was even an empirical proposition; most others followed him in this. As much as others, Carnap of course disliked this mixing of logic with empirical considerations when the derivation of arithmetic depended on it, and he figured out in his Logical Syntax of Language (1934, §§3, 15) a way of verifying the Axiom of Infinity in ordinary Type Theory without assuming any knowledge of the number of individuals: let the level-0 objects simply be positions in dimensions (whose metric properties are left unspecified, e.g. the units used).19 This avoids making specific empirical existence claims, yet makes the Axiom of Infinity logically plausible – because our scales of measurement can have infinitely extended dimensions without the existence of infinitely many or infinitely large bodies. Be that as it may, Carnap had little influence with his idea, although it really deserved consideration. Thus, primarily because the Axiom of Infinity was regarded as not logically valid, Logicism was held to have failed. Gödel flatly regarded it as a failure, AI being partly responsible, in addition to Gödel’s strict interpretation of Logicism as requiring completeness. Wittgenstein also had become very negative towards Logicism, but for quite different, possibly ideological or aesthetic reasons which moved him towards ultrafinitism in logic and opposition to higher types and quantifiers, a position I find hard to accept as a serious one for scientific research because it utterly weakens logic. Other mathematicians, in particular those in the Hilbert school, shared Gödel’s view.

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5. C ARNAP THE L OGICIST? By the mid-thirties, after Carnap’s famous Logical Syntax of Language appeared in Vienna in 1934, a reassessment of Logicism seemed to be called for even by its most enthusiastic partisans. I can’t go into much detail here, but fortunately Herbert Bohnert (1975) has written a profound and fascinating review of the development of Carnap’s thinking about Logicism, so this is not necessary. Suffice it to say that Carnap (1934, §84) unfortunately withheld the core claim of classical Logicism: that of reducing numbers to concepts – despite his effort in making the Axiom of Infinity a logically plausible proposition.20 Carnap followed Hilbert’s idea – which was also Gödel’s practice in his Incompleteness Theorem (1931) – of putting natural numbers at type level 0, and so all number theory automatically became available at level 1 of Carnap’s version of Type Theory already, and Peano’s axioms followed straight out of the syntactic metatheory. This made a Frege–Russell definition of numbers superfluous, which is only mentioned in passing (in the context of a discussion of Russell’s elimination of classes as entities separate from concepts in § 38). As Bohnert has it, Logicism then shriveled to the mere statement that logic and mathematics simply belong together. In a sense, this confirms Poincaré’s suspicion that Logicism hijacked mathematics; but Poincaré was an old-fashioned Kantian and insisted that mathematics differed essentially from logic by having content and being based on intuition, whereas logic was empty, as Kant thought. Carnap’s rejection of intuition in his Syntax made it easy for him to put logic and mathematics together. It should be stressed, as Hao Wang has done, that Frege (like Bolzano and Leibniz before him) already emphasized that the basic reason why mathematics is logical is the simple fact that its laws, like the laws of logic, apply to everything: they are in a precise sense super-universal. Mathematics shares with logic the semantic function of theoretically representing reality; both equally involve truth. They are super-universal: they are more universal than physical laws because they are valid no matter what the empirical universe is like. (In Leibniz’s classically baroque image, they hold in all possible worlds!) Wittgenstein totally deflated this hyperbole and persuaded Schlick, Hahn and Carnap that both mathematics and logic are universal because they have no empirical content, i.e. are tautologies. I would like to point out that this claim, which in the first instance seems rather trivial, in fact amounts to a crucial correction of a bad error of traditional Platonism: this often was made to say that mathematics (e.g. geometry21) only applied (strictly) to “ideal objects”, whereas “real objects” never satisfy mathematics exactly. Despite Frege, and despite protests from Occam to Berkeley against the illegitimate dualism of both ideal and real objects, very many naïve people22 say that mathematics deals with a special domain of abstract (“ideal”)

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objects like sets and numbers, often called “structures”, in distinction from empirical science, which deals with concrete (“real”) objects. This is all nonsense, and Berkeley’s (and Neurath’s) scathing criticism of Platonist Verdopplung is quite correct. Plato himself knew this criticism perfectly well, as the second part of Parmenides makes clear; but he had no suitable answer. Frege did. To see the point, just consider that all the allegedly non-empirical (“abstract”) sets and numbers magically gain empirical content the minute mathematics is applied: in a mathematically formulated physics, all functions and all sets used are physical concepts. For example, the n-tuple assigning magnetic field strength to points in space is the magnetic field itself. Classic authorities have claimed logical validity, or analyticity, to be characterized by what I have labeled “super-universality”. Leibniz of course did so; Bolzano as well. More recently Schröder (1890) did so, and Skolem followed him. Carnap also tried very hard to define Allgemeingültigkeit as the culmination of his Syntax-program, being helped by Gödel in his effort. Unhappily the definition failed, in my opinion, because he failed to realize how unlimitedly powerful the theoretical apparatus he needed for his syntax language. Two remarks are apropos. First, there is an even more fundamental reason why Carnap’s characterization misses the essence of logical validity, and why his approach opened him to the criticism of Quine against the analytic /synthetic dichotomy. Universality plainly and simply cannot be used to distinguish between logic and mathematics, because mere universality does not give us any modal distinction between logical and physical necessity. Second, it is an interesting and deep historical fact that the Leibniz–Schröder–Skolem concept of universal validity in all subject domains is identical with Aristotle’s main characterization of metaphysics! Therefore, to find out what logic is, it will pay to look at Platonic Dualism in more detail, this time avoiding the errors of Verdopplung which even Plato made fun of in his dialogues (e.g. the second part of the Parmenides). 6. T HE R EAL N ATURE OF L OGICAL V ALIDITY The distinction underlying Platonist Dualism which really makes sense is the modal distinction for which Hume became famous in his analysis of “Is” and “Ought”. (Concrete) Reality is Hume’s “Is”, (Abstract) Ideality is Hume’s “Ought”. No “Is” implies an “Ought” and vice versa : the Naturalistic Fallacy is to be avoided. The Abstract and Concrete by themselves are not the basis for Platonic Dualism, since both always appear together anyway; as Aristotle observed for matter and form, they’re inseparable. Example: we have electromagnetic fields all over Nature, they are part of empirical reality. Yet if we look carefully at the definition of such fields, we cannot escape that they involve highly theory-laden structures associating vector forces with positions in space in a complex, multi-dimensional manifold.

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This distinction between Ought and Is (or between the Ideal and the Real) quickly sets the stage for clarifying the famous analytic /synthetic distinction. As we all know, discussions of the true nature of logic typically devolved to a search for some acceptable characterization of analyticity. The Vienna Circle thought it lay in Wittgenstein’s Tautology. Carnap struggled to extend the domain of tautologies to all mathematics in his Logical Syntax program, an effort which famously earned the negative judgment of Quine (1951), who complained that Carnap had only promulgated a Dogma of Empiricism without providing a plausible common feature of analyticity in all the various systems allowed by the Principle of Tolerance. There is no general, non-arbitrary non-modal characterization of logical truth for all of the infinitely many language systems. Quine’s critique was justified up until 1965, when Carnap finally revealed to an astounded group of top philosophers of science that (rational) intuition was what justified belief in logical principles and rules. Alas, Bohnert, whose papers on Carnap’s views on both Logicism in particular as well as on the analytic/synthetic distinction in general are wonderfully perceptive, almost completely missed this epoch-making event in the history of Logical Empiricism. Bohnert refers to intuition mainly to affirm the well-known fact that it was “officially” rejected among orthodox Logical Empiricists ever since the twenties, when Hahn and Schlick determined that Kant’s Anschauung was too broken to fix, a view almost everyone followed, from Richard von Mises to Hans Reichenbach (except the phenomenologist Kaufmann and the Platonist Gödel). Nevertheless Bohnert has several quite interesting things to say regarding intuition. He expresses an interesting opinion, although rather non-commitally, that Carnap had a “significantly ambivalent attitude toward the power of man’s intuition, both in general and as applied to matters of logic and mathematics”.23 Bohnert is exonerated from not paying attention to a paper Carnap (1968) held under the heading of inductive logic, because Carnap apparently didn’t mention intuition to Bohnert during the several-day-long discussions they had on logicism in 1968, two years before Carnap’s death. In connection with learning logic, Bohnert does stress the pragmatic aspects of learning meanings – the foundation for judgments of logical validity: first that they begin in childhood, and they “seep” gradually wider and wider, interlocking more and more. Bohnert discovers that this very pragmatic view was already hinted at in the Logical Syntax of Language §38a, p. 142. This insight is important, since it captures exactly how intuition develops from vague gropings to gradually increasing certainty and precision. The history of mathematics, logic and statistics – to say nothing of that of law, accounting, measurement standards and many other areas of norms – confirms that at every stage in their development, intuitions are refined, corrected, polished, extended more and more. What’s more, pragmatic criteria for acceptance of principles and rules were always admitted by Carnap throughout his career, even in the medium-rigorous, apparently anti-semantical period of the Syntax program. The best discussion of this key aspect is by Bryan Norton (1977), who argues (correctly in my opinion),

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that it is precisely with reference to pragmatic motives and interests which lend objectivity to formal abstractions, grounding discriminations of validity of logic. 7. A FFINITIES BETWEEN R AMSEY AND C ARNAP In the 1950s, Carnap came much closer to Ramsey’s philosophy after reading Shimony (1953, 1955), who first discovered extremely important connections between Carnap’s confirmation theory and the subjective probability theories of de Finetti (1937) and Ramsey (1926), particularly between Carnap’s concept of regularity and the Ramsey–de Finetti concept of coherence. The culmination of that work was Carnap (1962). Another area where Carnap was deeply influenced by Ramsey was empirical theory construction, for which Carnap (1958, 1966) made famous use of so-called “Ramsey-sentences”, first introduced in Ramsey (1929). But Carnap had something much more deeply in common with Ramsey, and that was his general approach to philosophy and its relation to human concerns. Ramsey (1929, pp. 263, 269) wrote In philosophy we take the propositions we make in science and everyday life, and try to exhibit them in a logical system with primitive terms and definitions, etc. Essentially a philosophy is a system of definitions … The chief danger to our philosophy, apart from laziness and woolliness, is scholasticism, the essence of which is treating what is vague as if it were precise and trying to fit it into an exact logical category. A typical piece of scholasticism is Wittgenstein’s view that all our everyday propositions are completely in order and that it is impossible to think illogically.

Carnap (1950, Ch. 1) wrote a famous chapter on the methodology of explication, which is central to all good philosophy, and nothing could be closer to the spirit of Ramsey (1929) than that! In particular, although both were strongly influenced by Wittgenstein, both opposed him on the exactly the same issue: both participated in the ideals of the Enlightenment of using reason constructively to improve knowledge and thereby to improve man; whereas Wittgenstein was a pessimist in the tradition of Schopenhauer and Spengler, cynical about improvements through reason (something Wittgenstein and Heidegger have in common). Ramsey was no doubt much more gifted than Carnap in mathematics and in original theorizing, but their principles were in remarkable harmony. One may say the underlying principle behind this harmony was applying a strong faith in the Testability Criterion of meaning to solve philosophical problems, strongly emphasizing the scientific nature of philosophy: Neurath’s Unity of Science !? The central question always was: What is the “cash value” of a concept? What kind of data can we find to decide an issue? This implied a strong motivation to look for technical tools and willingness to legislate new norms (Carnap’s explications and “language engineering”) and assuming responsibility for solid craftsmanship. It was the Testability Criterion that was behind their

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common interest in betting quotients to measure strength of belief; simultaneously both gladly imposed norms of rationality on how beliefs should evolve. Similarly, both worked hard to find the occult power of theoretical concepts, both fully appreciating the key role they play in science and its evolution. It’s unfortunate they didn’t meet in Vienna, but Ramsey was too much involved with the always troubled Wittgenstein, as well as with his own intensive psychoanalysis. (Apropos psychoanalysis: Ramsey, Carnap and Gödel all got deeply involved in extended psychoanalyses of dubious value.) A major difference between them was their divergent attitudes towards Realism, which Carnap (1928; 1934, § 17), the ultra-Conventionalist, was famous for opposing, whereas Ramsey was quite at home with Russell’s post-Idealistic Platonist Realism.24 For example, Ramsey (1925; 1931, p. 21) insisted that Richard’s paradox is not merely linguistic, as Peano claimed, but constitutes a violation of logical rationality. Calling it a mere linguistic faux pas amounts to an invitation to both logicians and linguists to dismiss it; it absolutely requires solution, and The only solution which has ever been given, that in Principia Mathematica, definitely attributed the contradictions to bad logic, and it is up to opponents of this view to show clearly the fault in what Peano calls linguistics, but what I should prefer to call epistemology, to which these contradictions are due. (1931, p. 21)

Thus Ramsey, the champion of Realism. (No wonder he became disenchanted with Wittgenstein.) Carnap’s Conventionalism of the Syntax period would seem to put him opposite Ramsey here, but in fact he was much closer than it appears. First of all, Ramsey’s Realism is strongly qualified by what may be called Cognitivist Behaviorism – to be seen in Ramsey’s explication of knowledge in terms of reliable causal chains, where observation and inference “behave” correctly; also to be seen in Ramsey’s (1929a) “explication away” of theoretical concepts as otherwise unspecified existentially quantified variables.25 Conversely, Carnap was passionately interested in logical legislation just like Ramsey. Carnap was a language engineer, but for him language was the tool of epistemology subject to pragmatic criteria which were not nearly as arbitrary as Carnap pretended at the time. Like Ramsey, Carnap was willing to put ordinary language and “conventional” philosophical traditions far behind. In a specific sense, Carnap was a closet Platonist all along. Prime evidence for this is his insistence on separating all logical sentences in the Syntax (and throughout his later research) into analytic and synthetic ones, tertium non datur, leading to his debate with Quine in defense of just this distinction. Carnap’s Kantian Apriorism was holding tough (and Kant was a Platonist in this sense). Surely Ramsey would have been on Carnap’s side here. Carnap himself ultimately solved his confusion by acknowledging fallibilistic intuition, a position whose consequences he had no time left to explore.

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Both Ramsey and Carnap side with Realism on the issue of permitting nonconstructive concepts, the logical core of the issue concerning Mach’s banishment of theoretical concepts. Ramsey (1925; 1931, p. 37) wrote My method, on the other hand, is to disregard how we would construct them, and to determine them by a description of their senses or imports; and in so doing we may be able to include in the set propositions which we have no way of constructing, just as we include in the range of values of ij x propositions which we cannot express from lack of names for the individuals concerned.

This is exactly the approach Carnap takes to his “indefinite” Language II in his Syntax (1934, Ch. III), where e.g. “indefinite” (impredicative) concepts are admitted, and the non-constructive notion of logical consequence (Folgerung) is admitted, in contrast to the constructive notion of proof (Beweis) of his Language I. Similarly, Carnap (1956) freely allowed all logical methods in his work on empirical concept formation. Incidentally, Carnap was encouraged by Gödel to use just such concepts in his Syntax; and once, after a talk Carnap gave at Neurath’s apartment on psychological (psychoanalytic) concept formation, Gödel reminded Carnap that he (Gödel) had recommended dropping the usually imposed restriction on empirical concepts that they be directly (constructively) definable. As Russell had pointed out during the heyday of Platonistic Logicism, a “robust sense of Realism” forms an essential component of the motivation for successful research. Judging by the logical conventions he actually made, Carnap was behaving for all the world like a robust Realist. 8. G ÖDEL ’ S I NCOMPLETENESS , AND H IS E QUATION OF S ET T HEORY WITH P ROOF Finally, I’d like to close with a comment on Gödel’s position. I mentioned that Gödel thought Logicism was a failure because of problems with Type Theory. But there has arisen the widely held view that Gödel’s Incompleteness Theorems refute Logicism as much as they refuted Hilbert’s Program; the reason being that no specifiable theory can ever provably contain all mathematical truths. Indeed, Gödel showed earlier than Tarski that truth for any theory is not definable within that theory; this follows easily from the first Incompleteness Theorem. Many decades later, for example, Harris and Parrington (1977) first showed that combinatorial complexity theorems of the sort first discovered by Ramsey (1928), including Ramsey’s Theorem itself, although patently number theoretical, cannot be derived in second-order Peano arithmetic. Even certain much weaker Diophantine equations require assumptions from higher analysis to prove. Wilder’s recent proof of Fermat’s Last Theorem also used, apparently essentially, some very exotic higher analysis. Hence it seems the Frege–Russell program of formalizing a logic and then deriving mathematics from that formalization is doomed to failure.

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I disagree. Logicism is not inalienably anchored to any specific formalization of logic, nor is it necessarily forced to derive all possible mathematics. The Harris–Parrington proof did not show that Peano arithmetic failed tout court, it showed rather that Peano arithmetic needs extension: “generally accepted” proof rules or logical axioms need strengthening – whereby it is a pragmatic matter whether to strengthen rules or axioms, which Frege emphasized. Although Frege strove of course for high rigor, he made no absolute claims that his axioms and proof rules were definitive. He did not even specify any absolute criterion for formalization; in particular, he did not demand finite axiomatizability, an assumption made by those who think Gödel’s Incompleteness destroys Logicism. After experimenting with exactly one proof rule in (1879, §6), namely modus ponens, Frege later settled for a small number of finitary rules in (1893, § 48) of a sort which Hilbert regarded as models for his finitary approach. However, Frege had no clear idea whether his list of rules were “complete”, only that they seemed strong enough to prove the theorems he aimed for. Frege (1879, §6) explicitly declares it a practical matter how many proof rules should be used and what strength they should have: In logic, one counts a whole series of inference rules (Schlussarten), according to Aristotle; I use just this one – at least in all cases where, from more than one theorem (Urtheil), a new one is derived –. … Thus, an inference following any kind of inference rule can be reduced to our case. Since it is accordingly possible to make do with a single inference rule [modus ponens], the need for perspicuity [Übersichtlichkeit] therefore calls for doing so. Moreover it may be added that there would otherwise be no reason to stop with Aristotelian inference rules, but that we could indefinitely continue adding new ones: out of every theorem expressed in a formula in §§ 13 to 22, a special inference rule could be made.

The demand for perspicuity could subsequently be contravened by the contrary demand for brevity, therefore this does not rule out, writes Frege (1879, p. xiii), that, at some later time, transitions from several theorems to a new one which are possible only indirectly with this single rule could be transformed into a direct one by taking a shortcut. This may indeed be recommendable in later applications. New inference rules would thereby come about.

Thus, Frege (1893, §48) lists eight inference rules, four substitution rules and six rules for parentheses. It is clear that, in general, Frege allowed theorems to be transformed into proof rules. But then what can stop a follower of Frege from converting any transfinite axiom or theorem into a corresponding transfinite rule? Now we face the issue of finitary methodology in the sense of Hilbert, which was definitively formulated only in Gödel (1931). (NB: Hilbert never gave an exact characterization of his “finitary standpoint”; that “homework” waited for Gödel to accomplish – thus Bernays.) Frege famously demanded that all assumptions be made explicit and all proofs be without gaps, in order to guarantee

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that nothing extraneous slips in. But Frege was not as much fixated, as Hilbert was, on the Phenomenalist ideal of concretely surveying proofs. In particular, I can nowhere find wording in Frege which explicitly excludes transfinite rules. As far as I can see, Frege would merely require that rules have to be explicitly stated, just like transfinite axioms – the position Carnap (1934, §43; 1935) takes, where Carnap employs transfinite rules to divide all logical statements into analytic and contradictory classes. More specifically, I think Frege would have agreed with Ramsey (1925; 1931, p. 37), quoted in the previous section, that “indefinite” concept formation, and hence transfinite axiomatizations, are permissible. The arch-finitist Hilbert (1931) himself was the first to publish such a transfinite rule, now called the Ȧ-rule, claiming it was finitary; Carnap was the first to put transfinite (“indefinite”) rules to work, although he (1934, §48) denied Hilbert’s claim, backed up by Tarski, who had already broached a similar transfinite rule in 1927. Methodologically, Hilbert’s Program was officially restricted to “concrete intuition” (roughly Kant’s sinnliche Anschauung), but as the perceptive Bernays pointed out, there is nothing much concrete left of even second-order Peano arithmetic, which Hilbert needed to arithmetize his desired completeness proof for mathematics. But Frege was a Platonist who rejected psychologistic / phenomenalistic restrictions to concrete observability and who believed that logical insight enables us to grasp logical structures beyond that which our mere empirical senses give us. So despite the prominence in Frege of what looks like a strictly finitary syntax (Begriffschrift), I think it more likely than not that Frege would have supported Carnap’s tolerant attitude toward transfinite syntax rules – so long as they are clearly specified, so that their applicability is definite. But this utterly changes the picture for those who think Logicism requires finite axiomatizability. It is true that Frege wanted his logic to cover all of classical arithmetic. Frege excluded geometry from this reduction, which he thought to be based on intuition and for this reason not logical – although he did mention geometry as a candidate for formalization using a Begriffschrift, together with physics and chemistry. It is hence not clear after all whether Frege wanted to reduce all analysis to logic, for analysis might contain some geometrical intuition. (Brouwer took a radically different path from Frege’s: whereas Frege “displaced” arithmetic intuition by logical insight (Einsicht, the same term Bolzano used) but preserved a special place for geometrical intuition, Brouwer discounted geometrical intuition as well as logical insight and gave arithmetic intuition the dominant role for all analysis! This is why the Intuitionists’ analysis contradicts most people’s geometric intuition, with its highly unexpected propositions about the continuum). Russell was famously quite cavalier about rigor compared to Frege; but on the other hand, Russell was doubtless much more insistent on reducing everything that mathematics could ever contain to logic – e.g. Cantor’s transfinite numbers. Frege had seen no need for a consistency proof, which would have required completeness. For this reason as well, it is rather unclear “how complete” Logicism is required to be. Of course philosophers of mathematics are well-

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acquainted with the famous debate between Frege and Hilbert on whether mathematics had to specify content: Hilbert’s Formalism “officially” ignored content, so Hilbert shifted the responsibility of mathematics to merely checking for consistency. But as Gödel (1953) emphasized, if one is convinced of the correctness of one’s theory based on one’s intuition of some content, a consistency proof is not crucial. Moreover, a misconception should be cleared up which is frequently overlooked (but not by Gödel): so long as we don’t have a proof of the contrary, a set of axioms in what Gödel called “subjective mathematics” can very well be complete despite the Incompleteness Proof ; we simply can’t prove it’s complete! There are after all, as Gödel (1951, p. 313) emphasized, other ways to “know” mathematics than by deduction, namely by induction or, to avoid equivocation, by probable reasoning, which can after all attain what Gödel calls “empirical certainty” – not to be sneezed at! We can gradually become convinced that a theory is complete if we use it for a time and find it covers everything we can imagine. This is what proponents of a “theory of everything” in physics hope for – spiting Gödel’s first Incompleteness Theorem (which after all holds for any physical theory containing arithmetic as well). There is something very deep in a certain observation of Gödel’s. He pointed out that every particular set theory is equivalent to some proof theory of a particular strength, and vice versa. Ultimately this insight must go back to Hilbert’s (Leibniz’s!) original vision of proof theory as a kind of arithmetic, in order to accomplish his consistency proof – which is what “metamathematics” is for. But Göttingen had to wait for Vienna to embarrass it with Gödel’s negative result before it was clear how the proof concept is to be mathematized: with Gödel-numbering! In the following sense Gödel equates mathematics with logic, although he still distinguishes them: Gödelization actually shows, through coding, how arithmetic is equivalent to a particular proof system – essentially the one Frege had in his more-or-less finitary Begriffschrift. Once Hilbert (1931) began to consider stronger proof rules, we can understand how theories stronger than arithmetic – e.g. various stages of analysis and on to higher set theories – would be equivalent to transfinite proof systems. Such an equivalence by encoding should not be misunderstood: Although proofs can be interpreted as set theoretical calculations, proofs still remain different from sets and are governed by different intuitions. But the equivalence by encoding allows enterprising mathematicians to go from one area to the other to double-check results, or to find results at all, just as Descartes’ discovery of analytic geometry allowed a deeper interchange and mutual enrichment of algebra and geometry. Now Gödel shared with Carnap and many others the idea that logical truth is what follows from the meaning of concepts alone. But meaning is inherently intensional and thus mental – like proofs.26 At the same time, as mentioned before, sets are taken to be non-mental, and indeed quasi-physical, as Gödel repeatedly remarked! 27 But if Gödel’s encoding equivalence is valid, we thereby discover that something mental is equated with something physical. This tantalizing problem I will leave the reader to ponder.

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A PPENDIX : A L ETTER FROM R AMSEY TO F RAENKEL [The following manuscript is contained in Carnap’s papers at the University of Pittsburgh Library (where Ramsey’s papers also are). The formulas are in Carnap’s handwriting. Fraenkel had influenced Carnap to clarify the foundations of mathematics, which ultimately led to Carnap’s Logical Syntax (1934). According to his diary, Carnap met with Fraenkel in Vienna on 14, 15 and 25 March 1928, and Fraenkel presumably allowed him to copy some of the text at that time. We can be rather sure Carnap showed the text to Gödel by the following fall, who studied with him in the winter semester 1928/29 in a seminar on “Philosophische Grundlagen der Arithmetik”, where just the themes of the letter were treated, and regularly met with him in cafés. Others whom Carnap may have shown it would have been Waismann and Kaufmann, but also Schlick, who told Carnap about his correspondence and visits with Ramsey.]

081-38-01 Abschrift Ramsey an Fraenkel. Jan. 26 1928. Howfield, Buckingham Road, Cambridge. ... the fact that the so called "non-predicative processes" were of several essentially different kinds. It has always seemed to me very unfortunate that Russell's use of his Vicious-Circle Principle tended to conceal the fact that the circles he wished to eliminate were of two quite distinct kinds. I think that in a general discussion of non-predicative processes there are three things which should be clearly distinguished. First there is the entirely harmless process of describing an object by reference to a totality of which it is a member; an instance of this is "the tallest man in the room". To this process I do not see that objection can reasonably taken; certainly Russell has no such objection, and it seems to me that perhaps some slight alteration ... [X-d over by Carnap] Secondly there is the process of forming a class which is a member of itself. It seems to me that the objection to this is not that it is circular, since (if the Theory of Types be sound) it is equally wrong to suppose that the class is not a member of itself, but simply that it is nonsense. .. (It has always seemed to me that the arguments by which Russell deduced this part of the theory of types from his vicious circle principle were fallacious, but that the theory was nevertheless right in spite of the reasoning being wrong). Thirdly there is the formation of the non-elementary property of having all properties of a certain sort. It is this that makes the real difficulty, because it does seem as if the property arises subsequently to the collection of properties involved in its definition and so cannot be a member of it; whereas in the first or harmless kind of non-

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predicative process the object described is evidently independent of that way of describing it. Your first example of "the maximum value of a function" is, of course, the harmless kind. In Russell's symbolism it is

( x) (( ∃ y ) ƒ x = f (y) ƒ;;ƒ (z) ƒ x • f (z)) This is a ij1 (x) first-order [formula] and no sort of circle can arise, because only individual and not functional apparent variables are involved. The third or serious non-predicative process only comes in in the proof that the maximum value or upper bound exists, not in the notion of such a value. In Russell's symbolism we can say that if a function ȥ x is defined thus

ȥ x ƒ;;ƒ = ƒ;;ƒ (ij) ƒ f (ij ˆ;;z , x)

Df

we get a danger of a circle. But a definition of the form

a=

{(

x) (ij x)

Df;; the ij

is not circular whatever the form of

ijx. ———————————— ... I thought that by using Wittgenstein's work the need for the axiom of reducibility could be avoided, but he had no such idea and thought that all those parts of analysis which use the axiom of reducibility were unsound. His conclusions were more nearly those of the moderate intuitionists; what he thinks now I do not know. ... I am very glad to find that you still regard Russell as a possible alternative to Hilbert and Brouwer. I had the impression that in Germany he was regarded as entirely superseded. ... Sheffer's Theory of Notational Relativity was a manuscript which he sent to Russell; he said he was going to publish a book but has not yet done so. This manuscript dealt with various problems of combinatory analysis, from which he promised to make most remarkable applications to logic. Neither Russell nor I could in the least understand how the applications were to be made.

———————————— 8th March 1928. P.S. I ought to confess that what I said in my paper about the Axiom of Infinity doesn't now seem to me satisfactory, nor do I know what ought to be said. But the Multiplicative Axiom I don't feel to be so difficult.

C OMMENT ON R AMSEY : I NTUITING I MPREDICATIVE P ROPERTIES It may be possible to understand Impredicativity better using some ideas discussed between Gödel and Wang (1974, Ch. VI; 1996) concerning “idealized intuition”. Boolos (1971) and Parsons (1977) later took up the topic. The main idea is that of “running through” an infinite set somehow in finite time and making a judgment based on the result of applying some operation. I would like to suggest various ways to strengthen the rather simple, quasi-constructive

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approach used by resorting to what may be called “hyper-processes” of perception and inference.28 For example, instead of somehow sequentially running through an infinite set, we might imagine an infinitely large array of receptors (sensory organs) with denumerably many or even continuum receptors wired together in parallel yielding an output in finite time and viewing infinitely much during one observational cycle. Not only that, we could also imagine what may be called “hyper-feedback”, such that the global state of a system is input back into one of the local registers of the system with no wait-cycles (action at a distance). In such a way, we could “amplify” the perceptual and cognitive powers of an ideal mind, obtaining perhaps what Laplace had in mind with his demon. Notice that Maxwell’s demon seems to be like Laplace’s in the first instance, but according to Szilárd’s (1929) explication of Maxwell’s demon, it obeys the laws of thermodynamics actually thought to hold in our universe; whereas Laplace’s clearly cannot; cf. Frank (1932, Ch. II, 1). This is because Maxwell’s demon, although very quick and accurate, is not assumed by Szilárd to execute “supertasks”, whereas Laplace’s demon clearly must to be able to predict the entire future of the universe from a single time-slice: Laplace’s demon must have a continuum of infinitely fast receptors, and he must solve infinitely many differential equations infinitely quickly to make his infinitely complex prediction in finite time. By the way, Parsons (1977) makes an interesting point that the ideal intuition used to iterate sets seems to be incompatible with a widely held view of theology that God cannot be active but is static, because he must be eternal, beyond time and space. This may be solvable by considering cosmologies with non-Archimedean time orderings: a “hyper-universe” could occasionally gain access to a sub-universe through worm-holes, such that, from the point of view of an observer in the hyper-universe, an infinitely long activity in the sub-universe appears to occur instantaneously in the hyper-universe. Of course, succession remains normal. But there are also cosmologies such as Gödel’s where there are closed time loops here and there. These could also be considered in configuring mental processes for ideal intuition. (Our imagination need only be limited by the demand for consistency.) Such hyper-physical processes (super-tasks) would become mental merely by being used in brain functions of an infinite mind in a hyper-world. Intelligence is just process, as Whitehead was wont to say. My guess is that, by proper configuration of suitable processes such as “hyper-feedback” and closed time-loops, an ideal intuition could be established with the capacity to “run through” any impredicative concept, so long as it’s consistent, thereby “verifying” the principle it’s based on. Constructivists won’t be pleased, because their restrictions will be violated; but their restrictions arbitrarily restrict concept formation to what they think is humanly possible – even thought they all go way beyond human capacities anyway. As Ramsey said, human ability to perceive (and reason with) only finitely many objects at once is a merely empirical accident which logic should not be bound by.

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N OTES 1.

2.

3.

Thus the authoritative Barwise (1977, p. vii): “Mathematical logic is traditionally divided into four parts: model theory, set theory, recursion theory and proof theory”. It should also be noted, however, that computer science departments have requisitioned logic as well: most courses taught and textbooks written in the areas of model theory and recursive functions are doubtless taught and written there. And lest anyone surmise that these courses and textbooks are restricted to finitary theories such as Turing Machines, he will be disabused of his error by glancing at IEEE Transactions periodicals in engineering libraries brimming with articles on transfinite machines and continuum game-playing automaton scenarios! All too few know that, in addition to his famous definition of cardinal number, in his Grundgesetze II, Frege (1903) also made considerable progress towards a very beautiful theory of real quantities (Positivklassen). The relation-theoretical notion of the ancestral of a relation (independently discovered by Russell 1901) was already published in Frege (1879). The explication of cardinal number was published in Frege (1884), who first stated all the Peano (1889) axioms, also independently anticipated by Dedekind (1888). Dedekind (1888, p. iii) is clearly a Logicist: “In science, nothing capable of proof ought to be accepted without proof. As reasonable as this demand seems, yet has it by no means been held to in recent expositions [a footnote refers to Schröder, Kronecker and Helmholtz], even in the foundations of the simplest science, namely that part of logic dealing with number theory. In speaking of arithmetic (algebra, analysis) as a part of logic, I frankly mean to say that I hold the number concept to be entirely independent of our notions or intuitions of space and time, but rather that it proceeds immediately from the laws of thought.” [Translation E.K.] So much for Kant! But Dedekind, on p. x of the 2nd edition of his Was sind und was sollen die Zahlen?, also selflessly gives Frege priority to his own discovery of the number axioms. Despite Dedekind’s demand for a reduction of arithmetic to logic, it needs to be emphasized that Dedekind did not actually prove his number axioms from (some version of) set theory, as Frege (1893) did! In the 3rd edition of 1911, well after the appearance of Russell’s Paradox, Dedekind movingly upholds his conviction in Logicism: “When I was requested about eight years ago to replenish the 2nd edition, already out of print, by a 3rd, I had reservations in complying because meanwhile doubt had been thrown on the reliability of important foundations of my viewpoint. Even now I make no mistake about the importance and legitimacy, in part, of this doubt. But my trust in the inner harmony of our logic has not been shaken thereby; I believe that an exact investigation of the creative power of our mind to create out of specific elements a specifically new object, their system [set], which necessarily differs from every one of its elements, will certainly lead to the design of a flawless foundation of my work [Schrift].” This important statement, so different from Frege’s disappointed withdrawal of his Logicist program, was unfortunately not included in the 1963 Dover reprint of Dedekind’s famous brochure. Dedekind seems to indicate some kind of Vicious Circle Principle in his concluding sentence to eliminate membership loops like that of Russell’s Paradox, but Dedekind’s formulation, as it stands, would exclude e.g. even his own Dedekind cuts from analysis. Ramsey (1926, § III) provided a proposal using “predicative functions” to solve this famous problem (see also Ramsey’s letter to Fraenkel in the Appendix of the present article); but ZF going back to Zermelo (1908), unfortunately not referred to by Dedekind, is already thought to have dealt adequately with the problem. Finally, Dedekind’s frank Idealism is notable in his reference to the “mind’s creation of systems”, which clearly distinguishes his Logicism from the strictly anti-psychologist Logicism of Bolzano and Frege. However, in Köhler (2000), I show that neither Bolzano nor Frege can escape psychologism, as the reference to “pure thought” in the subtitle of Frege (1879) veritably concedes. A treatment expressing rather standard negative views was Black (1933), who had studied in Göttingen and who argued that Whitehead & Russell’s Principia Mathematica made essential use of non-logical axioms in its “reduction” of mathematics to logic, violating Logicism’s original intention. In particular, the Axiom of Infinity remains even after Ramsey’s attempt to elimi-

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nate the Axiom of Reducibility. Gödel basically agreed with similar lines of criticism in discussion remarks with Carnap in 1929. 4. Davis (1965) is the standard source book on the original recursion-theoretical (computability) literature, containing classic pioneering papers by Gödel, Church, Turing, Rosser, Kleene and Post, which now form a central part of the computer science curriculum. 5. Ramified Type Theory didn’t even contain the principal semantic relations of naming or satisfaction, to say nothing of logical consequence, so it was very paltry in semantics, and we may really ask: why bother with it at all? In Russell’s Foreword to the English version of Wittgenstein’s Tractatus, he famously called for the creation of a metatheory to solve Wittgenstein’s “puzzles” about allegedly not being able to say what could at best be shown about logical truths. Wittgenstein disavowed Russell’s Foreword; and metatheory had to wait for Hilbert, Carnap and Tarski. 6. The literature on set theory is extremely rich and variegated, ranging e.g. from “topoi” to “fuzzy sets”! For an authoritative treatment of much of set theory, mainly from Zermelo’s point of view, see Fraenkel, Bar-Hillel & Levi (1973); the system of von Neumann–Bernays (NB) is also discussed here, but the main source is Bernays (1958). The standard source for NBG is Gödel (1940). The early history of Z’s reception is dealt with in Moore (1977). For Quine’s NF and ML see Quine (1937) and (1940); but these systems are widely considered non-standard. 7. Originally, Quine (1937, p. 70 f.) put set theory in logic, but later, e.g. Quine (1963a, § II) and Quine (1970, pp. 64–74), he pulled it out again, for methodological reasons: first-order predicate logic is decidable and accepted as a universal standard since Hilbert & Ackermann (1928), whereas set theory is a quasi conventionalist, experimental science; cf. Orenstein (1977, p. 94) and Romanos (1983, § 2.4). This is like punishing logic for bad behavior by cutting off one of its main branches. To be sure, mathematics has misbehaved often, worst of all before Bolzano and Weierstrass cleaned up analysis. If we let the puritan Quine have his way, much of mathematics should, by parity of reasoning, be demoted to the status of accounting rules! 8. The “iterative concept of set” developed by Boolos (1971), Wang (1974, Ch. 6) and Parsons (1977) makes sets seem independent of concepts, but in fact the set-iteration procedures themselves constitute concepts! Remember that concepts are any procedures usable to classify objects, and are identical with Frege’s intensions (Arten des Gegebenseins von Gegenständen). 9. Oddly enough, Frege’s concepts (Begriffe) are in fact not any more intensional than Frege’s objects, as Pavel Materna has shown (they are “functions-in-extension”, in Russell’s terminology), whereas Bolzano’s concepts are truly intensional. 10. Leibniz’s fixation on monadic predicates is of course the main story in the pioneering book by Russell (1900): much of Leibniz’s metaphysics of monadology was dominated by his logic of monadic predicates. Couturat (1901, 1905) followed Russell. Körner (1979) thinks Russell went awry, but authoritative Leibniz-scholars such as Mates (1986), Rescher (1967) and Wilson (1989) largely side with Russell. BocheĔski (1956) and Kneale & Kneale (1962) provide fragments of relation-theoretical logic scattered throughout the history of logical texts; but (of course aside from Frege 1879), the first systematic treatment of relations was Schröder (1890– 1905), based mainly on the incomplete researches of Peirce; and, independently of Frege, Russell (1901). Structural features of relations such as symmetry, transitivity or reflectivity were naturally always felt to belong to logic, despite the tenacious fixation on syllogistics. Of course Poincaré is right in feeling that the same structures are dealt with by mathematical functions (which Frege, Peirce and Russell easily defined in terms of relations) and the algorithms they represent. One may perhaps say logic and mathematics have “equal claim” to relations. 11. The most detailed and authoritative history of logic for this period is Grattan-Guinness (2000); Coffa (1991) also provides very solid philosophical guidance, especially on semantic aspects; Hylton (1990) provides a very illuminating account of Russell’s transition from Idealism to the logic of the Principia and discusses in particular Russell’s approach to comprehension and perception of classes (i.e., sets). NB: Only after von Neumann–Bernays set theory was introduced in the thirties were classes distinguished from sets in the way now widely accepted by mathematicians. Earlier on, Bolzano had a place for them in his ontological panoply of logical objects, identifying sets with manifolds, i.e. Mannigfaltigkeiten. 12. This is close to the classic view of Cantor (1895, p. 481): “By a set ( Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) M of definite and sepa-

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rate objects m (which are called the ‘elements’ of M) of our intuition or our thought.” Dedekind was also a quasi-Idealist. The “collection” follows a “law” according to Cantor (1883, fn. 1): “Theory of Aggregates. With this word I denote a theoretical concept encompassing very much, which I have previously tried to develop only in the special guise of an arithmetic or geometric set theory. By an ‘aggregate’ or ‘set’ I understand in general every multiplicity which can be thought of as a unity, i.e. every quintessence of specific elements which can be combined by a law to a whole; and I believe I am thereby defining something which is related to the Platonic İȚįȠȢ or ȚįȑĮ , as well as with that which Plato calls μȚțIJȩȞ in his dialogue Philebus, or the Highest Good. He contrasts this with the ȐʌİȚȡȠȞ, i.e. the unbounded, indefinite, which I call the improper infinite (Uneigentlich-unendliche), as well as with ʌȑȡĮȢ , i.e. the bounded, and declares it to be an ordered ‘mixture’ of the latter two. That these concepts are of Pythagorean origin is indicated by Plato himself … .” This was not so for Cantor and Dedekind, however, insofar as both let sets be “legislated” by a mind. In Köhler (2001), I argue that the subject–object (or mental–physical) dichotomy is relative, and that both mental and physical interpretations can be found for both intensional and extensional entities in logic. The standard treatment of this famous episode in the history of logic and mathematics is Moore (1982). Immediately concerned is the famous “Vicious-Circle Principle”, first promulgated by Poincaré; see the authoritative treatment by Hallet (1984). This had of course been developed most fully in Whitehead & Russell (1910–13), but Russell originated its main ideas already in 1903 and developed them in detail in 1908 – the same year as Zermelo’s main publication on axiomatic set theory. Linsky (1997) thought the notorious Axiom of Reducibility could indeed be considered logical. The case is not closed. Ramsey’s ideas were discussed inside and outside the Schlick Circle meetings on Thursday evenings, but unfortunately none of the protocols by Rose Rand mention this. Of course, when Ramsey (1931) appeared, it was immediately read and discussed, in particular by Carnap and Gödel, but also by Schlick and Waismann. In correspondence, we have a better idea of what was discussed, and I reproduce a letter by Ramsey to Fraenkel of which Carnap copied a part in an appendix to this paper. Or infinitely wide dimensions. But once we consider “adjusting” dimensions, we will quickly arrive at Carnap’s clever solution of letting the individuals of type level 0 be – not physical bodies but – positions on scales of measurement. See below. At this Point in my original paper, I tendered the idea of proving the Axiom of Infinity – rather than Russell’s assuming it – by adding to the positive types negative ones descending infinitely deep. I thought this plausible on the ground that pure logic should not prejudice the empirical question of how deep the foundation of real objects goes, including whether the foundation is finitely deep. With infinitely many descending types, every type level automatically obtains infinitely many infinitely large propositional functions (or classes, or sets). In the discussion after my paper, Wolfgang Degen objected that no known proof rule could show this. I now conjecture that the situation is far worse: the cardinality of every type probably is immeasurably large, presumably exploding any and all limitations of size which are characteristically employed in the known axiom systems to avoid the paradoxes. So much for my idea of negative types. Carnap obviously came up with this idea through his familiarity with coordinate systems assigning numbers to objects through scales of measurement such as Relativity Theory presupposes; cf. Carnap (1922, 1926). Nowadays we would justify the “logical necessity” that infinitely many indexical positions exist with Carnap’s much later idea of “Meaning Postulates” (1952), which essentially state measurement norms as logic postulates. For example “nothing blue is red” or “nobody is taller than himself ”. Carnap (1963, p. 11 ff. ) was one of the very few students Frege had at the University of Jena; according to Gabriel (1996), only Carnap provided complete sets of lecture notes. Flitner (1986, p. 126f.) describes how Carnap even kept Frege’s lectures alive by motivating other students to attend. Carnap (1929, § 21b), (1950, §6), (1953, §34c) presented Frege’s famous explication of cardinal number as a model for all philosophy and mathematics of how to explicate a concept by characterizing it in terms of more elementary ones. Carnap emphasized that, in contrast to Peano’s

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treatment of number, only Frege’s definition clarified exactly how cardinal numbers are used to count things, thereby proving that the cardinal number concept has to be defined the way he did. One wonders why Carnap didn’t prominently say this in his Syntax (1934) as well. Perhaps he skirted the issue because of the general disaffection with Logicism. Knowing that both Wittgenstein and Gödel rejected it, it was difficult to present a view that did not represent important members of his circle of colleagues, as Carnap strove to present his circle’s position. Geometry is a bad example, since it should properly be reckoned to physics; but it is the example foremost in the minds of Platonists, including Plato himself. Arithmetic, on the other hand, clearly is mathematical; unfortunately it’s not at all a plausible example, as Frege discovered. In Köhler (2002), I provide an explication of Platonism partly based on Gödel’s own definition, although I criticize Gödel for repeating the ancient error of assuming a Verdopplung of object domains and failing to realize that Platonic Dualism is instead based on the fact that mathematical truth has a different modality, namely normative validity. Using this simple insight, I could also solve the famous dispute between Carnap and von Neumann on information theory and entropy in Köhler (2001). My apologies to Gödel; even Frege sometimes says this. Bohnert adds the significant counter that “On the other hand, he had great hope for man’s ability to use formalism to construct an instrument whereby he could double-check intuitions by making them explicit … Indeed, he saw in formalism an instrument capable of leading man over intellectual chasms where his intuition seemed to fail altogether.” These comments are well-motivated but misleading, as “formalism” is also based on intuition, as Gödel insisted and as Carnap should have admitted. After all, the leading formalist, Hilbert, gave concrete intuition the central rôle in his epistemology, a fact consistently avoided in Carnap’s Syntax (1934). Here Ramsey was much closer to the ultra-Platonist Gödel, the “Mozart of mathematics” (Karl Menger), with whom he also shared a great and multifaceted mathematical creativity. A really “robust” Realist would claim that our intuition is powerful enough to see and identify the theoretical entities specifically ; like Russell’s “the king of France”, the variables used for theoretical entities aren’t assumed to denote anything specifically identifiable anymore. Still, by Quine’s famous existence criterion, the mere use of the variables at least implies ontological commitment to something not directly observable. But meaning is also normative, embedded in norms of rationality constitutive of mind, a fact which even Gödel missed, as a consequence of which he and Quine and many others failed to capture the true difference between logic / mathematics and natural science. Davidson’s philosophy makes essential use of just this modal distinction without realizing it. E.g. in Wang (1996), 8.1.9, 8.5.6. In Köhler (2001) and (2006), I treat the relation between the mental and the physical in much detail. On the one hand, I argue that all proofs or any other mental acts can somehow be “physicalized”; but conversely that all sets – as well as all physical objects – can be “mentalized”. The key to understanding this is simply to attend to whether one is on an object-level or a meta-level. The reason why semantics always involves mind is that semantics always uses a metatheory – whose purpose is to describe entire conceptual frameworks in order to reflect reality on some “higher level”, i.e. in some mind. Any attempt to separate theory from mind, as Bolzano, Frege and Husserl tried to do, is impossible, because theories constitute (part of) the knowledge of a mind. What Bolzano, Frege and Husserl really wanted was to distinguish empirical from normative psychology, but they didn’t realize this because they didn’t realize that logic is normative. Only in the title of Frege (1879) may the embarrassing implication be made that “pure thought” is the object of logic; and only later did Kantians widely realize that “pure” means normative. Benacerraf (1963) calls these intellectual processes “super-tasks” in honor of Zeno and the Eleatics. It is clear that the foundations of analysis and the infinitesimal calculus presuppose their existence in some sense or other; and that is a central problem for the epistemology of mathematics. To help understand super-tasks, I propose in Köhler (2002b) that they are to be situated in what I call “hyperworlds”, i.e. quasi-physical universes whose laws permit instantaneous communications which allow infinitely large receptors to observe infinite data, and infinitely fast processors to allow infinite proofs, etc.

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Frank Plumpton Ramsey (1929): “Theories”, in Ramsey (1931, 1978). Frank Plumpton Ramsey (1929a): “Philosophy”, in Ramsey (1931). Frank Plumpton Ramsey (1931): The Foundations of Mathematics and Other Logical Essays, ed. by R.B. Braithwaite with an Introduction by G.E. Moore, Routledge & Kegan Paul, London. Frank Plumpton Ramsey (1978): Foundations. Essays in Philosophy, Logic, Mathematics and Economics, re-edited version of Ramsey (1931) with a different selection of texts, including especially the two papers on taxation and savings, by D.H. Mellor with introductions by D.H. Mellor, L. Mirsky, T.J. Smiley and Richard Stone, Routledge & Kegan Paul, London 1978. Miklós Rédei and Michael Stöltzner (eds.) (2001): John von Neumann and the Foundations of Quantum Mechanics, Vienna Circle Institute Yearbook 8, Kluwer, Dordrecht. Nicholas Rescher (1967): The Philosophy of Leibniz, Prentice-Hall, New York. George W. Roberts (ed.) (1979): Bertrand Russell Memorial Volume, Allen & Unwin, London. George Romanos (1983): Quine and Analytic Philosophy, MIT Press, Cambridge MA. Bertrand Russell (1900): A Critical Exposition of the Philosophy of Leibniz, Cambridge University Press, Cambridge; 2nd ed. Allen & Unwin, London 1937. Bertrand Russell (1901): “Sur la logique des relations avec des applications à la théorie des séries”, Revue de Mathématique (Rivista di Matematica) VII, 115–148; translated in Russell (1956). Bertrand Russell (1903): The Principles of Mathematics, Allen & Unwin, London; 2nd ed. 1937. Bertrand Russell (1908): “Mathematical Logic as Based on the Theory of Types”, American Journal of Mathematics 28, 222–262; reprinted in Russell (1956). Bertrand Russell (1914): “On the Nature of Acquaintance”, The Monist XXIV, 1–16, 161–187, 435– 453; reprinted in Russell (1956). Bertrand Russell (1956): Logic and Knowledge: Essays 1901–1950, edited by Robert Charles Marsh, Allen & Unwin, London. Paul Arthur Schilpp (ed.) (1944): The Philosophy of Bertrand Russell, Library of Living Philosophers V, Northwestern University Press, Evanston IL. Paul Arthur Schilpp (ed.) (1963): The Philosophy of Rudolf Carnap, Library of Living Philosophers XI, Open Court Publishing Co., La Salle IL. Ernst Schröder (1890–1905): Vorlesungen über die Algebra der Logik I–III, Leipzig. Abner Shimony (1953): A Theory of Confirmation, Ph.D. dissertation at Yale University. Abner Shimony (1955): “Coherence and the Axioms of Confirmation”, Journ. o. Sym. Log. 20, 1–28. Friedrich Stadler (ed.) (2000): Elemente moderner Wissenschaftstheorie, Veröffentlichungen des Instituts Wiener Kreis 8, Springer-Verlag, Vienna. Leo Szilárd (1929): “Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen”, Zeitschrift für Physik, 53, 840–856; translated as “On the Decrease of Entropy in a Thermodynamic System by the Intervention of Intelligent Beings” by Anatol Rapoport and Mechthilde Knoller in Behavioral Science, 9, 301–310; the latter reprinted in Szilard (1972). Leo Szilárd (1972): The Collected Works of Leo Szilard: Scientific Papers, ed. by B.T. Field and G. Weiss, MIT Press, Cambridge MA. William W. Tait (ed.) (1997): Early Analytic Philosophy: Frege, Russell, Wittgenstein, Open Court Publishing Co., La Salle IL. Hao Wang (1974): From Mathematics to Philosophy, Humanities Press, New York; Ch. VI “The Concept of Set” is reprinted in Benacerraf & Putnam (1983). Hao Wang (1987): Reflections on Kurt Gödel, MIT Press, Cambridge MA. Hao Wang (1996): A Logical Journey: From Gödel to Philosophy, MIT Press, Cambridge MA. Alfred North Whitehead and Bertrand Russell (1914): Principia Mathematica, Cambridge University Press, Cambridge; the 2nd edition of 1925 included a long new Introduction largely influenced by Ramsey. Catherine Wilson (1989): Leibniz’s Metaphysics: A Historical and Comparative Study, Princeton University Press, Princeton. Ernst Zermelo (1904): “Beweis, dass jede Menge wohlgeordnet werden kann”, Mathematische Annalen 59, 139–141; translated in van Heijenoort (1967).

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Ernst Zermelo (1908): “Neuer Beweis für die Möglichkeit einer Wohlordnung”, Mathematische Annalen 65, 107–128; translated in van Heijenoort (1967). Ernst Zermelo (1908a): “Untersuchungen über die Grundlagen der Mengenlehre I”, Mathematische Annalen 65, 261–281; translated in van Heijenoort (1967).

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