Incompleteness
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Schlick was sympathetic to the drift of the Viennese über-conversation and his presence at the university soon attracted like-minded thinkers from across many disciplines. At first they gathered in an old Vienna café. But the numbers of those participating gradually grew and, in 1924, Schlick agreed to make the gatherings somewhat more formal, moving the group to a room at the university.
Though all (or almost all) in the Circle held positivist views and everyone (even the clandestine Platonist) had either a connection to or a deep sympathy for the exact sciences, there was a diversity of interests and personalities and opinions among them. There was, for example, Rudolf Carnap, who had been trained as a physicist and mathematician at Jena, where he had been influenced by the logician Gottlob Frege (1848–1925). Carnap was “especially interested in the formal-logical problems and techniques,” and would have been a happy man indeed to have seen every question reduced to a straightforwardly technical one—the recalcitrantly irreducible of course declared meaningless. He was said to have had a face, especially in his youth, “that almost seemed to exude sincerity and honesty.” His intellectual earnestness impressed his fellow positivists; he worked and learned constantly. When anything came up in conversation that was new to him or that he wanted to follow up, he would produce a little notebook and jot down a few words. His ease in writing soon made him the leading exponent of the Circle’s ideas.
Otto Neurath was a social scientist and economist, a great big elephant of a man (he signed his letters with a picture of an elephant) with elephantine resources of energy and capacities for enjoying life. Both Carnap (who was an introvert) and Neurath (who was not) had the instinct for political utopianism; and Neurath, in particular, tried to push the Circle in political directions, often making it seem, perhaps unintentionally, that there was more political homogeneity within the Circle than there in fact was. “Schlick especially seemed to resent this since in Vienna, the Circle was named after him, the Schlick-Kreis.”
Neurath and Carnap felt also that the Circle was intimately connected with other cultural movements, in particular arguing for an affinity between their point of view and the industrial-design-inspired ideology of the Bauhaus. Both were an expression of the neue Sachlichkeit, the “fact-mindedness” that received the seal of approval from the sciences. And then in Germany there was the “Berlin Group,” centered around the philosopher of science Hans Reichenbach, whose outlook was all but identical with that of the Schlick Circle.
Neurath’s sister, the blind, cigar-smoking Olga Neurath, was also an active member of the Circle. She was a mathematician with wide tastes that extended into logic. In her youth she had written three papers, one of which, on the algebra of classes, is described by Clarence I. Lewis in his Survey of Symbolic Logic as “among the most important contributions to symbolic logic.”
Olga Neurath was married to Hans Hahn, who was also an important member of the Circle. Hahn had been responsible for bringing Schlick from Germany to Vienna. He was a first-rate mathematician, whose name prominently lives on in the useful Hahn-Banach extension theorem in functional analysis. Hahn’s mathematical interests were wide, and eventually he became interested in logic. It was he who brought the work in mathematical logic of the German Gottlob Frege and the English Bertrand Russell to the forefront of the Circle’s attention. He had an unbounded admiration for Russell and did the Vienna Circle the great service of saving them the difficulty of reading through the monumental three-volume Principia Mathematica, explaining it all to them in his seminar of the academic year 1924–25.
Hans Hahn is of particular interest in our story because when Gödel decided to switch his focus from number theory to mathematical logic, Hahn became his dissertation advisor. Though Hahn’s specialty was not logic (though he had done some significant work in set theory) his mathematical interests were certainly flexible enough to accommodate Gödel’s new interest. Gödel had first come into contact with Hahn in 1925 or 1926, and he told Hao Wang that Hahn had been a first-rate teacher, explaining everything “to the last detail.”
Hahn’s intellectual interests went far beyond mathematics as well. One of Hahn’s extra-mathematical interests was in the empirical evidence for parapsychological phenomena, which was a hot topic in Vienna at this time; a large number of reputed mediums had appeared in the postwar years and eventually a committee was formed, which included Schlick and Hahn and other scientifically oriented thinkers, for the purpose of investigating their claims, the validity of which became a much-vexed bone of contention within the Circle. It was not that Hahn was a believer, but he kept an open mind, which was enough to enrage other members, for example his brother-in-law, Otto Neurath. “Who looks into these matters?” Neurath once demanded with his characteristic vigor, answering his own question in the sociological terms he favored: “Uncritical, run-down aristocrats and a few supercritical intellectuals such as Hahn. Studies of the kind further the belief in supernatural forces and serve only reactionary groups.”
Then there were also two of Schlick’s young students, Frederich Waismann and Herbert Feigl, who as “favored students” were invited by Schlick to join the Circle. Hahn, too, would get two of his most talented students, Karl Menger and Kurt Gödel, invited into the select company of Schlick’s Circle.
Noch Einmal: Man Is the Measure of All Things
In 1929, when Schlick rejected an offer of a prestigious and lucrative professorship in his native Germany, the other members of the Circle decided to celebrate by publishing, in Schlick’s honor, a booklet setting out the tenets and aims of their joint point of view. The result was a sort of positivist manifesto entitled Wissenschaftliche Weltauffassung: Der Wiener Kreis, or The Scientific Worldview: The Vienna Circle. “Everything,” it proclaimed, “is accessible to man. Man is the measure of all things.” The ancient Sophist’s words were reiterated verbatim, only now given a scientifically minded twist: whatever question is, in principle, not susceptible to measurement, that is, empirical procedures, is no question at all. Since the limits of knowability are congruent with the limits of meaning, no meaningful matter can escape our grasp. We are cognitively complete.
A few years later, Herbert Feigl (who went on to become a prominent philosopher of science in America) co-authored an article in the American Journal of Philosophy entitled “Logical Positivism: A New Movement in European Philosophy.” The article, Feigl writes, provided “our philosophical movement with its international trade name.”
The term “positivism” had long been in circulation, always connoting a pro-scientific attitude used as a standard for meaningfulness. It was first applied to the ideas of Auguste Comte (1798–1857) and Herbert Spencer (1820–1903). The Viennese physicist Ernst Mach (1838–1916) had demanded, in the name of positivism, that all meaningful propositions be reducible to constructs of sense impressions, thus adding much more substance to the positivism of Comte and Spencer. The “Introductory Remarks” chapter of his book Ccntributions to the Analysis of Sensation (1885) has the subtitle “Antimetaphysical.” His positivism had led him to denounce both the reality of atoms and Einstein’s relativity. The Schlick positivists acknowledged Mach as one of their guiding lights, although tempering his denunciation of relativity sufficiently to admit Einstein, too, as one of their inspirations (though ignoring the realist interpretation that Einstein conferred on his theory. The positivists, rather, were inspired by Einstein’s redefinition of the concept of “the simultaneity of events” in terms of the speed of light).
“Logical” was appended to “positivism,” explains Feigl, to emphasize that the Viennese positivists excluded logical propositions (among which they included mathematics) from the otherwise exclusive disjunction: empirical or meaningless.
The truths of pure mathematics (i.e. not including physical geometry or other branches of the factual sciences) are a priori indeed. But they are a priori because they are . . . validated on the basis of the very meaning of the concepts involved in the propositions of mathemati
cs. The Vienna Circle regarded, for example, the identities of arithmetic as necessary truths, based on the definition of the number concepts—and thus analogous to the tautologies of logic (such as “what will be, will be”; “the weather will either change or remain the same”; “you cannot eat your cake and not eat it at the same time”).
In other words, the logical positivists believed that mathematics, just like logic, was devoid of any descriptive content. Mathematical propositions, if not quite tautologies, are analogous to them. (It’s hard to make out this middle ground, but never mind for now.) Another way to put this point is that mathematics is merely syntactic; its truth derives from the rules of formal systems, which are of three basic sorts: the rules that specify what the symbols of the system are (its “alphabet”); the rules that specify how the symbols can be put together into what are called well-formed formulas, standardly abbreviated “wff,” and pronounced “woof”; and the rules of inference that specify which wffs can be derived from which.
To get a sense of what it means to describe mathematics as syntactic (though we will look more closely at the notion in the next chapter), it helps to contrast the view with Platonism. For those who believe that mathematics is syntactic, the phrase “is true” takes on a special meaning when it is applied to a mathematical proposition: A mathematical proposition is true relative to the stipulated rules, the syntax, of a formal system. Analogously, for a moral relativist, like Protagoras, “is true,” when applied to an ethical statement, is shorthand for “is true relative to x,” where x is a person or, more likely, a conglomerate of ethically agreeing persons. Moral truths are only true relative to the stipulated rules of a society. They are, in the academic terminology du jour, social constructs. Similarly, according to the view under consideration, mathematical truths are formal constructs.
A mathematical Platonist, on the other hand, uses the word “true,” even when applied to mathematical statements, in exactly the same way as we normally use the word, not as a shorthand for “relative to x,” but to represent existing states of affairs. For a Platonist, mathematical truth is the same sort of truth as that prevailing in lesser realms. A proposition p is true if and only if p. “Santa Claus exists” is true if and only if Santa Claus exists. “Every even number greater than 2 is the sum of two primes,” is true if and only if there is no even number greater than 2 that isn’t the sum of two primes (even if we can never prove it).8
The view, then, of the syntactic nature of mathematics—its lack of any descriptive content—was indicated in the very name “logical positivism.” (And Gödel, impassioned Platonist that he was, sat among the positivists and spoke not a word of dissent.)
The scene for the meetings of the Vienna Circle was “a rather dingy room,” on the ground floor of the building in the Boltzmanngasse that housed the mathematical and physical institutes of the university. (It is now the meteorological institute.) The room was filled with rows of chairs and long tables, facing a blackboard. When not being used by the positivists, it was a reading room sometimes used for lectures. Those who arrived first at the Thursday evening meetings would shove some tables and chairs away from the blackboard, which most speakers used. The chairs would be arranged informally in a semicircle before the blackboard, and there was a long table in the back used by anyone who wanted to smoke or take notes. People would stand around talking in groups until the signal from Schlick—a sharp clap of his hands. The conversations halted, everyone taking a seat, Schlick at the end of the table closest to the blackboard. He would announce the topic of the paper to be read or the discussion to be pursued, sometimes first reading communiqués from colleagues, and the formal proceedings of the night would begin.
At any one meeting there were usually no more than 20 Viennese members, with foreign visitors sometimes attending. For example, John von Neumann (who, among his other prodigious abilities, managed to inhabit various far-flung points on the globe simultaneously, including Budapest and Princeton) might grace the Circle if he were anywhere in the vicinity; the young Willard van Orman Quine, from America, who went on to dominate Anglo-American analytic philosophy for many decades, from his position at Harvard; Carl Hempel, from Germany, who, among other distinctions, was my first-year advisor when I was a graduate student; the great Polish logician Alfred Tarski (né Teitelbaum), from Poland; and the philosopher Alfred Jules Ayer, from England, all spent time with Schlick’s group. Ayer, after spending some months in Vienna and imbibing the doctrines, went back to England and wrote up his imbibition in his highly influential polemic Language, Truth, and Logic, thus disseminating the ideas of Vienna’s positivists in the English-speaking world:
We shall maintain that no statement which refers to a “reality” transcending the limits of all possible sense experience can possibly have literal significance; from which it must follow that the labours of those who have striven to describe such reality have all been devoted to the production of nonsense.
By far the most influential figure connected with the Vienna Circle was not even a member of it, and in fact steadfastly refused membership. This was the philosopher Ludwig Wittgenstein. Wittgenstein, at least according to the interpretation that I will propose, plays a significant, if ambiguous, role in the story of Gödel’s incompleteness theorems. Wittgenstein’s almost mystical influence on the members of the Vienna Circle, the esteemed thinkers among whom the young logician first came to think rigorously about the foundations of mathematics, must have struck a person of Gödel’s persuasion as highly dubious. There are still-smoldering remnants of Gödel’s resentment of the philosopher to be found in the Nachlass, written (though never exposed to the public) many decades after the Vienna Circle had ceased to be, only a few years before the logician’s death.
Gödel’s and Wittgenstein’s views on the foundations of mathematics were, as we will see, at loggerheads, and neither could acknowledge the work of the other without renouncing what was most central in his own view. Each, I believe, was a thorn deep in the other’s metamathematics.
Wittgenstein and the Circle
Wittgenstein came from one of the wealthiest and most culturally elite families of Vienna, “the Austrian equivalent of the Krupps, the Carnegies, the Rothschilds, whose lavish palace on Alleegasse had hosted concerts by Brahms and Mahler, Clara Schumann, and the conductor Bruno Walter.”9 He was, in his intensity, preoccupations, ambitions, and conflicts, indelibly stamped by the sensibilities of that intense, preoccupied, ambitious, and conflicted city. While studying aeronautical engineering at the Technische Hochschule in Berlin, he had learned of Russell’s paradox, and became interested in the foundations of mathematics.
Russell’s famous paradox is of the self-referential variety. The liar’s paradox—this very sentence is false—is of the same variety. We get into trouble because some linguistic item talks about itself, at least potentially, and by reason of this self-referentiality we end up both asserting that some statement is true and that it is also false, which is logically impossible if anything is.
Russell’s paradox concerns the set of all sets that are not members of themselves. Sets are abstract objects that contain members, and some sets can be members of themselves. For example, the set of all abstract objects is a member of itself, since it is an abstract object. Some sets (most) are not members of themselves. For example, the set of all mathematicians is not itself a mathematician—it’s an abstract object—and so is not a member of itself. Now we form the concept of the set of all sets that aren’t members of themselves and we ask of this set: is it a member of itself? It either is or it isn’t, just as the problematic sentence of the liar’s paradox either is or isn’t true. But if the set of all sets that aren’t members of themselves is a member of itself, then it’s not a member of itself, since it contains only sets that aren’t members of themselves. And, if it’s not a member of itself, then it is a member of itself, since it contains all the sets that aren’t members of themselves. So it’s a member of itself if and only if i
t’s not a member of itself. Not good.
Paradoxes have often been found lurking about in the deepest places of thought. Their presence is often the signal (like the canary dying?) that we have managed, sometimes unwittingly, to stumble on a deep and problematic place, a fissure in the foundations. Russell’s discovery of his paradox had grievous consequences in the foundations of mathematics, and for one man in particular: Gottlob Frege. Frege had only just finished his monumental two-volume Grundgesetze der Arithmetic (The Fundamental Laws of Arithmetic), which was the first attempt to reduce arithmetic to a formal system of logic. The logic that Frege employed also includes set theory; in other words (to speak in the language of a logician) sets are included as individuals in the universe of discourse, over which the bound variables of the system range. What this basically means is that the system can be interpreted as talking about sets. Numbers are then defined in terms of sets and the arithmetical laws are derived from the axioms and rules of set theory and logic.
Frege’s axioms of set theory allowed for the formation of the set of all sets that are not members of themselves, and since this set involves a contradiction—since that set both is and is not a member of itself—there was something fundamentally wrong with Frege’s system. Although the system was adequate for the expression of arithmetical truths, it was also inconsistent, which is the very worst thing that a formal system can be. One can prove absolutely anything (and hence effectively nothing) from a contradiction.10 So an inconsistent system is worthless as a tool of proof.
Russell and his collaborator, Alfred North Whitehead, devised a new formal system for expressing arithmetical truths, accomplished in their Principia Mathematica (the very work that appears in the title of Gödel’s 1931 paper, setting forth the proof for the first incompleteness theorem). However, to secure consistency, Russell and Whitehead had imposed ad hoc rules governing set formation. Their Theory of Types decrees that there are ascending orders in the universe of discourse—the types of things we interpret the formal theory as speaking about. Basic individuals constitute Type I; sets of individuals, Type II; sets of sets, Type III; sets of sets of sets, Type IV; etc. An item can be a member only of an item of a higher type. The question then of whether a set is a member of itself can’t even arise. The rules of Principia Mathematica bar the formation of such paradox-breeding sets as the set of all sets not members of themselves. Russell and Whitehead called their rules the “Theory of Types,” but the problem was that there was no real theory behind the rules at all, as they themselves ruefully acknowledged; there was no explanation at all as to why certain sets were allowable and others not,11 other than that if one allowed the unallowable very bad things would happen to one’s system. Their formal system is consistent by fiat.