Past Masters of the Postmodern

If anxiety about truth is very much in the air, it has been in the air for a very long time. Well before the advent of tweets and trolls, George Orwell expressed the fear that “the very concept of objective truth [was] fading out of the world.”¹ Orwell’s anxieties were political. Truth was under attack by politicians and their propagandists. Our anxieties are cultural, the widespread sense that a number of philosophical theories have escaped the academic world, and, in the wild, are doing great harm. Writing in the otherwise staid European Molecular Biology Organization Report, Marcel Kuntz affirmed that “postmodernist thought is being used to attack the scientific worldview and undermine scientific truths; a disturbing trend that has gone unnoticed by a majority of scientists.”²

This view of postmodernism as an intellectual villain is very common.

Whatever postmodernism is in the arts, I am concerned with its philosophy of language, its view of truth. Familiar figures have, by now, become infamous: Jean Baudrillard, Jacques Derrida, Michel Foucault, Martin Heidegger, Frederic Jameson, Douglas Kellner, Jean-François Lyotard, and Richard Rorty. Philosophical postmodernists emphasized the possibility of different perspectives on things, or different interpretations of them. They denied the existence of unmediated or innocent observation, and were dubious about the distinction between facts and values, the analytic and the synthetic, the mind and the world. Skeptical about truth, and sensitive to shifts in culture, society, and history, they thought of words as tools, and believed that vocabularies survive because they enable us to cope, to meet our goals.

They mistrusted authority.

The Positivists

The philosopher A. J. Ayer published Language, Truth, and Logic in 1936. It is a book that served to express the movement that came to be known as logical positivism, or logical empiricism, a congeries of doctrines associated with the Vienna circle, and figures such Rudolf Carnap, Herbert Feigl, Philipp Frank, Kurt Gödel, Hans Hahn, Victor Kraft, Otto Neurath, Moritz Schlick, and Friedrich Waismann. Although members of the Vienna circle suffered many points of disagreement, they were like-minded to the extent that their goal was a form of philosophy adequate to the demands of modern science.

Our system of knowledge and belief, they argued, is a building resting on secure foundations. The senses justified the facts of science; the rules of language, the facts of mathematics and logic. Beyond these facts, the one scientific, the other conventional, there was nothing but nonsense. If exhilarating, this position was also awkward. In the Tractatus Logico-Philosophicus, Ludwig Wittgenstein recognized that philosophy, since it was neither science nor logic nor mathematics, might be on the side of nothing.³

Logical positivism flourished in the decades between the end of the First World War and the beginning of the Second World War. Countercurrents were already forming. In 1935, Karl Popper accommodated David Hume’s skeptical conclusions about induction by agreeing with them.⁴ Theories could be refuted but not confirmed. In the same year that Popper’s book appeared, the Polish biologist and historian Ludwik Fleck developed the idea that scientists inevitably become locked into a Denkstil, or style of thought.⁵ A shared Denkstil characterized a Denkkollectiv, a group, or perhaps a sect, locked into the same style. Fleck’s views foreshadowed Thomas Kuhn’s idea that normal science proceeds in terms of a shared paradigm or governing style. Like all governments, a paradigm remains stable until pressures generate a revolution.⁶

The logical positivists took for granted a very crude representational theory of meaning. The human mind holds up the mirror of language to the world; the world is reflected in the mirror. John Dewey provided a very perceptive critique of this idea:

The basic fallacy in representative realism is that while it actually depends upon the inferential phase … of enquiry, it fails to interpret the immediate quality and the related idea in terms of their functions in inquiry. On the contrary, it views representative power as an inherent property of sensations and ideas as such, treating them as “representations” in and of themselves. Dualism or bifurcation of mental and physical existence is a necessary result, presented, however, not as a result but as a given fact … psychical or mental existences, which are then endowed with the miraculous power of standing for and pointing to existences of a different order.⁷

Dewey rejected the idea that if the word “apple” designates an apple, it is because it stands for some mental power that has antecedently identified the apple, and lacks only a convenient way of expressing what it has identified.

The epistemology of meaning must have a different focus.

Trapped in Translation

Derrida made famous the idea that there is nothing beyond the text—il n’y a pas de hors-texte. The idea that there is nothing beyond language to explain language is a common postmodern theme. It is itself a postmodern irony, that this postmodern doctrine has an echo in modern mathematical logic. A formal language is a regimented language, its parts of speech listed explicitly. A formal language L_A adequate to the demands of ordinary arithmetic contains variables x, y, z, … , such as the x in 5x = 25; and predicate and relation symbols, F, G, H, … of every finite order. It contains logical symbols to denote implication, conjunction, negation, and alternation; and it contains quantifiers for everything and something. It contains a numeral for every natural number, and symbols P, Q, R, S, … standing for sentences. A variable is bound if it falls under the scope of a quantifier, and free otherwise. Grammatical rules and rules of inference are explicit and effective. A machine could check on them. The well-formed formulas are those passing grammatical muster; the sentences, well-formed formulas with no free variables: Fx is a well-formed formula, but not a sentence; it makes no determinate claim; but ∀xFx is a sentence. It says that everything is F.

L_A is a language about the natural numbers 1, 2, 3, … . The natural numbers, in turn, comprise a model M = <D, F> of L_A, where D is a nonempty set, the domain of M, and F, a function assigning to the predicate variables of L_A relations of corresponding rank on D. The values of F are relations on M. An assignment α is a function that maps the individual variables of L_A onto individuals in the domain D. The notion

α satisfies S(x, …) in M

can then be defined by recursion on the length of the formula S(x, …). An assignment α satisfies a formula if the formula holds in M, when its predicate variables are interpreted by F, its free individual variables are interpreted by α, and its bound variables range over D.

What follows is the all-important notion of definability. A relation is definable within a model M of L_A if and only if there is a formula in L_A defining it. A number x is even if it is the sum of two identical numbers. Evenness is definable in M. The requisite formula is ∃y (x = y + y). A relation in M has now been defined by a formula of L_A.

In his paper demonstrating the incompleteness of arithmetic, Gödel introduced an ingenious code endowing the natural numbers in M with the queer power to refer to their own descriptions in L_A. To the variables, predicates, logical connectives, and parameters of L_A, Gödel assigned a unique natural number. So, too, the sequences of such symbols in L_A. An ordinary statement of arithmetic, by this means, acquired the double power to refer to the natural numbers and to refer to the language describing them. A number with this double power is now known as a Gödel number.

Let Tr be the set of Gödel numbers of all of the true sentences of L_A. Is Tr definable in M? If so, there must be a sentence S in L_A with a single predicate variable F, such that S is satisfiable in the natural numbers only for the interpretation of F as Tr.

There is no such formula.

Alfred Tarski demonstrated this in 1931. Tr is not arithmetically definable. There is no difficulty whatsoever in defining the concept of truth for L_A. A sentence in L_A is true if and only if it is satisfied by every assignment in M. There it is: truth itself. But the definition cannot be expressed in L_A. It must be expressed in a system notably stronger than L_A—stronger in containing concepts denied to L_A itself. There is not the slightest difficulty in defining such a language, and, indeed, defining it as a formal system, Meta(L_A)—the metalanguage of L_A. The definition does just what one might expect. It entails infinitely many sentences of the form

P is true in M if and only if P,

where P is the name in Meta(L_A) of the sentence P in L_A.⁸ An example in ordinary English conveys the same effect:

“Snow is white” is true if and only if snow is white,

where quotations serve as names; and M is understood implicitly as the real world, the model of things.

No language rich enough to express the properties of the natural numbers can define the conditions under which its statements are true. The defect is ineliminable, and the process of ascending to ever richer languages unavoidable.

There is no escaping the tyranny of words.

The Slingshot

In the logician’s account of truth, there is no matter of fact, and no mention of them, either. This is at odds with the very old idea that a statement is true if and only if it corresponds to the facts. Facts are facts. No doubt. But how many of them are there? It would appear that there are only two: the one true fact, and the one false fact.

In his magisterial Introduction to Mathematical Logic, Alonzo Church offered an informal account of this famous argument. Consider the statement that Sir Walter Scott is the author of Waverley. Whatever the fact to which this statement corresponds, the statement that Sir Walter Scott is Sir Walter Scott must correspond to the same fact, since the name “Sir Walter Scott” has replaced “the author of Waverley,” both names referring to one and the same object. By the same token, “Sir Walter Scott is the author of Waverley,” and “Sir Walter Scott is the man who wrote 29 Waverely novels altogether,” must also correspond to the same fact. What is “the man who wrote 29 Waverley novels altogether” but another name for Sir Walter Scott? But “Sir Walter Scott is the man wrote 29 Waverley novels altogether” and “The number such that Sir Walter Scott is the man who wrote that many Waverley novels altogether is 29” are logically the same. Whereupon there is “The number of counties in Utah is 29,” when “the number of counties in Utah” is substituted for “the number such that Sir Walter Scott is the man who wrote that many Waverley novels altogether.” They refer to the same thing, after all.

The conclusion follows: if “Sir Walter Scott is the author of Waverley” corresponds to a fact, it corresponds to precisely the same fact as “The number of counties in Utah is 29.” All true sentences refer to the same fact.

Following Gödel, Donald Davidson offered a formal version of the slingshot. Consider any two sentences S and P, such that S and P are both true. The notation (ιx)(x = d & S) designates x, such that x = d, whatever d, and S. The argument proceeds in four steps:

1. S

which is true by assumption.

Whereupon

2. (ιx)(x = d & S) = (ιx)(x = d),

which is logically equivalent to 1.

Next

3. (ιx)(x = d & P) = (ιx)(x = d),

where 3. is derived from 2. by substituting (ιx)(x = d & P) for (ιx)(x = d & S). And why not? They refer, both of them, to whatever it is that (ιx)(x = d) refers to.

And, finally,

4. P

in virtue of the fact that P and (ιx)(x = d & P) = (ιx)(x = d) are logically equivalent.

This is not a conclusion that enhances the correspondence theory of truth.

Coherence

Is it possible our words distort our thoughts? This was a concern of classical philosophers, one of whom, Cratylus, was so concerned about it that he lapsed into silence and would only communicate by wagging his finger. In writing about the Peloponnesian War, Thucydides expressed similar concerns:

Words had to change their ordinary meaning and to take that which was now given them. Reckless audacity came to be considered the courage of a loyal supporter; prudent hesitation, specious cowardice; moderation was held to be a cloak for unmanliness; ability to see all sides of a question incapacity to act on any. Frantic violence became the attribute of manliness; cautious plotting a justifiable means of self-defense.⁹

Words can betray us. Consider the American Psychiatric Association’s Diagnostic and Statistical Manual of Mental Disorders (DSM). Published in 1952, DSM 1 comprised 132 pages; published in 2013, DSM 5 comprised 947. Have psychiatric disorders increased sevenfold? Classifying syndromes is like classifying the shapes of waves or clouds. Cratylus thought it was always like this, but he was unable to say so.

There is no independent view of language and the world. The definition of truth seems to reduce to a triviality: “Snow is white” is true if and only if snow is white. What then is left? Many philosophers in the nineteenth century, often swimming in Georg Wilhelm Friedrich Hegel’s turbulent wake, promoted a coherence theory of truth. If coherence is all we can expect, what of schemes that are both coherent and yet in conflict? Sober philosophers of science have asked this very question.

In 1906, Pierre Duhem argued that theories are underdetermined by their data. If this is so, there is nothing to impede the multiplication of coherent theories.¹⁰ “If two theories are available, both of which are compatible with the given factual material,” Albert Einstein remarked, “then there is no other criterion for preferring the one or the other than the intuitive view of the researcher.” Chastened by these arguments, some philosophers have suggested that we repose our confidence not in our theories, but only in their observational consequences.¹¹

This view has gained some support from Craig’s theorem. There it is again, the odd link between postmodernism and mathematical logic. Any recursively enumerable theory, William Craig demonstrated, is equivalent to a recursively axiomatized theory. Within mathematical logic, a theory is a set of sentences closed under logical deduction; an axiomatic theory is one in which the theory follows from a finite set of axioms. Euclidean geometry is an example. There are for the most part no axiomatic theories governing social, personal, or political life, but there are plenty of theories. A theory is recursively enumerable if its sentences can effectively be generated, one after the other, by a computer, or something machinelike. A theory is recursive if it is recursively enumerable and if there is also some effective procedure determining its complement—the sentences that do not belong to the theory. A recursive theory has a calculus determining which sentences are in and which sentences are out. It is one of the achievements of modern logic to show that some recursively enumerable theories are not recursive.¹²

Craig’s theorem establishes that every recursively enumerable theory may be derived from a recursive set of axioms. Do note the double play: a recursively enumerable theory but a recursive set of axioms. The theorem marks an ascent in strength. To every recursively enumerable theory T, Craig showed, there is a recursively axiomatizable theory T* such that T and T* coincide. They coincide as theories, but only T* follows from a recursive set of axioms.

Craig published his theorem in the late 1950s, well after logical positivism had commenced its decline. The theorem came as a welcome surprise. The logical positivists were committed to the ancient idea that an axiomatic system comprised the ideal form to which scientific theories were tending. This idea is still current. A language, L(Th,Ob), expressing an axiomatic system in the sciences typically contains both theoretical and observational terms. Casting a cold eye on L(Th,Ob), the remaining positivists were all for getting rid of the theoretical terms. They wished for a direct confrontation with experience. Craig’s theorem offered them what they needed. With theoretical terms purged from L(Th,Ob), what remains is the observational language L(Ob)—a sub-language of L(Th,Ob), a part of the whole. To ask whether a given sentence belongs to L(Ob) is to ask for a mechanical way of deciding the matter, and, surely, one is forthcoming. It is one experiment after the other, and so the sentences that that are in L(Ob) are plain enough. The complement of L(Ob) consists of those sentences that have nothing to do with observation and experiment. One look at them suffices. Out they go. Whatever is in L(Ob) is in; whatever is out, is out. The matter, and so the property of being within L(Ob), is recursively decidable.

Now consider a recursively enumerable theory T over L(Th,Ob). A language is one thing. It comprises everything that can be expressed within its ambit. A theory is quite another. It consists of a set of sentences closed under logical deduction. At issue now is a theory T over L(Th,Ob). The empirical content of T is the set sentences in L(Ob) that may be derived from T. This is a subtheory T* of T, a restriction to a purely observational language. And that property is recursive. It follows that T* is recursively enumerable. By Craig’s theorem, it now follows that there is a recursive set of sentences in L(Ob) whose theory T* exhausts the empirical content of T. But what is this other than a recursive axiomatization of the empirical content of T?

Theoretical terms have vanished. Only observational terms remain.

Power-Mad

In a coherent world, some coherencies are more coherent than others. The postmodernists sometimes wrote as if the sciences dealt with social constructions rather than the real world, or invented categories of thought rather than discovering facts about nature. There is a nice conspiratorial edge to these social constructions. They admit obviously of sinister political or commercial manipulation. Foucault’s is the representative voice:

We must cease once and for all to describe the effects of power in negative terms: it “excludes,” it “represses,” it “censors,” it “abstracts,” it “masks,” it “conceals.” In fact, power produces; it produces reality; it produces domains of objects and rituals of truth. The individual and the knowledge that may be gained of him belong to this production.¹³

A scientist spending a lifetime to discover the shape of a protein molecule is not apt to take kindly to the idea that things and facts are the products of power. He will quickly admit to some contingencies. Shifts in power, he will acknowledge, do affect the reception of his work. Beyond these concessions, concessions cease. Once he is on the case, he is not influenced by reward nor rhetoric.

He is listening to nature’s voice.

Analytic philosophers are likely to sympathize. We can accept that social, economic, political, or even military factors supported and motivated Captain James Cook’s voyages of exploration. But that had nothing to do with the unprecedented accuracy of his naval charts. There are no political or social or economic variables affecting whether the contours of the shoreline, the depths of bays and anchorages, or the range of the tides and the speeds of their currents are just as Cook said they were.

The example of mapmaking confers certain benefits. One postmodernist target is the idea that nature speaks with one voice, or demands to be described in one particular way, so that there is such a thing as the truth, the whole truth, and nothing but the truth. Maps support no such idea. And there is no end to maps: rainfall maps, population maps, geological maps. Reality does not constrain the choice. But once the cartographer has chosen what to map, and once the conventions governing the map’s symbols are settled, then reality has its say. It was after deciding to show depths or currents that Cook needed to observe and register those things, and it was then that he laid himself open to charges of inaccuracy or falsity.

Maps remind us that we may have multiple perspectives on reality. They do not encourage the idea that those perspectives are in conflict. Interpretation is one thing, the truth another. A landmark such as the Eiffel Tower admits many perspectives, but they are all compatible, and they all may be reconciled. There is no contradiction between the view from Montmartre and the view from the Panthéon. And when there is a contradiction between two perspectives on, say, an historical event, we can often enough resolve the issue in favor of one. If anything, an emphasis on perspective works to the benefit of objective fact, reminding us that it may take work and care to ensure that our own interpretation of things is likely to survive a challenge.

As well as mocking the ideal of a complete map written in nature’s language, postmodernists have been apt to suppose, in Rorty’s words, that language is for coping not copying. Following William James, Rorty saw language as a tool at the service of human aims and ends, so that its prime virtue is not truth but utility. Cook’s charts helped generations of sailors to cope, certainly. There is a simple explanation of why this was so: they helped sailors to cope because of their accuracy, and they were accurate because they were true. If Cook had gotten it wrong, sailors went aground. Scientists seeking a suitable vocabulary for mental aberrations have nothing so straightforward to go on. Interpretation and vocabulary choice are unavoidable, and this is certainly something that can be ceded to the postmodernists.

Late in his career, Derrida offered his postmodern readers an exercise in mellowness:

[I]t will be understood that the value of truth (and all those values associated with it) is never contested or destroyed in my writings, but only reinscribed in more powerful, larger, more stratified contexts. And that within interpretive contexts … that are relatively stable, sometimes apparently almost unshakeable, it should be possible to invoke rules of competence, criteria of discussion and of consensus, good faith, lucidity, rigor, criticism, and pedagogy.¹⁴

What more could any traditionalist want?