Wittgenstein on Philosophy of Logic and Mathematics
Abstract and Keywords
Ludwig Wittgenstein (1889–1951) wrote as much on the philosophy of mathematics and logic as he did on any other topic, leaving at his death thousands of pages of manuscripts, typescripts, notebooks, and correspondence containing remarks on (among others) Brouwer, Cantor, Dedekind, Frege, Hilbert, Poincaré, Skolem, Ramsey, Russell, Gödel, and Turing. He published in his lifetime only a short review (1913) and the Tractatus LogicoPhilosophicus (1921), a work whose impact on subsequent analytic philosophy's preoccupation with characterizing the nature of logic was formative. Wittgenstein's reactions to the empiricistic reception of his early work in the Vienna Circle and in work of Russell and Ramsey led to further efforts to clarify and adapt his perspective, stimulated in significant part by developments in the foundations of mathematics of the 1920s and 1930s; these never issued in a second work, though he drafted and redrafted writings more or less continuously for the rest of his life.
Keywords: Ludwig Wittgenstein, philosophy of mathematics and logic, analytic philosophy, Vienna Circle, nature of logic, formativity
Ludwig Wittgenstein (1889–1951) wrote as much on philosophy of mathematics and logic as he did on any other topic, leaving at his death thousands of pages of manuscripts, typescripts, notebooks, and correspondence containing remarks on (among others) Brouwer, Cantor, Dedekind, Frege, Hilbert, Poincaré, Skolem, Ramsey, Russell, Gödel, and Turing. He published in his lifetime only a short review (1913) and the Tractatus Logico‐Philosophicus (1921), a work whose impact on subsequent analytic philosophy's preoccupation with characterizing the nature of logic was formative. ^{1} Wittgenstein's reactions to the empiricistic reception of his early work in the Vienna Circle and in work of Russell and Ramsey led to further efforts to clarify and adapt his perspective, stimulated in significant part by developments in the foundations of mathematics of the 1920s and 1930s; these (p. 76) never issued in a second work, though he drafted and redrafted writings more or less continuously for the rest of his life.
After his death large extracts of writings from his early (1908–1925), middle (1929–1934), and later (1934–1951) philosophy were published. These include his 1913 correspondence with Russell (see CL), 1914 dictations to Moore (MN), his Notebooks 1914–1916 (NB), transcriptions of some of his conversations with the Vienna Circle (WVC), student notes of his Cambridge lectures and dictations (1930–1947), and many other typescripts and manuscripts, the most widely studied of which are Philosophical Investigations (PI) and Remarks on the Foundations of Mathematics (RFM). Not until 2000 did a scholarly edition of his middle period manuscripts (WA) and a CD‐ROM of his whole Nachlass appear (1993–, 2000), thereby making a more complete record of his writings accessible to scholars. Since Wittgenstein often drafted multiple versions of his remarks, reinserting them into new contexts in later manuscripts, and since his later, “interlocutory” multilogue style of writing is so sensitive to context, this new presentation of the corpus has the potential to unsettle current understandings of his work, especially since a fair number of remarks on mathematics and logic have yet to be carefully scrutinized by interpreters. ^{2} Nevertheless, because of the wide circulation of the previously edited volumes there is a substantial and fairly settled body of later writings that has exercised a significant and continuous influence on philosophy since the 1950s.
Wittgenstein's discussions of mathematics and logic take place against the background of a complex, wide‐ranging investigation of our notions of language, logic, and concept possession, and the fruitfulness of his work for the philosophy of mathematics has arisen primarily through his ability to excavate, reformulate, and critically appraise the most natural idealizing assumptions about the expression of knowledge, meaning, and thought in language on which philosophical analyses of the nature of mathematics, logic, and truth have traditionally depended. His probing questions brought to light tendencies implicit in many traditional epistemological and ontological accounts to gloss over, miscast, and/or underrate the complexity of the effects of linguistic expression on our understanding of what conceptual structure and meaning consist in. Though he held, like Frege and Russell, that a proper understanding of the logic of language would subvert the role that Kant had tried to reserve for a priori intuition in accounting for the objectivity, significance, and applicability of mathematics, his focus on the linguistic expression of thought in language ultimately led him to question the general philosophical significance of their logicist analyses of arithmetic. His resistance to logicism ultimately turned on a diverse, multifaceted exploration of the questions “What is logic?” “What is it to speak a language?” and, more generally, “What do we mean by rule‐ (or concept‐) governed behavior?”
Wittgenstein's overarching philosophical spirit was anti‐rationalist, in sharpest contrast, among twentieth‐century philosophers, to that of Gödel. For Wittgenstein, as earlier for Kant, philosophy and logic are quests for self‐understanding and self‐knowledge, activities of self‐criticism, self‐definition, and reconciliation with the imperfections of life, rather than special branches of knowledge aiming directly at the discovery of impersonal truth. These activities should thus aim, ideally, at offering improved modes of criticism, clarity, and authenticity of expression, rather than at a certain or explicit foundation in terms of general principles or an enlarged store of knowledge. Like Plato in the Meno, or Kant in the Critique of Pure Reason, when Wittgenstein discusses a particular logical result or a mathematical example, he is most often model‐ or picture‐building: pursuing, through a kind of allegorical analogy, not only a better understanding of the epistemology of logic and mathematics, but also a more sophisticated understanding of the nature of philosophy conceived as an activity of self‐expression and disentanglement from metaphysical confusion, for purposes of an improved mode of life. He is investigating how and why mathematical analyses affect and expand our self‐understanding as human beings, rather than assessing their cogency as pieces of ongoing mathematical research. Rather than adopting empiricism or a clearly formulated doctrine about mathematics and logic to combat rationalism, Wittgenstein fashioned a novel appropriation of the dialectical, Augustinian side of Kant's philosophy, according to which philosophy is an autonomous discipline or activity not reducible to, though at the same time centrally concerned with, natural science. This sets his philosophy apart both from Frege's and Russell's philosophies and from logical empiricism, though his thought bears important historical relations to these and cannot be understood apart from assessing his efforts to come to terms with these rival approaches.
Because of philosophers' tendency to focus on general questions of concept possession and meaning in connection with Wittgenstein's work, his philosophy of mathematics is the least understood portion of his corpus. At present there is no settled consensus on the place of his writings on mathematics and logic within this overarching philosophy, nor is there agreement about the grounds for assessing its philosophical worth or potential for lasting significance. His post‐1929 remarks have in particular met with an ambivalent reaction in contemporary philosophy of mathematics and logic, drawing both the interest and the ire of working logicians. ^{3} This is not surprising, in light of the controversial nature of Wittgenstein's overarching philosophical attitude and his having presented few clear indications of how to present a precise logico‐mathematical articulation of his ideas. The main divide in interpretations of Wittgenstein's philosophy of mathematics is between those who take his remarks on logic and mathematics to offer a restrictive epistemic (p. 78) resource argument (constructivist, social‐constructivist, anti‐realist, formalist, verificationist, conventionalist, finitist, behaviorist, empiricist, or naturalist) based on the finitude of human powers of expression, and those that stress his emphasis on expressive complexity to forward criticisms of all such global epistemic positions. Most agree that the heart of his philosophy of mathematics contains a recasting of traditional conceptions of a priori knowledge, certainty, and necessary truth‐an attempt, not unlike Kant's in his day, to find a way to resist naïve Platonism or rationalism without falling into pure empiricism, skepticism, causal naturalism, or fictionalism. Yet, unlike Kant, Wittgenstein rejected the idea that the objectivity of mathematical and/or logical concepts may be satisfactorily understood in terms of an overarching purpose, norm, or kind of truth to which they must do justice. Wittgenstein came to suggest that a detailed understanding of how human beings actually express themselves in the ongoing, evolving stream of life shows not the falsity, but the lack of fundamental interest, of this idea. The main interpretive debate about his philosophy concerns the basis and content of this suggestion.
One thing is clear. In his later writings Wittgenstein explicitly insisted that his own philosophical remarks were a propadeutic, an “album” of pictures, crisscrossing the same terrain in different directions from differing points of view for differing, localized purposes. On this view the philosopher's task is to investigate, construct, explore, and highlight the philosophical effects and presuppositions of many different possible models or “pictures,” each having a certain naturalness given actual linguistic and mathematical practice; the point is to explore problems rather than defend or enunciate general truths. In his later writings he suggested that we conceive of human language as a variegated, polymorphous, overlapping collection of “language‐games.” Such a “game” he took to be a simplified structure, model, or picture (Bild) of a portion of our language exhibiting various kinds of distinctive patterns and regularities of human action and interaction.
Wittgenstein's resistance to defending a single, unified conception, and his many appeals to the notion of Bild, reflect his philosophy's lineage from the dialectical side of Kant—a side ignored and/or minimized in the mature philosophies of Frege and Russell. Kant had insisted that human claims to possess unconditional truth about reality, especially when grounded in the apparently absolute and universal truths of formal logic, would inevitably end in conceptual contradiction and paradox. The notion of Bild has a long philosophical history, but in the German tradition it is connected with the general nature of symbolism, with the mathematical notions of image and mapping, with the intuitive immediacy and transparency to thought of a geometrical diagram,^{4} and with the notion of education or self‐development (Bildung). Wittgenstein's distinctive uses of this (p. 79) notion were stimulated, in part, by his observation of the use of scale models in engineering, but he was also especially taken with an idea he found in Hertz's Principles of Mechanics. Hertz had suggested that metaphysical confusions and contradictory conceptions of fundamental notions (e.g., of force) cannot be resolved by the discovery of new knowledge, since they are caused in the first place by the accumulation of a rich store of relations and associations around these notions in the context of present‐day knowledge. On this view, the best hope for ridding ourselves of vexing conceptual perplexity is thus to sort through and conceptually model, organize, or “picture” the store of relations and associations so as to resolve conceptual confusions and antinomies, thereby relieving ourselves of the need to summarize or join them in a general explanation of their fundamental nature. So conceived, activities of restructuring, picturing, and formulating the architecture of concepts (Begriffsbildung) provide new ways of conceiving fundamental problems and notions, and are to be contrasted with the activity of discovery of truths by reasoned deduction from first principles. As Hertz wrote in his introduction to his Principles, “When these painful contradictions are removed, the question as to the ultimate nature of [e.g.] force will not have been answered; but our minds, no longer vexed, will cease to ask illegitimate questions.” The allusion to Kant's critique of human claims to know the Ding an sich, the ultimate essence of things apart from human conditions of representation, is clear in this quotation.
Yet Wittgenstein pressed to a logical conclusion Kant's talk about “limits to knowledge,” his distinction between “things in themselves” and things conceived as appearances to us, subject to human forms of sensibility. Wittgenstein saw the Kantian attempt to fashion theoretically justified limits to thought and/or knowledge by articulating general principles of knowledge and/or experience as nonsensical, for there is no making sense of a “transcendental” standpoint on our knowledge that lies at or beyond its limits. For Wittgenstein the logical paradoxes indicated that no such maximally general standpoint on knowledge can be made sense of. The best that can be done is to engage in an immanent exploration of the very notion of a limit to thought or to language.
Thus Wittgenstein radicalized Kant's idea that the laws of logic and/or mathematics require philosophical analysis because of our tendency to misconstrue them as unconditionally true of an ahumanly conceived domain of necessary truth. For Wittgenstein, mathematics and logic are to be seen not only as sciences of truth or bodies of knowledge derived from basic principles, but also as evolving activities and techniques of thinking and expressing ourselves; they are complex, applied, human artifacts of language. The logicism of Frege and Russell was right to see the application and content of logic and arithmetic as internal to their nature; mathematics and logic cannot be accounted for wholly in terms of merely formal rules. Yet what it is to be part of arithmetic, grammar, or logic cannot, Wittgenstein believed, be understood in terms of an appeal to a set of primitive truths or axioms, or knowledge of logical objects. The best way to understand the (p. 80) apparently unique epistemic roles of logic and mathematics is to forgo the quest for a unified epistemology in favor of a detailed investigation of how these activities and artifacts shape and are shaped by our evolving language.
In the end, Wittgenstein's work must be measured not in terms of its direct contributions to knowledge—there were no theorems, definitions, or results of any special importance in it—but in terms of the critical power of the widely various arguments, analogies, terms, questions, suggestions, models, and modes of conceptual investigation he invented in order to understand the post‐Fregean place of mathematics and logic within philosophy as a whole. Wittgenstein offers us problems rather than solutions, new ways of thinking rather than an especially persuasive defense or application of any particular philosophical thesis about the nature of language, logic, or mathematics. Despite the “negativity” and “reactionary” flavor that such a statement may suggest to some,^{5} these ways of thinking mark a decisive step in the history of philosophy, in particular in relation to the legacies of Kant, Frege, Russell, and the Vienna Circle, and inaugurated new and fruitful ways of investigating phenomena of obviousness and self‐evidence in connection with the notion of logic.
After Wittgenstein, traditional logical and philosophical vocabulary involving fundamental, apparently a priori categorial notions such as proposition, sentence, meaning, truth, fact, object, concept, number and logical entailment has more often been relativized by philosophers to particular languages, rather than assumed to have an absolute or universal interpretation a priori. Fundamental logical notions have been conceived as requiring immanent explication, a localized examination of their “logical syntax” in the context of a working, applied language or language game, rather than assumed to reflect independently clear, universally applicable categories capable of supporting or being derived from universally or necessarily true propositions or principles. It is largely to Wittgenstein, especially as inherited by Russell, Carnap and Quine, that analytic philosophy owes its willingness to question the Fregean idea that objectivity requires us to construe collections of various declarative sentences and that‐clauses of declarative sentences, whether in the same language or in different ones, as reflecting a determinate content, thought, proposition, sense, or meaning.
In what follows, some major themes governing Wittgenstein's discussions of mathematics and logic are treated against the background of the evolution of his thought. A strong emphasis is placed on the early philosophy, both because this set in place the major problems with which Wittgenstein was to grapple throughout his life, and because it was the early work that placed Wittgenstein in direct confrontation with earlier philosophies of mathematics, those of Frege and Russell in particular. It was also the early philosophy that exerted the most direct and measurable influence on the Vienna Circle. Readers primarily interested in the (p. 81) later philosophy, which has received widespread attention since the late 1970s, are referred to the accompanying bibliography, which provides an overview of current literature.
Evolution of Wittgenstein's Thought
1. The Early Period
It seems that an early effort to solve Russell's paradox was partly responsible for turning Wittgenstein toward philosophy as a vocation. Trained as an engineer at the Technische Hochschule in Berlin (1906–1908), he studied aeronautics in Manchester (1908–1911), there encountering the works of Frege and Russell. ^{6} By April 1909 he had sent to Jourdain a proposed “solution” to the Russell paradox; Jourdain's response sufficiently exercised him that by 1911 he had drawn up plans for a book and had met with both Frege and Russell to discuss his ideas (Grattan‐Guinness 2000, p. 581). At Cambridge (1911–1913) conversations with Moore and especially with Russell stimulated Wittgenstein profoundly, and his final version of the Tractatus Logico‐Philosophicus was finished by 1918. It may be read, at least in part, as a commentary on the philosophies of Frege, Moore, and Russell, and the place of logicism in relation to the legacy of Kantian idealism. The Tractatus thus forms a bridge between the earliest phases of analytic philosophy and the logical positivism of the Vienna Circle (especially as expressed in the work of Carnap), which in many ways it helped to create.
Wittgenstein made his primary impact by pressing to the fore the broadly philosophical question “What is the nature of logic?,” a question that naturally emerges if we attempt to gauge the general philosophical significance of Frege's and Whitehead and Russell's “logicist” analyses of arithmetic. Given that it had been shown how formally to derive basic arithmetical truths and principles from basic logical principles, on what grounds may these principles themselves be held to be “purely logical”? Our grasp of Frege's basic laws seem, it is true, to involve no obvious appeal to intuition or empirical knowledge, and his formal proofs (Aufbauen) appear to be fully explicit, gap‐free logical deductions. On the surface, his basic principles express truths about fundamental notions (such as concept, (p. 82) proposition, extension) that had long been acknowledged to be logical in nature. Yet neither Frege, nor after him Whitehead and Russell, provided a satisfactory account of why their systematized applications of the mathematical notion of function to the logico‐grammatical structure of sentences should compel us to regard their analyses as purely logical in anything more than a verbal sense.
Frege's and Russell's own mark of the logical had been the explicit formulability and universal applicability of its truths: they conceived of logic as a science of the most general features of reality, framing the content of all other special sciences. ^{7} Their quantificational analysis of generality was what they had on offer to make this conception explicit. But there were internal tensions within this universalist view. Since the content and applicability of logic are assumed by the universalist to come built‐in with the maximally general force of its laws, it is difficult to see how to make sense of its application as application, for from what standpoint will the application of logic be understood, given that the application of logic is what frames the possibility of having a standpoint? Frege's and Russell's views led them to resist the idea of reinterpreting their quantifiers according to varying universes of discourse, for they conceived their formalized languages as languages whose general truths concern laws governing all objects, concepts, and propositions whatsoever. They had no clear conception of ascending to a metalanguage in the model‐theorist's way. And yet, as a result, much of the extrasystematic talk about the application of logic in which Frege and Russell engaged could not, by their own lights, be formalized within their respective systems of logic.
Thus, for example, Frege was explicit that he had to fall back on extraformal “elucidations”—hints, winks, metaphors—to explain his basic, purely logical distinctions. He took inferences applying universal instantiation—the move from “(x)f x” to “fa”—to reflect a universally valid rule of inference, but the universal validity of that rule itself could not be explicitly formulated, on pain of a vicious regress. He denied that “truth” is a genuine property word. The applicability of many of his most fundamental notions and analyses (e.g., the procedures of cut and assertion involved in modus ponens, such presuppositions as that “there are functions,” “no function is an object,” and “Julius Caesar is not the number one”) could not be enunciated in terms of purely logical truths and/or definitions within the language of his logical system. At best, they could be indicated or shown by regarding the use of the formal system as the use of a contentful, universally applicable language.
Even more worrisome, the unrestricted application of Frege's primitive notions mired him in contradiction, as Russell (and earlier Zermelo) showed. If the unrestricted application of basic logical principles engendered contradiction, how (p. 83) were the restrictions placed upon their applications to be defended as purely logical? It was difficult to see how Russell's theory of types could be conceived to be “purely logical” on the universalist view, for it fractured the interpretation of quantifier ranges into an infinitely ascending series of stratified levels, forcing readers to understand each statement of a logical law, and each variable, as a “typically ambiguous” expression. Moreover, Whitehead's and Russell's unlimited, ascending series of types cannot be spoken about as a whole, in terms of a meaningfully demarcated range of generality. The universal application of Russell's general requirement of predicativity (associated with the “vicious circle principle”) suffers from the same problem. And his axiom of infinity raised the question how any claim about the cardinal number of objects in the universe (whether finite or infinite) can possibly be seen to rest on considerations of logic alone. Either this question is one for physics, in asking what mathematics is needed to account for the cosmological structure of the universe, or it begs the question for the logicist in taking for granted that we already have mathematical knowledge of precisely those mathematical structures that the “logical” theorems will have to account for.
The universalist conception of logic's limitless application seemed in the end to leave no room for any model‐theoretical approach to logic. This left insufficient room for the kind of rigorization of the notion of logical consequence with which we are now familiar. Neither Frege nor Russell had any means of formally establishing that one truth fails to follow from another, because they had no rigorous systematization of what in general it is for one truth to follow by logic from another. All they had conceptual space for were explicit formulations of the logical laws and rules of inference that they regarded as universally applicable and the display of (positive) proofs in their systems. The completeness of these systems with respect to logically valid inference could not be assessed except inductively and/or in general philosophical terms, by pronouncing on the maximal generality of logic in its role of framing the content of all thought. ^{8} It would take nearly fifty years after Frege's 1879 Begriffsschrift for the question of completeness with respect to the notion of logical validity to be properly formulated, in part because of the universalist conception that informed his formalization of logic. ^{9}
In his early work, while responding to the internal conceptual tensions within the universalist view, Wittgenstein began to zero in on the project of isolating a notion of logical consequence. This is somewhat ironic, in light of the fact that Wittgenstein's own later remarks about the notion of following a rule appear, at least at first blush, to sit uncomfortably with the idea that we have a clear intuitive idea of one sentence's (p. 84) following with necessity from another. ^{10} Still, though his philosophy always remained indebted to its universalist origins, he took several decisive steps away from the universalist model, steps that were to influence many who followed him.
Frege, Moore, and Russell had each rejected as unclear the notion of necessity, replacing it with that of universality; the early Moore and Russell explicitly held that the notion of a “necessary” proposition was a contradiction in terms, arguing that all “necessary” or “analytic” truths were in fact tautologies; and since no tautology could express a proposition, all genuine propositions involving predication are synthetic (for a discussion of this prehistory, see Dreben and Floyd 1991). In his early philosophy, Wittgenstein brought necessity and apriority back into view, denying that the notion of universality could replace them in accounting for the nature of logic. At the same time he preserved the idea that necessity is not a clear property word or predicate; indeed, he took the Russell–Moore view one step further in calling all so‐called propositions of logic “tautologies.” This had the radical effect of denying that logic consists of genuine propositions (i.e., of sentences capable of being either true or false, depending upon the relevant state of affairs in reality), a sharp break with the universalist view of logic as a science of (maximally general) truths.
Wittgenstein's attempt to distinguish the concepts of necessity and generality while retaining the universalist idea that logic frames all thought was fraught with difficulty, and his struggles to make sense of this project dictated the course of his future discussions of logic and mathematics. The Tractatus countered Russell's oft‐repeated insistence on the importance to mathematical logic of the reality of external relations (i.e., relations independent both of the mind and of the relata related by the relations) by revisiting the idealist notion of an “internal” relation or property, a notion connected in the Kantian tradition with necessity understood as a reflection of human conditions and forms of knowledge.
All logical concepts and relations were for Wittgenstein internal, and vice versa. Internal relations and properties are necessary features of the objects, facts, or properties they relate in the sense that we cannot conceive these to be what they are, were their internal relations and properties to differ. An example is the coordinate (2, 2) in a Cartesian coordinate system: it is part of what it is to be this particular coordinated point that its first coordinate be 2 rather than 1, that it have two coordinating numbers, that it differ from the point (1, 1), and so on. ^{11} To ask, for example, whether its first coordinate must be the number 1 is an ill‐posed question, to evince a misunderstanding, both of this particular coordinate and of the framework within which it figures. In the Tractatus all such necessities are taken to reflect structures of possible thought or modes of representation of facts via propositions, not independent metaphysical facts or substances; they shape how we see facts holding or not holding in the world, but they are not articulable in terms of propositions telling us truly what objects really, ultimately are. The holding of an internal property or relation thus does not turn on any facts being or not being the case, but on logical features of a proposition internal to its modes of possible expression. These are not describable with a proposition, true or false, for they reflect conditions of possible description that must be reflected in any description. For this reason Wittgenstein calls “internal” or “formal” properties and relations pseudo concepts (i.e., not notions serviceable in the description of reality).
Wittgenstein used this idea of the internality of logical characteristics to reject the Hegelian tradition's insistence that the internal relatedness of subject and predicate in a proposition could be used to establish monism as a metaphysical truth. This was, in effect, to revitalize Kant's view that logic, in being a constitutive framework of thought as such, is factually empty, giving us no absolute or substantial knowledge of reality or thought as they are in and of themselves (the Hegelians had granted this for formal or “general” logic, but not for substantial, true logic). It was also to revitalize the Kantian idea that logic is in its nature a fertile source of dialectical illusion, constantly tempting us to confuse our forms of representation with necessary features of reality, and therefore demanding careful critique for its appropriate application.
Each of these ideas directly conflicts with the universalist standpoint of Frege and Russell, according to which logical laws are truths, full stop.
Wittgenstein's emphasis on internal relations was, however, also symptomatic of his departure from Kant's way of critiquing general logic. Kant set out a “transcendental” logic of appearances that displayed the conditions of human knowledge in a (p. 86) set of synthetic a priori principles. Wittgenstein's conception of logical relations as internal to their relata was intended to allow him to undercut any such independently given principled or mentalistic role for the transcendental. He aimed to display the limits of sense from within language itself, by spelling out that which is internal to the basic notion that we express ourselves meaningfully and communicatively, sometimes truly and sometimes falsely. This was the ultimate point at which Wittgenstein could hope to show that neither necessity nor universality can be seen aright without surrendering the idea that they are substantial concepts or properties.
Universal instantiation is as good an example as any. In the Tractatus the logical relation between “(x)fx” and “fa,” in being “necessary” and “universally applicable,” expresses an internal logical relation between these two propositions: neither proposition could be what it is apart from the second sentence's “following by logic” from the first. This indicates the idea that it is nonsensical to ask, given the assumption that “(x)fx,” whether or not “fa” is true—as nonsensical as it would be (on Wittgenstein's view) to ask whether the coordinate (2, 2) has as its first coordinate the number 2, or whether the number 2 is a natural number. The logical or formal character of the relation (or property) comes in, ready‐made, with our representation of its elements. Differently put, the logical content and force of generality is for Wittgenstein something shown in our language, but not said; it is applied, but not described or established by a truth.
Wittgenstein's chief philosophical difficulty lay in working out a conception of logic that would display all logical features of propositions as internal to them, and display all necessities as logical. His Notebooks 1914–1916 had begun with the remark that “logic must take care of itself,” an insistence intended to preempt as nonsensical any appeal to the extralogical in understanding the scope and nature of logic. This insistence incorporated strands of the universalist point of view, but pointed in a new direction. Wittgenstein was simultaneously rejecting an empty formalism about logic that leaves its application to a theory of interpretations (in the universalist spirit) and at the same time rejecting all theories—including certain aspects of Frege's and Russell's philosophies of logic—that rely on an appeal to purely mental activity—or to any extralogical facts—to account for the application of logic in judgment. ^{12} If logic could be shown to “take care of itself,” (p. 87) then the pseudo character of all attempts to justify its application and our obligations to it could be unmasked, precisely by showing them to purport to have achieved a perspective outside of logic from which to explain it. Of course, strictly thought through, this conception would impugn itself, in that its own remarks about logical relations would positively invite themselves to be read as substantial truths about a reality underlying logic. In the Tractatus Wittgenstein bites the bullet, declaring his own sentences as, by his own lights, nonsensical, to be surrendered in the end as themselves misleading, as neither true nor false to any kind of fact. Their use was to portray much traditional epistemic and metaphysical talk of necessity and reality as equally so by freeing us from misleading construals of logical grammar.
Wittgenstein's most basic conceptual move, expressed in his Tractatus conception of sentences as “pictures” or “models” of states of affairs, was to take the unit of the perceptibly expressed judgment—the Satz or “proposition” as an applied sentence, true or false—to be logically fundamental. The challenge was to contrive a presentation of logic that would allow this notion to remain basic to the presentation of what logic is, and to show how to unfold from it, without further ontological or epistemological appeal, all fundamental logical notions and distinctions. Logical form, on this view, was not to be construed as anything entity‐ or property‐like, but instead as expressed in the ways in which we (take ourselves to) apply and logically operate with propositions. If this could be accomplished, Wittgenstein could surrender as otiose the whole idea of a logical fact or law, and still view logic as in some way limitlessly applicable.
The analogy between pictures and propositions intuitively suggests this line of thinking. A picture (a proposition) is conceived not as an object, but as a complex structure, a fact laid up against reality as a standard without intervention of an intermediate entity—something like a yardstick. The variety of different ways in which one might model (or measure) the same particular fact, configuration of objects, situation, or state of affairs—for example, by means of colored dots on a grid, or by means of words, by means of pencils and toy people (by means of miles, feet, inches, or nanometers)—tempts us to conceive of a “something that these models have in common with one another” (a “real length” of an object independent of how it is measured on one or more occasions). Yet there is no such notion to be had apart from the applicability of some particular system of measurement (there is no notion of length in itself, unindexed to such a system). This leaves us with the idea that the “it” that is common, for example, to all possible depictions of a particular person's stance at a particular moment in time, is their common depiction of that particular person's stance, and no other, independent kind of fact. But then the “it” of a picture's content, the essence of its particular comparison with the way the world is—analogously, a proposition's logical form—is on this view not object‐like, or to be understood apart from our understanding and application of it (i.e., of the propositional signs that, in (p. 88) application, are true or false in expressing it). Thoughts and senses, identified by Wittgenstein with applied propositional expressions, are on this analogy equally unobject‐ or unentity‐like: they are shown or expressed in our activities of arguing, agreeing, and disagreeing on the truth of particular propositions. On Wittgenstein's conception, there is thinking without (Fregean) thoughts: what it is to take ourselves objectively to communicate is given through our acceptance of different complexes of signs (in different mouths of different speakers) to express the same propositions or symbols. But this is simply to redescribe or reiterate as basic the notion of a proposition as an applied sentence, true or false.
The weight of Wittgenstein's overarching conception of logic was thus not taken to rest so much on its ability to uncover a true logical structure of the world (e.g., an ontology of objects and facts suitable to justify the priority of truth‐functional and/or quantificational logic) or a model‐theoretic semantics of possible states of affairs to ground an intuitive notion of logical consequence or entailment (though, as Wittgenstein was aware, one may easily interpret some of his remarks in this way). Instead, he took his conception to rest wholly on its ability to display all such (very natural, indeed perhaps inevitable) talk of necessity in connection with the relation of logical consequence as in the end nothing but a repeating back or reflection of the acknowledgment—to be recognized as fundamental rather than defended as a further independent proposition—that we express propositions, some of which are true and some of which are false. This unmasked as nonsensical (unsinnig) the idea that the kind of categorical talk most natural to general philosophical discussion of logic could be viewed as contributing to an independent body of metaphysical truth about the most general features of reality.
Wittgenstein used the truth‐table notation—implicit in Frege's and Russell's logic—to set forth his conception of the logical. He construed a truth‐table as a fully complete expression of a proposition or thought, and aimed to unfold from this diagram the most fundamental logical distinctions. First, he identified the “sense” (Sinn) of a proposition with the truth‐functional dependence of the proposition's truth‐value upon the different possible assignments of truth and falsity to its elementary propositional parts. This extensionalized the notion, allowing him to dispense with Frege's (intensional) way of defining a thought as the Sinn of a proposition, the “mode of presentation” of a truth‐value. It also vividly displayed Wittgenstein's idea of bipolarity, the notion that what it is to be a proposition—to express a definite sense—is to be an expression that is true or false, depending upon how the facts are. It depicted as internal to the proposition its combinatorial capacity to contribute to the expression of further truth‐functionally compounded propositional structures. And it gave Wittgenstein a way to tie the notion of sense to that of logical equivalence in terms of purely truth‐functional structure: “¬p ∨ q,” “p ⊃ q,” and “(p & p) ⊃ q” all express the same sense in being logically equivalent, and vice versa, for the truth‐table form of (p. 89) writing each of these shows that they each express the very same dependence of truth‐value upon assignments to “p” and “q,” regardless of the fact that different truth‐functions appear in their expression. It is, in other words, the way in which the final column of the truth table relates assignments of truth and falsity to the elementary parts that ultimately (logically) is what matters, and not the classification of a proposition's logical form as that of a conditional, a conjunction, or a disjunction. Meaning and sense, in their logical aspects, are thus extensionalized.
This conception yielded a way to distinguish, on the basis of the truth‐table notation alone, the “propositions” of logic: “tautologies” are defined as having “T” in every row of the final column, “contradictions” as having “F” in every such row, and the “propositions” of logic are just these sentences. This portrays logic as empty of factual content, for the truth‐values of such sentential forms may be seen not to depend upon any particular truth‐assignment to their elementary parts; they hold no matter what assignment is chosen. (As Wittgenstein remarked, to say that it is either raining or not raining right now tells me nothing about the weather.) These purely logical “propositions” lack bipolarity, and hence sense: Wittgenstein declares them sinnlos, regarding them as limiting cases of propositions that are not really either true or false, that carry “zero” information.
This characterization of logic was purely “formal” or “extensional” that is, it worked without any independent appeal to extralogical meanings or truths, and solely on the basis of reflection on the propositional sign, the truth‐table notation itself. On Wittgenstein's view the logical status of logical sentences emerges solely through their internal logical structure, a structure that cancels out the depicting features of the sentence's elementary parts, but not through any general truths or through their gap‐free derivability from basic logical laws. The truth‐functional connectives are themselves reduced, in this notation, to what is displayed in a truth‐table, particular dependencies of truth‐values upon truth‐assignments. The interdefinability of the truth‐operations indicates that they do not contribute to any factual aspect of a proposition. Thus Wittgenstein remarks that his fundamental thought (Grundgedanke) is that the logical constants do not refer to or serve as proxies for logical objects. There is, then, no room for a genuine disagreement over the truth of individual logical laws. Any purported justification (e.g., of bivalence, excluded middle, or noncontradiction) would be at best another way of spelling out one's prior acceptance of the truth table as a suitable expression of the proposition itself, and hence no justification at all. In inviting us to suppose that one can step outside of logic to assess the truth of its laws, any such justification is nonsense. There is no question of generality here at all, only a question of one's understanding of the notion of proposition properly.
This understanding of the notion of proposition, built upon Wittgenstein's “extensionalism” about logic, was thus tied to a rejection of Frege's and Russell's respective applications of the distinction between function and argument (Frege's distinction between concept and object), a distinction intrinsic to the quantificational (p. 90) conception of generality that appeared to give the universalist conception its plausibility. As we have seen, Wittgenstein took the ability of a truth table to stand in as an expressive replacement for (or a definition of) a truth‐functional sign (for example, in showing the expressive adequacy of “¬” and “&” relative to the other truth‐functions) to indicate that the signs for the truth‐functions are not names of logical objects or functions, as Frege and Russell had held. Instead, he conceived the connections as part of a framework for expressing logical dependencies, “internal” (nongenuine) relations between sentences. This demanded, for a proper treatment of the logical signs, the introduction of a new grammatical category distinct from that of function and name, viz., that of an operation. The depiction of objects and functions by configurations of names and concept words is part and parcel of the content of propositions for Frege, Russell, and Wittgenstein. But for Wittgenstein operations, being merely formal, are simply a way of operating with propositional signs; they do not stand proxy either for concepts (functions) or for objects. They are instead like punctuation marks: an operation sign forms part of the way in which a particular sense happens to be expressed, but in no way can a particular logical connective be taken to reflect what a sentence means or is about. Wittgenstein takes part of his task to be to show how we may regard all of the usual logical signs as operation signs (i.e., as truth‐operational features internal to the expression of propositions) rather than features referring to that about which the propositions in which they explicitly figure speak. ^{13}
Wittgenstein's construal of logical features as emerging from what we do with propositions—reflecting internal features of our ways of seeing the world rather than properties or functions—remained a central leitmotif in all of his subsequent work. Over and over again he warns against the tendency to reify our means of expression, showing how the oversimplification of logical grammar naturally leads to the posing of pseudo‐questions. He saw it as all too easy to assume that objectivity demands there must be an object or meaning corresponding to every working sign in the language. Frege's so‐called context principle (“never ask for the meaning of a word in isolation, but only within the context of a proposition”) and Russell's notion of an incomplete symbol (his use of the theory of descriptions to analyze away apparently denoting phrases in favor of quantificational structures) were moving in the right critical direction, but did not go far enough in undercutting our tendency to confuse elements of our expressive powers with entities corresponding to elements of our language. For Frege and Russell failed to apply their own principles to the logical elements of language. To Wittgenstein, their uniform extensions of the distinction between function and argument across logical notions were conceptually procrustean, mired in an overly optimistic extension of (an updated form of) the ancient duality between subject and predicate.
Difficulties about the analysis of generality were, however, central to Wittgenstein's problem, and these could not be faced by means of the truth‐table and the picturing ideas alone. It was alien to Wittgenstein, as much as it had been to Frege and Russell, to conceive of the quantifiers as reinterpretable according to arbitrarily chosen universes of discourse; the whole point was to see logic growing seamlessly out of the given application of sentences in a language. But he conceived of this growth in radically new terms. The decision procedure (algorithm) for truth‐functional validity, satisfiability, and implication that Wittgenstein exhibited with the truth‐tables in the Tractatus allowed him to make this view plausible for part of logic: as he pointed out, in cases where the generality symbol is not present, we can see that so‐called “proof” within pure logic is a mere calculation, a “mechanical” expedient allowing us to display for ourselves the tautologousness (or contradictoriness) of a logical proposition simply by rewriting it. The idea that the truth‐functional connectives refer to separable entities, or that we can prove the logicality of a sentence (or the validity of a proof through its conditionalization) via rational inferences from truth to truth, justified step‐by‐step by the application of logical laws to general features of objects and concepts, is thus—for Wittgenstein—otiose.
But how was this conception to be extended to quantification theory, where (as Church was to establish in 1936) we have no decision procedure for validity, and where (as Gödel was to show in 1930) we do have a systematic search procedure for it?
The purely philosophical motivation for extending Wittgenstein's view across “all of logic” was clear. His emphasis on the mechanical aspect of proof in pure logic was motivated by his critical reactions to the wedding of Frege's and Russell's philosophical arguments for logicism with their success in having formalized logic. They had each argued, against formalists and traditional algebraists of logic, that logic and mathematics are more than merely mechanical, formal games of calculation—they are applicable and contentful sciences of truth. Frege explicitly insisted that arithmetic was more than a mere game like chess because its propositions express senses, thoughts [Frege 1903, §91]. For Wittgenstein, this kind of argument rang hollow. Arguably what Frege and Russell had done, in spite of (or perhaps because of) their universalist conceptions, was to show how the purely logical character of an argument can be verified mechanically, by tracing through its formalized structure. This appeared to reinforce the algebraical view that logic is essentially empty, a calculus with uninterpreted signs. Wittgenstein did not wish to fall back on an uncritical formalism. But he did wish to portray the debate between Frege and Russell and their formalist forebears as mired in confusions over what the application of logic and mathematics involves.
Wittgenstein understood quantification as something operational, that is, as belonging to the way a proposition is expressed and applied, rather than to a distinctive descriptive or functional element of a proposition. The Tractatus claims (p. 92) that all quantificational forms of sentence may be generated purely operationally, and uses for this Wittgenstein's operator N, a generalized form of the Sheffer stroke of joint denial that negates elementary propositions either singly or jointly. Since he was writing at a time when the distinction between first‐ and second‐order logic had not been formulated, we may assume that Wittgenstein intended to use operator N to express all quantificational structures. But his notational suggestions have never been made wholly precise. For the first‐order case, however, at least this much is clear: “fx,” a propositional variable indicating a function, stipulates a range of elementary propositions (i.e., those in which the function figures with an object); “¬(∃x)fx” is expressed through operator N by jointly (generally) denying the elementary propositions presented by “fx”; and “(x)fx” is generated by jointly (generally) denying the elementary propositions presented by “¬fx.” The logical step from “(x)fx” to “fa” is thereby seen to be an immediate, purely formal operation rather than an instantiation in accordance with a general logical rule or law of inference; generality contributes nothing in and of itself to the material content of a proposition. The sign for generality is thus not to be read as a function symbol, but as a way in which a collection of elementary propositions is operated upon, or displayed. In a sense, generality is already expressed in “fx,” and therefore, in a sense, the “(x)” part of the generality notation is unnecessary or misleading. ^{14} In virtue of the fixed domain of (elementary) sentences that are presupposed from the outset, this sign can at best be seen to contribute as an index to the applications of operator N at work in cases of multiple generality. But it does not express a genuine kind word, second‐order function, or concept in its own right.
This is very far from the universalist's conception of generality. Wittgenstein is conceiving generality to be expressed in something like the way a genre painting expresses an archetypal feature or scene. Such a painting is applicable to an aspect of each concrete situation exemplifying the relevant genre features. That is what its being a genre painting is, and that is what makes any exemplar an exemplar of it. Its own way of representing is, of course, not reducible to the depiction of any one such example, but each such example exemplifies on its own, without any intermediary principle. This is reflected in the fact that the very same picture could be used to depict (could be inspired initially as a depiction of) a particular scene, and also a genre scene; there is nothing in its internal structure that says how it should be interpreted. That comes out in our uses of it.
Wittgenstein's treatment is not intended to give a unified account of the notion of generality. His distinction between function and operation bifurcates it into two different notions, the material generality of concepts involved as functions at the atomic level in elementary propositions, and the operational generality of formal concepts. As we have seen, he conceives of formal concepts as pseudo (p. 93) concepts because of our tendency to assimilate them to genuine concepts or kind words when we ought, as he thinks, to focus on the operational, procedural aspects of our hold on them: formal notions do not sort independently given objects into kinds (they are neither true classifiers nor sortals), but instead are shown or expressed through the kinds of applications we make of them. Their applications are internal to them, and the applications are, in turn, what they are insofar as they successfully display or express the formal notions.
This gives Wittgenstein a way to dissolve Russell's paradox. According to the “picture” idea, the elementary proposition is a fact in which names serve as proxies for objects via their structural, configuration within the sentence as it is used to depict the holding of a state of affairs. Material (“propositional”) functions (concepts and relations) are expressed via fixed modes of positioning names within the projected propositional sign. This exhausts the articulation of functional and objectual material content (and generality) at work in language. Wittgenstein holds that no material function can apply to itself: no sentence can say of itself that it is true, for the sentence is a proposition only insofar as it is compared, through its functional articulation via a structure of names, with something else that it models truly or falsely. He is thinking here in terms of an analogy with the positionality notation of our decimal system: that our numerals are written in the decimal system, and not some other one, is presupposed in our understanding of how to read the positional notation. This is not something that the numerals say; it is something that our operations with them show. Given a fixed system of positionality to express material functions, writing down “f(f(x))” in effect gives rise to a claim about two different material functions, much as the digit “2” reflects something different in each of its occurrences within the (decimal) notation “222.”^{15} This reflects Wittgenstein's idiosyncratic reading of the generality of the variable: “fx,” on his view, presents a fixed range of elementary propositions (“fa,” “fb,” and so on) that cannot be changed by trying to reapply “f.” In “f(fx)” we thus have two occurrences of the same sign functioning as different (p. 94) symbols; in principle we could simply use different signs (say, with indices) to distinguish them. This is Wittgenstein's “solution” to Russell's paradox. ^{16}
Yet Wittgenstein does not rule out a purely formal, “selfreferential,” or recursive series of nested applications, as in O′x, O′(O′x), O′(O′(O′x)). … What he does insist is that if we imagine such a series being logically connected according to repeated application of a single formal rule, then we are treating “O” as a sign for an operation, not as a genuine (propositional, material) function symbol. Operations, unlike (material) functions, are in their very nature recursively iterable procedures generating collections of instances that are internally ordered in what Wittgenstein calls formal series. Wittgenstein's notation for this kind of (recursive) generality uses square brackets: in, for example, “[a, x, O′x],” “a” stands for the basis step of the series, x for an arbitrary member of the series, and “O′x” for the result of applying an operation O to x. This bracket notation is thus equivalent to writing down “a, O′x, O′O′x, O′O′O′x …,” an expression with an ellipsis. It expresses a rule whose applications are internal to its expression, but the generality of the rule is shown or indicated through the manner of presenting its instances, not said or described. Furthermore, Wittgenstein allows that the basis might itself be given through a formal series, so these bracketed expressions may themselves be iterated. On this view, the procedure of moving from type to type in Russell's hierarchy is operational, something done formally, without any appeal to the facts. It cannot be summed up in the quantificational manner of Frege and Russell without confusion, for the generality of the formal series, ordered by an “internal” rule of the series, is fundamentally different from the generality of, for example, the (material) concept horse.
We have seen that the notion of proposition is held by Wittgenstein to be a formal (pseudo)concept. This is evinced in his treating it as fully displayed through applications of the (formal) operator N. Wittgenstein holds that all propositions are results of applying iterations of operator N to a basis of elementary propositions. No proposition talks about the general form of proposition, for this is a formal notion, given through a recursive template or rule, not via the functionally articulate name of a class or totality. Beginning with some basis of elementary propositions, p, the idea is that by a finite number of applications of operator N, we see that we shall be able to generate all propositions. Wittgenstein writes what he calls “the general form of proposition” in terms of his recursive notation: “,” where “” is a schema for a basis propositions, “” a variable standing for all results of some finite number of applications of operator N to elements in that basis, and “N′()” for the totality of results of applying operator N.
The Grundgedanke of the Tractatus is that the so‐called logical constants (the logical expressions) do not refer: all logical expressions indicate operations, according to Wittgenstein. Now we see that all operations are logical in being expressed through purely formal, recursively iterable rules. And operations are always tied, directly or indirectly, to our application of propositions in language, via truth‐operations on the elementary propositions. Nothing that is a proposition fails to be subject to logical operations.
There has been a recent debate over the question whether and in what sense this Tractatus notation for the general form of proposition is expressively adequate, even for first‐order logic alone, and, therefore, whether and on what grounds Wittgenstein might have been committed, wittingly or unwittingly, to the existence of a decision procedure for all of logic. ^{17} Wittgenstein's situation with respect to decidability was vexed and complex. In 1913 he was apparently one of the first (wrongly) to conjecture the existence of a decision procedure for all of logic, writing to Russell that the mark of a logical proposition was one's being able to determine its truth “from the symbol alone” (cf. CL, pp. 59, 63). Yet it remains unclear on precisely what basis Wittgenstein might still have been committed to this as late as 1918. ^{18} It is usually held—and it was reported to have been said by Wittgenstein himself—that in the Tractatus he conceived of the quantifiers in terms of potentially infinite truth‐functional conjunctions and disjunctions, not realizing the intrinsic barrier to extending the truth‐table analysis into the unrestricted quantificational domain.
Yet Wittgenstein himself had no interest in setting out a smooth‐running codification of quantification theory, and in fact he is not explicit in the Tractatus about what semantics or notational details he would require. His central philosophical problem was, after all, to indicate how one might be able to dispense, conceptually, with the idea that general logical laws or rules of inference such as universal instantiation are true in virtue of meanings of the words “all” and “some” that go beyond what is expressed at the elementary level by their instances; he wanted to show that none of the logical constants must be conceived as going proxy for an object or function. Only in this way could he undercut the idea of a stance from which one might dispute the correctness or incorrectness of an interpretation of the quantifier's range, of the inference from “(x)fx” to “fa,” and show that logic is not a science of true propositions or laws at all.
Arithmetic remained, however, as a potential stumbling block. For in rejecting Frege's and Russell's quantificational conception of generality, Wittgenstein was in effect rejecting the means by which they carried through their account of (p. 96) arithmetic as logical in nature. The first difficulty he had to face was an account of the universal applicability of the cardinal numbers within extramathematical propositions. Here he proposed making cardinality and numerical identity internal features or forms of the elementary propositions. This was to construe cardinal numbers as undefinable, as internal aspects of our presentation of facts in propositions, rather than as objects or properties of objects. Falling back once again on his rejection of the adequacy of the function/argument distinction, Wittgenstein sharply distinguished (as algebraists of logic had traditionally done) between logical identities and mathematical equations. He then argued that the sign for identity is, in connection with propositional functions, expressively, and therefore logically, unnecessary. It would always be possible, he argued, to devise a language in which there were no redundant names: logic cannot bar this as an expressive option. This would allow us to read numerical identity between material concepts directly off from the expression of an elementary proposition; ascriptions of number were then not (as they had been for Frege and for Russell) logical identities.
On this view, “fa & fb” shows without saying that there are at least two F's; the number is a depicting feature of the symbolism, and not a separate object referred to by the proposition. ^{19} Wittgenstein adopts a convention for the variable that allows him to read “¬(∃x, y)(fx & fy)” as saying that there are not two f's. ^{20} Thus the fact that we use equations in mathematics cannot, for Wittgenstein, be used to argue that the numbers are objects, for though he argues that mathematics consists essentially of equations, he holds that equations are not objectual identities in which genuine (propositional) functions work. Instead, they are part of an operational calculus of signs, working via (what mathematicians had long called) the method of substitution. ^{21} Probability is also subjected to analysis in terms of the nation of operation in the Tractatus, via the truth tables.
In this way Wittgenstein revitalized in the Tractatus an older, algebraical way of conceiving mathematics and logic that Frege and Russell had hoped to wipe out as “formalist,” but precisely by tying this language itself back to the logicist idea that our cardinal number words reflect aspects of the logical form of propositions that we apply to the world. A mathematical equation, according to the Tractatus, is neither true nor false, it expresses no thought or proposition, but instead sets forth rules for the substitution of one numerical (operational) sign for another, either in other mathematical equations or in genuine propositions (extramathematical ascriptions of number). Ascriptions of number within mathematics (e.g., “there is only one solution to x + 1 = 1”) enunciate symbolic rules for the application of operation signs; in this way the “x” in such an equation is not read as a universally quantified variable, but as a formal operator. ^{22} Regarded from the perspective of mixed contexts, equations are taken to show us ways we may interchange numerical operations in genuine propositions without affecting the sense expressed. They reflect what we can do with these operations, not underlying general truths.
The pictorial character of number words is part and parcel of Wittgenstein's idea here. Operating on an equation via substitutions of one term (operation sign) for another yields different “pictures” or standpoints from which to consider the operation signs figuring in it: in mathematics we are shown different aspects of the logical syntax of number words (cf. TLP 6.2323, 2.173). Arithmetical “proofs” of equalities between particular number words are calculations, manipulations of a series of pictures using operation signs in accordance with “the method of substitution” (6.241). On this view, the numbers are construed neither objectually nor adjectivally, but practically, in terms of what we do with them. And Wittgenstein has no need to define a general notion of (mathematical) function.
On this view, second‐order logical principles such as Hume's principle^{23}—which essentially requires the notion of identity for its formulation—are otiose. The equinumerosity of two concepts (whether material or formal) is already (p. 98) expressed (applied) in the similarity of form shared by different first‐order propositions (pictures) involving these concepts. For example, given Wittgenstein's eliminative proposal about redundant names, “fa & fb” and “gc & gd” show without saying that there are at least as many f's as g's through their shared forms. Given Wittgenstein's exclusive interpretation of the variables “x” and “y,” “(∃x)(∃y)(fx & fy) & ¬(∃x)(∃y)(∃z)(fx & fy & fz)” and “(∃x)(∃y)(gx & gy) & ¬(∃x)(∃y)(∃z)(gx & gy & gz)” show equinumerosity through their shared forms. Cardinality, for material and for formal concepts, is thus taken by Wittgenstein to be given with a concept's expressive possibilities, possibilities whose applicability must, he believes, already be in place in the use of (nonpurely logical) propositions before talk of one‐to‐one correlation is cogent. The existence of a one‐to‐one mapping between numerals and objects may be understood either materially or formally, externally or internally. Externally and materially, this is a way in which individual objects may be associated, either with one another or with numerals. But the heart of Wittgenstein's picture idea is that a mere association between names and objects is, in and of itself, not something qualifying as propositional at all; only the internal features of a proposition give rise to a logical relation of equinumerosity. The proposition, conceived as an articulate, applied structure or fact, true or false, is fundamental. The upshot is that Wittgenstein treats the cardinal numbers and the notions of object and numerical identity as indefinable forms, expressive features of the symbolism, rather than as objects or names. ^{24}
Wittgenstein did not mention the principle of mathematical induction in the Tractatus—a remarkable thing, given that one of the primary achievements of Frege's and Russell's second‐order analysis of arithmetic was to show how to derive this (apparently “synthetic”) truth from pure logic alone. He did take the cogency of proof by induction to be a reflection of the logical structure of number words; it is just that his conception of the logical differed from the logicists'. For Wittgenstein an induction “principle” is one whose generality and applicability are shown in the general form natural number, which is a formal notion. On this view, mathematical induction's application to the natural numbers is part and parcel of what makes a number a number. Induction is thus not depicted as a separable general truth that could be deduced via a second‐order definition of number in terms of equivalence classes. Wittgenstein's own “definition” construes the (p. 99) numbers recursively, as “exponents” of operations (i.e., as forms common to the development of any formal series). ^{25} Every formal series has a first, a second, a third member, and so on; the notion of “operation” is equivalent, Wittgenstein remarks, to the notion of and so on. By writing down “[0, ξ, ξ + 1],” a variable ranging over exponents of all operations, Wittgenstein shows the general form of natural number (TLP 6.03). In effect, he simply writes down “0, 1, 1 + 1, 1 + 1 + 1, …,” without analyzing the ellipsis away explicitly, as is done with the ancestral construction of Frege and Russell.
To the objection that he was opening arithmetic up to an account in terms of synthetic a priori intuition of succession, Wittgenstein replied that language itself would provide the necessary “intuition”—which was just to dismiss any serious independent role for the notion of intuition. On his view, the individual arithmetical terms within this series are not freestanding, but emerge through their connection with the notation for the general form of proposition and, hence, through the expressive structure of the operations we perform with elementary propositions. Wittgenstein went so far as to deny (in the early 1930s) that there could be a notation for arithmetic in which numerals functioned as proper names (WVC, p. 226). And he always insisted that the theory of classes is “superfluous” for mathematics (TLP 6.031). Indeed, he considered the ancestral definition of successor to suffer from an expressive “vicious circle” (TLP 4.1273). This was not for him an epistemic argument against our claiming to have knowledge of abstract objects or a freestanding preference for predicativity. It formed part of an expressive argument about the philosophical advantages of one notation for the numbers over another, presupposing his treatment of generality.
Wittgenstein was thus always sympathetic to the kind of considerations Poincaré brought forward against logicism—namely, that in setting out the formalization of logic within which the logistic reduction would be carried out, Frege and Russell already depended upon their reader's ability to apply arithmetic and inductive inference. But this was not (as it was for Poincaré) intended to refute the notion that arithmetic is in some sense “analytic.” Instead, it was to show that the logistic reduction is not of any fundamental epistemic relevance. ^{26} Wittgenstein feared that Frege and Russell would reinforce the philosophical tendency to look at the logical characteristics of mathematics through the distorting lens of a (p. 100) unified account of generality. From his perspective, the logicistic reduction is likely to reinforce the tendency to look behind mathematical practice for an underlying account of a reality of necessities. The advantage of his recursive notation lay, he believed, in its showing on its face that which is logical in arithmetic, namely, the internality of the relations among arithmetical principles, arithmetical operations, and arithmetical terms, regarded as a reflection of our fundamental ability to ascribe number in empirical contexts. The general notion of natural number is not a material predicate or kind word, as he sees it, but, like the notion of proposition, is given through our application of recursive operations to expressive features of our language.
The universalist's conception is depicted here as wrongheaded, a needless wrapping up of logic and arithmetic in the guise of talk derived from general principles about objects, functions, and relations. Logic and arithmetic could certainly be developed axiomatically on Wittgenstein's view, but such a development would be, at best, just one mode of exposition among many other possible ones, and at worst, a philosophically misleading exposition. It could never demonstrate to us, as an analysis, the true nature of our concepts of logic, number, and arithmetic, or ground our knowledge of mathematics in absolutely certain or absolutely general principles. It would not show any deep facts of deductive determination, but would recapitulate in one style the very same expressive grammatical structure that we could look at differently; that, indeed, is what would make its derivations “purely logical.” As Wittgenstein sees it, logic “takes care of itself”: a proposition of logic (or, in mathematics, an equation) belongs to logic (or to arithmetic) just as it is, in virtue of its figuring in a calculus of operations that we apply to it. A universalist interpretation in terms of an extensional conception of classes or concepts thus distracts from what is to Wittgenstein's mind truly fundamental to arithmetic and logic—namely, our ability iteratively to operate with (to “follow”) a recursive rule in connection with internal necessities of language; and it is on this ability that the applications of logic and mathematics fundamentally rest. Frege and Russell gloss over this in eliding the crucial difference between accidental (material) and nonaccidental (operational, formal) generality, the difference between that which the sentences of our language depict, and that which comes built‐in with the logical syntax of any depiction.
2. “Middle” Life and Philosophy (1929–1933)
After the First World War, Wittgenstein largely withdrew from academic philosophy. He did speak with Ramsey in the mid‐1920s about the foundations of mathematics (about the notions of identity and cardinality in particular), and by the late 1920s he had decided to return to Cambridge to attempt a further (p. 101) articulation of his views. He continued to write and discuss philosophy until his death. His “middle” or transitional period (1929–1933) was a period of exploration of his earlier views in light of his own and others' reactions.
After 1935 Wittgenstein came explicitly to advocate philosophical investigation of grammar and concepts (i.e., of logic) in which concepts generally, and mathematical concepts in particular, are treated as a “motley,” a or many‐colored, evolving family of notions, notions that lack sharply definable ranges of application (cf. RFM III, §§46ff.). In a general way, his focus shifted from an emphasis on the notion of a calculus or system to the broader notion of a language game, but this shift was made against the backdrop of an underlying continuity in his thinking about the contributions and errors made by Frege and by Russell. During the initial phase of this latter part of his philosophical life, his ideas were constantly evolving under the pressure of his attempts to clarify his Tractarian ideas and shift them in response to recent developments in the foundations of mathematics and logic. What he had eventually to surrender was the idea that the notion of an “internal” property or relation (i.e., his unmasking of the idea that arithmetical and logical participles refer to independently given objects and functions) could be accomplished through the display of mechanically effective algorithms or the notion of a purely “internal” or “formal” relation or technique. In the end, however, his aim of thinking through the nature of logic, and the limitations of the universalist conception, always remained central to his work.
Feigl recounts that Wittgenstein returned to philosophy after hearing a lecture by Brouwer in Vienna in 1927, and some have inferred from this that Wittgenstein should be read as a Brouwerian, but this reading seems very doubtful, both in light of Wittgenstein's insistence on the importance of linguistic expression to our understanding of mathematics (his skepticism about a transcendental, solipsistic self), and in light of his denial (in a sense more radical than Brouwer's) that logic consists of any laws or principles. ^{27} In any case no one influence may be said to have governed his thinking: he experienced no radical conversion in his overarching conception of philosophy, but thought through his ideas in a new, far more complicated intellectual setting. By the late 1920s and through the early 1930s he was reacting critically to the Tractatus's positivistic, empiricistic appropriation by the Vienna Circle and attempting to come to terms with recent developments in the foundations of mathematics and epistemology: not only Brouwer's intuitionism but also Russell's Analysis of Mind, Ramsey's work on the foundations of mathematics, Hilbert's metamathematics and finitism, discussions by Weyl, and Skolem's analysis of quantifier‐free arithmetic (cf. here especially WA, PR, PG, WVC). By 1935 he began to try to come to terms with the diagonalization arguments of Cantor, Gödel, and Turing (Wittgenstein and Turing discussed (p. 102) philosophy and logic between 1937 and 1939). Beginning in 1929, when he first returned to Cambridge, conversations with Ramsey and Sraffa stimulated Wittgenstein, as did his reading of Nicod's Geometry and Induction; he was also influenced by his reading of Spengler, Frazer's Golden Bough, and Freud. In conversations with the Vienna Circle he discussed, among others, Frege, Russell, Husserl, Hilbert, Weyl, Brouwer, and Heidegger. The grammar of the concept of infinity, both within and outside of mathematics, was a central preoccupation from 1929 to 1934.
Given this complexity and ferment, there are many different ways of understanding the development of Wittgenstein's thought. Here we shall content ourselves with a brief description of the ways he developed just a few Tractarian ideas about philosophy, logic, and mathematics; in particular, we shall focus on his treatment of generality and necessity.
Not until he returned to England in 1929, and to sustained conversations with Ramsey, did Wittgenstein begin seriously to investigate whether and how far his Tractarian treatment of arithmetic might be adequate. The difficulty came in assessing how this treatment could be interwoven with his Tractarian conception of logic without glossing over his sharp distinction between material and operational generality, and without adopting a substantial metaphysical or epistemic position in discussing the nature of logic and its applicability. Wittgenstein had not been explicit in the Tractatus about which cardinalities he took to be built into the forms of the elementary propositions and, thus, what sort of elementary propositions he had in mind. He discussed the natural numbers and left all further discussion to the side. This left his attitude toward real and imaginary numbers—and the application of mathematics in physics—unclear, and thus, especially in connection with the topic of the infinite, Wittgenstein's elliptical uses of the notion of operation came to seem to him too coarse and nebulous. The fundamental problem was a clash of analogies at the heart of the Tractatus: he had linked his conception of finite cardinality much too tightly to his view of the truth‐table as an adequate display of the logical. This, as Russell and Ramsey helped him to see, left crucial conceptual questions open.
In the Tractatus Wittgenstein had suggested that the logical impossibility of an object's being both red and green all over at the same time, or both exactly three inches long and exactly four inches long at the same time, would be shown at the atomic level of analysis. Now he saw that he could not hold on to his views about the application of number without linking his conception of the logical to systems of propositions. If, in Wittgenstein's manner, we ascribe a cardinal number x to a material concept fx (e.g., through a proposition like “fa & fb”), then it is immediate (shown within the expression of this proposition) that the application of this number (viz., two) rules out as false the ascription of exactly one to the concept. This treatment of number thus pulled in the opposite direction from the image of an ultimately self‐sufficient elementary proposition; what was given (p. 103) already in the application of the truth‐table were systems of propositions, according to Wittgenstein's own construal of the number words as operation signs. Imagining real numbers used in the formulation of elementary propositions makes this point vivid: here we have infinite complexity of a certain kind within the elementary proposition, complexity we understand through our understanding of the system of real numbers itself. The yardstick analogy with the proposition pulled in this direction anyway. For if an object is presented, via a material concept, to be exactly one meter long (or green all over), then it is immediate (internal to the proposition's expression) that the object so described is not three meters long, not four meters long, and so on. A yardstick, in other words, figures in a system of measurement.
Wittgenstein's treatment of cardinal number left it unclear whether he wished to (or could) make room for the transfinite, as Russell noted in his introduction to the Tractatus. It was equally unclear how Wittgenstein's replacement for Russell's axiom of infinity was to work. How were we to be shown the cardinal number of all objects as all of them? If there were a finite number of objects, n, then on Wittgenstein's view it would be nonsensical to try to say that there were n + 1 objects; but on what ground could we possibly regard this as something purely logical? And if there were an infinite number of objects, then how would that be shown in language? No formal rule could set out the names in advance without destroying Wittgenstein's sharp distinction between names and operations signs, a distinction on which his conception of the universal applicability of the truth table depended. But without such a rule, the totality of elementary propositions remained expressively opaque, in principle impossible to list completely. Russell suggested in his introduction to the Tractatus that an infinite hierarchy of languages would defeat Wittgenstein's use of the show/say distinction—perhaps the first place in print where the contemporary idea of a metalanguage was broached. Wittgenstein had no way to rule this out formally, given his own discussion of the movement from type to type in Russell and Whitehead's hierarchy. But the picture of a series of languages raised with a vengeance the question of whether and on what basis one could make sense of the unity of language (i.e., the very notion of a speaker's grasp of a—his or her own—language.
Nearly immediately upon his arrival in Cambridge in 1929, Wittgenstein's exclusive reliance on the model of the independence of elementary propositions fell by the way, precisely in order that he could retain that which was fundamental to his philosophical conception (cf. WA, PR). For some time he attempted to think through what a logical analysis of the “phenomenological” language of first‐person experience would look like, focusing in his discussions with the Vienna Circle on the logical status of hypotheses and the holistic systematicity to be found in descriptions of experience (WVC). He accused himself of having erroneously construed the quantifiers in terms of conjunctions and disjunctions. (see G. E. Moore [1954–1955]). He admitted that he had failed clearly to show how one could view the application of the quantifier in pure number theory (cf. PR p. 130).
Wittgenstein did not surrender the show/say distinction; he elaborated and adapted it to his evolving standpoint. Confronted with recent work on the Entscheidungsproblem, he retained, but softened, his earlier tendency to connect this distinction with the display of algorithms and the notion of a calculus. Conversations with Ramsey in 1929–30, along with his reading of Hilbert, Weyl, and others, made him critical of the vagueness of his earlier appeal to the notion of operation. He now explicitly retreated from embracing algorithmicism as a general account of proof in logic and mathematics. Ramsey had obtained an important result (1928) for a partial case of the Entscheidungsproblem, and was engaged in trying to develop an “extensional” account of the foundations of mathematics in terms of the notion of arbitrary function, conceiving of his own work as a development of the Tractatus's extensionalized standpoint. In sharply differentiating his philosophical outlook from Ramsey's, Wittgenstein came to radicalize his previous tendency to resist a unified account of generality. By 1934 he was explicitly rejecting two ideas that, as he now saw it, the Tractatus had, in its vagueness, invited: (1) the idea that mathematics and/or logic have a unified core or nature; (2) the idea that a philosophical understanding of logic or mathematics could rest upon the solution to a leading or single fundamental problem (such as the Entscheidungsproblem). The articulation of these rejections became the hallmark of his later discussions of mathematics and logic.
The evolution of Wittgenstein's thought was, however, piecemeal and uneven during this “middle” period. In 1929 he began to speak (as he had not in the Tractatus) of the “sense” of a mathematical proposition or equation as something that is shown in its proof (cf. WA, PR). He still sharply distinguished between equations and propositions, but now began to try to do better justice to the distinctive generality at work in mathematical proof. This comes through in his examination of Skolem's quantifier‐free treatment of primitive‐recursive arithmetic (PG, pp. 397ff.). Wittgenstein still wished to deny that an induction could be conceived on the model of the demonstration of a true general proposition based on general principles. Now he insisted that such a proof shows on its own the “internality” of the connection between universal quantification and instances by constructing a kind of algorithm or template. In other words, the meaning of “all” in an inductive proof is fully expressed in the giving of the argument itself—it is a reflection not of the existence of a further function or rule, but of the grammar of the formal series of natural numbers. On this view it would be nonsense to ask for a general principle of mathematical induction to justify the generalization: once a property has been shown to hold for zero and to hold for the successor of an arbitrarily chosen k, in some sense the application of “all” is fully expressed or exhausted. ^{28} For Skolem to claim that in an inductive argument a general proposition (p. 105) is proved to be true is, as Wittgenstein sees it, just dispensable and potentially misleading “prose,” gas surrounding the hard and genuine core of the proof, which consists in nothing but the construction of a diagram for an algorithm. Thus Wittgenstein conceives the recursive proof as a (schematic) picture, not as a proposition: it shows or directs us in the way an algorithm, table, or rule does. In an inductive proof, something is expressed that cannot be satisfactorily accounted for in terms of the notion of an arbitrary function or an explicit second‐order principle of logic. Instead, the generality comes out in our applications of the template or schema. We see from the proof (the picture) how to go on. ^{29}
Wittgenstein's resistance to Ramsey's extensionalist conception of (propositional) functions—a major theme of his middle period—also turns on his adaptation of the Tractarian conception of showing. The heart of his unwillingness to follow Ramsey's approach to the foundations of mathematics was that he could not see what made the notion of function‐in‐extension a logical notion. Logic unfolded what could be conceived of as given in the use of propositions, true or false. But then a function‐in‐extension à la Ramsey could not be taken to be a propositional function: it lacked the kind of pictorial complexity Wittgenstein associated with propositions. For Ramsey to allow any arbitrary output whatsoever for an input to such a function left us at sea about the applications of this notion within language, within the expression of propositions. From 1929 through 1934 Wittgenstein set his face against Ramsey's extensionalism about the infinite by speaking of what he called the “intensional” character of the concept of infinity, by which he meant a conception of the potential infinite as given through our possible applications of a rule, in contrast to a conception of it as a list or abstract object such as a set. From the Tractatus Wittgenstein retained the idea of conceiving the indefinitely extendable as a grammatical rule or elliptical expression we apply in language, as opposed to a property, object, or totality, as well as the idea that the notion of infinite is not a specifically logical notion; infinity, he argued, is not expressed in a picture or isolated fact. During his early middle period he tried to investigate in far greater detail than he had before different contexts in which the notion of the infinite shows its grammatical face. His discussions, though often read as endorsing a straightforward form of verificationism or intensionalism, are perhaps better seen as a link between his earlier and his later philosophy: a further, more detailed effort to reject an overarching or homogenous categorial structure for all of logic and mathematics. By 1935 Wittgenstein came to see his talk of an “intensional” conception of infinity as too vague and coarse, as liable (just as his earlier talk of operations had been) to misconstrual in (p. 106) the hands of intuitionists, finitists, and verificationists. He became explicit that revision of mathematical practice in the light of epistemic or empirical constraints on human modes of knowledge was not his aim—thereby more or less admitting that his earlier discussions had invited such misuses. And he also began to zero in on the ways in which his earlier discussions of operations and intensions had rested on far too general and uncritical an appeal to the notion of following a rule. While the latter topic was to become an explicit focus of concern by the mid‐1930s, Wittgenstein was never to surrender the idea that the theory of classes (the extensional conception of [propositional] function) was “superfluous” for (i.e., parasitic upon) the working applications of mathematics in language, in its role as part of our framework for giving empirical descriptions. For him, set theory was a strange hybrid of traditional logical notions (concept, extension, proposition) and purely mathematical notions about whose mathematical applications he always remained confused and about whose purely philosophical applications and metaphysical motivations he always remained doubtful. This was not to reject set theory's results as genuine parts of mathematics, to but insist that these results require detailed scrutiny (cf. PG, pp. 460ff.; RFM II, §§19ff.; V, §§7ff.; VII, §§33ff.).
To rid himself of spurious philosophical appeals to meanings and objects, Wittgenstein had always emphasized the central role of algorithms and calculation in mathematics, the image of mathematical activity epitomized by the solving of equations according to calculation techniques. He saw no reason to give this strategy up, or to try to justify it. But as a picture of mathematics generally, it was misleading. Mathematics consists in more than the construction of algorithms, procedures, and axiomatic systems; it seems to give insight into structure. In particular, the metamathematics of Hilbert seemed to be able to show us how to mathematize certain notions and philosophical questions that had not been mathematized before. In response, Wittgenstein insisted that metamathematics is not a theory establishing true, independently grounding principles for otherwise incomprehensible practices. Instead, it gains its point and purpose from its application to our practices of arithmetical calculation. Metamathematics was for Wittgenstein just another branch of mathematics, an extension of it in a new direction, not a more fundamental supertheory (cf. WVC).
In one way, metamathematics appeared to Wittgenstein to be just a sophisticated way of formally picturing or modeling the grammar already in place in arithmetic. In another, it struck him as misleading prose inessential to the working core of everyday mathematical practices such as devising techniques for solving equations, engineering solutions for the construction of machines, reckoning calculations, and applying geometry. The trouble with metamathematics, for Wittgenstein, is that it tends to mislead philosophers into thinking that the metamathematical language gives us a single way of surveying the core, or interpreting the meaning, of apparently fundamental mathematical and logical (p. 107) notions. But ascent to the metalanguage is just another perspective on practices that gain their character within language from their working applications in human life. Such ascent may change our perspective on our own language, but it grows from our current practices, and is parasitic upon them: it cannot make them more epistemologically certain.
3. The Later Philosophy
Throughout his life Wittgenstein mounted many arguments designed to question how far we may legitimately hold that logical or mathematical necessity (or truth) is lodged in the purely deductive implications of prior intellectual commitments (e.g., commitment to the truth of certain general arithmetical principles, to definitions and laws of inference, or to general rules of grammar previously accepted). His point was not to question arithmetic or the formal derivations to be found in Frege's Grundgesetze or Principia Mathematica, nor was it to insist, in an irrationalist or mystical vein, that its bases are not in any way cognitive—though he did often highlight the arationality of the sometimes rote, unreflective aspects of training on which teaching of logic and mathematics depend. His primary focus was on the misleading pictures of language, understanding, and rationality that emerge from naïve interpretations of such accounts of mathematics. Like the logicists, Wittgenstein took the kind of apparently unrevisable, impartial, universally agreed‐upon truth of mathematics to arise from the very grammar of our understanding of the terms involved. At the same time he insisted more and more stridently over time on the philosophical relevance of the fact that this grammar arises in tremendously complicated, partly contingent ways that evolve over time within the natural world, depending upon how a particular sentence is learned, taught, applied, and contextualized within larger mathematical and extra‐mathematical linguistic contexts.
A central analogy remaining from Wittgenstein's earliest work was between mathematical theories and systems of measurement. An equation such as “2 + 2 = 4” may be conceived to serve speakers of the language as an operational recipe, a standard or “paradigm” to which certain activities, if they are to count as arithmetical, must conform (RFM, I). As a purely arithmetical tool licensing the substitution of one numerical term for another, the equation serves a role much like that of a conversion ruler showing how to switch between inches and meters; in its applied role, it is used like a ruler for measuring particular episodes of calculating and counting, giving us something like a rule or model of grammar rather than a particular belief. If one counts two apples and another two apples, without overlap, then there must be four apples counted, and if there are not four on a subsequent count, we say either that there has been an error in counting or (p. 108) that a surprising physical event has occurred. The equation's “truth,” if we wish to speak this way, holds as much in virtue of our own, contingently evolved commitments to certain methods of representation and ongoing communal practices and needs as it does in virtue of the nature of things. Like a system of measurement, mathematics (like logic) is for Wittgenstein a complex human artifact, situated and created in and for an evolving natural world, and its claims to objectivity and applicability ultimately turn upon our human ability to find one another in sufficiently constant agreement about results of its application to make the practice prove its worth (RFM VI, §§39ff.).
If we insist on speaking of discovery in mathematics, Wittgenstein suggests that we ought to allow ourselves to think in terms of an analogy with technological discoveries, which are perhaps better conceived of as inventions, like the steam engine or the wheel or the decimal notation or the computer (RFM I, §§168ff.; II). As we have seen, algorithms and fixed procedures of calculation were for Wittgenstein—as they are in any system of measurement—of central importance here, and he always conceived tables, algebraic representations of constructive procedures, algorithms, and calculations to play a central role in mathematics. ^{30} As time went on, he relied more explicitly on his looser analogy between applied mathematical structures (systems of numerals, geometrical diagrams, equations, episodes of counting, and proofs) and pictures or models (Bilder). He explicitly regarded proofs and episodes of establishing equinumerosity as conceptual paradigms, models giving us ways of synoptically expressing reproducible routines for describing and/or constructing empirical events. Insofar as these routines allow us to take in, understand, and make sense of empirical situations, they are to be counted as part of the logical syntax or “grammar” of our language; they fix concepts and their applications precisely by making them synoptically surveyable. Such artifacts are, like systems of measurement, as much invented as discovered, because to understand them, as to understand any human artifact, we must be able to understand their uses in concrete situations we encounter and wish to describe.
The use of such pictures in communicating and establishing mathematical conviction is tied to empirically conditioned perceptual and intuitive aspects of our nature as human beings, aspects that shape their qualities of design. A theorem that may be communicated with an easily surveyable model gives us a kind of understanding and insight (an ability to “take it in,” to appreciate and communicate (p. 109) the result) that a long, intricate, formalized version of the proof does not. That is not to say that there are no uses for formalized proofs (e.g., in running computer programs). It is to say that a central part of the challenge of presenting proofs in mathematics involves synoptic designs and models, the kind of manner of organizing concepts and phenomena that is evinced in elementary arguments by diagram. This Wittgenstein treated as undercutting the force of logicism as an epistemically reductive philosophy of mathematics: the fact that the derivation of even an elementary arithmetical equation (like 7 + 5 = 12) would be unperspicuous and unwieldy in Principia Mathematica shows that arithmetic has not been reduced in its essence to the system of Principia (cf. RFM III, §§25,45–46).
Of course, our ability to use a picture or diagram to express and/or communicate a mathematical principle or concept depends upon our understanding of how to apply it. A central difficulty Wittgenstein raises here is, the question of what is to count as an application (correct or incorrect) of a logical or mathematical rule, concept, or procedure, construed as such a model or picture. Wittgenstein is to be credited with having tied this question to more wide‐ranging ones about the nature of concept‐possession. In his later work he often drew an analogy between the ability to project a noun or adjective into new contexts and the computation of instances of an elementary arithmetical function, or the drawing of a purely logical inference (cf. RFM I; PI §§186ff.). This was intended to soften his earlier Tractatus distinction between material and formal generality. Now he emphasized that the difference between the mathematical and the empirical (or the psychological) was not given by the grammatical structure of sentences: the very same sentence might in one way be viewed as empirical, and in another way as merely conceptual or mathematical (cf. RFM VII, §§20ff.). A fortiori mathematics and logic were to be seen to have no monolithic core, just as—and indeed because—our notion of that which belongs to language does not. All we have to rely on in expressing the general applicability of a model, an episode of empirical counting, or a diagram is our language, broadly construed. And yet no image, picture, sentential form, or model applies itself. Not any arbitrary application or use one might make of a sentence or diagram can be held to be fitting or appropriate to it if there is to be any coherent notion of applying it. Yet how deviant from the communal norm can an application be before we say that it is no longer an application? The question is just as difficult (and perhaps just as unanswerable) as the question of when the use of an artifact—such as a knife—ceases to be a use of the artifact as the artifact that it is. One of Wittgenstein's main points, throughout his discussions of rule‐following, is to show how an absolute notion of logical necessity, treated without qualification, is as much a will‐o'the‐wisp as an absolute or general notion of rule. ^{31}
Indeed, in drawing analogies between equations, proofs, numerals, and concepts and rules of grammar, Wittgenstein was suggesting that our cognitive verbs such as “know” and “believe” ultimately work against the grain of our usual traffic in logical and mathematical statements (cf. RFM I, §§106ff.). For instance to say that a person believes that 2 + 2 = 4 appears to leave open the possibility that the sentence “2 + 2 = 4” might not be true, suggesting that one could understand the equation quite apart from agreeing with it. Wittgenstein's view, in depicting the equation as a rule of language, is that understanding and assent work so closely together in the case of this kind of accepted logical and mathematical sentence that one cannot step outside one's own understanding of the notions and terms at work in them to prescind from one's commitments with in our language when considering the equation's truth. To understand “2 + 2 = 4” in the way we expect an adult speaker of our language to understand it is to be unwilling to grant the sense of circumstances in which this sentence might turn out to be false. To suppose that its truth is a matter of very well confirmed or perhaps even indubitable professional opinion, or that only a proof from more general principles can, ideally, bring about belief or certainty in its truth, is to fail to give this conceptual aspect its proper place.
Proof is thus not sufficiently understood, according to Wittgenstein, by maintaining that it yields firm conviction or certain truth dependent upon knowledge of derivations from prior truths, for it equally involves acceptance of a sentence (or proof) as a standard in one's own language of what does and does not make sense. Proof, on this view, is not a series of sentences meeting purely formal deductive requirements, but an activity of achievement and acceptance constitutively shaping one's conceptions of language and the world, sustained in complicated ways by psychology, natural regularities, and communal practice. To appreciate the necessity and universal applicability of an equation like “2 + 2 = 4,” then, Wittgenstein asked his readers to investigate the role it plays in shaping what does and does not make sense to us, our very notion of understanding.
Wittgenstein tries to display the importance of this idea in his later writings by sketching simplified language games or imagined forms of life—often usefully pictured as those that might be played with children—in order to illustrate how much varied, active cultural training, both rote and reflective, must be in place for human objectivity in mathematics to take the forms that it does. This brings out the relativity to particular practice of such apparently fundamental notions as object, name, reference, number, proof, and so on. There is in fact a great variety of empirical, contingent human factors—historical, aesthetic, anthropological, pedagogical, psychological, and physiological—on which the evolution and interest of mathematical objectivity depend. Wittgenstein's point is an anti‐reductive one, though it is easy to slide from his remarks to the notion that he was offering a psychologistic, radically empiricistic, purely conventionalist, ethnomethodological or relativistic account of mathematics: indeed, Wittgenstein is often hailed as a (p. 111) hero by social constructivists. ^{32} His primary aim was, however, to diffuse (without refuting) assumptions about knowledge shared by various forms of skepticism and those anti‐skeptics (such as Frege, Russell, and Gödel) who hold that a universally applicable, eternal framework of contents, thoughts, or senses, possibly transcending the reach of humanity's current concepts, must be postulated to account for mathematical objectivity and truth.
Wittgenstein came to treat mathematics and logic as a “motley,” then, partly because he came to emphasize the importance to philosophy of the idea that there is no single property, criterion, mental state, fact, or characteristic feature common to all cases of the expression of understanding. Indeed, it is part and parcel of his later investigations of logic to make this image of our concept of understanding plausible. Understanding is manifested in many different kinds of ways, even in the case of a particular linguistic form such as “2 + 2 = 4”: sometimes in a characteristic experience of the moment (“Aha!! Now I get it!!!”); sometimes in the ability to pronounce it unhesitatingly (as in the memorization of multiplication tables by children); sometimes in the ability to apply it to solve minimally difficult word problems; sometimes in the ability to set it into a more general, systematic context of definitions and proofs; sometimes in the ability to successfully teach, communicate, and/or defend it to another. Wittgenstein suggested, famously, that the notion of understanding—along with such notions as proof, truth, meaning, mathematics, number, language, and game—has a “family resemblance” character, in that, though no one characteristic might belong to every instance of the general notion, overlapping similarities from case to case would suffice to tie the instances together. His discussion was intended to explore the extent to which the notion of full or complete understanding of a sentence is itself a potentially misleading idealization, especially for the logician. ^{33} In this way he retained his early willingness to question the relevance to philosophy of the idea that to every predicate or concept of the language we may associate in the same way a function or an extension (i.e., a sharply defined property or concept‐word). He also retained his idea that knowledge and/or understanding ought never to be conceived of as a relation between a person and a proposition without regard to the particular context of utterance or the particular sentence affirmed (cf. RFM I, appendix III).
A major difficulty here is, of course, how to do justice to the intuitive notion of mathematical and logical truth, for mathematics is not just a game. Wittgenstein's resistance to any doctrine of contents or knowledge that pretends to universal validity, coupled with his insistence that the applications of mathematics are internal to its character, led him to deny that there is a substantial (i.e., more than merely formal or family resemblance) characterization of the notion of mathematical (or even just arithmetical) truth. His conception aimed to deny that (p. 112) the objectivity, sense, and applicability of the most basic notions of logic and mathematics (e.g., concept, proposition, elementary arithmetical truth, proof, number, and so on) may be explained by setting forth an axiomatized theory in which truths involving the said notions are explicitly derived from fundamental principles. From this perspective it is not surprising that by the mid‐1930s Wittgenstein became fascinated with understanding the fundamental basis of Cantor's and Gödel's limitative arguments, as well as Turing's analysis of computability (cf. RFM I, appendix III; RFM II; RFM VII). Of course, his resistance to making a definition of truth central to philosophy was, as we have seen, based on purely philosophical considerations that predated by over a decade his encounter with Gödel's work; in general historical terms it was a hallmark of the Kantian tradition—shared by Frege, among others—that philosophers could learn nothing important from analyzing or defining the concept of truth. ^{34} But Gödel's incompleteness theorems forced Wittgenstein to rethink the role of the notion of truth in mathematics.
Wittgenstein's emphasis on the image of the mathematician as inventor or fashioner of models, pictures, and concepts was, in the main, directed at the philosophical talk of those, like Hardy and Russell, who insisted on speaking of mathematical reality in a freestanding way, picturing the logician or mathematician as a zoologist embarked on an expedition to new, hitherto unseen lands, analogous to an empirical scientist. Wittgenstein himself explicitly said he did not wish to deny that there is a “mathematical reality.”^{35} But on his view the Hardy–Russell picture of truth tended to preempt as irrelevant to mathematics its evolution as a language, the importance to it of problems of expression, intention, formulation, and construction. For Wittgenstein the mathematician is an inventor, not in the sense of making up truth willy‐nilly as he or she goes along, as a pure conventionalist would suppose,^{36} but in the sense of engaging in the activities of fashioning proofs, diagrams, notations, routines, or algorithms that allow us to see and accept (understand, apply) results as answering to what does and does not make sense to us. We “make” mathematics in the sense that we (p. 113) make history: as actors within it (cf. WVC, p. 34, n. 1). The “what” in “what makes sense to us” is then evolving, to be understood locally for present purposes, and not in terms of any theory of content or a priori known domain of fact. Wittgenstein still took philosophical considerations ultimately to rest on terms and structures of the language we take ourselves to be speaking right now, but in the end he emphasized the variety of perspectives we may bring to bear on our understanding of language's internal necessities and requirements. That which “belongs to language” is something open for current investigation, not something to be taken as determined, stipulated, or fully circumscribable in advance.
Bibliography
Bibliographic Essay
The bibliography that follows contains a representative, though by no means exhaustive, sampling of literature relevant to Wittgenstein's philosophy of logic and mathematics. The following remarks are intended to provide an overview of themes treated in recent literature for the interested reader.
Those who wish to learn the biographical details of Wittgenstein's life are advised to consult McGuinness (1988) and Monk (1990), as well as the brief Malcolm (2001).
New Logics
Wittgenstein's philosophy has helped to inspire the construction of new logics that serve to question the philosophical hegemony of first‐order logic. Hintikka's development of systems of modal logic, the logic of knowledge, and game‐theoretic semantics was inspired in part by the Tractatus, and in part by Wittgenstein's notion of a language‐game (see Hintikka (1956, 1962, 1973, 1996a, 1996b, 1997). Parikh (1985, 2001) has also stressed connections between logic and games in his formal treatments of logics of knowledge and information. For the development of an inconsistency theory of truth and paraconsistent logic that is relevant to Wittgensteinian questions about whether an inconsistency in arithmetic can be logically contemplated, see Priest (1987, 1994). Tennant (1987) develops a constructive logicist foundation for number theory based on intuitionistic relevance logic, describing his system as a generalization from Wittgenstein. Sundholm (p. 114) (1994) subsumes an intuitionistic view of truth as existence of proof under a truthmaker analysis, situating his analysis explicitly in relation to logical applications of Wittgenstein's Tractarian ideas. Shapiro (1991) makes an interesting application of the rule‐following considerations to argue for the expressive superiority of second‐order logic over first‐order logic.
Rule‐Following
The centrality of what are now called “rule‐following” considerations to the interpretation of Wittgenstein was initially emphasized by Fogelin—see Fogelin (1987), Wright (1980), and Kripke (1982)—building on Wittgenstein's later writings exploring the analogy between the application of an arithmetical principle and concept‐projection; each formed a response to the view, voiced in Dummett (1959), that Wittgenstein was a “full‐blooded” conventionalist about logical necessity. Different responses to Dummett may be found in Stroud (1965) and also in Diamond (1989, 1991), who is simultaneously reacting to Wright. A range of differing reactions to Kripke's exegesis of Wittgenstein may be found in Baker and Hacker (1984), Goldfarb (1985), Cavell (1990), Floyd (1991), Minar (1991, 1994), Steiner (1996), and Wright (2001, ch. 7); Holtzman and Leich (1981) is a useful anthology exploring general philosophical implications of this theme. More focused discussions of the implications of these considerations for the notion of content and the ascription of meaning via linguistic behavior may be found in Boghossian (1986) and Chomsky (1986). Proudfoot and Copeland (1994) and Shanker (1998) have connected their readings of Wittgenstein on rules with questions about Church's thesis, Turing's mechanism about the mind, and the foundations of cognitive science. “Rule‐following” considerations have also been brought to bear on the presumption (e.g., by Quine and Putnam) that the Skolem–Löwenheim theorem can establish inscrutability of reference and/or “internal” realism: on this, see Hale and Wright (1997), Benacerraf (1998), and Wright (2001, part IV), as well as Parikh (2001), which connects these questions with Searle's Chinese Room thought experiment and the theoretical foundations of computer science.
Wittgenstein's Place in the History of Analytic Philosophy
A more detailed historical approach to Wittgenstein's philosophy has characterized much recent work on his philosophy of mathematics, sparked by a growing interest (p. 115) in developing an enriched account of the origins and development of early analytic philosophy in relation to the history of logic. Baker (1988), Coffa (1991), Simoris (1992), Friedman (1997), and Hacker (1986, 1996) offer readings of early Wittgenstein through the broad lens of a larger philosophical history of neo‐Kantianism, realism and logical positivism; Potter (2000) offers an especially concise technical reconstruction of the Tractatus treatment of logic and number in light of the history of philosophies of arithmetic from Kant to Carnap. Dummett (1991b) examines Wittgenstein's relation to Frege in connection with the theory of meaning. Marion (1998) places Wittgenstein into context against the background of the history of finitism as a mathematical tradition, contrasting Wittgenstein with Brouwer and drawing connections between Wittgenstein's middle‐period views on generality and remarks by Weyl and Hilbert. Aside from this work, the only other recent book‐length treatments of Wittgenstein's philosophy of mathematics in its historical development, from early through later philosophy, are Shanker (1987) and Frascolla (1994); Shanker (ed., 1986) is a useful anthology of articles covering all phases of Wittgenstein's philosophy of mathematics. Shanker rejects both Kripke's rule‐following skepticism and Dummett's use of the realist/anti‐realist distinction to interpret Wittgenstein on mathematics, elaborating an alternative reconstruction inspired by the grammatical conceptualist reading of Baker and Hacker; he explores in detail Wittgenstein's conception of proof and his reactions to Skolem and Hilbert, and also explores the significance of computer proofs for Wittgenstein's point of view. Frascolla offers the most detailed formal reconstruction yet of Wittgenstein's Tractatus account of arithmetic—cf. Frascolla (1997, 1998) and Wrigley (1998) for a rejoinder; and, like Marion and Shanker, questions the attribution to Wittgenstein of an epistemic perspective that could support strict finitism.
Van Heijenoort (1967) is the locus classicus laying out the “universalist” conception of logic, attributing this view to Frege, Russell, and the early Wittgenstein and contrasting it with the algebraic tradition of “logic as calculus”; this article has profoundly affected how the Tractatus show/say distinction has been read, along with the history of the quantifier and the notion of completeness (cf. Goldfarb [1979, 2000]; Dreben and van Heijenoort [1986]). The extent of this conception's accuracy to Frege has been disputed by, among others, Dummett (1984); for a discussion of the debate, see Floyd (1996). Van Heijenoort's reading remains especially influential among those interested in the history of early analytic philosophy, and in Wittgenstein in particular. Among others, Hintikka and Hintikka (1986), Weiner (1990, 2001), Ricketts (1985, 1986a, 1986b), and Diamond (2002) have deepened and generalized Van Heijenoort's analysis in interpreting Wittgenstein's relation to Frege. Reck (2002) and Crary and Read (2000) contain further essays interpreting Wittgenstein's philosophy of logic and mathematics in the historical context of early analytic philosophy.
Boghossian and Peacocke (2000) contains several essays touching on Wittgenstein's philosophy of mathematics in relation to the evolution of twentieth‐century conceptions of a priori knowledge as they shape contemporary discussion.
(p. 116) Wittgenstein and Gödel
An abiding interest in locating Wittgenstein's philosophy in relation to Gödel began with nearly unanimous condemnation of Wittgenstein: reviews of the earliest published excerpts from his writings on Gödel were roundly dismissed, and skepticism about his remarks on Gödel still remains (see Hintikka [1993]). Klenk (1976) offered an early attempt to defend the cogency of some of Wittgenstein's remarks. In recent years there has been a revisiting of the topic, both philosophically and historically, in light of newly detailed historical questions about the evolution of Wittgenstein's appreciation of the incompleteness theorems and the significance of diagonalization methods. Assessment of the nature and implications of Wittgenstein's attitude toward Gödel's incompleteness theorems has been taken up by Shanker (1988), Floyd (1995, 2001b), Rodych (1999, 2002), Steiner (2000), and Floyd and Putnam (2000), following earlier discussions by Goodstein (1957, 1972) and Wang (1987b, 1991, 1993); Kreisel (1983, 1989, 1998) has also elaborated thoughts on the topic. Webb (1980) and Gefwert (1998) situate remarks of the on middle Wittgenstein within the larger history of formalism, mechanism, the Entscheidungsproblem, and the effect of Gödel's and Turing's work on Hilbert's program. Maddy (1997) contains a useful discussion of Wittgenstein and Gödel's attitudes toward the philosophy of mathematics. Friedman (1997, 2001) probes Carnap's attitude toward the relevance of Gödel's arithmetization of syntax to Wittgenstein's Tractatus. Tymoczko (1984) invokes Gödel's epistemological remarks on the incompleteness theorems to argue for a socially based “quasi‐empiricism” he associates with Wittgenstein.
Recent Trends in Interpreting Wittgenstein: New Approaches to the Philosophy of Mathematics
General Themes.
Large‐scale shifts in the interpretation of Wittgenstein's general philosophy since the mid‐1980s have affected and been affected by accounts of his philosophy of logic and mathematics. The presentations of Wittgenstein in Wright (1980) and Kripke (1982) radicalized Dummett's (1959) view of Wittgenstein as a conventionalist, reading him as offering a form of skepticism about the rational necessity of applying a rule. Among those who questioned Dummett's reading while rejecting the attribution to Wittgenstein of skepticism were Stroud (1965) and, beginning in the 1970s, Diamond (see [1989, 1991]); she fashioned a rival account of Wittgenstein's philosophy of logic and mathematics in the context of a wider recasting of discussions of realism in ethics and a stress on the continuity of (p. 117) Wittgenstein's development (compare Gerrard [1991]). Diamond's approach to Wittgenstein has been especially influential in the United States, along with work of Cavell (e.g., [1976, 1979]), which independently challenged both the then‐popular broadly empiricistic reading of Wittgenstein (see, e.g., Pears [1986, 1987, 1988]) and the reading of Wittgenstein's conception of grammar due to Baker and Hacker (e.g., their [1980], [1984], [1985]). Diamond (1999) and Gerrard (1999) reject traditional readings of Wittgenstein as a verificationist and explore Wittgenstein's treatment of the notion of proof in connection with his later discussions of the notion of meaning. Readers influenced by Cavell and Diamond have inspired recent heated interpretive debate about a “New Wittgensteinian” approach (see Crary and Read [2000]); this kind of reading has been applied to Wittgenstein's later discussions of mathematics by Floyd (1991), Conant (1997), and Putnam (1996). Questions surrounding this “new” interpretation in relation to the Tractatus conceptions of analysis and realism have been pursued by Goldfarb (1997), Diamond (1997), McGinn (1999), Floyd (1998), Proops (2001), Ostrow (2002), and Sullivan (2002a); Floyd (2001) and Kremer (2002) examine its implications for our understanding of the Tractatus treatment of arithmetic. Wright (1993) sets the verifiability principle into conceptual context against the backdrop of different strands of Wittgenstein's philosophy of mathematics, and outlines in (2001) challenges he takes to be faced by the “noncognitivist” approach to mathematical proof he associates with the later Wittgenstein.
Constructivism.
The grounds on which Wittgenstein might be taken to have held a substantial constructivist attitude toward mathematics—despite his claim not to be forwarding “theses” in philosophy—have received substantial investigation in recent years. Marion (1995), Sahlin (1994), Wrigley (1995), and especially Sullivan (1994) analyze in useful detail Wittgenstein's debates with Ramsey in the late 1920s with an eye toward understanding the character of Wittgenstein's interest in the infinite; Noë (1994), Hintikka (1996a), and Marion (1998) offer general philosophical interpretations of Wittgenstein's transitional period writings on the philosophy of mathematics. Mancosu and Marion (2002) examine Wittgenstein's effort to “constructivize” Euler's proof of the infinity of primes in light of the immediate historical context of constructivist discussions in Vienna in the late 1920s and early 1930s.
Finitism.
Kreisel's (1958) and Wang's (1958) association of Wittgenstein with “strict finitism” or “anthropologism” about mathematics spawned in the hands of Dummett (1970), a “finitist” reading of Wittgenstein that has received scrutiny and detailed development since the mid‐1980s. Kielkopf (1970) associates the term “strict finitism” with a view of mathematical necessities as a posteriori, but more frequently the term is associated, as in Dummett (1970), with the form of constructivism that rejects constructions that outstrip what is too complex or too lengthy for an individual or (p. 118) community to actually carry out in practice—a restriction on methods of proof more severe than any suggested by the intuitionists. It is unclear what precise form such a strict finitism should ideally take; Wright (1982) defends the program's internal coherence. Wang (1958) associated the notion of human “feasibility” with Wittgenstein's remarks on the surveyability of proof, suggesting that an anti‐reductionist investigation of complexity would be a fruitful foundational approach. He also emphasized connections between Wittgenstein's conception of the surveyability of proof and his critical comments about Principia's “reduction” of arithmetic to logic. Wright (1980), Steiner (1975), and Marion (1998) discuss the character and validity of these criticisms of logicism from three rather different points of view.
Despite the continued association of Wittgenstein with finitism, as well as his critical attitude toward set theory, there are some (e.g., A.W. Moore [1990] and Kanamori [2003]) who have suggested that Wittgenstein's grammatical investigations of infinity and the limits of sense may be the best kind of philosophical approach to a proper conception of set theory and the higher infinite.
Social Constructivism.
Wittgenstein's emphasis on the social aspects of language use have inspired some to develop his ideas into a social constructivist philosophy of mathematics: Bloor (1983, 1991) and Ernest (1998) offer recent accounts with a strongly sociological orientation that take up the relation of this approach to philosophy of mathematics education. Tymoczko ([1993]; ed., [1986]) are also relevant, rejecting foundationalist programs and calling for a view of mathematical practice as a community‐based art rather than a science.
Abbreviations of Wittgenstein Works, Lecture Notes, and Transcriptions

CL Cambridge Letters (1995)

LFM Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge 1939 (1989)

MN “Notes Dicated to G.E. Moore in Norway, April 1914,” appendix II in NB (in 1979a)

NB Notebooks 1914–1916 (1979a)

PG Philosophical Grammar (1974)

PI Philosophical Investigations, 2nd ed. (1958a)

PR Philosophical Remarks (1975)

RFM Remarks on the Foundations of Mathematics, rev. ed. (1978)

SRLF “Some Remarks on Logical Form” (1929)

TLP Tractatus Logico‐Philosophicus (1921)
(p. 119)

WA Wiener Ausgabe (1993–)

WVC Ludwig Wittgenstein and the Vienna Circle (1973)
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Notes:
(1) There were, strictly speaking, also a lecture published, though never delivered and immediately disowned by Wittgenstein (SRLF), a dictionary designed to teach spelling (1926), and a letter to the editor of Mind (1933).
(2) Some recent discussions of these issues may be found in Marion (1998), Floyd (2001b), Rodych (2002), and Mancosu and Marion (2002).
(3) See, in particular, the critical discussions of Bernays, Goodstein, Gödel, Kreisel, and Dummett.
(4) In The World as Will and Representation, a book that influenced Wittgenstein in his teenage years, Schopenhauer emphasized the superiority of proof by diagrams in geometry, contrasting them to the axiomatic, deductive style, which he associated with Euclid.
(5) See, e.g., remarks in works of Kreisel and in Hintikka (1993).
(6) He also evinced skills as an applied mathematician, patenting a propeller engine that was later used for helicopter propellers by the Austrian army in World War II.
(7) See the annotated bibliography below for some key articles explaining the universalist conception and its significance for the history of logic.
(8) Frege (1903, § 17). Compare Whitehead and Russell (1910, p. 95n) and Russell (1919, p. 191).
(9) On this, see Dreben and van Heijenoort (1986), and compare the reply of Dummett (1985).
(10) It is not certain that Wittgenstein ever read Gödel's completeness theorem (of 1930; in Gödel 1986), and quite certain that he did not appreciate its mathematical significance, given that he died in 1951, before the theorem's significance and fertility were well understood. No explicit discussions of the theorem are so far known in Wittgenstein's works, though it is not impossible they might be found in the future. It would be nice if they were, for it is the completeness theorem, much more than Gödel's incompleteness theorems, that would seem to be most difficult for Wittgenstein to interpret philosophically, given his later rule‐following discussions.
Wittgenstein may have learned of the completeness theorem from Turing and Watson, with whom he had discussions in 1937 that we have good reason to believe touched on Church's and Turing's 1936 results concerning the undecidability of first‐order validity (cf. Watson [1938]). Goodstein (1957, 1972) mentions that by the early 1930s Wittgenstein knew that the notion of finite could not be expressed in an axiomatic system; this is a consequence of compactness in Gödel's 1930 paper on the completeness theorem (cf. Gödel [1986]).
(11) Another example is the expression of happiness in a particular facial expression: the facial expression could not be the expression of happiness it is apart from expressing what it expresses. Yet another example is the numberhood of the number 2: 2 could not be what it is and fail to be a natural number, nor could the concept natural number be what it is without applying to the number 2.
(12) The theories he had in mind to undercut included Kant's transcendental doctrine of judgment as synthesis, Frege's notion of assertion as an act working on a content, Moore's and Russell's analyses of judgment as constituted by a relation between a mind and a proposition, and Russell's later multiple relation theory of judgment. All these theories fall back on an appeal to an extralogical mental activity to account for the nature, structure, and objectivity of truth in judgment. The effects of Wittgenstein's reactions (1913 and earlier) to Russell's theories of judgment on the Tractatus notion of picture are greatly illuminated by Pears (1977), Ricketts (1996b), and Proops (2002), and analogous effects of his reactions to Frege by Ricketts (2002) and Sullivan (2001b).
(13) Hylton (1997) and Floyd (2001a) examine the distinction between functions and operations in the Tractatus in relation to Frege's and Russell's philosophies of logic.
(14) Compare Kremer (1992).
(15) Wittgenstein's idea that functionality might be expressed through positional structure within a sentence resembles the positionality conventions at work in our decimal notation in another way: a decimal numeral is not, from this perspective, just a name, but a structure or an aspect of pictorial form. This is to rework Frege's analogy between the functional articulateness of the complex arithmetical term (such as “2^{2} + 3^{5}”) and the sentence, using the analogy to reject Frege's view that numerals and sentences are object expressions (names).
On Wittgenstein's analogy, the formally characterizable difference between an elementary proposition and a molecular proposition emerges through applications of truth‐operations in just the way that the difference between decimal numerals and complex arithmetical terms emerges through the application of arithmetical operations (addition, multiplication, subtraction, and exponentiation).
(16) Sullivan (2000) shows how understanding the cohesion of the Tractatus's technical and philosophical treatment of this distinction is crucial for understanding his responses to Russell's type‐stratification of the universe. For another useful treatment, see Potter (2000).
(17) See Fogelin (1982), Geach (1981), Soames (1983), Sundholm (1990), and Floyd (2001a).
(18) Certainly by the early 1930s he was insisting, in response to conversations with Ramsey, that no such “leading problem” of mathematical logic would help us understand logic's basic nature. But this was in part a criticism of his earlier conception's having (apparently) made it a central concern.
(19) The nonlogical character of Russell's Leibnizian second‐order analysis of identity in terms of coincidence of all properties is intended to be exposed by this thought experiment: Wittgenstein is arguing that it is perfectly possible that we could conceive that two different objects fall under precisely the same concepts. Furthermore, what Russell tries to assert in his axiom of infinity would not be asserted, but shown in the use of infinitely many different proper names. And if—as one would expect Wittgenstein to have assumed, given the obvious demands of physics—real numbers were added to the forms of the elementary propositions, this approach would yield elementary propositions with potentially infinite formal complexity. Moreover, such pseudo statements as “(∃x)(x = x)” and (∃x)(x ≠ x)” could not be written in this language as propositions, true or false; that which they try to express as general truths would be shown in the application of names in genuine propositions.
(20) The precise interpretation of this proposal is complex, and still not worked out with full formal precision; for two discussions, see Hintikka (1956) and Floyd (2001a).
(21) The terms in which Wittgenstein treats arithmetic in the Tractatus are precisely those used by Whitehead (1898). See Floyd (2001a) for a discussion.
(22) Wittgenstein (1993–), vol. IV, p. 239: x ^{2} = 1 has two roots versus on the table are 2 apples. The former is a grammatical rule of the variable. … Can I determine a variable by saying that its values should be all objects which satisfy a certain function? Not if I don't know this some other way—if I don't, the grammar of the variable is simply not determined (expressed).
(23) “Hume's principle” is Frege's second‐order contextual definition of sameness of number, so dubbed in work by Wright (1983) and Boolos and Heck (cf. Boolos [1998, part II] for a discussion). Intuitively speaking, for concepts U and V the principle is that the number belonging to the concept U is the same as the number belonging to the concept V if and only if there exists a one‐to‐one correlation between the objects falling under each concept.
(24) Such considerations against the priority of Hume's principle were forwarded by Wittgenstein (cf. WVC) and later, under his influence, by Goodstein (1951) and Waismann (1951, 1982, 1986). Dummett (1978, 1991a) responded critically to Waismann's version of these considerations, which lacked Wittgenstein's more or less explicit reliance on the idea of the proposition as a picture. Marion (1998) contains a useful survey of the issues raised here, which are obviously connected with the notion of a function‐in‐extension (i.e., with the notion of a function as an arbitrary mapping between individuals or a set of ordered pairs).
(25) Detailed reconstructions of this idea may be found in Frascolla (1994, 1997), Marion (1998), and Potter (2000).
(26) An assessment of the legitimacy of this kind of objection, and the differences between Wittgenstein and Poincaré, lies outside the scope of my discussion here; the adjudication of these issues is complex. For two especially relevant treatments of Poincaré's arguments, see Goldfarb (1988) and Mclarty (1997); on Wittgenstein's criticisms of Frege and Russell's logicism, see Steiner (1975) and, for a contrasting interpretation, Marion (1998).
(27) Three interesting (and contrasting) discussions of Wittgenstein and intuitionism may be found in Fogelin (1968), A.W. Moore (1989), and Marion (1998).
(28) For a range of slightly different readings of these remarks, see Waismann (1951), Shanker (1987), and Marion (1998).
(29) This approach to mathematical induction, not unlike Weyl's, held heuristic value for one of Wittgenstein's students, Goodstein, who went on to develop a “logic‐free” system of quantifier‐free arithmetic in the 1940s. For relevant discussions, see Goodstein (1951) and Marion (1998).
(30) Marion (1998) argues that Wittgenstein was an algorithmicist about mathematics, the kind of finitist or constructivist who sees the construction of algorithms as the core of mathematics. In light of Wittgenstein's later discussions of proof and rule‐following, this is not, I believe, all there is to his philosophy of mathematics (see Floyd [2002]), but there is no doubt that he insisted on construing calculation as something extremely important and central to mathematics—something not every mathematician or philosopher does.
(31) And hence, that no notion of intention is likely to be able to determine the applications of these notions. Compare Floyd [1991].
(32) See, for example, works by Bloor.
(33) For an especially lucid article on this idea, see Goldfarb (1992).
(34) Cf. Kant on truth at (1781/1787, A58/B82). Frege's argument that truth is a primitive undefinable notion, and not a genuine property word, may be found in Frege (1918). This was an essay that Wittgenstein always disliked, as we now know from the Frege correspondence with him (1989), but Wittgenstein seems to have objected to Frege's handling of idealism, not his treatment of truth (cf. Floyd [1998]). For two useful papers on the role of the concept of truth in this tradition, see Ricketts (1996a) and Diamond (2002). So far as we know, Wittgenstein never discussed Tarski's theorem on the undefinability of truth. Ramsey and Turing appear to have agreed with him that the concept of truth was not of central importance.
(35) See, e.g., LFM, pp. 136–141.
(36) For a conventionalist reading of Wittgenstein, see Dummett (1959); compare the discussions of Dummett in Stroud (1965) and in Wright (1980)