Picturing as Mapping: A Mark of Continuity in Wittgenstein’s Notion of Representation
Picturing as Mapping:
A Mark of Continuity in Wittgenstein’s Notion of Representation


Throughout his philosophical development, Ludwig Wittgenstein has been using an identical paradigm when accounting for the nature of representation: namely the paradigm of projective translation. In the Tractatus logico-philosophicus, Wittgenstein describes propositional pictures as projective mappings of a given multiplicity into another. After repudiating this early “picture-theory” of proposition, he nonetheless maintains the paradigm of representation as a projection, especially in his exposition of the business of philosophy. He now claims that the goal of philosophy is to provide “synoptic” representations of language and of its grammatical rules. Such synoptic representations aim at translating the logical multiplicity of a given grammatical system into a perspicuous symbolism, in a way that is commonly compared by Wittgenstein to a kind of “map-making”. On that respect, the notion of representation he has in mind in his later philosophy is quite comparable to the one he had in mind in his early philosophy, except that this projective paradigm is now used to characterize philosophical elucidations rather than the nature of propositions.

Table of contents

    It is rather commonplace to speak of a turning-point in Wittgenstein’s philosophy from 1929 onwards, and to describe such a switch by focusing on the withdrawal of the early “picture-theory” of language. The aim of this paper is to reevaluate such a description, and to show how the concept of picture Wittgenstein had in mind in the Tractatus logico-philosophicus (TLP) still pervades his later notion of representation. More precisely, I will claim that the Tractarian understanding of pictures as projections is still implicit in Wittgenstein’s later notion of “synoptic views” intended as the goal of philosophical clarification. According to the later Wittgenstein, philosophy aims in fact at providing “perspicuous presentations” [übersichtliche Darstellungen] that are supposed to map the connections and jointures of language and its grammar. On that respect, his later approach to philosophy rests upon a theory of representation whose very core is that same paradigm of pictures as projections that was already crucial in the Tractatus. When it comes to the conception of pictoriality, the difference between the “early” and the “later” Wittgenstein (if any) is therefore only the following: whereas the early Wittgenstein had been using the paradigm of map-making in order to account for propositional language, the later Wittgenstein switches to a higher level, and now uses this projective paradigm in a description of the task of philosophical elucidation.

    1. Renouncing the picture-theory?

    The Tractatus logico-philosophicus is known to have introduced a “picture-theory” of language, whose main claim is that “[t]he proposition is a picture of reality./ The proposition is a model of reality as we think it is” (Wittgenstein 1922, 4.01). In other words, the proposition shows the state of affairs it is meant to describe while speaking about it. Now, the notion of “picture” that is at stake in this early picture-theory of proposition is obviously not the picture in a basic or “mimetic” sense of the word. As a matter of fact, one should take seriously the reference to picture as a “model” in the passage mentioned above, and read it as a highly probable echo to the description of mechanical models provided by Ludwig Boltzmann in his celebrated entry for the Encyclopedia Britannica (Boltzmann 1974). Such a reference obviously implies that propositional pictures as seen by the early Wittgenstein do not seek to reproduce the external aspect of their objects, but rather focus on a restitution of their internal structure. Consequently, the crucial condition for the success of propositional representation is the existence of a relation of “coordination” [Zuordnung] between the elements of the proposition and those of the state of affairs it depicts (Wittgenstein 1922, 2.1514):

    The representing relation consists of the co-ordinations [Zuordnungen] of the elements of the picture and the things.

    Now, it is a well-known fact that such a notion of coordination has its roots in the characterization of “dynamical models” introduced by the physicist Heinrich Hertz in his Principles of Mechanics (Hertz 1895, esp. §418 ff.). Hence Wittgenstein’s insistence, in some later descriptions of his early picture-theory, on the mathematical or mechanical dimension of his former notion of picture (Wittgenstein 1967a, p. 185):

    This concept of picture is something I have inherited from two sides: on the one hand, from the drawn picture [ausgezeichneten Bild]; and on the other hand, from the picture of the mathematician, which is a general concept. For the mathematician still speaks of representations, where the painter would no longer use this expression.

    In the TLP, the mathematical dimension of this early notion of picture is explicitly stated when Wittgenstein refers to the concept of projection, and regards it as the very condition of propositional representation (Wittgenstein 1922, 3.11):

    We use the sensibly perceptive sign (sound or written sign, etc.) of the proposition as a projection of a possible state of affairs.
    The method of projection is the thinking of the sense of the proposition.

    This “mathematics-based” picture-theory is actually quite convenient, since it provides the early Wittgenstein with a unitary account of any proposition whatsoever. Yet, its main failure also lies in this very “over-comprehensiveness”. As Wittgenstein stresses after 1929, such an unwarranted generalization of the picture-theory prevents in fact the philosopher from gaining awareness of the multiple figures of language (Wittgenstein 1967b, §444):

    The tendency to generalise the case seems to have a strict justification in logic: here one seems completely justified in inferring: “If one proposition is a picture, then any propoposition must be a picture, for they must all be of the same nature”. For we are under the illusion that what is sublime, what is essential, about our investigation consists in its grasping one comprehensive essence.

    That’s why Wittgenstein was eventually to renounce his early picture-theory, in favor of a conception of language focusing on the multiplicity of language-games. Nevertheless, this renouncement does not imply that Wittgenstein should have repudiated every aspect of his early picture-theory. Quite the contrary: although he is no longer committed to the claim that any proposition is a projection of reality, the later Wittgenstein keeps adhering to the mathematical paradigm of pictures as projections. As I will try to show right now, the point is that he no longer relies on this paradigm in order to explain how propositions work, but rather uses it in his description of the kind of representations philosophy is expected to produce.

    2. Synoptic views as translations of a logical multiplicity

    According to the later Wittgenstein, the right way to avoid the dogmatism involved in the early-picture theory is to be aware of the multiple figures language is likely to assume. But this approach also implies the necessity of an orientation among such a multiplicity: and this is precisely what “synoptic presentations” are to accomplish. This point is made, for instance, in a dictation to Moritz Schlick (Wittgenstein 2003, pp. 124-125):

    Philosophical description gets its importance from the fact that it affords us an overview [Übersicht] which guards us against adopting a different system only because we do not see the right one. We yearn for a perspicuous representation, i.e., for a system, we do not see the right one, we are seduced into adopting a false system by the language or by some circumstance or other, and philosophy saves us by offering us the correct perspicuous representation.

    Wittgenstein’s interest for synoptic views is of course a major theme in his later philosophy, a theme whose motto may be found in the Philosophical Investigations, §122:

    A main source of our failure to understand is that we do not command a clear view of the use of our words. – Our grammar is lacking in this sort of perspicuity. A perspicuous representation produces just that understanding which consists in ‘seeing connexions’. Hence the importance of finding and inventing intermediate cases.

    But what exactly is a “synoptic” or “perspicuous” view? Obviously, it is a way of presenting things by outlining their structure and relations. A synoptic view aims at showing directly to the eye the connections and correlations that pervade a given system. In other words, the synoptic views Wittgenstein focuses on in his later philosophy seem to be doing the same job as the propositional pictures of his earlier philosophy, namely the job of translating a logical multiplicity into another.

    This reading is supported by a fascinating passage from Wittgenstein’s Remarks on Logical Form. In this passage, philosophy is described as a kind of therapeutics whose main device is the elaboration of an elucidating symbolism. This symbolism seeks to devise a “clear picture” of language, the picture in question being a translation of its logical multiplicity (Wittgenstein 1929, p. 16):

    The idea is to express in an appropriate symbolism what in ordinary language leads to endless misunderstandings. That is to say, where ordinary language disguises logical structure, where it allows the formation of pseudo-propositions, where it uses one term in an infinity of different meanings, we must replace it by a symbolism which gives a clear picture of the logical structure, excludes pseudo-propositions, and uses its terms unambiguously. Now we can only substitute a clear symbolism for the unprecise one by inspecting the phenomena which we want to describe, thus trying to understand their logical multiplicity.

    This reference to the logical multiplicity of linguistic phenomena is not to be explained by mere chronological arguments to the effect that the Remarks on Logical Form are a text of transition between Wittgenstein’s early and later philosophy. As I will now insist, the reference to philosophical representations as projective pictures goes far beyond 1929, and keeps underlying Wittgenstein’s notion of synoptic views until the Philosophical Investigations.

    3. Synopticity as a mapping

    To support my claim that the notion of representation involved in Wittgenstein’s later concern for synoptic views is similar to the notion of representation already involved in the Tractatus, I would like to stress how, in both cases, propositional pictures are described as mathematical projections of a given state of affairs. I have already presented the mathematical implications of the notion of picture in the TLP: my point is now that that the synoptic views of the later Wittgenstein also rest on the same mathematical paradigm. For instance, Wittgenstein’s insistence on the diagrammatic nature of synopticity is explicit in the Philosophical Remarks, when he elaborates on the so-called “color octahedron” (Wittgenstein 1964, §221). For the color octahedron is literally a diagram that clarifies the grammar of colors by means of a mapping of chromatic relations (Wittgenstein 1964, §1):

    The space of colors is, e.g., incidentally described by means of the octahedron, on the vertex of which are the pure colors; and such a representation is a grammatical one, not a psychological one. […] The octahedron-representation is a synoptic representation of grammatical rules.

    On that respect, synoptic presentations that consist of such diagrams are quite comparable to the mathematical mappings of the TLP. No wonder, then, that this understanding of synoptic views as diagrammatic presentations should be associated to a recurrent reference to cartography. See, for instance, the Cambridge Lectures (Wittgenstein 2001, p. 43):

    One difficulty with philosophy is that we lack a synoptic view. We encounter the kind of difficulty we should have with the geography of a country for which we had no map, or else a map of isolated bits. The country we are talking about is language, and the geography its grammar. We can walk about the country quite well, but when forced to make a map, we go wrong.

    The claim that the synoptic task of philosophy amounts to a kind of map-making of language is also explicit in the Big Typescript, where philosophy is openly compared to geography (Wittgenstein 2005, §90):

    Teaching philosophy involves the same immense difficulty as instruction in geography would have if a pupil brought with him a mass a false and falsely simplified ideas about the courses and connections of rivers and mountains.

    This metaphor is of course connected to the celebrated paradigm of orientation that underlies Wittgenstein’s description of philosophy, for instance in Wittgenstein 1953, §23 (“a philosophical problem has the form: ‘I don’t know my way about’”), or again in a remark from a manuscript where Wittgenstein asserts that “[his] duty is to teach you the geography of a labyrinth, so that you may completely find your way about it” (Wittgenstein 2000, 162b): 6v). A similar analogy between philosophy and sign-positing is again mentioned in a 1931 remark (Wittgenstein 1994, p. 18), to the effect that:

    Language sets everyone the same traps; it is an immense network of easily accessible wrong turnings. And so we watch one man after another walking down the same paths and we know in advance where he will branch off, where walk straight on without noticing the side turning, etc. etc. What I have to do then is erect signposts at all junctions where there are wrong turnings so as to help people pass the danger points.

    Now, granted that map-making is a form of projective depiction, one may propose a new reading of Wittgenstein’s evolution with respect to the notion of picture. Although the later Wittgenstein has ceased to regard the proposition as a projection of a given multiplicity into another, he does reintroduce the projective paradigm at a higher level: namely, when he insists on the necessity of synoptic presentations, the latter being intended as structural projections of a system of rules. Consequently, it is correct to assume that Wittgenstein never lost his interest for a conception of pictures as projections. The notion of picture Wittgenstein has in mind after 1929 when he asserts that philosophy should produce synoptic depictions of language is identical to the notion of picture that was underlying his early pictorial theory of language; the only point is that this concept now switches from the level of propositional pictures to the level of philosophical representations.


    1. Boltzmann, Ludwig 1974 “Model”, entry in the Encyclopedia Britannica (1902), reproduced in Theoretical Physics and Philosophical Problems, Selected Writings, ed. B. McGuinness, Dordrecht: Reidel
    2. Hertz, Heinrich 1895 Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt, in Gesammelte Werke, ed. Ph. Lenard, vol. III, Leipzig: Barth
    3. Wittgenstein, Ludwig 1922 Tractatus logico-philosophicus, tr. C. K. Ogden, London: Routledge
    4. Wittgenstein, Ludwig 1929 Some Remarks on Logical Form, in Proceedings of the Aristotelian Society, Suppl. vol.9, 162-171
    5. ––– 1953 Philosophical Investigations, tr. G. E. M. Anscombe, London: Blackwell
    6. ––– 1964 Philosophische Bemerkungen, Frankfurt: Suhrkamp
    7. ––– 1967a Wittgenstein und der Wiener Kreis, ed. B. McGuinness, Frankfurt: Suhrkamp
    8. ––– 1967b Zettel, ed. G. E. M. Anscombe and G. H. von Wright, Oxford: Blackwell
    9. ––– 1994 Culture and Value, ed. G. H. von Wright and H. Nyman; transl. by P. Winch, London: Blackwell
    10. ––– 2000:Wittgenstein’s Nachlass, The Bergen Electronic Edition, Oxford: Oxford University Press
    11. ––– 2001 Cambridge Lectures 1932-1935, ed. A. Ambrose and M. McDonald, Prometheus Books
    12. ––– 2003 The Voices of Wittgenstein. The Vienna Circle, ed. G. Baker, London: Routledge
    13. ––– 2005 The Big Typescript, TS 213, ed. C. G. Luckhardt and M. A. E. Aue, Oxford: Blackwell
    Sabine Plaud. Date: XML TEI markup by WAB (Rune J. Falch, Heinz W. Krüger, Alois Pichler, Deirdre C.P. Smith) 2011-13. Last change 18.12.2013.
    This page is made available under the Creative Commons General Public License "Attribution, Non-Commercial, Share-Alike", version 3.0 (CCPL BY-NC-SA)


    • There are currently no refbacks.