wittgenstein/literatur/kraft-sequential.md

# Tim Kraft — How to Read the Tractatus Sequentially

*Nordic Wittgenstein Review* 5 (2) 2016, pp. 91–124.
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ARTICLE SECTION

Tim Kraft

tim.kraft @ ur.de

How to Read the Tractatus Sequentially
Abstract
One of the unconventional features of Wittgenstein’s Tractatus LogicoPhilosophicus is its use of an elaborated and detailed numbering system.
Recently, Bazzocchi, Hacker und Kuusela have argued that the
numbering system means that the Tractatus must be read and
interpreted not as a sequentially ordered book, but as a text with a
two-dimensional, tree-like structure. Apart from being able to explain
how the Tractatus was composed, the tree reading allegedly solves
exegetical issues both on the local (e. g. how 4.02 fits into the series of
remarks surrounding it) and the global level (e. g. relation between
ontology and picture theory, solipsism and the eye analogy, resolute
and irresolute readings). This paper defends the sequential reading
against the tree reading. After presenting the challenges generated by
the numbering system and the two accounts as attempts to solve
them, it is argued that Wittgenstein’s own explanation of the
numbering system, anaphoric references within the Tractatus and the
exegetical issues mentioned above do not favour the tree reading, but
a version of the sequential reading. This reading maintains that the
remarks of the Tractatus form a sequential chain: The role of the
numbers is to indicate how remarks on different levels are
interconnected to form a concise, surveyable and unified whole.


Tim Kraft


1. Introduction: How to read the Tractatus
One of the unusual features of Wittgenstein’s Tractatus LogicoPhilosophicus 1 is the elaborated numbering it uses instead of more
conventional ways of arranging a philosophical text. For several
decades the dominant attitude towards the numbering system
appears to have been half-ignoring it: The numbering system is
dutifully mentioned, but it is not given much relevance to
interpreting the Tractatus either. 2 In a fascinating recent
development Bazzocchi (2010a, 2010c, 2014a, 2014b, 2015),
Hacker (2015) and Kuusela (2015) argue against this attitude by
construing the Tractatus as a text with a two-dimensional, tree-like
structure, which must be read and interpreted accordingly. 3 As
advertised by Hacker, the tree reading explains
how the Tractatus Logico-Philosophicus was composed, precisely what the
numbering system of the book signifies, and hence, how the book was
meant to be read. […] Read correctly, […] the line of argument
becomes clear, the anaphoric references, baffling in a consecutive
reading, become evident, and the interpretation of the text becomes
much easier. The Tractatus must be read in accordance with the numbering
system, and that demands that the reader follow the text after the manner
of a logical tree. (2015: 648 f.)

If correct, the tree reading has wide-ranging consequences: It
promises solutions both to problems of detail, e. g. what “this” in
4.02 refers to, as well as to central exegetical problems, e. g. how
ontology and picture theory are related, how the eye analogy fits
into the discussion of solipsism and whether the Tractatus must be
read resolutely. In this paper I argue against the tree reading on two
fronts: On the destructive side, I argue that the tree reading offers
The Tractatus Logico-Philosophicus (originally published 1921/1922) is abbreviated TLP, the
Prototractatus (Wittgenstein 1971) PT. Quotations are from the critical edition (Wittgenstein
2001) and from the translation by Pears and McGuinness (Wittgenstein 1961).
2
Notable exceptions are Stenius 1960, Ludwig 1975, Mayer 1993 and Gibson 1996 who
provide extended discussions of the numbering system. For interpretations that fall
somewhere within the range between extraordinary overinterpretation and esoteric
numerology see Finch 1971 (4 as the centre surrounded by the pairs 1/7, 2/6 and 3/5,
zero remarks indicate what is unsayable), Scheier 1988 (boustrophedon reading), Lange
1989 and 1996 (Tractatus must be read in 141 sequences of seven remarks each).
3
Hacker mentions some precursors of the tree reading (Schroeder 2005: 23–26, White
2006: 16–18), but neither of them defends it explicitly.
1


Nordic Wittgenstein Review 5 (2) 2016

neither an adequate account of the structure of the Tractatus nor a
convincing solution to any of the exegetical issues it allegedly
solves. On the constructive side, I argue that there is a version of
the sequential reading – the sequential chain reading – that offers a
better account of the structure of the Tractatus and fares at least as
well with respect to the exegetical issues. After introducing the
problems surrounding the numbering system of the Tractatus
(section 2), I sketch two interpretations of the numbering system,
the tree reading and the sequential chain reading (section 3). In the
sections that follow I discuss three central issues: how to
understand Wittgenstein’s own explanation of the numbering
system (section 4), how to resolve anaphoric references within the
Tractatus (section 5) and how the two readings differ with respect to
the broader exegetical issues mentioned above (section 6). (Since
sections 4–6 contain a lot of short discussions of sample passages
of the Tractatus, readers can skip those sections they are not
interested in without loss of continuity.) Finally, I summarise my
case for the sequential chain reading (section 7).
2. Interpreting the numbering system of the Tractatus
As is well known, Wittgenstein did not take the numbering system
lightly. When Ludwig von Ficker, whom Wittgenstein had
contacted as a potential publisher, inquired whether the Tractatus
could be printed without the numbers (von Ficker 1988: no. 477),
Wittgenstein replied on 5 December 1919:
(By the way: the decimal numbers of my remarks absolutely must be
printed alongside them, because they alone make the book
perspicuous and clear: without the numbering it would be an
incomprehensible jumble.) (PT: 18 f./1980: no. 115)

Yet, beginning with the earliest published reactions to the
Tractatus interpreters have been rather critical of its style and
numbering system (Ramsey 1923, de Laguna 1924). A similar
attitude can be found in several of the early monographs on the
Tractatus (e. g. Black 1964, Favrholdt 1964). Black even went so far
as to suggest that the numbering system may be “a private joke on
the reader’s expense” (1964: 2).


Tim Kraft


But what explains this dismissive attitude towards the
numbering system? It is difficult to tell with any certainty, but apart
from being unconventional a number of reasons seem to
contribute their share. The numbering system easily creates
expectations that are never met by the Tractatus. It reminded many
readers of similar numberings systems of logical and mathematical
treatises, e. g. Russell and Whitehead’s Principia Mathematica.
Evidently, presenting a logical system with axioms, definitions,
theorems and lemmas is not the role of the Tractatus’s numbering
system. The numbering system may also be expected to mark
chapters, sections, subsections and so on. Again, this is not how
Wittgenstein uses the numbering system. Judging from their
‘headings’, the first two ‘chapters’ should be about ontology, but
starting with 2.1 the predominant topic are pictures. Similarly,
judging from its ‘heading’, the sixth ‘chapter’ is about
Wittgenstein’s claim that all sentences share a logical form.
However, the remarks of this ‘chapter’ neither explain the
symbolism nor offer an argument for this particular general form.
All of that happens in the preceding ‘chapter(s)’. To understand
this ‘chapter’ readers have to go backwards searching for
Wittgenstein’s account of variables, the N-operator and so on. Of
course, readers who are already familiar with the book will easily
find the relevant passages, but not because the alleged headings tell
what is where. 4 In addition to its function being obscure, the
numbering system also has maze-like aspects. While the
hierarchical arrangement of the first eight remarks may be
perspicuous, starting with the ninth remark (2.01) things get more
complicated. Here readers come across the first zero remark with
the first major comment (2.1) being postponed until the 50th
4

In this context a comment Wittgenstein made in a conversation with Drury may seem
relevant: “the Tractatus [is] highly syncopated. Every sentence in the Tractatus should be seen as

the heading of a chapter, needing further exposition.” (Drury 1984: 159) But Wittgenstein’s
comment is not evidence for the table of contents interpretation of the numbering system. For his
point is that every remark stands in need of further explanation, not that they were intended as an
ersatz table of contents. This remark fits more with an episode reported by Moore: “Ramsey told
me that, in reply to his questions as to the meaning of certain statements, Wittgenstein answered
more than once that he had forgotten what he had meant by the statement in question.” (Moore
1954: 2 f.)


Nordic Wittgenstein Review 5 (2) 2016

remark. Moreover, starting with the 14th remark (2.0123) – again
quite early in the text – there is a sequence of eleven remarks (until
2.0211) containing no two neighbouring remarks that are at the
same level. Finally, there are also signs of a hidden double meaning
behind the numbering system that can easily strengthen the
impression that its role is predominately an aesthetic one. For
example, it has been suggested 5 that it is no coincidence that 7
consists of two clauses of seven syllables each, that ‘chapter’ 1 and
‘section’ 6.5 consist of seven remarks each and that 3.5 as seven’s
half and 4 as the midpoint in a series of seven items form the twin
centre of the Tractatus.
In addition to these qualms about the numbering system as
such, Wittgenstein’s actual implementation poses several challenges
to any interpretation of the numbering system. A first challenge is
the comment problem: According to Wittgenstein’s own
explanation (in the footnote to 1), decimal remarks are “comments
on” the remark with one digit less. This explanation has proven to
be elusive: Are remarks comments individually or do siblings form
a comment together? For example, is 4.5 individually a comment
on 4 or does it continue the sequence 4.1, 4.2, 4.3, 4.4 to form a
(unified) comment on 4? Some remarks do not even seem to be
comments at all. For example, 2.1 introduces a new topic instead of
merely commenting on 2.
A second challenge is the logical weight problem: According to
Wittgenstein’s own explanation, the numbers are supposed to
indicate their “logical weight (logisches Gewicht)” and the “stress
(Nachdruck) laid on them”, i. e. important remarks have less digits
than less important ones. But that does not always seem to be the
case. For example, 4.0312 (“my fundamental idea is that the
“logical constants’ are not representatives”) and 4.111 ff. (on
philosophy) appear to be more important than the number of digits
suggests.
A third challenge is the zero remark problem: The Tractatus
contains fifteen sets of zero remarks, i. e. sets of remarks
5

For references see fn. 2. Let me emphasise that I am not endorsing any of these suggestions. To
the contrary, I am mentioning them solely to explain why its numbering system may strike readers
of the Tractatus as off-puttingly strange.


Tim Kraft


containing at least one zero. Wittgenstein’s explanation of the
numbering system does not cover zero remarks explicitly, but a
natural interpretation is the identity thesis: 2.01, for example, is a
comment on 2.0 which is identical with 2 (Stenius 1960, Ludwig
1975). But how are the two (or three) different sets of comments
on a given remark related to each other? If the number of digits
indicates importance, zero remarks are less important than the nonzero remarks. However, since zero remarks are always prior in
order, they are always closer to their parent remark – and more, not
less informative – than the non-zero remarks. Compare, for
example, the informativeness as comments of 2.01 and 2.1 or 4.001,
4.01 and 4.1.
A fourth challenge is the anaphoric reference problem: The
Tractatus contains only two explicit cross references (5.151, 5.31),
but plenty of implicit cross references (anaphoric pronouns,
demonstratives, indexicals, inferential particles, etc.). In some cases,
e. g. 4.02, 5.64 and 6.01, interpreters struggle to resolve these
references. These remarks contain anaphoric pronouns that can
refer back to something mentioned in the immediately preceding
remark (e. g. 4.016, 5.634 or 6.002) or in the preceding same-level
remark, if there is one (e. g. 4.01, 5.63), or in the higher-level
remark they are a comment on (e. g. 4, 5.6 or 6).
To sum up, readers should be forgiven for thinking that
whatever the merits of the numbering system are, Wittgenstein has
plainly overdone it. The numbering system appears to introduce
only an additional, distracting layer to an already difficult to
understand text. In the absence of a constructive account of the
numbering system and its role, the Tractatus hands out too many
invitations to dismiss it, or so it seems. Thus, why not put their
numbers to the background and focus solely on the interpretation
of the remarks themselves?
3. The tree reading and its rivals
There are two principled ways of responding to this question and
the challenges that give rise to it: Whereas orthodox, sequential
approaches take the Tractatus to be a linear text, non-sequential
approaches take it to be a non-linear, two-dimensional text. In this

Nordic Wittgenstein Review 5 (2) 2016

section I introduce these two approaches to the numbering system
of the Tractatus.
The tree reading, as defended by Bazzocchi, Hacker and
Kuusela, promises to explain the numbering system in a systematic
way that clarifies the general outline of the Tractatus and avoids
errors of interpretation that stem from ignoring the numbering
system. To read the Tractatus as a two-dimensional tree means to
read it not as a sequentially ordered text, but as non-linear text
consisting of a root (1, 2, 3, …, 7) from which fourteen branches
(1.m, 2.0m, 2.m, 3.00m, 3.0m, 3.m and so on) originate and which in
turn lead to even further branches until the leaves or endpoints are
reached. These branches are in a strict hierarchical order, but not in
a sequential order. The branch 2.m, for example, does not come
before or after the branch 2.0m or the branch 3.m, but is just
independent of these other branches. Although branches are not
ordered, within a branch order matters. For remarks are comments
jointly with their siblings. 2.1 and 2.2, for example, are not
independent comments on 2, but belong to the same branch and
form a comment together.6 The result is a text arranged like a tree:
The Tractatus now consists of 183 branches containing on average
2.9 remarks with 1 branch containing the maximum number of
nine remarks and 67 branches containing only a single remark. 72
branches lead to, usually multiple, further branches and 111
branches consist only of uncommented on remarks. Thus, when
reading the Tractatus not a particular, predetermined order is to be
followed, but the text is to be explored by following branches and
nodes until an endpoint is reached and the reader must return to
one of the branches or nodes she came from. There is no obvious
(non-cheating) way of ensuring a complete reading, but that is
presumably considered a feature, not a bug.7
6

The Tractatus is, therefore, not a tree in the logical or mathematical sense, even on the
tree reading. Mathematical trees contain only one type of relation between nodes, but the
figures by Bazzocchi (2013: 40, 2014a: 256, 2014b: i) and Hacker (2015: 655) need two
kinds of lines, solid and dashed, to represent the ‘comment on’ relation and the ‘belongs
to the same set of comments as’ relation.
7
A printed tree version of the Tractatus printing every branch as a single page is Bazzocchi
2014b.


Tim Kraft


Equipped with this account of the structure of the Tractatus, the
tree reading can answer the challenges raised above. 8 Regarding the
comment problem, the tree reading takes the idea that all decimal
remarks are comments on and only on the remark with one digit
less seriously. This is the idea which motivates the tree reading in
the first place. Regarding the logical weight problem, it
distinguishes two senses of ‘important’: A remark can be important
in the sense of being a remark on which other remarks depend
(“weight-bearing”) or it can be important in the sense of being a
‘fruit of the tree’, that is, an important result of other, more central
remarks. The “fundamental idea” of 4.0312 is important in the
latter, but not the former sense (Hacker 2015: 649 n. 2). Regarding
the zero remark problem, the tree reading bites the bullet: The
different sets of comments are “equipollent” (Hacker 2015: 655)
and it is arbitrary which set gets the zero numbers and which one
the non-zero numbers. Regarding the anaphoric reference problem,
the tree reading solves it in a principled way: All anaphoric
references are to the preceding same-level sibling, if there is any, or
to the preceding higher-level.
For interpreting the Tractatus the tree reading has several
consequences. Most importantly, the unit of interpretation should
be branches, which often consist of remarks that are not printed
close to each other in traditional editions. For example, 4.1, 4.2,…,
4.5 must be read and interpreted together, not with their sequential
predecessors and successors. Moreover, there is not a single end or
conclusion to the text, but a multitude of endpoints. Since all
remarks are connected to a higher-order remark and their siblings,
many remarks must be read and interpreted independently of
remarks they are usually read and interpreted with.
It is tempting to object to the tree reading that it credits
Wittgenstein with an invention that is plainly too advanced for the
8

For discussing the tree reading it is useful to distinguish between the thesis that the
Tractatus can be read and interpreted as a tree and the thesis that it must be read and
interpreted that way. Since the weaker thesis is too non-committal to give helpful
responses to the challenges, I take the tree reading to defend the stronger thesis.
Unfortunately, tree readers are not always clear on that point. According to one of
Hacker’s descriptions, the tree reading is adopted “once one avoids reading the work only
consecutively, and also reads it tree-wise” (2015: 649, emphasis added).


Nordic Wittgenstein Review 5 (2) 2016

pre-computer age. Fortunately, the tree reading has a good
response to this objection, namely that it fits rather well with (what
we know about) how Wittgenstein actually composed the Tractatus.
The tree structure is a natural byproduct of the genesis of the
Tractatus as MS 104 shows, i. e. the manuscript from which the
Prototractatus was derived (Bazzocchi 2010b, 2015). The philological
details are complex, but some key recent findings are the
following: 9 Wittgenstein composed the Prototractatus by first writing
down a core structure consisting of (some of) the cardinal
remarks 10 to which he added further remarks of which many are
selections from his (often diary-like) notebooks. Even if we do not
know how exactly Wittgenstein kept the numbering system
consistent, it is clear that the Tractatus was not written in the order
as it appears now, but was composed by adding layer after layer to
a bare outline. Thus, relying on a numbering system was an
important writing tool. It allowed Wittgenstein to organise his
thoughts in a systematic way and to take precautions against being
unable to complete his work. At any stage of composition there
existed a more or less detailed, yet comprehensive summary of his
ideas. At the same time the role of the numbering system is clearly
not limited to its instrumental role. As Wittgenstein stated in the
letter to von Ficker, the surveyability, clarity and unity of the
Tractatus depend on it – the focus of this paper is how it achieves
that, not how it helped Wittgenstein composing the Tractatus.11
9

In addition to the papers by Bazzocchi see also Potter 2013, Pilch 2015 and Hacker
2015: 653–657. Examples of controversial issues are the dating of PT, especially when
Wittgenstein began writing it, how exactly Wittgenstein kept the numbering system
consistent over an extended period of time and the identity of various manuscripts and
typescripts mentioned in so-called Hermine’s list and in various letters and recollections.
10
Due to the Pilch’s (2015) impressive philological work we know now that the original
first paginated, now missing, page of MS 104 contained cardinal remarks 1–6 (with 7 and
the formula in 6 being added later). The first surviving page of the MS repeats the cardinal
remarks 1–6 and adds some of the first-level remarks.
11
Several conjectures have been made as to which texts may have inspired Wittgenstein’s
use of a numbering system. Since texts with numbers are not rare – legal documents,
manifestos, modern editions of the bible, textbooks come to mind –, it is not clear how
much inspiration Wittgenstein needed, but Russell and Whitehead’s Principia Mathematica
(e. g. McGuinness 1988: 265, 301) and Tolstoy’s Gospel in Brief (Westergaard 2009) are the
most likely candidates. Wittgenstein was familiar with both and there are at least some
similarities. But similarities are not proof of an actual influence, of course.


Tim Kraft


3.2 The Tractatus as a sequential chain
The tree reading is a powerful challenge to orthodox, sequential
readings of the Tractatus. At the very least, it forces proponents of
an orthodox reading to explain and defend their approach. But the
arguments for the tree reading also suffer from underestimating the
sequential reading. This reading is not to be confused with a
nihilistic reading that ignores the numbering system altogether. To
the extent that arguments for the tree reading are plainly arguments
in favour of paying attention to the numbering system at all, they
may have a point against particular, isolated interpretations that fail
to do so, but not against the sequential reading. Understood
correctly, the sequential reading claims that the numbering system
serves a different purpose than the tree reading assumes, not that it
has no function at all. To distinguish this reading from the nihilistic
reading, I also call it the ‘(sequential) chain reading’. Since its main
tenets were not clearly stated in the literature before the tree
reading was presented as an alternative to it, it is difficult to tell
who does or would adopt it, but by taking up some ideas it is
possible to at least sketch what a sequential reading can and should
look like.
The sequential chain reading takes as its starting point the
observation that the cardinal remarks are linked by overlapping
elements: With the exception of 7, each cardinal remark picks up a
topic from the preceding remark and introduces a new one (e. g.
Stenius 1960: 6, Dietrich 1973: 6):
world (1) – what is the case (1, 2) – facts (2, 3) – thoughts (3, 4) –
propositions (4, 5) – truth-functions (5, 6) – general form (6)

A central motivation of the chain reading is that this
observation can be generalised to the Tractatus as a whole. Decimal
remarks, too, are usually both backwards- as well as forwardsdirected insofar as they continue the topic under discussion and
link it with the (sub-)topic to be discussed in the following remarks.
As an example, consider 2 and 3: What they have in common is
that they are both partly about facts. But what is new in 3 (thoughts
as logical pictures of facts) is prepared by 2.1 (pictures of facts) and
2.2 (logical pictures and shared form). Both 2.1 and 2.2 are, of

Nordic Wittgenstein Review 5 (2) 2016

course, also connected to 2 with facts and states of affairs being the
common thread. Thus, the sequential reading and the tree reading
disagree about the basic function of decimal remarks: Whereas a
decimal remark is just a comment on a particular higher-level
remark according to the tree reading, a decimal remark connects
remarks before as well as after it according to the sequential
reading. The numbers are meant to reveal the unity and
interdependence of the remarks, not their separation into distinct
branches. The sequential chain reading also takes seriously what
Wittgenstein has to say about the book’s main topic and thesis in
the preface: Its topic is to “to draw a limit […] to the expression of
thoughts” and its basic conclusion is that “what can be said at all
can be said clearly, and what we cannot talk about we must pass
over in silence” (TLP: preface). Hence, the Tractatus does have a
direction and leads somewhere. The chain structure combines
hierarchy with a kind of progression absent in a tree: The remarks
are at different levels in a hierarchy, but they also advance the
discussion towards a conclusion. Of course, how exactly the
Tractatus does so cannot be answered by looking solely at its
structural features. Looking at the structural features is necessary,
but by no means sufficient for understanding why the Tractatus is
not an “incomprehensible jumble”. Not any series of remarks with
overlapping elements forms a unified whole. Since it is advisable to
look at examples when wanting to understand how a unity is
achieved by a chain of remarks, see below my discussion of
2.063/2.1 (transition from ontology to picture theory), 5.53 (on
identity) and 5.63 (eye analogy) for more on advancing a discussion
with the help of such chains. 12
With that account at hand, we can return again to the four
challenges raised above. Regarding the comment problem, the
12

Although the chain metaphor is very loosely inspired by 2.03, that does not mean that
the chain reading is committed to drawing an analogy between the structure of facts and
the structure of the Tractatus itself. It is merely a coincidence that the chain metaphor can
be used for both. For the same idea can be expressed by other metaphors as well, for
example a train ride in a foreign country. We can take the high-speed train only stopping
at major cities (seven stops) or the express train stopping at minor cities as well (32 stops)
or the slow train calling at every village (526 stops). Either way we will get to know
something about the country and reach our destination.


Tim Kraft


sequential chain reading accepts the objection and rejects the idea
that decimal remarks are best understood as comments on their
parent remark. Instead, their basic function is to connect several
remarks with each other. Regarding the logical weight problem, the
sequential reading differs from the tree reading. It assumes, as does
the tree reading, that remarks can be important when looked at in
one way and not that important when looked at in a different way.
But 4.0312, for example, owes its importance not to its being a
result or ‘fruit’, but to its function and context. Its importance
derives from which other ideas it links with each other and how it
does so. That logical constants do not refer is a “fundamental idea”
because it suggests looking more closely at other parts of
propositions and whether they represent something or not. It is an
idea that opens up a new perspective (to put it even more
metaphorically, it is a key or seed), but it is neither a premise or
axiom nor a result ending an investigation. Regarding the zero
remark problem, the sequential chain reading sees their function
differently, too. There must be zero remarks because otherwise
there would be no way to link a cardinal remark with its first
supporting link. Thus, zero and non-zero remarks are not
independent and equipollent. The first non-zero remark is
connected to its higher-level predecessor via zero remarks that
both comment on the higher-level predecessor and prepare the
next step, the first non-zero remark. Regarding the anaphoric
reference problem, the sequential chain reading predicts that
anaphoric references need to be resolved by looking at the context.
Anaphoric expressions usually do not pick up something
mentioned several remarks or even pages ago.
3.3 An objection
With the two readings of the numbering system outlined, I turn to
the arguments that have been advanced in favour of the tree
reading in the next three sections. But before doing so, I need to
address an objection: Why is there even a need to decide between
the two interpretations? The bottom line is that they make
incompatible predictions when applied to individual remarks:
Either 2.1 continues 2.063 (as the sequential chain reading predicts)

Nordic Wittgenstein Review 5 (2) 2016

or it does not (as the tree reading predicts), either 4.02 continues
4.016 or it does not, and so on. Of course, it may turn out that
both interpretations make some correct and some incorrect
predictions. But even if that were the case, it would be instructive
to know which interpretation is closer to Wittgenstein’s intentions.
I will argue below that one interpretation – the chain reading – is
clearly superior in that regard, but that does not mean that it is a
bad idea to keep in mind that both interpretations of the
numbering system exist and can both be applied whenever
discussing specific (series of) remarks.
4. Wittgenstein’s explanation of the numbering system
The first argument for the tree reading is that it can make better
sense of Wittgenstein’s own explanation of the numbering system
by taking it at face value. It is contained in the footnote to the first
remark of the Tractatus:
The decimal numbers assigned to the individual propositions indicate
the logical weight 13 of the propositions, the stress laid on them in my
exposition. The propositions n.1, n.2, n.3, etc. are comments on
proposition no. n; the propositions n.m1, n.m2, etc. are comments on
proposition no. n.m; and so on.

What we find here is a functional claim (numbers indicate logical
weight and stress) and a structural claim (numbers indicate to
which other remark a remark is related). Starting with the
functional claim, both terms used – “logical weight (logisches
Gewicht)” and “stress (Nachdruck)” – are standing in need of some
clarification. As I understand him, Wittgenstein does not define
logical weight as stress or vice versa, but thinks of them as two sides
of the same ideal: He hints at a supposed harmony between form
(“stress laid on them in my exposition”, how the remarks are
presented) and content (“logical weight”, what their relevance or
importance is). Although one may wonder what “logical” means
here, Wittgenstein’s use of “logical” should not be overinterpreted.
13

Following Hacker (2015: 649 n. 2) I have altered the translation from “importance”
(Wichtigkeit, the term used in PT) to the more literal “weight” (Gewicht, the term used in
TLP).


Tim Kraft


What he has in mind are not only or even predominately logical
relations. The cardinal remarks are neither a system of axioms from
which the other remarks follow deductively nor are they
conclusions with the decimal remarks as their premises. Instead I
take it that “logical” contrasts with “psychological”. By using the
term “logical” Wittgenstein clarifies that importance is not meant
subjectively.
The second part of Wittgenstein’s own explanation of the
numbering system is the structural claim that decimal remarks are
“comments on (Bemerkungen zu)”. On the one hand, the term used
by Wittgenstein is rather unspecific: Instead of “elucidation
(Erläuterung)”, “explanation (Erklärung)” or “annotation
(Anmerkung)”, he uses the generic “comment (Bemerkung)”. 14 In
ordinary usage a “Bemerkung” on a statement can be anything
from the loosely related idea to the tightly organised argument. By
using a generic term Wittgenstein does not put a severe constraint
on how to understand the relation between remarks. Presumably,
he thinks that the precise role of a decimal remark need not be
explained, but is made transparent by the remarks themselves –
understanding their role is left as a case by case exercise to the
reader. Yet, the tree reading asserts not only that remarks are more
or less weighty comments, but also that the comment structure
precludes a sequential ordering. Wittgenstein’s explanation
establishes the former, but not the latter: the comment-relation is
not in competition with the successor-relation. Nothing in his
explanation rules out that one function of the numbering system is
to order decimal remarks by arranging them with their respective
higher-, same- and lower-level remarks.
Further support for the thesis that Wittgenstein’s explanation is
not sufficient to establish a tree structure are two other comments
on the numbering system. First, contrary to the footnote in TLP,
14

The common translation of “Bemerkung” is “remark”: The English title of
Wittgenstein’s Philosophische Bemerkungen, for example, is Philosophical Remarks, not
Philosophical Comments. In the main text I follow Ogden and Ramsey’s and Pears and
McGuinness’s decision to translate “Bemerkung” as “comment”, but in both translations
“Bemerkung” is also translated as “observation” (3.331) and “remark” (5.62).


Nordic Wittgenstein Review 5 (2) 2016

the corresponding note in PT explicitly mentions as one function
of the numbering system ordering the remarks:
The numbers indicate the order of the propositions and their
importance. Thus 5.04101 follows 5.041 and is followed by 5.0411,
which is a more important proposition than 5.04101. (PT: 42)

A straightforward explanation of why Wittgenstein does not
mention ordering the remarks as a function of the numbering
system in the published version of the footnote is that there is no
need to mention it anymore since the remarks are by now already
in their intended order. Moreover, the PT version does not assert
that remarks with decimal numbers are comments on other remarks,
but that they simply follow upon each other. That supports the
impression that “is a comment on” need not mean more than
“something or other that follows on”.
Second, in the letter to von Ficker in which Wittgenstein claims
that the point of his book is an ethical one (undated letter to von
Ficker, October or November 1919) he also explicitly mentions the
“conclusion (Schluß)” of the Tractatus:
For now I would recommend you to read the preface and the conclusion,
because they contain the most direct expression of the point. (PT:
16/1980: no. 107)

It is highly unlikely that with “the conclusion” Wittgenstein is
referring to just remark 7. For it would not be a very helpful
recommendation to read only a single sentence of the book in
addition to its preface. Wittgenstein is most likely referring to the
last several pages of the typescript and, thus, has no objection to
reading them as a continuous unit.
Finally, the absence of any further comments on the numbering
system (in the Nachlass, in letters, in recollected conversations)
provides further evidence against the tree reading. As far as we
know, he did not discuss the numbering system with Ogden or
Ramsey, for example (cf. Wittgenstein 1973). Even when stressing
the importance of the numbering system to von Ficker, he does
not bother to explain why or in which way it is important. If the
numbering system was meant as an instruction for a nonconventional way of reading a book, it would have been easy to

Tim Kraft


give von Ficker, who explicitly asked him about it, at least some
hints about this alleged role of the numbering system.
5. Anaphoric references
Let us turn to the second argument for the tree reading: The tree
reading is supposed to be needed to get anaphoric references
between remarks right (Bazzocchi 2014b, Hacker 2015: 660 f.,
Kuusela 2015: 229). The Tractatus contains a lot of remarks that
contain anaphoric pronouns, demonstratives, inferential particles
etc. that refer back to earlier remarks, but 4.02 is allegedly a
particularly telling example. In this section I argue for two claims:
With respect to the primary example 4.02, I argue that the
sequential interpretation of 4.02 did not get a fair hearing and that
the tree interpretation of this remark generates its own problems.
With respect to the anaphoric references in the Tractatus in general,
I argue that independently of the interpretation of 4.02 a systematic
study of all anaphoric references reveals that the tree reading has
more and not less problems with resolving anaphoric references
than the sequential reading. The argument for the tree reading is
based on a one-sided selection of examples.
5.1 “We can see this from the fact that…” (4.02)
Let us turn to the tree readers alleged smoking gun first:
4.01 A proposition is a picture of reality. […]
[…]
4.016 In order to understand the essential nature of a proposition, we
should consider hieroglyphic script, which depicts the facts that it
describes.
And alphabetic script developed out of it without losing what was
essential to depiction.
4.02 We can see this from the fact that we understand the sense of a
propositional sign without its having been explained to us.
4.021 A proposition is a picture of reality: for […]

4.02 is an interesting test case not because the tree reading
offers a new interpretation of it – that “we can see this from the

Nordic Wittgenstein Review 5 (2) 2016

fact (dies sehen wir daraus)” refers back to 4.01 is the majority view 15
– but because it can explain why it is correct: What other
interpreters arrive at only in a cumbersome, ad hoc way, is the only
possible interpretation according to the tree reading. “This” in 4.02
can refer either to something mentioned in its sibling 4.01, as the
tree reading predicts, or to something mentioned in its immediate
predecessor 4.016, as the sequential reading expects. According to
the tree reading, 4.02 can only refer to something mentioned in
4.01 because 4.02 is neither a comment on nor otherwise related to
4.016. Accordingly, a fully spelled out 4.02 reads ‘We can see that a
proposition is a picture of reality from the fact that…’. According
to the sequential reading, 4.02 is expected to refer to something
mentioned in its immediate predecessor 4.016 because an
anaphoric expression refers to something mentioned several
sentences earlier only under very special circumstances.
Accordingly, a fully spelled out 4.02 reads ‘We can see what the
essential nature of a proposition is from the fact that…’. 16
Let us consider the tree reading’s interpretation first. An
important argument for it is the original context of 4.02: In PT 4.02
is on the same page as 4.01 – between 4.01 and 4.02 are what are
now 4.05 and 4.06 –, while the 4.01s including 4.016 are later
additions (PT: 86, 88). 17 It seems to be rather unlikely that a
reference shift was intended. Yet, such shifts have occurred
elsewhere: In PT “this remark (diese Bemerkung)” in 5.62 refers to 5.6
and not to 5.61 (PT: 184, 192). Since the possibility of a reference
shift cannot be ruled out in advance, the genetic argument must be
supplemented by an interpretation of the role of 4.02 in the final,
authoritative text. With this the sibling interpretation struggles: On
the one hand, it offers no plausible interpretation of 4.016. This
15

It is defended by Stenius (1960: 11), Mayer (1993: 115), Lange (1996: 93 f.), Proops
(2000: 103–105), Schroeder (2006: 24), White (2006: 17), Morris (2008: 148), Bazzocchi
(2010a: 180–182), Hacker (2015: 652). It is rejected (as far as I know) only by Favrholdt
(1964: 222), Scheier (1991: 116) and Frascolla (2007: 14). Agnosticism is chosen by Black
(1964: 164).
16
The sequential reading is compatible with other ways of spelling out 4.02 as referring to
something or other mentioned in 4.016. I discuss only (what I consider to be) the best
candidate in the main text.
17
Since the second sentence of 4.01 does not occur in PT at all, this argument also
supports the hypothesis that “this” in 4.02 refers only to the first sentence of 4.01.


Tim Kraft


remark now ends abruptly. Instead of making good on his promise
to explain the essence of propositions, Wittgenstein would limit
himself to an empirical claim about the evolution of scripts before
changing the topic. It defies belief that all Wittgenstein has to say
about the essence of propositions concerns the evolution of scripts.
On the other hand, the sibling interpretation also runs into trouble
with the subsequent remark 4.021. If “this” refers to the first
sentence of 4.01, two consecutive remarks (4.02, 4.021) offer an
argument for the same claim: But why should Wittgenstein use an
anaphoric expression in the first remark when he then spells out his
claim in full in the very next remark? If 4.02 and 4.021 were about
the same claim, Wittgenstein should have used an anaphoric
expression in 4.021, but not in 4.02.
The immediate predecessor interpretation avoids these
problems. It avoids the second problem because it rejects that both
remarks (4.02, 4.021) are about the same claim. Instead, 4.02 is
about the essence of propositions while 4.021 offers the
superfluousness of explaining their sense as an argument for the
pictorial character of propositions. It avoids the first problem
because it does not consider 4.016 to be an endpoint. Instead, the
topic of 4.016 is continued in 4.02. The argument why ordinary,
alphabetic propositions are pictures is also different. Wittgenstein
does not offer a shallow evolutionary argument, as the tree reading
must assume, but argues that they are pictures because of their
shared essence with other propositions (4.016) and explains what
that shared essence is in the following remarks (4.02 ff.).
Thus, although the tree reading’s interpretation is supported by
the original context of 4.02 in PT, it does not account for the flow
of the argument in TLP. The sequential interpretation at least takes
seriously the task to explain how the various claims are connected
and thus avoids treating 4.016 as a loose end and 4.021 as a
pointless repetition. 18
18

The reason why none of this has been discussed in the literature (see the list in fn. 15) is
presumably that it does not make much of a difference for the interpretation of
Wittgenstein’s account of propositions as a whole. I do not discuss the interpretation of
“this” in 4.02 because it is central to understanding Wittgenstein’s account of


Nordic Wittgenstein Review 5 (2) 2016

5.2 A systematic survey of anaphoric references
A systematic survey of all anaphoric references in the Tractatus
reveals that of approximately seventy anaphoric references, i. e. of
remarks containing at least one expression referring back to
something mentioned in an earlier remark (demonstrative,
indexical, pronoun, inferential particle, phrase introducing an
example etc.), almost all are without doubt to their immediate
sequential predecessor(s). 19 The remaining cases (see below) are all
controversial and open to rival interpretations. The general
impression is that Wittgenstein does not follow strict rules: When
giving examples for a claim made in a preceding remark, the remark
containing the example(s) is sometimes on the same level (4.1272,
4.442, 5.12, 5.252, 5.5352, 6.372) and sometimes on a lower level
(3.3441, 4.0411, 4.1211, 6.1201, 6.1221, 6.241, 6.341, 6.3751).
When using indexicals like “here” and “now”, he uses them
sometimes in a specific way, i. e. to refer to a single remark (4.1213,
4.1251, 6.1223/6.1224), and sometimes in an unspecific way, i. e. to
refer not to a particular claim, but to the present state of the
discussion in general (4.5, 5.4, 5.534, 5.55, 6.342). 20 Anaphoric
expressions usually refer to something mentioned not just in the
immediately preceding remark, but even mentioned in its last
sentence, but again there is at least one (uncontroversial) exception
(4.1272, “thus (so)” refers to the first sentence of 4.1271). These
observations should suffice to counter the idea that anaphoric
propositions, but because it is an illustrative test case for accounts of the numbering
system.
19
To avoid inflating the number of anaphoric references I have counted some remarks
only once, namely consecutive remarks using the same anaphoric pronoun (e. g.
4.114/4.115) and remarks containing more than one anaphoric expression (e. g. 5.231).
The full list is: 2.0212, 2.023, 2.025, 2.1512, 2.1513, 2.1515, 2.172/2.174, 3.201, 3.312,
3.313, 3.321, 3.324, 3.325, 3.331, 3.3441, 4.02, 4.0411, 4.0412, 4.114/4.115, 4.1211,
4.1213, 4.1251, 4.1272, 4.242, 4.28, 4.442, 4.4611, 4.5, 5.12, 5.156, 5.21, 5.231, 5.252,
5.2522, 5.4, 5.41, 5.441, 5.531, 5.532, 5.5321, 5.533, 5.534, 5.535, 5.5352, 5.542, 5.5421,
5.55, 5.5521, 5.5542, 5.62, 5.6331, 5.634, 5.64, 5.641, 6.001, 6.01, 6.11, 6.1201, 6.122,
6.1221, 6.1222, 6.1223/6.1224, 6.1251, 6.241, 6.341, 6.342, 6.3631, 6.372, 6.3751 and 6.42.
20
Hacker mischaracterises the sequential reading as being committed to interpreting
dialectical indexicals as referring to a single remark (2015: 660). As spatial uses of “here’
and temporal uses of “now” can refer to areas and time intervals of different sizes
depending on context, dialectical uses of them can refer to smaller or larger parts of the
discussion as well.


Tim Kraft


references between remarks are governed by strict rules. Hence, it
should not come as a surprise if no account can resolve all
anaphoric references without exception.
It is a judgement call how many exceptions are too many, but
the number of clear exceptions to the tree reading is not negligible.
A telling case against the tree reading is 6.01 and its predecessor:
6.002 If we are given the general form according to which
propositions are constructed, then with it we are also given the general
form according to which one proposition can be generated out of
another by means of an operation.
6.01 Therefore the general form of an operation […]

The reference of “therefore” poses no problem for the sequential
reading. The immediately preceding remark explains how to derive
the general form of an operation from the general form of a truthfunction or sentence. The tree reading, however, rules out a
reference to 6.002 for the same reason it rules out that 4.02 refers
to 4.016: As a lower-level remark 6.002 is on a different branch.
Hence, “therefore” must refer to 6 which would leave 6.002
hanging in the air as a useless endpoint leading nowhere. Moreover,
since 6.002 is a late addition (both this remark and “therefore” in
6.01 are missing in PT), Wittgenstein knew exactly what he was
doing. If he had intended to follow the guidelines set by the tree
reading, he should have placed 6.002 after 6.01 (as 6.011). Alas,
Wittgenstein opted in favour of a different arrangement.
There are even more cases that provide further evidence against
the tree reading, although their interpretation is not as obvious as
the interpretation of 6.01. First, in Wittgenstein’s discussion of
identity in 5.53 and its two subsequent series (5.5301 ff. and 5.531
ff.) the remarks are intricately linked with each other: In 5.533
“therefore” refers not just to its siblings 5.531 and 5.532, but to
these remarks as well as the lower-level remark 5.5321. Otherwise
Wittgenstein could not claim to have shown that the identity sign
can be eliminated from logic. Moreover, 5.5321 (“thus, for
example”) in turn cannot be a consequence of 5.532 because these
two remarks have different topics: 5.532 is about reflexivity claims
and 5.5321 about ‘only’ claims. Instead, 5.5321 follows from the
general claim made in 5.53 together with its elucidation with respect

Nordic Wittgenstein Review 5 (2) 2016

to the same example in 5.5301 (which is on a different branch,
according to the tree reading) together with the technique for
eliminating the identity sign applied already in 5.531 and 5.532.
Similarly, 5.531 (“thus”) shows how to put into action the
programmatic remark 5.53 that is defended in detail in 5.5301–03.
The zero branch is as much an argument for 5.53 as it is a
preparation of 5.531 and it is rather implausible to connect 5.531
only with 5.53. Second, in 2.1513 “conceived in this way (diese
Auffassung)” refers to the picture theory as a whole. Everything said
so far about pictures, including 2.131, 2.141 and 2.15121, is part of
the picture theory and not merely what has been said in the branch
leading here. Third, in the context of discussing laws of nature
Wittgenstein asserts in 6.342 that “and now we can see the relative
positions of logic and mechanics”. “and now” cannot refer only to
the remarks in the branch leading to this remark since logic is
barely mentioned in this part of the tree. The comparison of
mechanics with logic must draw on resources from elsewhere in
the book.
What these examples have in common is that in all these
remarks Wittgenstein presupposes knowledge of remarks that
according to the tree reading belong to different branches. In such
cases the tree reading fares worse than the sequential reading when
resolving anaphoric references. It does not fail in that regard due to
some anomaly, but because it requires splitting up Wittgenstein’s
ideas into distinct branches, although his aim to all appearances has
been to weave them together into a tight, organic whole.
6. Exegetical issues
The final argument for the tree reading is that it helps to solve
broader exegetical issues which are not only about interpreting
single remarks or even words: These are the relation between
ontology and picture theory, solipsism and the eye analogy and the
resolute reading and its interpretation of 6.54.


Tim Kraft


6.1 Ontology and picture theory (2.1)
When read sequentially, the Tractatus begins with outlining an
ontology (world, facts, states of affairs, simple objects) and then
turns to a general theory of representation, the picture theory. The
tree reading resists the temptation to conclude that Wittgenstein’s
ontology is prior to the picture theory because priority claims based
on the order in which topics are discussed are ruled out by it.
Ontology is one branch of a tree, but there is no such thing as the
first branch of a tree. Hence, ontology does not precede the picture
theory, but is on the same level (Hacker 2015: 656).
This is all correct as far as it goes, but the priority claim is
neither here nor there. Neither is the sequential reading committed
to a priority claim nor is the tree reading committed to rejecting it.
It is a bad rule of interpretation in general to conclude priority from
the order in which topics are discussed by the author. In fact, I do
not know of a single interpretation of the Tractatus that argues for
the priority of ontology based on the order of the remarks. The
relation between the theory of representation and ontology,
between language and world is a complex one in the Tractatus – that
should be obvious to anyone who has read the whole book, no
matter whether from beginning to end or from root to leaves.
There is, however, a residual point of interest, namely the
transition from 2.063 (the last ontological remark: “the sum-total of
reality is the world”) to 2.1 (the first remark on pictures: “we
picture facts to ourselves”): Whereas the tree reading treats 2.063 as
an endpoint of a branch that is separated from the branch 2.1 ff.
(“this is where the path ends and there is nothing after it”,
Bazzocchi 2010c: 332), the sequential reading tries to avoid such an
abrupt break. The sequential reading therefore expects 2.063 to
have both a backwards- and a forwards-directed role. Suppose
2.063 was missing and the line of thought ended with 2.062 which
is about the independence of states of affairs. In that case the
relation between 2.062 and 2.1 would be strained indeed. However,
with the help of 2.063 the transition is much smoother: 2.063
summarises the preceding discussion by highlighting its central
topic, i. e. the world, and prepares the next remark which is partly
about facts, i. e. the world. Again there is, as the sequential reading

Nordic Wittgenstein Review 5 (2) 2016

predicts, a topic that is shared by consecutive remarks, that is, 2.063
is not a dead end, but a link in an intricate chain.
6.2 Solipsism and the eye analogy (5.6s)
The 5.6s (limits of language and world) are a particularly difficult to
interpret passage of the Tractatus. One of the exegetical problems is
to understand the analogy Wittgenstein draws between
(metaphysical) subject and its world on the one hand and the
(metaphysical) eye and its visual field on the other. 21 This analogy is
inserted between the startling claim that I am my world (5.63) and
the equally surprising claim that understood properly solipsism is a
form of realism (5.64). How it is supposed to fit in between these
remarks is the exegetical issue to be discussed here. Tree readers
promise that their approach allows them to make considerable
progress on it (Bazzocchi 2014a, 2014b: xi–xiv, Hacker 2015:
660 f.). For the relevant remarks contain several anaphoric
expressions that connect these remarks with each other: (a) In
5.6331 with “for” an explanation is offered – but an explanation of
what? (b) In 5.634 Wittgenstein states that “this is connected with
the fact that” – but what does “this” refer to? (c) In 5.64 a
conclusion is introduced by “here it can be seen that” – but what
does “here” refer to?
To answer the first structural question, another issue needs to
be mentioned first: The figure in 5.6331 is drawn differently in
Wittgenstein’s manuscripts and typescripts than in the various
printed editions (Lampert and Graßhoff 2004: 103, Bazzocchi
2014a: 260–265). His drawing of the (mistaken) form of the visual
field does not show an eye in- or outside of a visual field, but uses a
bold dot to mark the part of the visual field that is meant to
indicate the presence of an eye. 22Accordingly, 5.6331 expands on
21

The metaphysical eye is to be distinguished from the biological eye. For (part of) the
biological eye is in the visual field (eyelid and -lashes) and the distinction between subject
and body (5.631) has to be applied to the eye as well for subject and eye to be analogous.
22
I follow Lampert and Graßhoff’s description of the figure here. Bazzocchi describes it
slightly differently. They agree that the eye is not drawn inside the visual field. But,
whereas Bazzocchi describes the eye as being outside instead, Lampert and Graßhoff
describe it as follows: “Wittgenstein never depicted the eye as an extended circle inside or
outside the visual field. Instead, in the original drawings a position is assigned to the eye


Tim Kraft


the last sentence of 5.63 and not on its middle paragraph. It does
not explain why “you do not see the eye”, but why “nothing about
the visual field allows you to infer that it is seen by an eye” 23 .
Wittgenstein does not argue that the eye is not part of the visual
field, but makes the stronger claim that its presence is not indicated
by any formal features of the visual field either. This settles the first
question. But note that this answer to the first question is
consistent with both the tree and the sequential reading. For both
readings predict that 5.6331 expands on 5.633. 24
The tree reading still promises principled responses to the
second and third question. Ad (b), 5.634 is connected to 5.633, but
not to 5.6331: Both “this” in 5.633 and “for” in 5.6331 refer to the
last sentence of 5.633, i. e. to the claim that nothing about the
visual field shows that it belongs to an eye. Ad (c), 5.64 is the final
remark of the sequence 5.61/5.62/5.63/5.64: “here” refers to this
sequence, but not to the eye analogy at all. Thus, the tree reading
puts 5.634 within the discussion of the eye analogy and 5.64 outside
of this discussion. The sequential reading offers somewhat different
answers to these questions. Ad (b), 5.634 is meant to draw a lesson
from the eye analogy and is as much about the subject as it is about
the eye: “this” refers to the main claim in this discussion, namely
that both the subject and the eye are neither in the world/visual
field nor indicated by their form. Ad (c), in 5.64 “here” does not
refer to a particular remark, but refers in an unspecific way to the
current discussion as a whole. Thus, the sequential reading treats
5.634 as connecting the two sides of the analogy, subject and eye,
with each other and 5.64 as drawing the main conclusion of the
discussion that began at 5.62.
by a dot at the tip of the visual field. […] The point of this is to make clear from where
one sees.” (2004: 103, my translation)
23
I have slightly changed the translation: The phrase used by Wittgenstein – “nichts an”
instead of “nichts in” – means nothing about or no feature of, but not nothing in (Bazzocchi
2014a: 259 n. 11). This may be an additional reason – the misleading reproduction of the
figure being the other one – why some interpreters do not distinguish properly between
the two claims made in 5.633.
24
Bazzocchi claims that the sequential reading is responsible for misunderstanding 5.633
and 5.6331 (2014a: 258 f., 265, 2014b: xiii n. 6), but it is the reproduction of the drawing
that is to be blamed here.


Nordic Wittgenstein Review 5 (2) 2016

Beginning with (b), proposals that treat “this” as referring to a
particular statement made in either 5.633 or in 5.6331 are
problematic. They are so even for the same reason. Since
Wittgenstein is drawing an analogy, a lesson from that analogy is to
be expected. If one of these proposals is correct, however, 5.634
connects a remark specifically about the visual field with a discussion of
apriority. Wittgenstein is evidently not interested in such a limited
lesson. Therefore, it is mistaken to require “this” to refer to a
particular statement and, instead, “this” is short for “this analogy”,
“this question” or “this issue”. Assuming that this interpretation is
plausible, neither the tree reading nor the sequential reading turn
out to be in a better position to explain the reference of “this”.
Since the issue under discussion is the same in 5.633 and 5.6331, it
does not matter whether “this” refers to the immediately preceding
remark(s) or only to the preceding sibling(s). Yet, looking at the
immediately preceding remarks, as the sequential reading
recommends, is useful for recognising what the analogy between
subject and eye is supposed to be. To see this, we need to take a
look at the body of 5.634. Wittgenstein argues for an incompatibility
claim: If something is “part of our experience” (i. e. it is
experienced itself or its existence can be inferred from experience),
it is not apriori. 25 And what is part of experience is only
contingently so, but, as Wittgenstein presupposes, apriority entails
necessity. Thus, Wittgenstein offers a general argument for his
claim that neither the existence of the eye nor of the subject can be
inferred from experience. In particular, no matter what specific
form the visual field is supposed to have, even if the visual field
actually had that form, one could at most know that an eye exists,
but not that it must exist – mutatis mutandis for the subject. This
argument is lost if 5.634 is read only in connection with 5.633, as
the tree reading suggests.

25

Contrary to Bazzocchi’s suggestion (2014a: 258 n. 7), this incompatibility is not trivial.
Wittgenstein is committed to a strong conception of apriority: Whereas on a weak
conception what is apriori can (but need not) be known independently of experience, on
the strong conception what is apriori can only be known independently of experience.
Only the strong conception licences the incompatibility claim.


Tim Kraft


Turning to (c), Bazzocchi’s argument is based on a misleading
characterisation of the sequential reading. The sequential reading
does not claim that “here it can be seen” in 5.64 must refer to and
only to the immediately preceding remark 5.634. Instead, this
phrase introduces a conclusion of the ongoing discussion (i. e. a
whole series of preceding remarks) while leaving it somewhat open
how many of the preceding remarks belong to this series. 26 As
outlined above, Wittgenstein uses indexicals (e. g. “here”, “now”)
and inferential expressions (e. g. “therefore”) to refer to the
overarching line of thought several times in the Tractatus. Hence,
my hypothesis about 5.64 is not an ad hoc hypothesis. Moreover, it
avoids the pitfalls of the interpretation suggested by the tree
reading. 5.64 is not a conclusion that one can draw from just the
sequence of its siblings. Although Bazzocchi claims that “it’s easy
to see the relationship” between 5.64 and 5.63 (2014a: 258), that is
blatantly not the case: If subject and its world are identical, as 5.63
suggests (“I am my world”), and this world is extended, the subject
cannot lack an extension. Realism is also usually a claim about the
world, not about someone’s world. To see what is wrong with
these objections the clarifications of 5.63 in the remarks between
5.63 and 5.64 are badly needed. Fortunately, the sequential reading
casts a wider net and includes the explanatory remarks that are
inserted between these remarks as being referred to by “here”. The
tree interpretation of 5.64 also faces the challenge of explaining
Wittgenstein’s renumbering of the remarks involved. In PT 5.64’s
(= PT 5.3355) preceding sibling used to be 5.631a (= PT 5.3354):
“there is no such thing as the subject that thinks or entertains
ideas”. This remark is now a comment on 5.63 and, therefore, on a
different branch of the tree. If, however, “here” refers to the wider
context, Wittgenstein’s renumbering of the remarks did not
threaten to make “here” incomprehensible because both 5.63 and
5.631 fall within the scope of “here” anyway.

26

See footnote 28.


Nordic Wittgenstein Review 5 (2) 2016

6.3 Nonsense and the ladder (6.54)
(Some) Proponents of the tree reading, argue that the tree reading
conclusively rules out a recently much debated interpretation of the
Tractatus, the resolute reading (Bazzocchi 2010a, 2010c, 2013,
Hacker 2015, but not Kuusela 2015). For present purposes the
resolute reading can be characterised as being defined by two
interconnected claims: 27 On the one hand, the distinction between
different kinds of nonsense is rejected. There are no ineffable
truths that cannot be said, but only shown. On the other hand, 6.54
is taken at face value so that the Tractatus consists literally only of
nonsense – with the sole exception being the so-called ‘frame’
including at least the preface and 6.54/7. This means, crucially, that
the Tractatus does not contain a theory of meaning that is then
applied to the Tractatus’s attempt of stating this theory to get to the
conclusion that any such attempt does not meet its standards of
meaningfulness.
As far as I can see, the claim that the resolute reading is ruled
out by the tree reading is backed up by two arguments: First,
according to the tree reading, 6.54 and 7 are unconnected remarks.
According to the resolute reading, however, they form a unified
sequence of remarks, which is the conclusion of the Tractatus.
Together with the preface they put a frame around the other,
nonsensical remarks. But if the Tractatus is not sequentially ordered,
it neither has a ‘conclusion’ nor can there be such thing as a ‘frame’
(Bazzocchi 2010a: 193–195, 2010c: 333, Hacker 2015: 662–666). 28
Second, proponents of the resolute reading put much weight on the
ladder metaphor, but according to the tree reading this metaphor
does not extend beyond comparing the Tractatus to a tool that is
27

See Diamond 1988, Conant 1989, 2002 for the resolute reading in general and Conant
and Diamond 2004: 47, Bronzo 2012: 53 for (roughly) this way of characterising it.
28
Bazzocchi (2013) also argues that connecting 7 to 6.54 is similar to referring into
parentheses which he deems to be impossible. His argument is that in, for example, “My
article is boring. (The Tractatus, too, is boring.) It is even mistaken.” “it” cannot refer to
the Tractatus, but only to the utterer’s article. This is correct as far as this example goes,
but the general impossibility thesis is false. To disprove it this paper contains a sentence
(marked with fn. 26) in which a demonstrative refers into parentheses. If any reader
stumbled over that sentence, this would support Bazzocchi’s objection to some extent –
but I trust that no one did.


Tim Kraft


not needed anymore once it has served its intended purpose. The
metaphor of the ladder invites the idea of a higher point of view to
be reached by climbing the ladder rung by rung. But if the Tractatus
is a tree, there is no higher position to be reached eventually
(Bazzocchi 2010a, Hacker 2015: 666–668).
Tree readers are surprisingly quick to dismiss the resolute
reading (Kuusela 2015: 231). As far as Bazzocchi’s and Hacker’s
objections are concerned, their criticism of the resolute reading is
rather unconvincing.29 The first argument misses that the resolute
reading does not presuppose a structural claim. One might think
that the distinction between frame and body would require 6.54
and 7 to form a unit. But in fact the distinction between frame and
body is about content and function (frame remarks contain reading
instructions or are metaphilosophical in nature), not about the
order of the remarks. That is so because the Tractatus contains
frame remarks in many places, for example, the metaphilosophical
remarks 4.111 ff. (Kuusela 2015: 231, Conant 2002: 457 n. 135,
Gunnarsson 2000: 51–53). The resolute reading does not even
claim that it is fixed or detectable in advance which remarks belong
to the frame and which belong to the body. This distinction is
supposed to emerge only in the process of working through the
book (Conant and Diamond 2004: 68 f.). And even if it is conceded
that 6.54 and 7 must not be read together, the resolute reading is
not doomed. Since 6.54 stands in need of an interpretation
independently of whether it is considered only as a comment on 6.5
or as a preparatory remark towards 7, it is still an advantage of the
resolute reading that it offers an interpretation of this puzzling
remark that takes it at face value. If the resolute reading is
mistaken, it is so because it offers an unconvincing interpretation
of 6.54, not because it reads 6.54 in conjunction with 7.
The second argument is based on an uncharitable
characterisation of the resolute reading. Although resolute readers
regularly use the ‘climbing the ladder rung by rung’ analogy, not too
much should be read into it. As far as the analogy goes, a
29

My discussion does not affect Hacker’s other objections to the resolute reading that are
independent of the tree reading and much more convincing (Hacker 2000).


Nordic Wittgenstein Review 5 (2) 2016

horizontal ladder used to cross an abyss would do as well. After all,
it is just an analogy. Crucially, the resolute reading does not claim
that there is a higher point of view from which one can “see the
world aright” (6.54). Seeing the world aright after having read the
Tractatus does not mean that a mistaken view of the world is
replaced by a correct one. For by giving up nonsense one does not
advance from an old position to a new one, but realises that there
never even was a position to be had, but merely nonsense.
None of this is intended as a defence of the resolute reading.
My aim is only to point out that the tree reading and the resolute
reading are not inconsistent. That being said, the tree reading’s
suggestion that 6.54 and 7 are unconnected is rather surprising
given that even Wittgenstein himself thought of his book as having
a conclusion. Apparently, Wittgenstein wanted to end his book
with a bang – a remark that is testament to the finality and
unassailability of his philosophical system. Obviously, such a
remark should not be accompanied by any comments placed after
it and whatever comments or elucidations Wittgenstein wanted to
add he had to place before it. Since he is relying on preparatory
remarks elsewhere (for example, 2.1–2.2 as preparations of 3, 3.5 as
preparation of 4), it is not unexpected to find preparations of 7 in
its preceding remarks.
7. Conclusion: How to read the Tractatus sequentially
In this I paper I have argued that the Tractatus should not be read
as a tree, but as a sequential chain. My proposal does not promise
new deep insights into the Tractatus. The main exegetical nuts
remain as hard to crack as they used to. Just paying attention to the
numbering system will not solve our exegetical problems for us.
Yet, the sequential reading makes sense of Wittgenstein’s
comments on the numbering system in the Tractatus footnote and
elsewhere, resolves anaphoric references within the Tractatus and
fares at least as well as the tree reading regarding three prominent
exegetical issues. The central recommendation of the sequential
reading is to pay attention to the numbering system when reading
the Tractatus by keeping an eye on how a given remark is connected
with earlier and later remarks. Wittgenstein never jumps from a

Tim Kraft


topic to another one, but develops his ideas in the form of an
intricately linked chain. Readers can follow his presentation by
taking all the tiny steps or by tracing only the general line of
thought, i. e. by taking only the major steps. Either way the
important question to ask is how Wittgenstein linked topics and
claims with each other, not how Wittgenstein separated them into
sub-topics and sub-claims. This way, in addition to offering
sensible interpretations of all the test cases discussed above, the
sequential reading can also explain how the Tractatus matches
Wittgenstein’s view of philosophy. His ideal of doing philosophy
was to write philosophy as concise (i. e. clear, brief and simple),
surveyable and unified as possible. This ideal is not just an aesthetic
one, but also plays an epistemic role insofar as meeting it is
evidence for the truth of what is said. Thus, understanding how
Wittgenstein attempted to write a concise, surveyable and unified
book is crucial for understanding why “the truth of the thoughts
that are here communicated seems to me unassailable and
definitive” (TLP: preface). Any interpretation of the numbering
system should make sense of this ideal and how he intended to
fulfil it. According to the sequential reading, the Tractatus forms a
unified whole because the remarks naturally lead from topic to
topic and hang together on several levels. Its unity is not derivative
of the unity of a core of cardinal remarks to which the other
remarks are simply attached. Conciseness and surveyability depend
on the numbering system because it allows Wittgenstein to display
context and connections without having to state them in a verbose
way. Thus the remarks and their contents take care of themselves.
Conciseness and surveyability are not achieved by dividing them
into branches with a multitude of leaves.
That the Tractatus is far from an “incomprehensible jumble”, as
Wittgenstein put it, but a “finely crafted work of art” (Hacker 2015:
649) is easy to assert, but difficult to explain. The sequential reading
is an attempt to offer such an explanation. 30
30

For discussion and criticism I am grateful to the participants of my graduate course on
the Tractatus (Regensburg 2015/16), Wilfried Keller, Oskari Kuusela, Tammo Lossau,
Adriana Pavić, David Stern and two anonymous referees.


Nordic Wittgenstein Review 5 (2) 2016

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quotations)

Biographical Note
Tim Kraft studied philosophy, mathematics and economics at
Göttingen and St Andrews. After completing his PhD with a thesis
defending semantic normativity (Göttingen 2011), he moved to
Regensburg as a “Wissenschaftlicher Assistant” (i.e. non-tenure-track
assistant professor). His research interests include philosophy of

Tim Kraft


language (rule-following, semantic normativity), epistemology
(doxastic attitudes, epistemic normativity, scepticism and agnosticism,
disjunctivism, epistemic closure) and the history of analytic philosophy
(Frege, early and late Wittgenstein).