The document provides an overview of Wittgenstein's Tractatus and some of its key influences and ideas. It discusses: 1) Wittgenstein was influenced by the exact sciences through thinkers like Boltzmann, Hertz, and Russell, which shaped his ideas around logical atomism, the theory of meaning, and truth tables. 2) In the Tractatus, Wittgenstein criticized the views of Frege, Russell, and others on topics like the nature of logical truths, the status of logical constants, and the character of logical connectives. 3) Wittgenstein proposed in the Tractatus that the world consists of facts and states of affairs, and that language pictures these through logical

1. Leiden University
MA Philosophy and History of Science
Wittgenstein's Tractatus (Prof. B.G. Sundholm)
The influence of the exact sciences on Wittgenstein:
theory of meaning, logical atomism and truth tables
Tractatus
Basil Evangelidis

2. Language and Experience
 Early logical atomism incubated the idea of an ideal
language (Russell & Whitehead).
 Wittgenstein criticized the disguise of logical
structure by ordinary language.
 He later pointed out the notion of vagueness.
 Common sense (Moore, from 1939 succeeded by
Wittgenstein in Cambridge).

3. The peculiarity of the Tractatus
 The late philosophy of Wittgenstein cannot be regarded as a
continuation of the Tractatus.
 The well-known narrative of this change distinguishes between:
 (1) a truth-conditional theory of meaning in the Tractatus, which
is realistic,
 (2) a later theory of meaning that consists in assertibility-
conditions that is considered as anti-realistic.
 However, there is also a THERAPEUTIC approach to
Wittgenstein’s philosophy, which uncovers continuity between the
Tractatus and the later works, facing the Tractatus as a
forerunner of Wittgenstein’s later writings (Crary and Read,
2000).

4. A world of facts / states of affairs
- The main theme of the Tractatus is "the connection between
language, or thought, and reality. The main thesis about this
is that sentences, or their mental counterparts, are pictures of
facts" (Anscombe, 1963: p. 19).
- It is worth mentioning that a logical product is defined as
conjunction of all the propositions of a given set, for instance,
p.q.r., whereas logical sum is defined as a disjunction of all
the propositions of a given set (Ibid. p. 23).

5. Russell's influence on Wittgenstein
 The Russellian contention that the classes and the descriptions are
incomplete symbols, namely their uses may be defined, but they
themselves do not mean anything at all, was highly influential for the
young Wittgenstein.
 Wittgenstein set as a goal the Sprachkritik, under the insight of Russell
that "the apparent logical form of a proposition need not be its real one"
(TLP 4.0031).
 Only in the Logical Investigations would he become totally convinced by
Mauthner, growing suspicious with logic, rejecting the representational
value of language, and the "logic of depiction" of the Tractatus.

6. Abstract thought
- Wittgenstein overstepped Russell’s tendency to abstractness by
insisting, as the various customary anecdotes relegate, that
“nothing empirical is knowable” and refusing “the existence of
anything except asserted propositions” (Blackwell, 1981).
- Around 1912, Wittgenstein sent letters to Russell on what he
called “the complex problem” and the “theory of symbolism.” He
criticized the theory of types as “superfluous,” arguing that “there
cannot be different types of things” (Blackwell, 1981: pp. 11-12).
- Wittgenstein refuted certain theories of Russell, such as the
theory of judgement. He also rejected Russell’s proposal to study
French prose, and he did that with a ferocious manner.

7. The words “and,” “or,” “two,” and “plus”
In proposition 4.0312 of the Tractatus, Wittgenstein declared
that his “leading idea” was that logical constants are
deprived of any referential understanding.
“My fundamental thought is that the ‘logical constants’ do
not represent. That the logic of the facts cannot be
represented.”
“In the Tractatus, he held a similar view concerning
arithmetic terms. This is one area where there is an
important continuity between Wittgenstein's earlier and
later philosophy” (Fogelin, 2009: p. 86).

8. “In the beginning was the deed”
“Frege and Russell were radically mistaken about:
- the nature of logical truths (conceiving of them as
essentially general),
- about the nature of logical necessity,
- about the content of logical truths,
- about the status of the axioms of logic,
- about the character of the logical connectives and
quantifiers, and
- about the relation between the truths of logic and
rules of inference” (Hacker, 2001b: p. 71).

9. The Influences
So haben mich Boltzmann, Hertz, Schopenhauer, Frege,
Russell, Kraus, Loos, Weininger, Spengler, Sraffa
beeinflusst. Kann man als ein Beispiel jüdischer
Reproduktivität Breuer und Freud heranziehen?
(Wittgenstein, 1980: p. 19).

10. LUDWIG WITTGENSTEIN in Tractatus was critical to his most
important influences:
- "All philosophy is a 'critique of language' (though not in
Mauthner's sense)" (4.0031),
- Frege's theory of meaning is "based on confusion" (5.02),
- Russell's view of the self represents "the superficial
psychology of the present day" (5.5421),
Sprachkritik I

11. - Russell's and Frege's laws of inference have "no
sense" and are "superfluous" (5.132), and
- their understanding of general propositions is
"incorrect" and "contains a vicious circle" (4.273).
- Wittgenstein agreed with Mauthner that the logical
truths are tautologies, while Frege and Russell
supported the substantive conception of logical truth.
Sprachkritik II

12. Fritz Mauthner
o Fritz Mauthner's Contributions to a Critic of
Language familiarized Wittgenstein with
Pyrrhonian skepticism.
o Mauthner was inspired by Ernst Mach and
Arthur Schopenhauer. The transcendence of
philosophy into silence was the conclusion of
Schopenhauer's book.
o "The critique of language must teach liberation
from language as the highest goal of self-
liberation."
"If I want to rise up in the critique of language, which is the most important
business of thinking mankind, I must destroy language step by step behind
me, before me, and within me, I must break the rungs of the ladder as I step

13. Ernst Mach
The metaphor with the ladder was first used by
Sextus Empiricus, then by Ernst Mach, by
Mauthner and Wittgenstein (Sinnott-Armstrong,
2004).
- As Einstein stressed, Mach was responsible for
a bold attempt to overstep classical mechanics.
From a phenomenalistic point of view, he
criticized the Newtonian notion of "absolute
space" on the ground of its unobservability
(Seaman, 1975).

14. Ludwig Boltzmann
One of his first scientific activities
concerned the thermodynamics and the
kinetic gas theory. Boltzmann
recognized the entropy as a measure of
probability, as to any increase in
entropy corresponds a tendency to shift
from improbable to probable situations,
from less to more disorder.
- The last years before his suicide he developed his Bildtheorie. The
fundamental idea was based (1) on Gustav Kirchhoff's kinematic
theory of the motions of particles, and (2) the non-explanatory theory of
electricity presented by Maxwell.

15. Heinrich Hertz
In the foreword of the Prinzipien der
Mechanik, Heinrich Hertz refers to his
picture theory of knowedge is presented
in the introduction of the book:
Wir machen uns Scheinbilder oder
Symbole der äußeren Gegenstände, und
zwar machen wir von solcher Art, dass
die denknotwendigen Folgen der Bilder
stets wieder die Bilder seien von der
naturnotwendigen Folgen der
abgebildeten Gegenstände.

16. Hertz and Wittgenstein
6.3431: “Through their whole logical apparatus the
physical laws still speak of the objects of the world.”
6.3432: “We must not forget that the description of the
world by mechanics is always quite general. There is,
for example, never any mention of particular material
points in it, but always only of some points or other.”
6.361: “In the terminology of Hertz we might say: Only
uniform connections are thinkable.”

17. Mathematical Multiplicity
4.04: “In the proposition there must be exactly as many
things distinguishable as in the state of affairs it
represents.“
“They must both possess the same logical
(mathematical) multiplicity (cf. Hertz’s Mechanics, on
Dynamical Models).
4.041: “This mathematical multiplicity naturally cannot
in its turn be represented. One cannot get outside it in
the representation.”

18. The logical picture of the facts
- Pictures of facts are produced by the human intellect (2.1), as
models of reality (2.12), in order to provide a linkage with reality
(2.1511) and a representation of the combination of things through
the connection of the elements of the picture (2.151).
- The structure of a picture is the connection of its elements (2.15).
Therefore, the models are composite, they must have form and
structure. The form of a model is the possibility of its structure. A
model stands for a state of affairs, an atomic fact (Sachverhalt).
- A picture is a fact, eine Tatsache (2.15); it applies a measure into
reality (2.1512). this is a logical form that renders the depiction
possible (2.18).

19. Wittgenstein and the Mind’s Eye
Sense - Reference
Saying - Showing
In Frege: “Sense determines reference, but reference does not determine sense”
(Dummett, 1981: p. 32).
A proposition shows its sense. (4.022)
Propositions show the logical form of reality. They display it. (4.121)
A visual, non-verbal process is crucial for the picture theory of language, which
originates in visual memory, spatial thinking and engineering design (Hamilton,
2001).

20. Modal atomism
- Wittgenstein’s metaphysics is atomistic, hence being “can be constructed
out of simple objects that have simple properties and stand in simple
relations to one another” (Bradley, 1992: p. xiv).
- His atomism is also modal, because every simple object has a “form”
(TLP 2.0141) or “logical form” (TLP 2.0233), an internal, necessary and
essential characteristic of the simple object.
- In the Notebooks, 1914-1916 (70(9-10)), Wittgenstein writes that
according to their logical forms simple objects belong to respective
“logical kinds”.
- Possible states of affairs (2.0124), which correspond to logical kinds and
logical forms, may be combined to render an “imagined” (2.022) or
“possible” (NB 83(2)) world. The real world is comprised by the totality
of the state of affairs that actually exist (2.04).

21. Three types of sentences
According to Nordmann (2005), Wittgenstein distinguishes in his
Tractatus three types of sentences, corresponding to three uses of
language:
(1) the ordinary and familiar descriptive sentences,
(2) the formalizing sentences such as the law of contradiction, the laws
and theorems of logic in general. The question is whether mathematical
sentences, tautologies, analytic statements, axioms and rules in formal
systems, or definitions belong to this kind.
(3) Nonsensical sentences which are not grammatical at all, "whereof
one cannot speak" (TLP, 7).

22. The senseless… and the sinnvoll
 A kind of solipsism in the sense of Schopenhauer and mysticism are
striking exceptions in the quest for meaningfulness in the Tractatus.
Even logic is classified as senseless as McGuinness (2002)
understands it.
 Cora Diamond (1991), however, observed that Wittgenstein did not
try to demarcate certain kinds of propositions that are meaningless.
Taking his distance from Frege, he writes that: “Every proposition is
legitimately constructed, and if it has no sense this can only be
because we have given no meaning to some of its constituent parts.
(Even if we believe that we have done so.)” (5.4733)

23. The meaning and the sign
“Frege ridiculed the formalist conception of mathematics
by saying that the formalists confused the unimportant
thing, the sign, with the important, the meaning… But if
we had to name anything which is the life of the sign, we
should have to say that it was its use” (The Blue Book, p.
4).
In the beginning of the Blue Book he distinguishes between verbal
and ostensive definitions, commenting that the words “one”,
“number”, “not” etc. do not seem to allow ostensive definitions.

24. Meaning is use
In his lectures in Cambridge, Wittgenstein insisted (1979:
p. 21) that meaning is use; meaning is the product of our
operating with words.
Therefore: “The meaning of a word is explained by
describing its use” (Wittgenstein, 1979: 48).
“Understanding a word” means “being able to use a word”
word” (Ibid. p. 78).

25. Occam’s razor
3.328: “If a sign is not necessary then it is meaningless. That
is the meaning of Occam΄s razor.”
“(If everything in the symbolism works as though a sign had
meaning, then it has meaning)”.
5.47321: “Occam’s razor is, of course, neither an arbitrary rule
nor one justified by its practical success. It simply says that
unnecessary elements in a symbolism mean nothing.”
“Signs which serve one purpose are logically equivalent, signs
which serve no purpose are logically meaningless.”

26. Wittgenstein’s aero-engine
The plans of Wittgenstein's aero-
engine.
Cater J , Lemco I Notes Rec. R.
Soc. 2009;63:95-104.
(Manchester University, School of
Mechanical, Aerospace and Civil
Engineering)

27. THE ENGINEER
George Stephenson
(born June 9, 1781, Wylam, Northumberland, Eng.
— died Aug. 12, 1848, Chesterfield, Derbyshire),
English engineer and principal inventor of the
railroad locomotive. Stephenson was the son of a
mechanic who operated a Newcomen atmospheric-
steam engine that was used to pump out a coal
mine at Newcastle upon Tyne. The boy went to
work at an early age and without formal schooling;
by age 19 he was operating a Newcomen engine.
(Sources: Britannica & Flickr).

28. Tractatus 6.1203

29. The cryptographic Wittgenstein
- Russell, criticizing the aphoristic style of the Tractatus, insisted that he
he should give arguments, but Wittgenstein responded that “arguments
“arguments would spoil its beauty.”
- Frege studied the manuscript and commented: “I find it difficult to
understand. You place your propositions one after the other mostly
without giving reasons for them, or without giving enough detailed
reasons” (Klagge, 2011: p. 16).
- Wittgenstein used reverse-alphabet code to record his thoughts. Among
Among his coded remarks, we know that on April 29, 1916, his life was
was in danger.

30. Truth tables
Truth tables are usually credited to Wittgenstein, but
Peirce and Schröder used something similar to truth
tables, while models were suggested by the
“interpretations” and “universes of discourse” of Boole and
and De Morgan (Dipert, 1998). Emil Leon Post (1921) is
also considered as an author of a work that introduced
truth tables, starting from the Principia Mathematica of
Russell and Whitehead.

31. Charles Sanders Peirce’s Truth Tables

32. Ludwig Wittgenstein and Alan Turing
As it is shown in the Blue Book 47, Wittgenstein exposed to
his students the problem of the feasibility of a machine that
thinks. The difficult part of that question is that the sentence
sentence “a machine that thinks (perceives, wishes)” seems
somehow nonsensical (Shanker, 1998).

33. Wittgenstein on machine intelligence
“If one thinks of thought as something specifically human and
organic, one is inclined to ask, “could there be a prosthetic
apparatus for thinking, an inorganic substitute for thought?”. But if
if thinking consists only in writing or speaking, why shouldn't a
machine do it? “Yes, but the machine doesn't know anything”.
Certainly, it is senseless to talk about a prosthetic substitute for
seeing and hearing. We do talk of artificial feet, but not of artificial
artificial pains in the foot. (P.G. § 47: p. 105).

34. “In mathematics everything is algorithm and nothing is meaning”
(Wittgenstein, 1974: p. 468)
As a Professor in Cambridge and teacher of Alan Turing, Wittgenstein’s technique
technique was to ask questions which brought words like proof, infinite, number,
number, rule, into sentences about real life, and to show that they might make
nonsense (Hodges, 2012).
I believe that mathematics, once the conflict about its foundations has come to
an end, will look just as it does in elementary school where the abacus is used.
The way of doing mathematics in elementary school is absolutely strict and
exact. It need not be improved upon in any way. Mathematics is always a
machine, a calculus. The calculus does not describe anything (Wittgenstein,
1979b: p. 106)