In Ludwig Wittgenstein’s philosophical text, Tractatus Logico-Philosophicus, he indicates thoughts and propositions are pictures, and these pictures are “a model of reality”. A picture, explained by Wittgenstein, is made up of elements and sates of (Biletzki). Moreover, for a picture to be a picture of the world, the picture should be verifiable in reality. One of the mathematical scripts, Euclid’s Elements, also have been regarded as a root of the world. In Euclidean Geometry: Its Nature and Its Use, J. Herbert Blackhurst indicated that “Known space is Euclidean and can be conceived in no other way”. What Blackhurst says is that objects and organization in our world are constructed in a Euclidean form. For instance, a table can be in a shape of rectangular or square, a cookie can be in a shape of circle. If this is the case, is Euclid’s Elements or Euclidean Geometry a picture of the world in the context of Wittgenstein’s “Picture Theory”? In this essay, I am going to analyze the structure, components, and implication of the Picture Theory, and then compare it with Euclidean Geometry to attain the answer for the question.
Even though Wittgenstein––who was one of the most influential philosophers and mathematicians of the 20th century––had provided insight mostly in the areas of linguistic, mathematics and logic, the “dramatic quality” of his life does not simply rest on academics, but also on his path of exploring epistemology and publishing Tractatus Logico-Philosophicus (Phillip 3). Born in a wealthy and influential family of Karl and Leopoldine in Vienna in 1889, Wittgenstein received distinctive education from private tutors in his early age. Because of this early vigorous and enlightening education, Wittgenstein was able to study engineering and occupied himself, in college education, extensively with experiments in aerodynamics––a study of motion of air––which required a thorough intellectual grasp of theoretical physics and mathematics (Phillip 4). In 1908, Wittgenstein left Berlin and went to England as a research student in the department of engineering at the University of Manchester (Phillip 4). Throughout his time in Manchester, Wittgenstein became increasingly interested in pure mathematics and the philosophy of mathematics (Phillip 4). He started reading mathematical works by Bertrand Russell, a British philosopher and mathematician, and was intrigued by Russell’s idea of building a formal logical system. Wittgenstein’s pursuit of mathematics and philosophy endeavored until the outbreak of the World War in 1914 when Wittgenstein volunteered for the Austrian army. In 1918, when the Austrian army was defeated in the war, Wittgenstein was captured by Italians and spent nine months in a prison (Phillip 5). During nine months in the prison, Wittgenstein wrote the Tractatus Logico-Philosophicus and “correspond[ed] with Russell and sent him the manuscript” (Phillip 5). Russell, with great surprise and admiration for Wittgenstein, helped him find a publisher for the book in 1922.
The purpose of Tractatus Logico-Philosophicus is to present “a comprehensive philosophical picture of the world” (Phillip 20). There are seven different propositions in the Tractatus, and each proposition has a number assigned with it; for instance, the first proposition will be 1, and the second be 2, and so on. To provide a glimpse of the content of Tractatus, the seven propositions are provided below:
“1. The world is everything that is the case. 2.What is the case, the fact, is the existence of atomic facts. 3.The logical picture of the facts is the thought. 4. The thought is the significant proposition.5. Propositions are truth-functions of elementary propositions. 6.The general form of truth-function is[p¯,ξ¯,N(ξ¯)][p¯,ξ¯,N(ξ¯)]. This is the general form of proposition. 7.Whereof one cannot speak, thereof one must be silent.”
Comments and elaborations for each proposition are numbered in different ways: the first proposition’s comments are labeled as 1.1, 1.2, and so forth; the second, similarly, is 2.1, 2.2, and etc. Throughout Tractatus, Wittgenstein mainly focuses on developing a notion of Picture Theory. Wittgenstein explicitly implies the Picture theory in comments 2.063 and 2.12 –– “the total reality is the world” and “the picture is a model of reality” (Wittgenstein). Both comments are suggesting pictures are the world. One might think, with Wittgenstein’s implication, a picture will show people the world visually. According to Derek L. Phillip’s remark on Tractatus, Wittgenstein’s pictures in the Tractatus are not, however, spatial pictures like maps or photographs; rather, they are what he called “logical pictures”. For instance, when reading a piece of music with musical notation at first sight, one might not visually regard the piece as a picture of music; for someone who has not yet studied musical theories, one will only regard the piece as a paper with bunch of uncanny symbols. Similarly, our alphabets will not be a picture of our speech as well (Phillip 24). Therefore, the similarity between a picture and what it pictures is not visual but formal––it does not refer to visual pictures that include various objects, like trees, books, cars, people, and cats, etc. The phrase does refer to, however, the picture of logic and its correlation with the world. In short, Wittgenstein’s definition of picture is based on the idea of language, and the language serves as a role of presenting a comprehensive philosophical picture of the world.
A picture, as it is defined in the Tractatus, has the following features: elements and states of affairs. In Tractatus, Wittgenstein mentioned that “To the objects correspond in the picture the elements of the picture”. What Wittgenstein means by this is the definition of an element is object. Objects are things in the world, such as a car, a tree and etc. In reality, one tends to think objects can be used to picture our world; however, a tree or a car, for example, cannot depict anything in the world, because they are meaningless for just having a name––it is not sufficient enough to constitute the world with a collection of these objects, but what is necessary is a knowledge of the combinations in which objects are related to one another (Phillip 23). Therefore, to give a proposition an ability to become a picture of the world, objects will need states of affairs which are statuses that hold objects together. The proposition “the book is lying on the table”, for example, is a picture because the proposition contains two features: the elements “book” and “table”, and a state of affairs “lying on”.
We learned from the picture theory that a proposition with elements and states of affairs can picture the world, but the proposition can be true or false according to its correspondence to the world. The proposition “the book is lying on the table” would be correctly depicting the world if and only if there is a book lying on the table which we can confirm in the reality––this is what Wittgenstein calls a true proposition (Phillips 25). If we cannot confirm the statement “the book is lying on the table”, then the proposition will still be a picture, but it is false. Therefore, the statement will not be a picture of the world. One aspect of the picture theory which Wittgenstein finds illuminating, which is we can see from a picture what situation it is depicting without knowing if it is true in the world. That being said, we can understand what a proposition is picturing, but we can still be uncertain whether the proposition is true or false. For instance, “unicorns exist in the world” is a proposition, and we know what a unicorn looks like, but we cannot confirm its existence in reality. This proposition is what Wittgenstein call a picture but not a picture of the world due to its correspondence to reality.
With a brief understanding of the Picture Theory in Wittgenstein’s Tractatus, we can now use it to approach whether Euclid’s Elements is a picture of the world.
Euclid’s Elements is one of the most influential works in the history of mathematics, and it has been used as a textbook to inspire people to think logically and deductively, from the time of its publication to the 21th century (Borceux). Euclid listed primitive and defined terms, postulates, and theorems in his book. Terms like “point”, “line”, and “straight lines” are primitive terms, which are introduced and explained but never precisely defined in Elements (Trudeau). For instance, Euclid defines a point as “which has no part” and a line “is breathless length”. These definitions are all interpretable for people, but they are not precisely defined and still having a sense of ambiguity. The defined terms, however, are more carefully stated. In definition 15, Euclid defined a circle as “a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within are equal to one another”. In comparison to primitive terms, the defined terms are slightly complex but descriptive. Based on the primitive and defined terms, postulates in Euclid’s Elements are geometrical statements that do not require extra proofs. Theorems, however, in Elements are statements which require proofs, and the process of proving will involve postulates, primitive and defined terms. The accomplishment of the Elements is to present these known results in a single, logically coherent framework, starting from an explicit system of axioms and proceeding via rigorous mathematical proofs (Borceux).
Parts in Euclid’s Elements, such as primitive and defined terms, postulates (except the fifth postulate), and theorems, have identical components––elements and states of affairs––with the picture as Wittgenstein described in Tractatus. Similar to Wittgenstein’s Picture Theory, terms, theorems and postulates in Euclid’s Elements are pictures. From the definitions about a point and a line, one can tell the elements (objects) “point” and “line” and their states of affairs: “has no part” and “is breathless length”. These components are exactly what a picture will need. Also, postulate 1––“it is possible to draw one and only one straight line from any point to any point” ––also beholds in the Picture Theory because it contains element (object) “straight line” and a state of affairs “draw…from”. Therefore, it is clear for me most parts Euclid’s Elements is a picture.
Even though it is verifiable that most parts Euclid’s Elements is a picture, Euclid’s Elements will not be a picture of the world. From previous comments about Picture Theory, one will know that people can see from a picture what situation it is depicting without knowing if the situation is actually true. The fifth postulate, namely the parallel postulate, in Euclid’s Elements has shown an uncertainty of infinity, which we cannot testify in reality. In Euclid’s Elements, the fifth postulate stated that:
“if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles”.
The fifth postulate is a complex picture because there are several objects “straight line”, “interior angles”, “right angles”, and etc., and states of affairs “falling”, “produced”, and etc., in between. Even though one might think the fifth postulate is a picture because of its fulfillment––containing elements and states of affairs––of being a picture, the fifth postulate is still not a picture of the world because the concept of infinity is unverifiable in the reality. For a picture to be a picture of the world, the picture will have to be verifiable in the reality. Frankly speaking, we cannot know, in the real world, where the infinite lines will intersect due to our limitation of measuring infinity. Therefore, in the context of Wittgenstein’s picture theory, the fifth postulate will be a picture; however, it is not a picture of the world speaking of infinity.
After examining the Picture Theory in Wittgenstein’s Tractatus Logico-Philosophicus and relating the theory’s core concept––picture is a model of reality” ––to Euclid’s Elements, I have reached the conclusion that Euclid’s Elements is a picture, but it is not a picture of the world because of the limitation of affirming the fifth postulate in reality. If this conclusion is valid, does it mean all the objects’ shapes, such as “circular”, “rectangular” that can be described in Euclidean geometry are inappropriate under Wittgenstein’s implication? The reason that causes the Euclidean geometry fail to become a picture of the world is the fifth postulate and our incompetence to verify it in our world. If one detaches the fifth postulate from the Euclidean geometry, then Euclidean Geometry will become a picture of the world. However, by doing so, the Euclidean geometry will no longer be “Euclidean Geometry”, and it will be called “Neutral Geometry” which is Euclidean Geometry without the fifth postulate. Regardless, examining whether Neutral Geometry will be a picture of the world in the context of Picture Theory will be a whole new topic.