Wittgenstein and Tufte on Thinking in 3d: 'escaping Flatland'

Before the philosopher Ludwig Wittgenstein became Bertrand Russell's protégé at Cambridge, his formal education was directed towards shaping the mind of a sophisticated research engineer. The engineer's ability to visualise inventions and solve design problems by creatively altering configurations of their elements calls for constructive, spatial, synthetic thinking. Training in engineering drawing builds visual thinking skills and teaches engineers how to represent visualised three-dimensional objects on a two-dimensional surface. Visualisation involves seeing projective relations in the mind's eye, which plays a critical role in the Bild theory of the Tractatus, Wittgenstein's early philosophical classic. He asked himself 'What is the ground of our-certainly well-founded-confidence that we shall be able to express any sense that we like in our two-dimensional script?' As Edward Tufte says, 'Even though we navigate daily through a perceptual world of three dimensions…the world displayed on our information displays is caught up in the two dimensionality of the endless flatlands of paper and video screens…Escaping this flatland is the essential task of envisioning information.' I will explore how Wittgenstein's Tractatus relates to Tufte's theories of envisioning information and could contribute to understanding the principles underlying three-dimensional visualisation. One time, we were discussing something – we must have been eleven or twelve at the time – and I said, 'But thinking is nothing but talking to yourself inside.' 'Oh Yeah?' Bernie said. 'Do you know the crazy shape of a crankshaft in a car?' 'Yeah, what of it?' 'Good. Now tell me: how did you describe it when you were talking to yourself?' So I learned from Bernie that thoughts can be visual as well as verbal.


INTRODUCTION
Ludwig Wittgenstein presents his famous picture (Bild) theory of language in the Tractatus Logico-Philosophicus. Thinking it through in his notebooks, he asked, 'What is the ground of our---certainly well-founded---confidence that we shall be able to express any sense that we like in our twodimensional script?' (Wittgenstein 1961).As Edward Tufte says In Envisioning Information, 'Even though we navigate daily through a perceptual world of three dimensions…the world displayed on our information displays is caught up in the two dimensionality of the endless flatlands of paper and video screens…Escaping this flatland is the essential task of envisioning information' (Tufte 1990).In the Tractatus, Wittgenstein develops a law of projection to translate between states of affairs and their representations, understanding his Bilder as standing in a 'projective relation' to reality.His method of projection calls for the visual thinking of the engineer and the descriptive geometer.
Visualisation involves seeing projective relations in the mind's eye, and this kind of visual thinking and representation plays a critical role in the Bild theory of the Tractatus.Could principles developed in the Bild theory help us understand visualisation in ways that could contribute to the development of professional disciplines in which visual thinking and representations are central?In this paper, I will explore how principles laid out in Wittgenstein's philosophy relate to Edward Tufte's work on visualising information.I argue that Tufte's visual displays can be understood as exemplars of Wittgenstein's propositional signs, functioning according to principles laid out in the logic of depiction in the Tractatus.To frame the argument, I will first give an account of Wittgenstein's background, and then I will present the Bild Theory of language.

RUSSELL'S GERMAN ENGINEER
Wittgenstein was an iconic philosophical figure, perhaps most famously as a dominating presence in philosophy at Cambridge University.But, in 1911, when he showed up unexpectedly in Bertrand Russell's rooms in Cambridge, Russell did not know what to make of him.Very soon, he felt Wittgenstein had learned all that he could teach him and gone further.In the Cambridge community, he was Russell's designated philosophical heir.
Wittgenstein's prior education had been technical and scientific, however.For the youngest son of one of the wealthiest and most cultured families in Vienna, the prestige of an elite gymnasium, then university, would have been normal.His father, Karl Wittgenstein, had financed Olbrich's Secession building, paid for Klimt's famous university mural, and had the young Casals and Brahms perform in his magnificent palais.The family's cultural life was deeply felt and intensely lived, particularly in their devotion to music.For generations, Wittgensteins had patronised important musicians and performed music to exacting standards themselves.But Ludwig had shown an aptitude for the practical, building a small sewing machine when he was only ten.His father, a self-made industrialist trained as an engineer, supported his son's practical bent.
Thus, Ludwig attended an Oberrealschule, a technical high school, and received his engineering certification from the Technische-Hochschule in Berlin, then the MIT of Europe.Engineering training there focused on mathematical drawing, hands-on drafting, and experimental experience.Wittgenstein was trained to think like a design engineer.
1 Brian McGuiness wrote He liked to think with the machine -to understand every detail of its functioning -and this accounts both for his interest in the older and more perspicuous types of mechanism . . .and for his success in repairing mechanisms that had gone wrong.Always it was achieved by the most careful observation of the machine from every side and deep and concentrated reflection until he had internalised the principle of the machine.It was thus that he had observed the distrustful sempstress in the Alleegasse and it was thus that he would examine the searchlight or boiler or whatever it was, sometimes rather to the irritation of onlookers, only intervening . . .when the course to be followed was absolutely clear to him (McGuinness 1988).
Wittgenstein retained his love of machines, as well as a practical engineering ability in experimental research, throughout his life.Professor G. H. von Wright, Wittgenstein's close friend and literary executor, told Allan Janik that it was important to remember 'firstly, that he was a Viennese and, secondly, that he was an engineer with a thorough knowledge of physics,' and this important insight was at the heart of their brilliant account of Wittgenstein's life and philosophical work as rooted in his Central European heritage, Wittgenstein's Vienna (Janik and Toulmin 1973).
After this first-class formal engineering education, he spent four years pursuing aeronautical engineering research at the University of Manchester, designing and patenting an innovative propeller.
2 Experimental data used in research and design in this area was mostly obtained from wind tunnel experiments with scale models, but such data presented serious difficulties for designers due to problems with scale effects.Did the experimental situation in the wind tunnel faithfully reflect the flying conditions of the actual airplane?How do you translate from the scale models of the engineer to the real life, full-scale engineering problem?Accurate translation between scale model experiments and real life situations depended on dimensional analysis, and propeller research was thus highly mathematical.Shifting his interests to the foundations of mathematics, he read The Principles of Mathematics, encountering the challenge to solve Russell's paradox.He then turned up in Lord Russell's rooms, too nervous to speak English, and Russell's 'German engineer' soon became the leading light in Cambridge philosophy.
What kind of changes would have taken place in Wittgenstein's mind as he learned to think like a design engineer?As Eugene Ferguson writes, Many features and qualities of the objects that a technologist thinks about cannot be reduced to unambiguous verbal descriptions; therefore, they are dealt with in the mind by a visual, nonverbal process.The mind's eye is a welldeveloped organ that not only reviews the contents of a visual memory but also forms such new or modified images as the mind's thoughts require.As one thinks about a machine, reasoning through successive steps in a dynamic process, one can turn it over in one's mind.The engineering designer, who brings elements together in new combinations, is able to assemble and manipulate in his or her mind devices that as yet do not exist (Ferguson1992).The engineer's ability to visualise an invention, to solve design problems by creatively altering configurations of its elements, calls for a kind of constructive visualisation that is also important for scientific practice.As Brook Hindle argues, 'synthetic-spatial thinking is, of course, involved in most intellectual activity including science, but in technology it has to be central' (Hindle 1981).Engineering drawings are representations of these thoughts, communicating the engineer's ideas from the level of tentative creativity in thinking sketches to the precision of technical blueprints for the construction of a completed mechanism.Wittgenstein brought this mindset to the Tractatus.

THE BILD THEORY
The Tractatus is an unusual, brilliant, and profoundly difficult little book.The major work of the early Wittgenstein, in contrast to the later Wittgenstein of the Philosophical Investigations, it is the only book he published during his lifetime.It was written under the duress of life on the Eastern Front in WWI, and it was a deeply personal statement, the product of years of intense reflection.
A profound influence on European philosophy between the wars, it has been a continuing source of philosophical debate, controversy, and insight to this day.
The structure is bewildering at first sight.It consists of carefully arranged numbered propositions, and it has been aptly described as a series of aphorisms.
The decimal numbers assigned to the individual propositions indicate the logical importance of the propositions, the stress laid on them in my exposition.The propositions n. 1, n.2, n.3, are comments on propositions no. n…(Wittgenstein 20101, n.2, n.3, are comments on propositions no. n…(Wittgenstein /1922) ) It is elegant, spare, and impossible to understand without careful study.Ordinary words are used in ways that will perplex intelligent readers outside their professional disciplinary ambit.It is not even a case of knowing their technical meanings in the context of the field.His use of words, development of propositions, and presentation of their connections repay careful study as much as the information in any dense Tuftean visual display.Indeed, I would argue that in many ways, the Tractatus is a superb, dense, Tuftean visual explanation.It is certainly 'self-exemplifying' in the manner of Tufte's books, as the object itself also embodies the ideas written about.As it is impossible to deal with the depth and richness of the work of either of these thinkers adequately in this brief space, I will consider some of the important principles of visual, analytical thinking developed in their work, showing critical connections between them.In-depth consideration must await space for a more complete, detailed analysis.
The first set of propositions in the Tractatus presents an ontology that describes the structures of the world as we know it according to his theory.He asserts that 'the world is all that is the case'(1), specifying that 'the world is the totality of facts, not of things'(1.1).His definition of states of affairs specifies that 'What is the case -a fact -is the existence of states of affairs' (2), and 'a state of affairs (a state of things) is a combination of objects (things)'(2.01).Thus, it is not objects standing alone, unrelated to each other, that constitute the world, but objects as they exist in combination, related to one another in states of affairs.Existing states of affairs make up what is the case, or the world as we know it.
What makes it possible for things to combine in states of affairs?'The possibility of its occurring in states of affairs is the form of an object'(2.0124).2.033 Form is the possibility of structure.

2.034
The structure of a fact consists of structures of states of affairs.
The form created by the configuration of objects combined with one another can be the structure of a state of affairs.To determine what actual structure exists in the world, one must check to see which of all the possible combinations of objects did, in fact, occur.
Returning to Wittgenstein's engineering training, using models of working parts of machines to teach machine construction to engineers was standard.Polhem's 'mechanical alphabet' represented mechanical movements necessary for the design of complex machines.The five 'powers' of Hero of Alexandria: the lever, the wedge, the screw, the pulley and the winch, were the vowels of his mechanical alphabet.No 'machine limb (could) be put into motion without being dependent on one of these ' (Ferguson 1992) This alphabet of objects, whose configurations produced various working inventions, gave engineers a tacit understanding of the component parts of machines and the principles underlying the forms of machines, allowing them to visualise these elements combined in new configurations.

Figure 1a: Polhem's Mechanical Alphabet
Figure 1b: Reuleaux of Machine Element (Hamilton 2001, 71 and 72 Wittgenstein's coursework used models elements intensively.A famous and widely used set of kinematic models was actually created by Professor Franz Reuleaux of the Hochschule in Berlin.Such machine parts have also been beautifully represented historically in engineering drawings, perhaps most famously in the engineering notebooks of Leonardo DaVinci. In terms of the Bild theory, I ask you to imagine, to visualise in your mind's eye, individual machine parts.See what combinatorial possibilities exist between them.Obviously, the forms of the parts determine how they can stand in rel another in a machine that will work.Obviously, again, there are many possible, yet predetermined (as only certain combinations are possible), structures that can exist.The forms of the (objects) determine the possibilities of differe

Figures 2a and 2b: Leonardo da Vinci's Engineering
Drawings (Hamilton 2001, 74 and 81) structures of machines (possible states of affairs).The actual combination of parts (configuration of objects) realised depends on which of those possibilities was realised in the construction of the machine.'If I know an object, I also know all its possible occurrences in states of affairs.(Every one of these possibilities must be part of the nature of the object)' (2.0123).'If all objects then…all possible states of affairs are also given' (2.0124).
Engineers can 'see' the structure of a projected machine; they actually think through how principles will work in the mind's eye in a n

1). (Wir machen uns Bilder der Tatsache). David Stern warns
Wittgenstein used the German word 'Bild' to talk about the model, a term usually translated as 'picture'; as a result, the theory of meaning it inspired is generally known as the picture theory.While both words cover such things as images, film frames, drawings, and paintings, the idea of a three-dimensional model is more readily conveyed by the German 'B English 'picture…it is important not to be misled: the theory involves generalizing from what models, pictures, and the like are supposed to have in common, and treats two pictures as just one kind of Bild 36).
'A picture presents…the existence and non existence of states of affairs model of reality'(2.12), and ole of spatial thinking, seeing the form or configuration of a structure in space, is crucial for understanding the possibilities of the structure being designed.Analyzing Morse's invention of the telegraph, Hindle developed this th remained a quality of mind that permitted him to manipulate mental images dimensional telegraph components as well as complete telegraphic systems, altering them at will and projecting various possibilities for change and development.Although he had used this mode of thinking in his art, his telegraph in no way depended upon his art.Conspicuous success in each, however, absolutely required conspicuous ability in spatial , 'painting and her sister arts of design rely upon form displayed in space' Leonardo da Vinci is perhaps the example of the combination of artistic and David Pye feels 'are really different expressions of one potentiality' (Pye orm displayed in space' is an excellent characterization of propositional structure Seeing machine elements in various possible configurations would have been second nature to Wittgenstein given his engineering training.Viewing theory from this perspective is unusually that utterly simple thing, which we have to formulate here, is not a likeness of the truth, but the truth….(Our problems are not abstract, but perhaps the most concrete that there Moving to what Tufte characterises in Galileo's eye of the mind' envisions,' 'we picture facts to Wir machen uns Bilder der .David Stern warns Wittgenstein used the German word 'Bild' to talk about the model, a term usually translated as '; as a result, the theory of meaning it inspired is generally known as the picture theory.While both words cover such things as images, film frames, drawings, and paintings, dimensional model is more readily conveyed by the German 'Bild' than the English 'picture…it is important not to be misled: the theory involves generalizing from what models, pictures, and the like are supposed to have in common, and treats two-dimensional ctures as just one kind of Bild' (Stern 1995, 35-…the existence and nonf states of affairs' (2.11).'A picture is a , and 'in a picture objects have the elements of the picture correspondi them' (2.13).'In a picture the elements of the picture are the representatives of objects 'What constitutes a picture is that its elements are related to one another in a determinate way Most importantly, thinking of a state of affairs means modelling it to yourself.'A Bild reality as we imagine it (4.01).'A logical world is a thought.'We communicate our thoughts.'We use the perceptible sign of a proposition (spoken or written, etc.) as a projection of a possible situation.The method of projection is to think of the sense of the proposition proposition is a propositional sign in its projective relation to the world' (3.12).It has that projective relation because of our thinking it, modelling it to ourselves.'A propositional sign applied thought out is a thought'(3.5),and 'a proposition with a sense'(4).'The total propositions is language'(4.0001).In a picture the elements of the tives of objects' (2.131).hat constitutes a picture is that its elements are her in a determinate way' (2.14).
Most importantly, thinking of a state of affairs Bild is a model of A logical Bild of the We communicate our thoughts.We use the perceptible sign of a proposition (spoken or written, etc.) as a projection of a method of projection is to sense of the proposition' (3.11).'A proposition is a propositional sign in its projective It has that projective g it, modelling it to A propositional sign applied and a thought is a The totality of The general concept of the proposition carries with it a quite general concept of the co ordination of proposition and situation: The solution to all my questions must be simple.
In the proposition a world is as it were put together experimentally (As when in the law court in Paris a motor represented by means of dolls, etc. This must yield the nature of truth straight away (if I were not blind).
Let us think of hieroglyphic writing in which each word is a representation of what it stands for.Let us think also of the fact that of situations can be right If the right-hand figure in represents the man A, and the left stands for the man B, then the whole might assert, e.g.: 'A is fencing with B proposition in picture writing can be true and false.It has a sense independent of its truth or falsehood.It must be possible to demonstrate everything essential by considering this case It can be said that, while we are not certain of being able to turn all situations into pictures on paper, still we are certain that we can portray all logical properties of situations in a two dimensional script.This is still very much on the surface, on good ground… It can be said that in our picture the right figure is a representation of something and also the left-hand one, but even if this were not the case, their relative position could be a representation of something.

Figure 3: Fencers graphic
In a proposition, we combine names (representing objects) to represent concretely states of affairs.To put propositions together experimentally, we construct their sense and test them against reality to s They answer yes or no, true or false, in order to represent reality.This is their truth that captures a possible logical form that exists, that is realised as a structure of a sta has represented something true about reality.reaches right out to it' (2.1511) . . .reality like a measure' (2.1512)He thought this through in the Notebooks as The general concept of the proposition carries with it a quite general concept of the coordination of proposition and situation: The solution to all my questions must be extremely In the proposition a world is as it were put er experimentally (As when in the lawcourt in Paris a motor-car accident is esented by means of dolls, etc.) [Cf. 4.031.]This The proposition in picture writing can be true and false.It has a sense independent of its truth or falsehood.It must be possible to demonstrate everything essential by considering this case.
It can be said that, while we are not certain of being able to turn all situations into pictures on paper, still we are certain that we can portray all properties of situations in a two-This is still very much on the surface, but we are It can be said that in our picture the right-hand figure is a representation of something and also even if this were not the case, their relative position could be a representation of something.
(Namely a

Fencers graphic (Wittgenstein 1961)
In a proposition, we combine names (representing represent concretely three-dimensional To put propositions together experimentally, we construct their sense and then test them against reality to see if we have a match.
answer yes or no, true or false, in order to This is their truth-value, and a Bild possible logical form that exists, structure of a state of affairs, has represented something true about reality.'It (2.1511) . . .'it is laid against (2.1512).design.Serious visual displays give that level of insight to careful consumers.Above all, they help us think through the data to understand the represented state of affairs, for evidence displays is to assist the thinking o producer and consumer alike.powerful metaphor is important in Tufte's thought: 'The metaphor for evidence prese analytical thinking' (Tufte 2006 analytical thinking are universal, and t logic of all of our languages.

Wittgenstein would have liked Tuf
The world is complex, dynamic, multi dimensional; the paper is static, flat.How are we to represent the rich visual world of experience and measurement on mere flatland?information…is to work at the intersection of image, word, number, art…and the standards of quality are those derived from visual principles that tell us how to put the right mark in the right place (Tufte 1983).
Putting the right mark in the right place, only those marks necessary to communicate data, is critical to the quality and truthfulness displays.The marks are the elements of evidence displays as propositional signs.relationships among those marks as accurately as possible constitutes the truth displays.
This insight informs Tufte's ratio.Data-ink is the non graphic, the non-redundant ink arranged in response to variation in the numbers represented.through the data to understand the represented state of affairs, for 'the point of evidence displays is to assist the thinking of oducer and consumer alike.'Thus, another powerful metaphor is important in Tufte's thought: The metaphor for evidence presentations is (Tufte 2006).The rules of ytical thinking are universal, and they cover the of our languages.
Wittgenstein would have liked Tufte's questions.
The world is complex, dynamic, multidimensional; the paper is static, flat.How are we to represent the rich visual world of experience d measurement on mere flatland?To envision mation…is to work at the intersection of art…and the standards of quality are those derived from visual principles that tell us how to put the right mark in the right Putting the right mark in the right place, and using marks necessary to communicate data, and truthfulness of visual marks are the elements of evidence displays as propositional signs.Presenting the relationships among those marks as accurately as ossible constitutes the truth-value of data informs Tufte's principle of data-ink is the non-erasable core of a redundant ink arranged in n the numbers represented.
ink ratio (Tufte 1983) The goal is to 'maximise the data-ink ratio, within reason' and 'erase non-data ink, within reason (Tufte 1983).Denser and more perspicuous information enhances the truth-value display.Observing this principle calls for the inclusion of more high-resolution, significant data and less chartjunk masquerading as data critical to the quality of the visual evidence.Sins against this and related such as distorting data, presenting it ways, cherry picking data, communication with non-representing diminish the truth-value of data displays Science plays a prominent role in both of t theories.It is the paradigm for both propositions and Tufte's evidence presentations.Tufte's gold standard for data display is the best scientific journals, particularly Nature He would say that scientists and engineers deal with facts, and no one gets their own fa his best sources of elegantly crafted visual arguments is the work of Galileo.embedded arguments concerning sun spots, Jupiter's satellites, mountains and craters on the moon etc., in his drawings.According to Tufte, Tufte removes the arrows in the analysis to make a point information display.Scattered, unconnected nouns are left, and the sense of the original evidence presentation is lost, along with the significance of the flow chart.As Wittgenstein says affairs can be described but not named.(Names resemble points; propositions resemble arrows, they have sense.)'(3.144).(Tufte 2006, 66) As the conclusion to this discussion of links and arrows, Tufte asks how Galileo would have handled linking lines.'This 1610 drawing shows carefully articulated links, as detailed annotation describ and delineates most links.We should do as well.(Tufte 2006)  The virtues or vices of such evidence can change the world for the better or worse, for the results of failure can be serious.The truth can be liberating, increasing our understanding and ability to navigate in the world, dishonest, or misleading evidence displays ca lead to serious errors of judgment on the part of those making decisions.Failure of visual representations can be tragic in practice, critique of the thin, misleading data PowerPoint presentations that crippled communication during the Columbia space shuttle tragedy shows.Underlining the realism of mathematical drawings in 'Wittgenstein and the Mind's Eye,' I wrote What is important for the Tractatus… concrete character of the manner in which points on the surfaces of figures…are connected to real world figures.When the subject of these drawings is a sophisticated invention that can fail, or a working bridge, the realism of these elegant pictures is immediately obvious.A faulty invention won't work.Wittgenstein once drew sketches for a machine based on a principle that had occurred to his father when he was a young boy.The principle was faulty, and the mach would not have moved.When force is added, as in graphical statics, it is even more concrete.Mistakes might mean that someday a bridge could collapse with your car on it 2001).This principle is even more poignant his critique of PowerPoint and the Columbia tragedy.The failure to communicate effectively contributed flawed decisions and the death of the Columbia astronauts.As Tufte argues, The wordlike qualities of sparklines create the writing with data graphics.'Elements of data displays, such as the dataword where upward whiskers represent wins and downward elements of the outcomes of the s sequence for the ) vidence presentations can change the world for the better or worse, for s of failure can be serious.The truth can increasing our understanding and world, but flawed, or misleading evidence displays can lead to serious errors of judgment on the part of ailure of visual can be tragic in practice, as Tufte's thin, misleading data displays in presentations that crippled Columbia space shuttle the realism of Wittgenstein and the ctatus…is the concrete character of the manner in which points are connected to When the subject of these drawings is a sophisticated invention that can fail, or a working bridge, the realism of these elegant pictures is immediately obvious.A faulty invention won't work.Wittgenstein once drew chine based on a principle that had occurred to his father when he was a young boy.The principle was faulty, and the machine is added, as graphical statics, it is even more concrete.Mistakes might mean that someday a bridge could collapse with your car on it (Hamilton in Tufte, given his critique of PowerPoint and the space shuttle dy.The failure to visualise and contributed to tragically flawed decisions and the death of the Columbia There are right ways and wrong ways to show data, there are displays that rev displays that do not.And, if the matter is an important one, then getting the displays of evidence right or wrong can possibly have momentous consequences' (Tufte 1997 An example of a flawed evidence presentation/ needing rethinking comes from the Wittgenstein's thought.The physicist Heinrich Hertz's The Principles of Mechanics important influence on Wittgenstein's thinking particular quote from it philosophically throughout we have accumulated aroun and 'electricity' more relations than can be reconciled with themselves . . .We have an obscure feeling of this and want to have things cleared up.Our confused wish finds expression in the confused question as to the nature of force and electricity….It is not by finding more and fresh relations it can be answered; but by removing the contradictions between already known (Hertz 1899 Hertz was troubled by the obscurity surrounding the notion of force in the canonical axiomatization mechanics.More relations had accumulated around the notion of force than could be reconciled.In Tufte's terms, there was too much confusing non-representing data ink.Wittgenstein could quote this famous passage by heart his entire life: these painful contradictions have been removed…our minds, no longer vexed will cease t ask illegitimate questions Principles, Hertz construct Darstellungen of mechanics, eliminating flaws in the representations of As the foundation of mechanics describes reality, however indirectly.Wittgenstein expresses this as follows: 6.341…Mechanics determines the form of the description of the world by saying: all propositions in a descriptio be capable of being got in a given way from a number of given propositions mechanics.In this way it supplies the stones for building up natural science and says: Whatever building you want to erect you must construct it somehow with these and only these stones.
He uses a mesh analogy to describe mapping propositions of mechanics to reality, mirroring Tufte's discussion of the effectiveness of grid maps.

6.341
Newtonian mechanics, for example, imposes a unified form description of the world.Let us imagine a white surface with irregular black spots on it.We then say that whatever kind of picture these make, I There are right ways and wrong ways to show data, there are displays that reveal the truth and And, if the matter is an important one, then getting the displays of evidence right or wrong can possibly have ntous consequences' (Tufte 1997).
evidence presentation/Bild needing rethinking comes from the heart of Wittgenstein's thought.The physicist Heinrich Principles of Mechanics was an important influence on Wittgenstein's thinking.One quote from it resonated for him throughout his life.
we have accumulated around the notions 'force' more relations than can be reconciled with themselves . . .We have an obscure feeling of this and want to have things cleared up.Our confused wish finds expression in the confused question as to the nature of force and electricity….It is not by finding out more and fresh relations it can be answered; but by removing the contradictions between those already known (Hertz 1899).
Hertz was troubled by the obscurity surrounding the notion of force in the canonical axiomatizations of mechanics.More relations had accumulated around the notion of force than could be reconciled.In Tufte's terms, there was too much confusing ng data ink.Wittgenstein could quote by heart his entire life: 'When painful contradictions have been removed…our minds, no longer vexed will cease to imate questions' (Hertz 1899).In the constructs his own bildliche of mechanics, eliminating those flaws in the representations of mechanics. of the physical sciences, describes reality, however indirectly.Wittgenstein expresses this as follows: Mechanics determines the form of the description of the world by saying: all propositions in a description of the world must be capable of being got in a given way from a number of given propositions---the axioms of mechanics.In this way it supplies the stones for building up natural science and says: Whatever building you want to erect you must construct it somehow with these and only these stones.s a mesh analogy to describe mapping the propositions of mechanics to reality, mirroring of the effectiveness of grid Newtonian mechanics, for example, imposes a unified form on the description of the world.Let us imagine a white surface with irregular black spots on it.We then say that whatever kind of picture these make, I can always approximate as closely as I wish to the description of it by covering the surface with a sufficiently fine square mesh, and then saying of square whether it is white or black.In this way I shall have imposed a unified form on the description of the surface.The form is optional, since I could have achieved the same result by using a net with a triangular or hexagonal mesh.Possibly the use of a triangular mesh would have made the description simpler: that is to say that we might be able to describe the surface more accurately with a coarse triangular mesh than with a fine square mesh (or conversely), and so on.The different nets correspond to different systems for the world… Analyzing grid-square maps and mesh maps, Tufte notes the improvement in the quality of information conveyed by these more fine-grained, perspicuous maps in contrast to conventional blot maps.This is shown by the following two maps representing relative densities of the Japanese population.

Figure 13a:
Grid-square map of population census results in Japan (Tufte 1990) Figure 13b: of population census results In Japan (Tufte 1990, 41) These examples exemplify the principles of seeing and knowing with which we are concerned.reflects that Science and art have in common the seeing, the wide-eyed observing that generates empirical information.Beautiful Evi about how seeing turns into showing, how empirical observations turn into e and evidence.The book identifies excellent and effective methods for showing evidence 2006).
These principles are universal, dealing with the logic of all our languages, for evidence display are derived from universal principles of analytical thinking.'The quality of our visual thinking and representation truthfulness and integrity of our understanding, both of these thinkers are deeply concerned wit the integrity of thinking and representing Making an evidence presentation is a moral act as well as an intellectual activity.To maintain standards of quality, relevance, and integrity for evidence, consumers of presentations should insist that presenters should be held Wittgenstein and Tufte on Thinking in 3D: 'Escaping Flatland' Kelly Hamilton 115 can always approximate as closely as I wish to by covering the surface with fficiently fine square mesh, and then saying of every square whether it is white or black.In this way I shall have imposed a unified form on the description of the surface.The form is optional, since I could have achieved the same ith a triangular or hexagonal mesh.Possibly the use of a triangular mesh would have made the description simpler: that is to say that we might be able to describe the surface more accurately with a coarse triangular mesh than with a fine square mesh (or onversely), and so on.The different nets correspond to different systems for describing mesh maps, Tufte notes the improvement in the quality of information rained, perspicuous contrast to conventional blot maps.This is the following two maps representing the ese population.
map of population census 13b: Mesh map of population census results In Japan (Tufte 1990, the principles of seeing and knowing with which we are concerned.Tufte Science and art have in common the intense serving that generates Beautiful Evidence is about how seeing turns into showing, how empirical observations turn into explanations and evidence.The book identifies excellent and effective methods for showing evidence (Tufte l, dealing with the , for 'principles of evidence display are derived from universal he quality of our visual thinking and representations affects the of our understanding, and concerned with the integrity of thinking and representing.
Making an evidence presentation is a moral act as well as an intellectual activity.To maintain vance, and integrity for evidence, consumers of presentations should insist that presenters should be held intellectually and ethically responsible for wha they show and tell (Tufte 2006 Wittgenstein famously described an 'ethical deed.' Janik and Toulmin argue that the Tractatus was part of the Krausian critique of language, indeed that Wittgenstein the crisis of integrity of representation in fin-de siècle have deep moral implications. If the Tractatus is not understood as a whole, point will be lost.This is Wittgenstein's view famous passage in his letter to Ludwig Ficker.The book's point is an ethical one.I once meant to include in the preface a sentence wh in fact there now, but which I will write out for you here, because it will work for you.What I meant to write, then, was this: My work consists of two parts: the one presented here plus all that I have not written.And it is precisely this second part that is the important one.My book draws limits to the sphere of the ethical from the inside as it were, and I am convinced that this is the ONLY rigorous way of drawing those limits ( Toulmin 1973) The important part is what was not said.T makes the Tractatus self-exemplifying, as the book embodies the ideas written about.indeed, things that cannot be make themselves manifest be shown.
Given the accumulated force of these and arguments, and many more could be given, would argue that Tufte's propositional signs in Wittgenstein's sense would Wittgenstein think of a proposition sign as a visual display, say a chart or a diagr perhaps best to look to the question.One of Wittgenstein's frequently used contributions to logic table, first presented in the

Figure 14: Truth table as a propo
There are quite a few tables and diagrams presented as propositions in the Wittgenstein immediately uses the 4.462 to analyze other representat about the material, and Frege's 'logical marks' is unnecessary more non data ink.
intellectually and ethically responsible for what w and tell (Tufte 2006).
famously described the Tractatus as Janik and Toulmin argue that the was part of the Krausian critique of language, indeed that Wittgenstein's work reflects integrity of all our means of de siècle Vienna.These ideas implications.
understood as a whole, its t will be lost.This is Wittgenstein's view in the famous passage in his letter to Ludwig Ficker.The book's point is an ethical one.I once meant to include in the preface a sentence which is not in fact there now, but which I will write out for perhaps be a key to the work for you.What I meant to write, then, was this: My work consists of two parts: the one presented here plus all that I have not written.
it is precisely this second part that is the important one.My book draws limits to the sphere of the ethical from the inside as it were, and I am convinced that this is the ONLY rigorous way of drawing those limits (Janik and s what was not said.This exemplifying, as the book s the ideas written about.''There are, indeed, things that cannot be put into words.They selves manifest'(6.

Truth table as a propositional sign
There are quite a few tables and diagrams presented as propositions in the Tractatus.immediately uses the visual display of representations for thinking to explain why one of s unnecessary, eliminating The following is a gallery of Wittgenstein' displays as propositions in the Tractatus

BEAUTIFUL EXPLANATIONS
In conclusion, the argument presented from 4.01 to 4.015 should now fall into place.proposition is not a blend of words theme in music is not a blend of notes Identifying propositions with musical themes is significant, for this is a Wittgenstein, whose relationship to music was integral to his music to display principles at the heart of the logic of depiction.
4.01 A proposition is a picture of reality.proposition is a model of reality as we 4.011 At first sight a proposition---one set out on the printed page, for example----does not seem to be a picture of the reality with which it is concerned.But neither do written notes seem at first to be a picture of a piece of music, nor o phonetic notation (the alphabet) to be a picture of our speech.And yet these sign languages prove to be pictures, even in the ordinary sense of what they 4.012 It is obvious that a proposition of the form 'aRb' strikes us as a picture.In this case the sign is obviously a likeness of what is signified… 4.014 A gramophone record, the musical idea, the written notes, and the sound waves, all stand to one another in the same internal relation of depicting that holds between language and the world.They are all constructed according to a common logical pattern.
4.0141There is a general rule by means of which the musician can obtain the symphony from the score, and which makes it possible to derive the symphony from the groove on the gramophone record, and, using the first rule, to derive the score again.That is what the inner similarity between these things which seem to be constructed in such entirely different ways.And that rule is the law of projection which projects the symphony into the language of Wittgenstein and Tufte on Thinking in 3D: 'Escaping Flatland' Kelly Hamilton

117
(5.5423) In conclusion, the argument presented from 4.01 to 4.015 should now fall into place.For, 'A proposition is not a blend of words----(Just as a theme in music is not a blend of notes)' (3.141).
th musical themes is cant, for this is a Wittgenstein, whose relationship to music was integral to his life, using principles at the heart of the logic A proposition is a picture of reality.A proposition is a model of reality as we imagine it.one set out on does not seem to be a picture of the reality with which it is concerned.But neither do written notes seem at first to be a picture of a piece of music, nor our phonetic notation (the alphabet) to be a picture of our speech.And yet these sign languages prove to be pictures, even in the ordinary sense It is obvious that a proposition of the form his case the sign is signified… A gramophone record, the musical idea, the written notes, and the sound waves, all stand to one another in the same internal relation of depicting that holds between rld.They are all to a common logical 4.0141There is a general rule by means of which the musician can obtain the symphony from the score, and which makes it possible to derive the symphony from the groove on the record, and, using the first rule, to That is what constitutes things which seem to be constructed in such entirely different ways.And that rule is the law of projection which y into the language of musical notation.It is the rule for translating language into the language of gramophone records.

4.015
The possibility of all imagery, of all our pictorial modes of projection, is contained in the logic of depiction.
The law of projection enables us to translate from the gramophone record, to the musical idea, to the written notes, to the sound w because what is projected is the logical form, through the internal relation of depicting.beautiful instantiation of this principle I have encountered was in Tufte's notes of Chopin's 'Berceuse as the sound waves were dynam a diagram on the screen.(Tufte 2006, 46) widely used set tic models was actually created by Professor Franz Reuleaux of the Technische Such machine parts have also been beautifully represented historically in s, perhaps most famously in Leonardo DaVinci.you to imagine, to individual machine ee what combinatorial possibilities exist between them.Obviously, the forms of the parts determine how they can stand in relation to one another in a machine that will work.Obviously, again, there are many possible, yet predetermined (as only certain combinations are possible), structures that can exist.The forms of the parts determine the possibilities of different Vinci's Engineering structures of machines (possible states of affairs).The actual combination of parts (configuration of depends on which of those in the construction of the I know an object, I also know all its possible occurrences in states of affairs.(Every one of these possibilities must be part of the nature f all objects are given, ates of affairs are also the structure of a projected actually think through how the mind's eye in a nonverbal fashion.The role of spatial thinking, seeing the form or configuration of a structure in space, is crucial for understanding the possibilities of the structure being designed.invention of the telegraph, thought: His great strength remained a quality of mind that permitted him to manipulate mental images of three-dimensional telegraph components as well as complete telegraphic systems, altering them at will and projecting various possibilities for change and development.Although used this mode of thinking in his art, his telegraph in no way depended upon his art.Conspicuous success in each, however, absolutely required conspicuous ability in spatial thinking (Hindle 1981).As Morse himself wrote, arts of design rely upon form di (Hindle 1981).Leonardo d best example of the combination of artistic and practical talents that David different expressions of one potentiality 1978).Certainly, 'form displayed in excellent characterization of propositional structure in the Tractatus.Seeing machine elements in various possible configurations would have been second nature to Wittgenstein given his engineering training.Viewing the Bild theory from this perspective is unusually concrete, but 'that utterly simple thing, which we have to formulate here, is not a likeness of the truth, but the truth….(Our problems are not abstract, but perhaps the are)'(5.5563).Moving to what Tufte character words as 'what 'the eye of the mind' Wittgenstein asserts ' ourselves'(2. Propositions are put together experiment'(4.031).Instead of 'this proposition has such and such a sense, we can say, this proposition represents such and such a situation.'(4.03311)'One name stands for one thing, another for another thing, and they are combined with one another.Names as signs for concrete objects, objects such as tables, chairs, and books standing in relation to one another, for the essence of a propositional sign is very clearly seen if we imagine one composed of spatial objects (such as tables chairs and books) instead of written signs.Then the spatial arrangement of the things will represent the sense of the proposition (3.1431).Wittgenstein's insight into the Bild theory famously came in the context of his reflections on the use of a scale model of an accident in a Parisian law court.Professor Von Wright recounted the story as Wittgenstein described it to him.It was in the autumn of 1914 on the Eastern Front.Wittgenstein was reading in a magazine about a lawsuit in Paris concerning an automobile accident.At the trial, a miniature model of the accident was presented before the court.The model here served as a proposition, that is as a description of a possible state of affairs.It had this function owing to a correspondence between the parts of the model (the miniature houses, cars, people) and things (houses, cars, people) in reality.It now occurred to Wittgenstein that one might reverse the analogy and say that a proposition serves as a model or picture, by virtue of a similar correspondence between its parts and t (Von Wright 1984).Wittgenstein and Tufte on Thinking in 3D: 'Escaping Flatland' Kelly Hamilton 111 ments of the picture corresponding to together 'by way of this proposition has we can say, this ents such and such a One name stands for one thing, another for another thing, and they are combined with one another.Names as signs for concrete objects, objects such as tables, chairs, standing in relation to one another, for the essence of a propositional sign is very clearly seen if we imagine one composed of spatial objects (such as tables chairs and books) instead of written signs.Then the spatial arrangement of the things esent the sense of the proposition' theory famously came in the context of his reflections on the use of a scale model of an accident in a Parisian law court.Professor Von Wright recounted the story as 1914 on the Eastern Front.Wittgenstein was reading in a magazine about a lawsuit in Paris concerning an automobile accident.At the trial, a miniature model of the accident was presented before the rt.The model here served as a proposition, that is as a description of a possible state of affairs.It had this function owing to a correspondence between the parts of the model (the miniature houses, cars, people) and things lity.It now occurred to Wittgenstein that one might reverse the analogy and say that a proposition serves as a model or picture, by virtue of a similar correspondence between its parts and the world He thought this through in the Notebo follows: must yield the nature of truth straight away Let us think of hieroglyphic writing in which each is a representation of what it stands for.Let us think also of the fact that actual pictures right and wrong.[Cf.4.016.]hand figure in this picture represents the man A, and the left-hand one B, then the whole might A is fencing with B'.

Figure 4 :
Figure 4: 'When a Cyberstar Is Born' the first image in Beautiful Evidence(Tufte 2006, 12)

Figure 7 :
Figure 7: Galileo's Scientific Drawings (Tufte Galileo displayed his arguments and eviden showed their truth.Wittgenstein's doctrine from the Tractatus is the importance of what can only be shown and not said.proposition is a picture of reality, for I know the state of affairs presented by it, if I understand the proposition.And I understand the proposition, without its sense having been ex me'(4.021).'The proposition shows its sense.The proposition shows how things stand, if it is true.And it says, that they do so stand'(4.022).words, 'Graphics reveal data'(Tufte 1983 In a parallel instance, Tufte considers the funct of 'Links and Causal Arrows,' analysi diagram on the cover of the 1936 Cubism and Abstract Art.Alfred Barr created a flow chart for the cover that acted as a History genealogy…paths of artistic verbs of the analysis---are depicted by 51 arrows

Figure 9 :
Figure 9: Theorica specula concavi sphaeri (Tufte 2006) You can think and write in pictures, quite literally.Galileo did, and so did Wittgenstein, in image and text in the body of their scientific and philosophical investigations.sophisticated development of this incorporation of image into text is the dataword coined the term sparkline 'small, high-resolution graphics, usually embedded in a full context of words, numbers, images.Sparklines are datawords'

Figure 11 :
Figure 11: Sparkline for 'win/loss sequence for the season's first 39 games'(Tufte 2006) 522).They can only force of these examples and many more could be given, I information displays are in Wittgenstein's sense.What would Wittgenstein think of a proposition sign as a chart or a diagram?It is perhaps best to look to the Tractatus to answer that question.One of Wittgenstein's most famous and contributions to logic is the truth ,

Figure 15 :
Figure 15: Truth table as propositional sign

Figure 17 :Figure 18 :
Figure 17: Image integrated in the text of