The Forms in Wittgenstein’s Tractatus Logico-Philosophicus

It is the intention of this work to consider the concept ofthe forms in the evolution of Wittgenstein’s works. It is theauthor’s belief that the concept of the forms is, at least, oneof the most significant unifying themes in Wittgenstein’swritings, and central to his views on language, religion,mathematics, philosophy, philosophy of mind, ethics andaesthetics. This work will attempt to trace the concept of theforms in its development from the Tractatus Logico-Philosophicus,through Wittgenstein’s later works.

In the Tractatus and the Notebooks, Wittgenstein is clearlywrestling with the distinction and role of scientific inquiry andphilosophy. This distinction is expressed in the a priori and aposteriori distinction made by Kant (and referenced in severalplaces in the Tractatus): an a priori judgment is trueindependently from experience, and an a posteriori judgment is truebased on experience. For example, a mathematic equation (“2+2=4”) or a definition (“A circle is round.”) is true a priori,whereas the judgment “New York City had a population over 1.0million persons in 2001” represents an a posteriori judgment, itdepends upon some empirical fact being the case.

Wittgenstein, building on Kant, properly saw that philosophydealt with a priori judgments, and scientific investigationconcerned itself with a posteriori judgments. Moreover, scientificinvestigation principally concerns itself not with a posteriorijudgments (or propositions), but with the verification orfalsification of those judgments. In contrast, philosophy wasNOT interested in the verification of a priori propositions, as nosuch verification would be possible—a priori propositions are trueregardless of what experience demonstrates. Moreover, philosophyis not interested in a priori propositions for their own sake, itis not tasked with the job of compiling the list of allpropositions that are true a priori. Compiling mathematicalequations or analytical definitions of concepts was properly thework of mathematicians or the drafters of dictionaries. What

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then is the role of philosophy? As Wittgenstein states in thePreface to the Tractatus:

The whole sense of the book might be summedup in the following words: what can be saidat all can be said clearly, and what wecannot talk about we must pass over insilence.

Thus the aim of the book is to draw a limitto thought, or rather-not to thought, but tothe expression of thoughts: for in order tobe able to draw a limit to thought, we shouldhave to find both sides of the limit (i.e. weshould have to be able to think what cannotbe thought).

It will therefore only be in language thatthe limit can be drawn, and what lies on theother side of the limit will simply benonsense.1

Philosophy concerns itself with the investigation of meaning, andspecifically, the limits of meaning, the boundary betweenpropositions with a sense and nonsense. Further, althoughphilosophy deals with language, philosophy does not aim at theproduction of propositions, it fundamentally deals with vision, away of seeing things, not with, for example, composing the set ofaxioms whereby everything could be explained. We have to see thelimit of language, see that certain propositions have meaning andothers are (well-disguised) nonsense. But we cannot ultimatelyexplain the limit of language, as to do so we would have to beable to think or speak the impossible.

Wittgenstein states:

1 LUDWIG WITTGENSTEIN, TRACTATUS LOGICO-PHILOSOPHICUS, 3 (D. F. Pears & B. F.McGuinness trans., Routledge 1st paperback ed. 1974).

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4.111 Philosophy aims at the logicalclarification of thoughts. Philosophy is nota body of doctrines but an activity. Aphilosophical work consists essentially ofelucidations.

Philosophy does not result in ‘philosophicalpropositions’, but rather in theclarification of propositions.

Without philosophy thoughts are, as it were,cloudy and indistinct: its task is to makethem clear and give them sharp boundaries.2

The aim of philosophy is attainment of a type of vision: to see clearly, specifically, to see the forms of our language clearly. In Wittgenstein, corresponding to the a priori and the a posteriori is a distinction between saying and showing. A posteriori propositionssay that they are true (and are verified by empirical observation), a priori propositions show that they are true. What is shown is a form.

The Tractatus aims at elucidating the form of thought. Whatconforms to the form of thought has sense, and what does notconform is meaningless. For Wittgenstein in the Tractatus, theform of thought is nothing less than logic:

3.03 Thought can never be of anything illogical, since, if it were, we should have to think illogically.

Thus, science concerns itself with the investigation of what is,3

whereas philosophy concerns itself with the investigation of whatis possible, e.g. the a priori conditions that have to exist for aproposition to be meaningful. Science concerns itself withfacts, and philosophy meaning. Science ultimately produces a setof verified propositions that describe the world. Philosophy2 Id.3 T sec. 4.11 “The totality of true propositions is the whole of natural science (or the whole corpus of the natural sciences.)”

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ultimately produces a clear vision of every possible world, avision of meaning, but no propositions. As Wittgensteinconcludes the Tractatus:

6.54 My propositions serve as elucidationsin the following way: anyone who understandsme eventually recognizes them as nonsensical,when he has used them—as steps—to climbbeyond them. (He must, so to speak, throwaway the ladder after he has climbed up it.)

He must transcend these propositions, andthen he will see the world aright.

Wittgenstein has sometimes been viewed as an anti-philosopher,perhaps with some justice. Many people are familiar with ReneDescartes’s skepticism with respect to the objective world, andphilosopher’s proofs of the existence of God, and the apparentcontradiction between a scientific account of nature based onmechanism and a cultural commitment to something like freedom ofthe will. The public, and many philosophers, believe it is therole of the philosophers to prove or disprove something: theexistence of God, the existence of free will, the existence ofobjective reality. Wittgenstein’s corpus stands as a challengeto this conception of philosophy.

Modern philosophy, stemming from Descartes, derives itssense of importance from epistemological questions, how do weknow God/free will/ the external world exists? Yet in support ofWittgenstein’s point of view, what have these centuries ofarguments established? Has the existence God been definitivelyproved or disproved? Has the existence of the external worldbeen conclusively established (or has the question been shuntedaside)? Wittgenstein’s approach does not attempt to answer thesequestions, so much as attempt to understand why these questionsarise in the first place. If we can see clearly how our sense ofa philosophical problem arose in the first place, we

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simultaneously have the key that can free us from the problem.According to Wittgenstein, when we consider the limits of ourlanguage, and see that limit clearly, philosophical problems arenot solved but dissolved through this approach. The propositionsof the Tractatus are nonsense according to Wittgenstein. Theyare neither true nor false. However, they are elucidations, e.g.statements intended to promote insight or vision in the reader,and in this sense, they either hit the mark or they miss.Specifically, the insight that they produce is that they aremeaningless. Wittgenstein asserts that we need insight, notlogical arguments, to crack open the philosophical nut.

Having provided some introductory remarks intended toprovide an overview and a brief defense of the approach of theTractatus, I will now move to consider the philosophy of languageexpressed in the corpus of the Tractatus. So far, I haveuncritically reported on the primary conceptual assumptions ofthe Tractatus—this approach is not intended to suggest that thereader uncritically (or dogmatically) adopt those assumptions,but rather, as in reading a work of fiction, to suggest thenecessary suspension of philosophical belief required to approachand understand the work.

A. Propositions

According to Wittgenstein:

4.001 The totality of propositions is language.

Whether we accept this definition, it is clear that the projectof the Tractatus is concerned exclusively with the analysis ofpropositions, that is, the totality of descriptive language.What is possible to describe constitutes the realm of thepossible, the logical space of the world. It is not as ifWittgenstein were not aware of other forms of language—theTractatus speaks of questions, elucidations, pseudo-propositions,and nonsense as well as propositions. Wittgenstein writes aboutprayer in the Notebooks, a set of writings that lead to the

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production of the Tractatus. What is clear is that these othertypes of language are philosophically unimportant—philosophy aimsat grasping the nature of reality, of all possible realities, theessence of things, which can only be disclosed in propositions.

What is a proposition according to Wittgenstein? He writes:

3.1 In a proposition, a thought finds anexpression that can be perceived by thesenses.

Thus, a proposition consists of a propositional sign thatexpresses a thought.4 A proposition that expresses a thought hasa sense, whereas a propositional sign that fails to express athought is nonsense.5

What do propositions do according to Wittgenstein?Propositions are pictures of reality.6 A proposition with asense is a mental/linguistic picture of a possible state of theworld. It is easy to see within this scheme the role of science—verifying propositions. A proposition expresses a mental pictureof a possible state of affairs—and it falls upon the scientist tocompare it with the actual world—that is, assign it a truthvalue. Thus, propositions are truth functions, if the worldcorresponds to the proposition, it is true. If it does notcorrespond, the proposition is false.

A picture, for example, an oil painting of a landscape bythe sea side, has two distinct features: a material composition,the individual brushstrokes, and a pictorial form, that it is animage of a landscape. Propositions, if they are to be picturesof reality, must likewise have material components, specifically,words, as well as a pictorial form, expressed in their specificaggregate combination. Thus, the Tractatus is predicated on akind of holism—just as a picture represents an image in itstotality—each individual brush stroke does not create the image,

4 Cf. T sec. 3.4.5 Cf. T sec. 4.022.6 T sec. 4.021.

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the specific totality expressed by a particular combination ofwords differs from each word considered alone. Propositions mayalso be compared with chemical formulas for chemical compounds(H-O-H or H2O). Propositions are therefore hylomorphic, havingboth a compositional component (words) and a form, revealedthrough the specific combination of words.

If propositions are to be pictures of the world, then theremust be a correspondence between the form of the world and theform of propositions (just as there must be a resemblance betweena photograph and the subject of a photo.) For Wittgenstein inthe Tractatus, the world is composed of states of affairs. Thestates of affairs amount to a particular concatenation ofobjects. A state of affair is also hylomorphic, consisting ofobjects and their particular arrangement. If this were not thecase, then there would be no logical hinge between language andreality. Propositions could not be said to be pictures, norcould propositions be assigned a truth value.

If logic is the super-structure of the thought, language andthe world, then the principle of non-contradiction is theexpression of that foundational principle. If A is true, then ~Ais false. If propositions are to act as truth functions,propositions must always be capable of being rendered to have adeterminant sense. What does this mean? If we consider ordinarylanguage, we assign a specific sex to children, based on theirgenitalia at birth. Thus, we can assign a truth value to theproposition “Marty is boy” or “Marty is girl” based on empiricaltruth conditions. However, in certain rare occasions, childrenare born for any number of reasons with either ambiguous ormissing genitalia. In these cases, we cannot apply the principleof non-contradiction. In order to apply the principle, we needto develop a third or fourth category to deal with these cases.In this way, we are able to apply the principle of non-contradiction in any circumstance.

If logic is the essence of thought, and the principle ofnon-contradiction requires that concepts have a determinant

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meaning, then there is disconnect between our ordinary languageand the essence of thought, as in ordinary language, expressionsoften lack a determinant sense in certain context, such as inassigning the sex of a hermaphroditic child. Likewise, the ideathat the propositions of ordinary language are actually picturesof the world is not necessarily obvious. For example, “the redcat is seated on the red mat” does not look like a visualrepresentation of how that same scenario would look, for example,if we drew a picture of the scenario. However, we can imaginethe scenario in our minds when we hear the proposition, and thewords, the proposition, is logically connected with the picturein the mind. As Wittgenstein notes:

Language disguises thought. So much so, thatfrom the outward form of the clothing it isimpossible to infer the form of thoughtbeneath it, because the outward form of theclothing is not designed to reveal the formof the body, but for entirely differentpurposes.7

Thus, Wittgenstein was led to postulate a fundamental differencebetween the forms of ordinary language and the forms of thought.The purpose of philosophical elucidations is thus to remove theveil of ordinary language, so that we can understand what lieshidden within the structures of the mind.

There is a third reason to consider the possibility of adisjunction between our ordinary language and thought. However,to develop this strand, we need to develop the distinctionbetween meaning and sense in the Tractatus. For Wittgenstein, aproposition was composed of a special arrangement of names suchthat the proposition had a sense. If we think of chemicalformulas, H2O is a possible molecular configuration, but H6O isnot a possible molecular configuration on its own under ordinaryexperimental conditions8. Similar to chemistry, objects can onlycombine in specific ways, therefore, every concatenation of words7 T sec. 4.002.

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does not correspond to a proposition, only certain specialconfigurations of words can stand for certain states of affairs.The special configuration is what the proposition shows, whilethe proposition itself is composed of names. For example, theproposition “the six foot rod is lying flat on the floor of thethree foot cubic bin” is nonsensical. We cannot imagine a sixfoot rod lying flat on the floor of a container smaller thanitself. On the other hand, “the three foot rod is lying flat onthe floor of the six foot cubic bin” can be imagined.

The point of this discussion is important: the names inpropositions have a meaning, e.g. they refer to objects in theworld. The combination of names (when combined in a coherentmanner) is the form of a valid proposition that reveals the senseof a proposition, it shows what the proposition means as it were.That is, it gives the names, and shows the logical relationshipof names to one another. The meanings of names in propositionsare completely arbitrary. We can name something “dog” or“perro.” At the same time, the sense of a proposition isdictated by the logic of language. Once the referents of namesare established, the sense of the proposition is determined (inthe same way that once we select paint and commit paint to paper,the image is logically defined, notwithstanding the choice ofpaint being arbitrary.)

The Tractatus is committed to a form of reductionism withrespect to complex objects. We can name a complex thing with aname. The name of the complex can then be analyzed into adefinition, that is, a proposition composed of names that namesimpler objects. For example, if we take a human body, we candefine it by stating the major organ systems of the body andtheir relations to one another. The definition of a complex is aproposition expressing the relation of a set of simpler objects.Likewise, each organ system can be itself defined through aproposition stating the relation of simpler structures, for8 The author’s knowledge of chemistry is insufficient to rule out an independent compound H6O ever occurring, but sufficient to expect that such a compound could only occur under rare circumstances if ever.

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example, a proposition expressing the relations of varioustissues to one another. However, if language is ever intended torefer to reality in a determinant fashion, this process ofanalysis must come to an end. Thus, Wittgenstein postulatessimples (what he call objects) that are the ultimate substancesnamed by what he calls elementary propositions. The simples areundefinable, and the names composing elementary propositionssimply refer to their objects. Thus, in an elementalproposition, we can “point” to what the name means but there canbe no further analysis of what a simple name means.

The simples (or objects) are unknowable, in the sense of ourbeing able to recite true, justified propositions about thesimples. Their existence is dictated by the form of ourpropositions, and the fact that our language can picture reality.If the simples do not exist, then language continues in aninfinite loop of definitions, without ever picturing the world ina determinant way. The simples not only have to exist, they mustbe indestructible. If a complex did not exist, a propositionpicturing a complex would simply be false. If a complex weredestroyed, a proposition picturing a complex would simply becomefalse. In contrast, if a simple did not exist, then names thatreferred to the simple would be meaningless. If a simple weredestroyed, then a formerly possible state of affairs would becomeimpossible. The simples are beyond the limit of our language,they are a something about which we can say nothing, we may onlyshow what we mean by means of the process of naming. We cancompare this to the experience of watching a train pass beyond anunknown horizon. We know the train has not vanished, but wecannot speak of its destination. Wittgenstein is not saying whatlies beyond the horizon, he is showing that there is a horizon,and that something must be beyond that horizon for language to bemeaningful, and must reflect the same logico-syntactical formswhich we use to speak of complex objects.9

9 The Big Bang theory proposes that space and time originate from a singularity. Thus, although we have a grammatical form of expression (“What happened before the Big Bang?”), if we understand and adopt the Big Bang

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Thus, Wittgenstein postulates that the indeterminant formsof ordinary language can be, at least in theory, analyzed down,or defined, by a set of elementary propositions. This ideallanguage would correspond perfectly to the logical structure ofthought itself. In this way, it is clear that the metaphysics ofthe Tractatus reveals itself based upon certain simpleassumptions about the operation of language.

We can now consider our previous discussion of science andphilosophy. Science concerns itself with the verification ofpropositions. Philosophy does not concern itself with meaning,at least in the sense of the meaning of names in elementarypropositions. The act of naming is completely arbitrary, and amatter of empirical accident. Nor can philosophy essentiallyspeak to the nature of objects, which are unknowable, asunknowable as the nature of thought itself. Philosophy mustconfine itself to showing the forms of things, the forms ofobjects (their means of combining with each other) and the formsof propositions. That is, Philosophy is concerned with thelogico-syntactical structure of any possible system ofdescription.

B. The World.

I have attempted to elucidate the theoretical assumptions ofthe Tractatus with respect to language and meaning, beforeapproaching the metaphysical system outlined in the work. I viewthe Tractatus as centrally based in a certain view of language,and the relation of mind and language and the world, and thatthis viewpoint dictates what would be otherwise a strange anddifficult book. Many of the points expressed previously mainlydeal with elucidations found in the third and fourth sections ofthe book. I will now attempt to develop the main structure ofthe metaphysics of the Tractatus.

theory, we realize that the expression is meaningless. The Big Bang is the horizon of space and time, we cannot say anything about something occurring temporally prior to the Big Bang. In the same sense, there is nothing logically prior to elementary propositions.

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The first section of the Tractatus states:

1 The world is all that is the case.

1.1 The world is the totality of facts, notof things.

1.11 The world is determined by the facts,and their being all the facts.

1.12 For the totality of facts determineswhat is the case, and also what is not thecase.

1.13 The facts in logical space are theworld.

1.2 The world divides into facts.

1.21 Each item can be the case or not thecase while everything else remains the same.

Wittgenstein departs from the traditional Western Aristoteliannotion of substance-based metaphysics, that holds that thefundamental structure of reality consists of substances(things/objects). Traditionally, if there are no things, nosubstances, then nothing exists. At the same time, therelationship between substances is purely accidental.

In contrast, for Wittgenstein, the world is composed offacts, states of affairs, objects in particular configurations.Although not included in the above quote, Wittgenstein conceivesof what he calls “negative facts”—essentially logicalpossibilities that are ruled out by a particular state ofaffairs. For example, if “John has red hair” is true, then thepropositions “John has black hair” or “John has blond hair” wouldbe negative facts—logically related to one another. Thus,although logic dictates infinite possibilities, the worldconsists of certain facts, and, by virtue of those facts

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representing the totality, it renders all other possibilitiesfalse. In addition, the elementary facts are independent of oneanother. The fact that John has red hair does not determine thecolor of Bill’s hair.

The second section of the Tractatus begins to furtherelaborate on facts, states of affairs. Wittgenstein develops hisconcept of simple objects previously alluded to above. Things,or objects, are independent of each other (one object can combinewith other objects), yet the potential combinations of objectsdepends upon the internal form of objects. Each object hascertain possible modes of combining with other objects, itsinternal property or structural property, which is logicallydetermined, e.g. it is the logical form of an object given apriori.10 This is analogous to the atoms of chemistry, which maycombine in various ways, but which combine in accordance withcertain physical laws. Wittgenstein analogizes objects to pointswithin a multi-dimensional space. For example, there are colorpoints corresponding to “color” objects, sonic pointscorresponding to “sonic” objects, etc. Qualities, according toWittgenstein, emerge from the concatenation of objects. Thequalities represent the form of objects (color objects combinewith other color objects, sonic objects with other sonicobjects).11 Thus, a particular “seeing” is composed of a seriesof colorless “color objects” in combination, like the picture ona television screen.

Wittgenstein’s metaphysics is distinguishable fromAristotelian substance in that, while Wittgenstein viewssubstance as essential to the world, he also views relationsbetween objects as also essential. An object can only exist incombination with other objects. Thus, objects manifesting theform of color are not colored.12 Color is what is manifested inthe combination of colored objects. Objects of the same form areonly distinguishable from each other in their actual contingent10 T sec. 2.0121.11 T sec. 2.0131.12 T sec. 2.0232.

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relation to each other. They are essentially the same.13 Thus,the world consists of states of affairs, objects in logicalrelationship with each other, not things. The contingency of theworld reflects the accidental combinations of objects. Objectscan neither be created nor destroyed, but they can bereconfigured.

Analogous to our subjective experience of the world is theworld of imagination, the world of the mind, in which we picturefacts to ourselves. Wittgenstein does not discuss the objects ofthought, but as is clear, there must be an isomorphicrelationship between thought, language, and the world. Wepicture facts to ourselves, and then we compare our pictures withthe world. According to Wittgenstein, pictures share a form withwhat they picture. For example, the painting of the Duchess ofAlba and the Duchess of Alba share a form in common. If they didnot share something identical, then the picture could not picturethe subject. All pictures must share something in common withreality in order to be pictures, the pictorial form. Aspropositions are pictures, propositions share a pictorial form(or a logical form) with reality.

However, although we can see the form shared by the picturesand the reality, and propositions and reality, the form is notsomething we can talk about. Names can refer to objects.Propositions can show imagined states of affairs. That is to saythat reference in language is based upon fragmentation anddisjunction, A or ~A. The proposition “A” can only refer tosomething other than itself. What the proposition “A” shows issomething that “A” participates in, thus, logically speaking, “A”cannot say, e.g., refer to its form. Saying, referring, areonly possible when there is a real disjunction between thelinguistic expression and what the linguistic expression refersto. Showing is only possible when there is a real conjunctionbetween the subject of showing and the actual image, e.g. thereis a shared a pictorial form, a resemblance. For example, if A’

13 T sec. 2.0233 and 2.02331.

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represents A, we can say that there is a resemblance between A’and A. However, if we try to represent the relationship, we canproduce a picture A’’ of the resemblance. But the picture willnot be a resemblance, it will be simply be a new picture whichresembles both A’ and A. The form cannot be pictured, it canonly be shown in the resemblance. What is shown cannot bereferred to, as there is no disjunction between language and whatit pictures.

What unifies not only the world, but unifies the world,language, and thought, can neither be a thought, nor a state ofaffairs, nor a propositional sign. If the logical form was astate of affairs that could be the subject of a meaningfulproposition, then the logical form could be the subject of aproposition with sense. But in this case, then the logical formand the proposition would have to share an “uber-logical” form incommon for the proposition to have a sense. This is clearly acontradiction. If we conceive of a limited totality (such as astate of affairs), it is clear that to represent this limitedtotality our mode of representation must lie outside the limitedtotality (e.g. our language is distinguishable from the existenceof the state 0f affair, the fact). If we conceive of eachindividual state of affair as limited totalities, and the world(here including language and thought) as the totality of thesetotalities, then it is clear that what unifies all things cannotbe the subject of a meaningful proposition. The philosopherseeks a clear vision of the logical form of the world, but whathe or she sees cannot ultimately be spoken of, although it isexpressed in all manner of representation. The logical formshared by all propositions, by all pictures, and all state ofaffairs, cannot be named nor pictured. As stated previously,although Wittgenstein may be viewed as rejecting a major strandof modern philosophy (Cartesian and post-Cartesian Europeanphilosophy), his philosophical project is not without parallelsin Platonic and Neo-Platonic thought.14

14 Part of Wittgenstein’s diagnosis of what is wrong with philosophy is that it attempts to say what can only be shown, and thereby attempts to say what is

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Wittgenstein, having developed atomic facts, states ofaffairs, composed of objects in determinant arrangement governedby the internal forms of the objects, and thoughts, mentalpictures of these states of affairs, begins to develop therelationship between propositions, thought and reality. Themind imagines states of affairs, propositions are the expressionsof those thoughts, and reality is the place in which thosepropositions are verified or falsified. However, the focus ofphilosophy is on the analysis of propositions, as it is inlanguage that the relationship between thought and reality isrevealed. The meaning of propositions in ordinary language canbe analyzed into a set of elementary propositions about atomicfacts. To the extent that ordinary language cannot be analyzed,it is revealed to be nonsense. The heart of Wittgenstein’sanswer to the problems of philosophy is that the questions andthe answers of philosophy are meaningless:

4.003 Most of the propositions and questionsto be found in philosophical works are notfalse but nonsensical. Consequently wecannot give any answer to the questions ofthis kind, but can only point out that theyare nonsensical. Most of the propositionsand questions of philosophers arise from ourfailure to understand the logic of ourlanguage.

(They belong to the same class as thequestion whether the good is more or lessidentical than the beautiful.)

And it is not surprising that the deepestproblems are in fact not problems at all.

properly beyond the limits of language (e.g. beyond what is subject to the principle of non-contradiction). Note that many anti-metaphysical philosophical approaches are equally guilty of this feature as are blatantly metaphysical approaches, as they both, in effect, deny the limit. We cannot stand outside the totality and meaningfully opine on the condition of the totality.

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What can philosophy concern itself with in this case? Inlanguage, a distinction is made between syntax and semantics, thegrammatical forms of expressions and their meaning, or reference.Wittgenstein indicates that the reference of names is arbitrary,and not given a priori. As such, philosophy must concern itselfwith the logical syntax of language. But that syntax does notsay anything, it shows something about the nature of the world.Thus, Wittgenstein begins to consider the general form of thelogical syntax, the logical form of the proposition, of anyproposition, such that language is meaningful. Wittgensteinmoves from propositions to consideration of what he callsexpressions, namely the formal properties of a class ofpropositions. This is connected with what he terms apropositional variable. For example, the expression “X is white”is not a proposition with a sense, as the name “X” has not beenassigned a meaning. However, the expression shows the formcommon to a class of propositions. We can then stipulate to themeaning of the variable, and therefore give the form common toall the propositions sharing a common expression. Thus, the formis expressed in the logical syntax common to a class ofpropositions. These general expressions are later termed formalconcepts.

Wittgenstein distinguishes between concepts and formalconcepts in that he asserts that we can only construct pseudo-propositions about formal concepts, whereas we can constructpropositions about concepts. He writes:

4.1272 Thus, the variable name “X” is theproper sign for the pseudo-concept object.

Whenever the word ‘object’ (thing, etc.) iscorrectly used, it is expressed in conceptualnotation by a variable name. . .

Whenever it is used in a different way, thatis as a proper concept-word, nonsensicalpseudo-propositions are the result.

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So one cannot say, for example, ‘There areobjects,’ as one might say, ‘There arebooks’. And it is just as impossible to say,“There are 100 objects’, or ‘There are[aleph-null] objects’.

And it is nonsensical to speak of the totalnumber of objects.

In interpreting this passage, it is important to remember theconcept of “object” used here is not the ordinary concept, butthe indestructible simples discussed previously in the Tractatus.“Objects” are the indestructible, ineffable, unspeakablestructure of the world necessitated by the logic of our language.We can only “point” to the simples to assign a meaning to thenames in elementary propositions. The syntactical structure thatcorresponds to a name in our language is a variable. We can seethat the syntax of our language shows that there are names, andthe names can be given meaning. But we cannot speak meaningfullyabout what the names name, as this would amount to the furtheranalysis of an elementary proposition (which is a logicalcontradiction). Thus, we produce pseudo-propositions aboutobjects, whereas, we do not do so with respect to books.15 Thesyntax of elementary propositions points to something beyond ourlanguage, it shows the form of something outside our language,but what it points to, we cannot speak about meaningfully(referentially).

Wittgenstein relates the concept of cardinal numbers to theinternal forms of objects, suggesting that numbers are the formalexpression of a set of series ordered by the internal relationsof objects to one another. Thus, “number” is also a formalconcept related to the formal concept of “object,” and thepropositions of mathematics are in effect, syntacticalexpressions, or pseudo-propositions. We cannot formulatepropositions about objects, as elementary propositions (composed

15 Although we can ask the question how the concept “book” is meaningful if there is not an object that “book” refers to.

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of names that refer to objects) are the most logically primitiveform of expression, and a proposition about objects would, ineffect, introduce fragmentation into the simples. Becauseobjects are logically primitive, we can only extra-linguisticallyrefer to objects. Likewise, we cannot speak about numberseither, although numerical expressions are related to series ofobjects. Numerical expressions are generated by defining a firstterm, and then performing a set of operations on that first term.A number is the exponent of the operation.16 Mathematicalexpressions show a form, a series of relations, betweensyntactical expressions, but are by nature meaningless (they showthe formal structures governing a set of elementarypropositions). An equation shows the logical equivalence of twoforms. As a corollary, because logical forms are, in effect,numbers, it follows that the logical forms themselves can have nonumber (otherwise, numbers would be the formal cause of thelogical forms and not the expression of the forms). Thus,Wittgenstein rejects any attempt at philosophical monism ordualism. The world is not one or two, it exists as a numberlessunity in difference.17

Wittgenstein goes on to attempt to eliminate logicaloperators from logical expressions. This is important to thetask of the Tractatus, because complexes must be analyzed into aset of elementary propositions conjoined (or disjoined) bylogical operations. As logic is strictly dictated by syntax, andcan make no reference to the meanings of signs (as the meaning isarbitrary and established a posteriori), obviously, there is no wayto make a distinction between a propositional variable and/orname, and a logical operator. The logical operators correspondto ways of seeing the relations between propositions. Forexample, if we picture an elementary proposition, in our picture,we can make out another proposition contained within thestructure of our elementary proposition (and thus logicalconjunction). Likewise, if we can picture an elementary16 T sec. 6.021.17 T sec. 4.128.

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proposition, we can also picture contradictory propositions,propositions with a different structure (and thus disjunction andnegation). Wittgenstein claims there are no laws of inference.What are conceived of as laws of inference reflect merely seeingwhat is common and what is uncommon between elementarypropositions. Logical operators just serve to show the logicalrelationship between two elementary propositions.

In the penultimate section of the Tractatus, Wittgensteinstates the general form of the proposition, which is anexpression, the variables of which consist of the all thepossible elementary propositions, all propositions (which can beconstructed from and analyzed into elementary propositions), andthe negation of all propositions. The general form of theproposition expresses the logical form of reality, the form ofevery proposition (every picture) possible. The general form ofthe proposition represents the essence of all things. AsWittgenstein states:

6.124 The propositions of logic describe thescaffolding of the world, or rather, theyrepresent it. They have no ‘subject-matter’.They presuppose that names have meaning andelementary propositions sense; and that istheir connexion with the world. It is clearthat something about the world must beindicated by the fact that certaincombinations of symbols—whose essenceinvolves the possession of a determinantcharacter—are tautologies. This contains thedecisive point. We have said that somethings are arbitrary in the symbols that weuse and that some things are not. In logicits is only the latter we express: but thatmeans that logic is not a field in which weexpress what we wish with the help of signs,but rather one in which the nature of the

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absolutely necessary signs speaks for itself.If we know the logical syntax of any sign-language, then we have already given all thepropositions of logic.

Wittgenstein asserts that the tautologies of logic show thescaffolding of the world. If the world, thought, and language,lacked a logical structure, then there would be no tautologies,everything would be true or false a posteriori. At the same time,the propositions of logic are meaningless, as they show the formof the world, they do not say anything about the world, they haveno reference.

Wittgenstein asserts that tautologies and contradictions arepropositions that have a sense. That is to say, they show thatthey say nothing, that no empirical observation of the worldcould corroborate what they picture. Any state of the world isconsistent with a tautology, and no state of the world isconsistent with a contradiction. Tautologies are analogous to apicture frame, which can contain any picture, and contradictionsare similar to a black screen, through which no picture can beexpressed. On the other hand, Wittgenstein claims that thepropositions of the Tractatus are nonsense, and therefore,neither tautologies nor contradictions. What sense can be madeof this assertion? To address this issue, I think it will behelpful to distinguish between the relationship of science,logic, and mathematics.

C. Science

According to Wittgenstein, atomic facts are independent ofone another.18 If this were not the case, we should be capableof deducing another atomic fact from the existence of one atomicfact. If this were, in fact, possible, it would be possible todeduce the world from the existence of one particular atomic

18 T sec. 5.134-5.135.

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fact. Thus, Wittgenstein asserts that the concept of free willrelated to the fact that the future is not logically deduciblefrom the past.19 For Wittgenstein (like Plato), the connectionbetween knowledge and what is known is logical necessity.20

Scientific inquiry is based on experiment, measurement andtesting, not a priori deduction. Causality, to be real, mustfollow from logical necessity, which is clearly excluded byvirtue of the scientific method: we have to engage in anempirical inquiry to determine the relationship between twostates of affairs.

As a result, Wittgenstein concludes that belief in a causalnexus between one fact and another represents a superstition.21

Accordingly, there can be no scientific laws that govern theuniverse. In fact, Wittgenstein reduces the task of science tomerely empirically determining all true states of affairs:

6.372 Thus people today stop at the laws ofnature, treating them as somethinginviolable, just as God and Fate were treatedin past ages.

And in fact both are right and bothwrong: though the view of the ancients isclearer in so far as they have a clear andacknowledged terminus, while the modernsystem tries to make it look as if everythingwere explained.

I believe this proposition expresses the spirit of the Tractatus,to the extent that it such an assertion is possible. In ancientphilosophy, God is the ultimate end of philosophical inquiry, thelimit. In Neo-Platonism, God is not a being, God is theprinciple by which all beings comes into existence. In Aquinas’sways, he derives the existence of God from the contingent limit

19 T sec. 5.1362.20 Id.21 T sec. 5.1361.

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of the world. But the nature of God is ultimately a mystery, andthe universe is ultimately a mystery. In contrast, to the extentthat we see the scientific description of the world, the laws ofnature, as an ultimate explanation of things, then everything isexplained. The mystery of the world disappears.

In what sense are the laws of nature a superstition,according to Wittgenstein? Clearly, Wittgenstein makes adistinction between logical necessity and causal necessity, onebeing dictated by logic, the other dictated by nature as it were.Wittgenstein states:

6.32 The law of causality is not a law butthe form of a law. . .

6.34 All such propositions, including theprinciple of sufficient reason, the laws ofcontinuity in nature and of least effort innature, etc. etc.—all these are a prioriinsights about forms in which thepropositions of science can be cast.

Mathematical physics provides an abstract and idealized frame ofdescription for physical events. One can fire artillery shells,and measure where the shell lands based on the angle of thecanon. Using mathematics, we can represent the relationshipbetween the angle and the distance the ball travels along theearth. To the extent that we can determine a mathematicalrelationship between the observed angle and distance, we canpredict the possible distance at a particular angle. Themathematical equation does not dictate or control the trajectoryof the cannon ball, the cannon ball simply flies through spacewith a certain amount of uniformity (else no description would bepossible). Our equation does not explain the motion of thecannon ball, it simply describes mathematically the form of thecannon ball’s (probable) trajectory. But notwithstanding ourcalculation, the cannon could misfire, the cannon ball could endup somewhere totally unpredicted, for the calculation does not

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prophesize the trajectory, it merely forecasts it. Further, tothe extent that we consider mathematical expressions outside oftheir application in concrete contexts, they are meaningless.Newton’s laws of gravity, in the abstract, have no meaning. Onlywhen we input specific initial conditions, can mathematicalexpressions be used to make specific predictions in specificcontexts. In as much as mechanical laws are general, they aremeaningless. In as much as they are meaningful, they have to befeed specific inputs in order to predict specific outputs.

In a set of lectures recorded by his student Yorick Smythiesin all probability in 1939, Wittgenstein spoke in greater detailabout the notion of natural law and the picture created by ourlanguage:

Wittgenstein said you can explain the waypeople looked at natural laws by saying theyregarded them as if they were rails, alongwhich things had to move. The expression ofa natural law was in some way such a rail.

The language of natural law suggests a comparison with astatutory law: the king commands you to obey a certain way.Further, because natural laws are descriptive, they suggest aquality of an irresistible command, akin to the command of anall-powerful deity. One wants to say that you can scoff at theKing’s law, but the laws of nature compel you to obey. But thisconception is nothing more than a picture. The fact is that a“natural law” merely describes an observable regularity that wepresume will continue. A certain thing tends to behave regularlyin a certain way, and we compose an abstract description of thatbehavior. This is not a case of compulsion, and if thephenomenon does not develop as we expected, we do not view it asdefiance. In fact, we make predictions that are routinelyflaunted by observations. Sometimes the discrepancy isattributable to random error, sometimes experimental error, butsometimes the discrepancy results in the scrapping or revision ofthe law. Further, there are random mechanical events in the

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universe. Scientists, as yet, have not developed a method forpredicting the results of a dice roll in Las Vegas.

Clearly, we just happen, as a matter of accident, to live ina universe where certain general regularities are observable.The sun rises every morning. Cannon balls generally followcertain predictable trajectories. But there is no reason whythis is the case, and no reason why it will be the case tomorrowor the next day. Science does not explain the regularity ofnature, it describes it, and scientific predictions are notprophesy, they are forecasts (and sometimes incorrect forecasts).The regularity of the cosmos is not compelled by logicalnecessity. The fact that the regularity of the cosmos can bedescribed with mathematics does not entail that the regularity iscompelled by mathematical necessity. The brute fact is that theregularity simply is (or so it seems to be). Further, sincestates of affairs are logically independent of one another, weshould rightfully expect that states of affairs would be random,and there would be no regularity in the world. Or rather, ourorderly description of the universe is nothing more than theimposition of a pattern on what is actually something random.

If we consider the relationship between propositions andstates of affairs, a proposition is the expression of a thought.The names composing a proposition have a meaning, and thepropositions shows it has meaning, e.g. a sense. On the otherhand, a state of affairs is simply a concatenation of objects.If the world is to have a meaning, the world must be conceived ofas the expression of a thought, but obviously, not a humanthought. The world is independent of my will as Wittgensteinassets. Even if the world and my will overlap, the overlap isfortune, coincidence.22 Thus, if the world is to have a meaning,that meaning must lie outside of the world, presumably in themind of God. If the world is to be conceived of as a subject ofscientific inquiry, it must manifest a form. But to truly besaid to have a form, it must be a sensible expression of

22 T sec. 6.374.

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something, just a propositions can only be meaningful if theyhave a sense. The laws of nature, in as much as they describethe world, presuppose a meaningful relationship between logicallyindependent states of affairs. As a description, they do nothave the capacity for compelling the world to be this way or thatway. But if the world simply is, the success of the scientificdescription of the world is only a matter of coincidence. It isthe imposition of a pattern on something random and meaningless.Thus, our trust in order and regularity in nature is bestunderstood as an expression of our faith in a Creator.Otherwise, our search for a scientific description of nature is abeautiful illusion.

Thus, the way of the ancients, viewing the world as themeaningful expression of a transcendent divinity, is a much moresensible way of looking at things than ascribing causal powers tothe meaningless syntactical expressions of mathematics, themodern superstition. Wittgenstein states:

6.53 The correct method of philosophy wouldreally be the following: to say nothingexcept what can be said, i.e. propositions ofnatural science—i.e. something that hasnothing to do with philosophy—and then,whenever someone wanted to say somethingmetaphysical, to demonstrate to him that hefailed to give a meaning to certain signs inhis propositions. Although it would not besatisfying to the other person—he would nothave the feeling we were teaching himphilosophy—this method would be the onlystrictly correct one.

In reading this, it is important to remember remark 4.11, thatthe corpus of natural science is only the description of all trueelementary propositions. It does not include mathematics,mechanical laws, laws of nature, causality. The metaphysicalwould occur when the student attempts to show empirical

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regularities between contingent states of affairs. Theseregularities may be described in mathematical expressions, but wecannot ascribe causal efficacy to mathematical expressions.Further, the fact that the world shows that it is orderly,predictable, is completely inexplicable. We cannot ultimatelysay anything about the order of the world, it is something thatis shown by the world, and it being the world, the totality. Itis something we cannot ultimately talk about.

D. The Anthropology of the Tractatus.

If we consider the question of the nature of the self, thethinker, in the Tractatus, we can begin to develop some insightsinto the meaning of the work as a whole. Wittgensteindistinguishes between what he calls the metaphysical subject andthe psychological subject. The psychological subject is the onethat can be described in terms of empirical states of affairs(e.g. that I am 5’ 10” and happen to be hungry for dinner.) Incontrast, Wittgenstein states:

5.641 Thus there really is a sense inwhich philosophy can talk about the self in anon-psychological way.

What brings the self into philosophy is thefact that ‘the world is my world’.

The philosophical self is not the humanbeing, not the human body, or the human soul,with which psychology deals, but rather themetaphysical subject, the limit of the world—not part of it.

The world of the Tractatus is strictly the phenomenal world ofthe subject. It is a world of a “conscious self” and isessentially solipsistic. It is not your world, or the world, itis my world as I found it.23 What lies outside the world is the23 T sec. 5.64 Wittgenstein “solves” the problem of solipsism by pointing outthat realism and solipsism can ultimately only point to the same thing, e.g. my world, and are therefore, to the extent that they can be meaningful

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subject, experiencing the world, as well as the mind, not onlypicturing the world, but capable of picturing any possible world.The metaphysical subject is metaphysical in that it contains theworld, as well as conceiving of other possible worlds. Thesubject thus transcends the world, and well as transcending thepsychological self.

It should be pointed out that the nature of the mind forWittgenstein reflects the structure of the world. The mind isnot a substance, it is a mirror of the world. It is composed ofthoughts, concatenations of imaginary objects. It is a principleof unification and containment of the diversity of thoughts, asthe world unifies and contains states of affairs. The nature ofthe mind is logic itself, and the general form of the propositionshows the form of the metaphysical subject.

It is now time to broach the subject of ethics. Wittgensteinindicates that there is no hierarchy in the world, thatpropositions are all equal in lacking in value.24 Value is themanifestation of something higher in the world. That is, to seesomething in the world as possessing value, is to see somethingas the manifestation of the transcendent. Ethics comes in thealignment of the metaphysical self with the vision of thetranscendent. If we live in accordance to the transcendent, weare happy, we live true. If we defy the transcendent, then weare false and unhappy. Wittgenstein does not conceive of thisprocess in terms of reward and punishment in an afterlife, but inour specific life, in our attitude toward the world.

We are left with God, Logic, Ethics, and the MetaphysicalSubject. These concepts are not the same, but also not logicallydistinguishable (as logical forms are numberless). Thus, whatappears to be beyond the limits of language in the Tractatusturns out to be a vision of things very similar to Neo-Platonismas expressed by writers such as Pseudo-Dionysius the Aeropagite.The world is a theophany of God. The role of the metaphysical

expressions, are conceptually indistinguishable.24 T sec. 5.556 and 5.5561.

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subject is to discern God within the theophanies of world, butultimately to transcend the world, and to return to God. Themetaphor of mirrors and pictures in the Tractatus leaves out oneimportant component necessary for pictures and mirrors tooperate: light. Likewise, the Tractatus does not expresslydiscuss the light necessary for the mind to picture the world, orfor the world to picture the nature of God. It is the light thatmakes it possible to discern the picture, the image.

The propositions of the Tractatus are nonsense, nottautologies, because they are consumed with a vision of theworld, with meaning, with statements about the world. But it isthe light that is beyond the world, beyond meaning, beyonddescriptions with which we should be concerned. When we climbbeyond the scaffolding of logic, and see the world aright, wewitness the heavenly light and are reduced to speechlessness.The propositions of the Tractatus are not tautologies becauseunlike tautologies, which show the logical form of the world, thepropositions of the Tractatus attempt to say what is beyond theworld, beyond the limits of language. The propositions of theTractatus are elucidations, they are intended to reveal thetranscendent light. If the elucidations succeed, and weapprehend what lies on the other side, we realize that anythingthat we try to say must ultimately be nonsense, a description ofthe indescribable.

Wittgenstein states in the preface to the Tractatus:

If this work has any value, it consists intwo things: the first is that thoughts areexpressed in it, and on this score the betterthe thoughts are expressed—the more the nailhas been hit on the head—the greater will beits value. . . on the other hand, the truthof the thoughts that are here communicatedseems to me unassailable and definitive. Itherefore believe myself to have found, onall essential points, the final solution to

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the problems. And if I am not mistaken inthis belief, then the second thing in whichthe value of this work consists is that itshows how little is achieved when theseproblems are solved.

Wittgenstein, in writing his preface, clearly understands that hehas stated in the body of his work that the work is nonsense. Itis unclear how the author can then maintain that the work nowexpresses thoughts, and that those thoughts are the definitivetruth. It is possible to interpret Wittgenstein as being cleveror ironic, however, this is not necessarily consistent with mostbiographical depictions of Wittgenstein, who comes across as avery serious and intense person.

I think the remark bears further scrutiny. First,Wittgenstein does not definitively state the work consists ofthoughts at all. Further, to the extent that it does expressthoughts, Wittgenstein states that the work will have value. Inthe body of the Tractatus, Wittgenstein indicates that the realmof value is the transcendent, and the transcendent is preciselythat which is unthinkable. It is helpful to compareWittgenstein’s preface to the introduction of Pseudo-Dionysius’swork on the Divine Names:

. . . Now as I have already said, we must notapply words or conceptions to this hiddentranscendent God. . . In the scriptures theDeity has benevolently taught us thatunderstanding and direct contemplation ofitself is inaccessible to beings, since itactually surpasses being. . . And yet, on theother hand, the Good is not absolutelyincommunicable to everything. By itself itgenerously reveals a firm, transcendent beam,granting enlightenments proportionate to eachbeing, and thereby draws sacred minds upwardto its permitted contemplation, to

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participation and to the state of becominglike it. . . With our minds made prudent andholy, we offer worship to that which lieshidden beyond thought and beyond being. Witha wise silence we do honor to theinexpressible. . .

Pseudo-Dionysius contends that God is beyond language andconcept, and is beyond discursive reason, but that God can,through revelations, enlightenments, raise sacred minds upwardsinto participation in divinity. However, the ultimate expressionof God is found in silence. In Pseudo-Dionysius, we find thatalthough discursive reason cannot produce valid assertions aboutGod, through elucidations, we can be raised up to participatingmore deeply in God. Notwithstanding, the ultimate homage to Godis found in silence.

I believe Pseudo-Dionysius’s writings suggest the best keyfor interpretation of the Tractatus. To the extent that theTractatus expresses thoughts, those thoughts are not humanpropositions but divine enlightenments. To the extent thatthose thoughts hit the nail on the head, that is, induce a visionof the sublime dimension outside the limits of language, the workhas value, in that it reveals the Valuable. Likewise, like theauthor of the Tractatus, Pseudo-Dionysius concedes that thehighest path is silence. Last, to the extent that we witness thesublime, we come to understand that the obsession of theTractatus, to definitively establish the limits of logic,language and science, is ultimately trivial.

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