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Sign, Symbol, and Grammar - Earlier and Later Wittgenstein

The meaning of a word is not an object named by the word, which was what 'symbol' was supposed to designate. That notion was replaced by 'grammar' or meaning as use. These are logic of language studies.

Outline of this page ...

Context: with respect to philosophical problems, what is language with meaning, what nonsense? That question is the background of these early discussions.

"The Earlier and Later Wittgenstein"

Note: Words that follow "Query" are Internet searches that were directed (or misdirected) to this site, and to which I have responded on this page.

Query: is language all powerful for Wittgenstein?

The answer is: No, neither in religion, nor in aesthetics.

Language is not always the key to resolving philosophical problems: and Wittgenstein's objective in philosophy is to resolve philosophical problems, not to investigate language for its own sake. The Philosophical Investigations is philosophy, not Philosophy of Language.

On the other hand, maybe this query concerns the Tractatus Logico-Philosophicus. A distinction is often made between "the earlier Wittgenstein" (the TLP and anything before it) and "the later Wittgenstein" (everything after the TLP). Russell, for instance, made this distinction:

... for convenience I shall designate [the philosophy of Wittgenstein's Philosophical Investigations] as WII to distinguish it from the doctrines of the Tractatus which I shall call WI ... (Bertrand Russell, My Philosophical Development, New York: 1959, p. 216)

Maybe this distinction is in many ways -- although certainly not in all ways -- justified, because the sense that might be given to search queries often really does depend on whether they are placed in the context of WI or WII.

"Sign", "Symbol", "Rules of Grammar"

Query: Wittgenstein and the ideal clear language.

The ideal language [As always, that is, according to my own reading of the Tractatus Logico-Philosophicus] would essentially exclude all ambiguity: for example, for every unique object there would be a unique name. For example:

Query: are the rules of chess contained in the laws of physics?

There is equivocation here, because 'rule' does not equal 'law' here. Although you might say instead of 'rules of chess': 'laws of chess', these are legislative laws (commands) not descriptive laws (taking note of regularities of nature). In TLP 3.323-3.325, Wittgenstein characterizes an ideal language as one that would make such mistakes impossible: a language where every "symbol" would have a unique sign.

[Imagine a tribe who were taught chess by missionaries and who believed that the rule for moving the knight in chess was a law of nature [physics] -- i.e. a matter of knowledge rather than of convention. Can you describe this?]

If in the Tractatus 'clarity' means the absence of ambiguity [a one-to-one correspondence between names and objects named], is there also a selected meaning of 'clear' in the Philosophical Investigations (one chosen out of many others)? In the Philosophical Investigations the concept 'clear' connects up with other concepts, e.g. with the concept 'to understand'. 'To understand' meaning: 'Now I can go on' or 'Now I see what I must do' (PI §§ 151, 123), and the test of that is: whether the person does go on correctly or not. But what 'going on correctly' means may vary from particular case to case. (There is no reason to believe that there is an essence of our concept 'clarity'.)

Suppose someone said: "Maybe that is the meaning of the word 'clear' with respect to others, but how do I know whether or not something is clear to me -- i.e. with what right [justification] before myself alone (i.e. without its being put to the test by other human beings, but alone in my own room as it were) can I say, 'Yes, I understand this'? (Because saying 'I understand' is not like saying 'I have a toothache'; only the latter requires no justification.) If there is no practical test (and in some cases I can conduct that test myself), then am I doing anything more than expressing self-confidence?"

'sign' versus 'symbol'

Query: Wittgenstein and sign.

"A sign is what is perceptible in a symbol" (TLP 3.32; cf. 3.321). "A sign is what can be perceived of a symbol." (tr. Pears & McGuinness) "The sign is the part of the symbol perceptible by the senses." (tr. Ogden) In other words: by the word 'sign' is meant the part of a "symbol" that can be perceived by the senses -- i.e. this is the distinction between a sign and its meaning (use). 'Sign': the physical aspect of language; 'symbol': a sign with a common use [usage, acceptation] [In a few places I have defined the word 'concept' that way, maybe unwisely]. But doesn't "symbol" here suggest a ghostly essence -- as if the "meaning" of a sign could be completely divorced from the sign and allowed to float free? -- And so I would avoid this way of talking.

TLP 3.326: "In order to recognize the symbol in the sign we must consider the significant use" (tr. Ogden) suggests that for 'significant use' could be substituted 'symbolic use'; it makes nothing clearer until Wittgenstein develops the use metaphor [3.328a] in the Philosophical Investigations (and yes, it is a simile: we use a word like we use a tool or like we make a move in a game).

It may seem innocent to say of 'Mr. White turned white' that here the proper-name-word 'White' and the color-word 'white' -- that here the word 'white' is the "same sign but different symbols". But again, I would not say that, because it suggests that a symbol ["signification"] is a ghostly something [a halo that a sign, when it is a symbol, always wears] hovering around the sign [In Wittgenstein's jargon, 'sign' = spoken sounds, ink marks on paper, the purely physical aspect of language]; the word 'symbol' here suggests "meaning as an atmosphere" or a cloudy atmosphere. I think it is much better -- i.e. less misleading -- to talk about a sign and its grammar, because 'rules' suggests something you must do [follow], rather than some gaseous something that you must "mentally perceive" -- and because a rule can be given a physical form, e.g. it can be written down. (Wittgenstein's "symbol" is like the Greek "lekton".)

The Tractatus coincides with the end of WW1, but by the time Wittgenstein had returned to Cambridge University he had radically revised his views about the meaning of language; he no longer held that "the meaning is the object named":

All explanations of meaning are definitions. Proper names do not function differently as symbols from such words as 'and'. Their meaning is given by their use, by the rules applying to them. (Wittgenstein's Lectures, Cambridge, 1930-1932, ed. Desmond Lee (1980), p. 112)

Wittgenstein showed why the meaning of a name cannot be "the object the name stands for". And if that cannot be its meaning, then the question is: what is its meaning? That [its meaning] is, in Wittgenstein's revised "logic of language", given by rules for using the name -- or word, because words are no longer all presumed to be names --; or, in other words, meaning is given by what in Wittgenstein's jargon is called grammar.

"Ideal language", "Reform of language"

If the meaning of a word is not an object named, then the program of inventing a unique-sign-symbol-relationship language no longer appears to be a simple project. That notion was based on a too simple picture of the way language works [of its "logic"], on a picture such as Augustine imagined.

Making a distinction between a sign and its meaning remains essential to logic and therefore to Wittgenstein's philosophy. But the Tractatus's project of inventing an ideal -- i.e. an 'ideally clear' = 'unambiguous' -- language does not.

We want to establish an order in our knowledge of the use of language: an order with a particular end in view; one out of many possible orders; not the order. To this end we shall constantly be giving prominence to distinctions which our ordinary forms of language easily make us overlook. This may make it look as if we saw it as our task to reform language.

Such a reform for particular practical purposes, an improvement in our terminology designed to prevent misunderstandings in practice, is perfectly possible. But these are not the cases we have to do with [in philosophy]. The confusions which occupy us arise when language is like an engine idling, not when it is doing work. (PI § 132)

... the task of philosophy is not to create a new, ideal language, but to clarify the use of our language, the existing language. (PG i § 72, p. 115; cf. TLP 4.112 on the aim of philosophy, according to Wittgenstein)

In the Philosophical Grammar it is not language that needs to be reformed, but our categories for understanding language -- that is to say, our "parts of speech" -- that need to be reformed. Or again: it is not natural language that needs to be reformed, but our way of looking at it. "The logic of our language is misunderstood" (TLP, Preface, tr. Pears & McGuinness), and therefore what needs to be reformed is not our language but our understanding of the logic of our language.

The purpose of Wittgenstein's investigation of language is to resolve philosophical problems. But in order to resolve those problems it is necessary to "bring words back to the language-games that are their original homes" (PI § 116). But in order to do that we must first identify their original homes -- and that means we must describe our language as it actually is, not as it might be "reformed" to be. Given that the concepts [-- i.e. the rules for using words; the word 'concept' replaced the word 'symbol' in Wittgenstein's jargon --] of natural language are fluid -- i.e. natural language does not for the most part consist of language-games played according to strict rules --, were the philosopher to try to invent strict rules in order reform [i.e. disambiguate] our language, the philosopher would misrepresent [distort] our language -- i.e. to give a false account of its actual grammar [the grammar of the language of our everyday life].

Query: is Wittgenstein for or against the creation of ideal language?

If all words were names and the meaning of a name were the object the name stands for, then an "ideal language" would be one in which "one and only one name stood for one and only one object", in which case there would be no ambiguity in that language -- i.e. (1) no name would name more than one object, and (2) one object would not be named by more than one name, and (3) there would be no name that did not name an object, and (4) there would be no object that was not named. However, all words are not names and the meaning of a word is never "the object it stands for".

Cf. the sword Excalibur (PI § 39): if the meaning of the word 'Excalibur' were the object it names, then if Excalibur were broken in pieces, the word 'Excalibur' would thenceforth be meaningless; and therefore the sentence 'Excalibur has a sharp blade' would thenceforth be meaningless (nonsense), because that sentence would contain a meaningless word.

The word 'nonsense' in the TLP

The right method of philosophy would be this. To say nothing except what can be said, i.e. the propositions of natural science ... and then always, when someone else wished to say something metaphysical, to demonstrate to him that he had given no meaning to certain signs in his propositions. (TLP 6.53, tr. Ogden

[Note that, because the propositions that "can be said" are nothing more than strings of names of simple objects, they don't much resemble what we normally call 'the propositions of natural science'.]

But, instead, words are used in many different ways; their meanings are explained in many different ways. Thus Wittgenstein's original "ideal language" idea lost its point once he set aside his earlier false view of the logic = "grammar" (in his jargon) of our language for his post-Tractatus view.

Query: Wittgenstein, Tractatus, religious language is meaningless.

For Wittgenstein 'meaningless' = 'undefined', but there are many meanings of the word 'meaning' in our language, not just one. In the TLP, where all words that are not "nonsense" are names of objects, 'undefined word' = 'a word that names no object', because in the TLP 'meaning' = 'object named' [or more clearly: 'meaning' = 'physical object named', because the TLP account of language has no place in it for names of abstractions ("abstract objects")]; and thus to say that a word does not name anything is to say that it is "meaningless" -- in that sense of the word 'meaning' (if there is such a sense of that word: Wittgenstein's remark about 'Excalibur'), and thus that the language of values and religion is "meaningless" if its words are not names of objects.

But to go on to say that the language of values and religion is therefore absolutely meaningless -- because there is only one true meaning of 'meaning' -- is another thing entirely. As Wittgenstein later defined 'meaning' (in contrast to the TLP's metaphysics where he "defined meaning" or said "what meaning really is"), religious language is not meaningless. (As the Vienna Circle "defined meaning" -- its "real definition of meaning", that is, not how it chose to define the word 'meaning' -- religious language, because it does not consist of verifiable propositions, is "meaningless". But Wittgenstein never accepted the Circle's "verification principle" as a universal test of meaning.)

For later discussion, see The meaning of 'nonsense' in the TLP.

"Must there be a reason?" versus "Can there be a reason?"

Query: why the sky is blue, James Jeans.

I don't know if this query concerns the physicist James Jeans; and what he wrote or might have written about the color of the sky, I also don't know. Do you think there must be a reason why the sky is blue? Suppose we responded, in the manner of Drury's examiner (The Danger of Words (1973) p. xi): "Dash it all, the sky has to be some color; so why shouldn't it be blue." Why are we disinclined to say that in this particular case? I think: because in this case, this is a question which, given what has already been discovered about the refraction of light, we are in a position to answer [i.e. where what is to count as an answer is defined]. Whereas in the case of "Why is that bird sitting just there on that telegraph wire?" we are not.

Our question might be taken in two different ways. On the one hand, someone might answer: The sky is blue because of the angle of refraction; in this case there is a technique [method] for answering the question: a method of verification has been defined. But on the other hand, someone might reply: No, but why is just that the angle of refraction rather than some other? In which case, the only reply would be: "Why is the sky blue? Because it isn't green!" Which is a rejection of the question, as if to say: not for any reason [because there is no defined technique for giving a reason here], because it is not clear what you are asking, unless it is in the category of Questions without Answers.

The sky does have to be some color or we wouldn't be able to see it: if the sky were utterly clear [colorless], we would not have the concept 'sky' that we have.

Do you think there must be a reason? "Why is that bird sitting just there and not someplace else?" I don't know if that combination of words has any clearer sense than "Why should the sky be blue rather than brown?" But it would seem strange to me if a scientist did not want to explain why the bird was sitting just there, because seeking explanations without end is the program of science, what science aims to do, tries to do, and wants to do.

Wouldn't the most that Drury could say be: not everyone is drawn -- or exclusively drawn -- to science's program? For example, Drury, although he was himself a medical doctor, was at least as powerfully drawn to the program of religion, where the aim is: "to understand more and more that there is something that you are never going to understand" (Kierkegaard) -- i.e. that not everything can be explained ... at least partly because: here we make, or try to make, a queer use of language, to ask questions for which there is not a defined technique for answering. ["Running against the limits of language? Language is not a cage."]

"Must there be a reason?" There cannot be a reason for nonsense [undefined language].

"... it is easy to be wise after the event," said Holmes. (Conan Doyle, The Problem of Thor Bridge)

There are some aspects of Wittgenstein's work that are indeed easy to be wise about after he has shown them to you [after he has taught you to look at them a new way]. But before they were found [before they were conceived by him], it needed the insight of philosophical genius to see them, however plainly in view they may have been. (Cf. "percepts without concepts are blind" and "A philosopher says: Look at things this way!" [CV p. 61].)

Moore and the skeptic -- after Wittgenstein

Wittgenstein looked puzzled during G.E. Moore's lectures, Moore told Russell when asked what he thought of the young Wittgenstein:

I think very well of him indeed. Because at my lectures he looks puzzled, and nobody else ever looks puzzled.

It solves no philosophical problem to contradict the skeptic. The skeptic says "You don't know" and Moore replies "I do know". What should be said is that the word 'know' has uses in our life, but that both the skeptic and Moore fail to understand the "logic of our language" (what an objective distinction between sense and nonsense rests on).

It would be wrong to say that Moore fails to grasp the skeptic's philosophical problem, as if there really were a philosophical problem rather than a conceptual muddle here in the skeptic's case. Because it is not that questions about the limits of human knowledge are not philosophical questions -- but that the particular case of Moore and the skeptic is not about the limits of human knowledge: it is instead a logical-grammatical confusion about the role of language in our life (The concept 'knowledge' is a tool belonging to a community).

The skeptic must be asked: if nothing can be known, then what does the word 'know' mean according to you? If we use that word the way we normally do -- and how else can we use it (PI § 246) -- then it is incorrect -- i.e. it is a false account of the grammar of our word 'know' -- to say that nothing can be known. For example, there are many things that Socrates knows, e.g. his own name and the name of the city in which he lives: "absolute skepticism" or "absolute ignorance" is not what we mean by Socratic ignorance or wisdom. That is of course simply a "grammatical reminder": we are not disputing any extra-linguistic facts with the skeptic: indeed, we are not disputing any facts with the skeptic: we are simply reminding him of something [namely, grammatical rules] which he is already knows and must admit (Z § 211; PI § 599): Yes, in our normal way of speaking, it is correct to say that Socrates knows his own name, whether he is an Athenian, and whether he has hands.

What the skeptic wants to say is that nonetheless Socrates doesn't really know. (The word 'really' here is a clue that we are dealing with something metaphysical.) That the skeptic is not talking about language -- but about "knowledge" ... although he still owes us a definition of that word ('know', 'knowledge').

Objective versus Subjective Meaning, or, Logic versus "Whatever seems correct"

But how will the skeptic define the word 'know' for us? He cannot for instance point to examples of knowledge in our life -- because according to his account the word 'knowledge' has no application to our life because it is impossible to know anything. But the skeptic feels no need to define that word, because he is subjectively certain that he already knows its "meaning"; -- but he does not define the word 'meaning' objectively [He has no "logic of language" in my jargon]: for him the meaning of a word is a matter of whatever seems right or not right to him. (PI § 258)

We ask ourselves: what is the skeptic talking about: what does he mean by the word 'knowledge' -- does he mean something like the "abstracted essence of knowledge" or "knowledge in itself"? What's that when it's at home? is the only reply logic can make (ibid. § 116): what are you talking about? how do you mean?

If "nothing whatever can be known", then it must always be false to say 'I know'. But if it is always false to say 'I know', then the words 'I know' have no use in our language -- i.e. they are meaningless. A statement of fact cannot have less than two values: either true or false.

If a statement -- i.e. any construction of the form statement-of-fact -- is always true, it is called a 'tautology': "Either it is raining or it is not raining." And if it is always false, it is called a 'contradiction': "It is raining and it is not raining." But what can tautologies and contradictions tell us about reality? Because they cannot collide [come into conflict] with reality, they tell us nothing.

But the skeptic's 'I know ...' is not a formal contradiction. At most it might be a proposed new rule of grammar: Exclude the combination of words 'I know' from the language; -- which amounts to saying: Exclude the words 'know' and 'knowledge' from our language, because of course on the skeptic's account the assertion 'I don't know' is always true.

Now why would we wish to exclude the word 'know' from our language, when that word has many well-established uses? Think of the words 'I know' as a tool human beings use. (PI § 360) Why would we wish to discard that tool? The skeptic's fundamental mistake is not to see that the word 'know' has no meaning beyond the various roles it play in our life. [This the theme of the Fable of The Born-Blind-People]

If 'I don't know' could be applied to all assertions, it would be like a paper hat placed on a chessman's head: a useless addition to the game of chess [In fact because no rule governed its use, it would be no addition at all]. (BB p. 65) It would be like the words 'I am here' -- were it correct to utter those words in any and all circumstances.

Nonsense is produced by trying to express by the use of language what ought to be embodied in the grammar. (PP p. 312)

But "reforming language" -- i.e. making new rules for talking -- is not what the skeptic wants. Instead, the skeptic wants to present a philosophical argument -- i.e. to give reasons why it must always be false to make the assertion 'I know'. But before an assertion can be false, it must first not be nonsense -- i.e. an undefined combination of words cannot be true or false.

[The skeptic's way of philosophizing is an instance of what Wittgenstein called metaphysics's obliteration of the distinction between factual and conceptual investigations (Z § 458), or in other words between verbal and real definitions.]

The skeptic assumes that there is some ideal to which the words 'know' and 'knowledge' could be applied. But he never describes that ideal beyond uttering the words "It would be something I cannot doubt". But what can or cannot be doubted is a question of definitions -- i.e. of defining grounds for doubt -- and the grounds for the resolution of those doubts. A doubt that cannot be resolved is not an objective doubt. The skeptic is only willing to recognize the grounds for doubting: these grounds he regards as his philosophical reasons for claiming that 'I know' must always be false -- i.e. that the assertion of knowledge can never be justified.

But it is nonsense (i.e. undefined language) to assert doubt where there are no grounds for doubt or where investigation has eliminated those grounds. The skeptic wants to say that those grounds can never be eliminated -- but that is to misunderstand the logic of our language, because again: the word 'know' is a tool we use in our life, and it has no meaning beyond that. And it is precisely "the meaning beyond that" that the skeptic wants to make claims about. The skeptic feels he can take language for granted: as if language were the transparent skin of thought, and as if that "thought" were the meaning of language. [Logic's objective meaning of 'meaning'.]

Suppose it were suggested that: the skeptic is willing to allow that there is "relative knowledge" -- such as is shown by the use of words and practical matters of our life, but that "such knowledge" is not what he wants: what he wants is "absolute knowledge". You might say: he has a picture of "absolute knowledge"; -- but he has no such picture: he is willing to describe no conditions whatever that would serve as criteria for saying that something is 'known with absolute certainty'. Which is another way of saying that on his account the combination of words 'absolute certainty' is undefined [and therefore meaningless].

'Absolute knowledge' would be knowledge for which no grounds were needed, because the skeptic claims to doubt all grounds. But by definition -- in the context of philosophy -- 'to know' means 'to have sufficient grounds'. There are no "absolute grounds" because all foundations are ultimately groundless [They are all "absolute" (free standing)]: there is no bedrock beneath the bedrock. And that is what the skeptic "pictures" [The quotation marks mean: it is a picture without application; indeed, it is no picture at all]: the "bedrock beneath the bedrock", the bedrock of the gods or of God. But even the eye of God cannot see nonsense; if God cannot be mistaken, it must be because God has no knowledge -- i.e. that whatever the eye of God sees, it is not what we mean by the word 'knowledge', but must be something completely different.

Again, think of the words 'know' and 'knowledge' as tools that we use -- tools that have a place in our human "form of life", -- not of "knowledge" as some rarefied abstraction one somehow grasps the meaning of in an unobjective way -- i.e. in a way that cannot be stated in public rules, conventions, definitions.

After Wittgenstein, this criticism is easily enough made. But the genius of a philosopher was needed to make that criticism in the first instance: to see things in a new light, by which others could afterwards also see.

"Why are they called undefined terms when in fact we can define them?"

Note: this continues the discussions The meaning of 'undefined' in mathematics, which answers the question why "the three undefined terms" are said to be undefined in geometry, and the Philosophy of Geometry.

In the following section I have used the words 'word' and 'term' -- however, it would be clearer if instead I had used the word 'sign', which in Wittgenstein's jargon means: spoken sounds, ink marks on paper, hand gestures -- in a word, the purely physical part of language. The problem with the words 'word' and 'term' is that they already suggest "meaningful word" and "meaningful term", as if simply being recognizable as an English word guaranteed that a sign had meaning in all circumstances [contexts].

Query: why are they called "undefined terms" in geometry?

Why indeed? I think: Because no one questions it. If a word is undefined [not defined], then isn't that word meaningless? Isn't the meaning of a word what is given in a definition, and therefore if there is no definition, isn't that word meaningless? Why are definitions given? -- There are many reasons, not just one, even in geometry. Just as there are many types of definitions.

We might say that color-words are undefined terms, if by that we mean that they are not defined verbally -- i.e. by stating sign-for-sign substitution rules such as are found in a dictionary. They are instead defined ostensively -- i.e. by pointing to examples of colors (i.e. colored objects). We could not define these words as we normally use them for someone who was born blind or who was born only able to see in black and white. A completely color-blind person could only be told that people who are not color-blind are able to make distinctions that he cannot make. (Let us imagine that he is unable to correlate the distinctions made by color-seeing people to gray-scale -- that for him those who see colors are like the sighted boy in Fable of The Born-Blind-People).

On the one hand, we could say that "If the word 'red' is undefined, then it is meaningless" -- and indeed, from a purely verbal point of view we might say that it is; but to insist that the only type of definition is verbal is to misunderstand the logic of our language. If you want to know the meaning of a word, ask: how are we taught to use this word; how would we teach someone else to use this word? Perhaps we can imagine the Greeks defining 'point', 'line' and 'plane' in Euclidean geometry by drawing figures in the sand -- ostensively, that is. [Although even defined that way, further stipulations may need to be made -- i.e. further rules given for using these words -- because the ostensive definitions can be misunderstood, and have been misunderstood perhaps especially in the case of 'point'. (PI § 28)]

But "drawings are not essential to geometry" and in that sense those words would not be definable ostensively -- because then all ostensive definitions would be excluded from geometry. But on the other hand, I wonder: are we not deluding ourselves by saying that drawings are inessential? -- because the entire point of geometry is the drawings. The proofs of geometry, were there no drawings to accompany them, would be of no interest to us, just as chess would be of no interest to us were there no chessboard and chessmen. I want to say that geometry, unlike arithmetic, is essentially visual. -- But then to what am I applying the word 'essential' here? I am applying it only to our interest in geometry: it is essential to our interest in geometry that there be drawings (at least in Euclidean geometry it is; non-Euclidean geometry is something else entirely because it departs from our visual experience of the world). That is, what I want to say is not a point about the logic of our language, because it cannot be stated in rules that belong to a true account of the grammar of our word 'geometry'.

Query: the 3 undefined terms in geometry, point, line and plane.

If they were undefined, wouldn't they therefore be meaningless? How would we know what anyone was talking about if he used those words without defining them -- wouldn't we say that he was only making incomprehensible sounds? That is to say, if there were no rules -- [a 'definition' is a set of rules (There are many different types of such rules) for using a word] -- for using these terms, then they would have no role to play in geometry: they would be as superfluous as the "king's paper hat":

I want to play chess, and a man gives the white king a paper crown, leaving the use of the piece unaltered, but telling me that the crown has a meaning to him in the game, which he can't express by rules. I say: as long as it doesn't alter the use of the piece, it hasn't got what I call a 'meaning'. (BB p. 65)

Also, remembering that 'term' = 'jargon word', what would it mean to say that a jargon word was undefined -- other than that it was meaningless? Because classifying a word as jargon is to note that it has a use [meaning] that is at variance with our normal, everyday use of that word.

Query: meanings of undefined geometric terms.
Query: undefined terms in geometry and their meanings.

If those terms are undefined, then how are they to be given meanings -- except by defining them (and how strange that we should be able to give them meanings -- because where are those meanings to be sought if not in geometry, the very place where they are said to be "undefined"). If those terms have meanings, then they are not undefined -- or what would it mean to say that a word had a meaning but that the word was nonetheless undefined? Which meaning of the words 'meaning' and 'undefined' would that be? This is an example of where you can say "the logic of our language is misunderstood" because there is more at issue here than merely an eccentric use of the word 'undefined' (although of course there is that); because the notion of "indefinable signs" fosters a lot of confused thinking in philosophy.

[The word 'grammar' in Wittgenstein's jargon: suppose Wittgenstein had told Moore that he was unable to define his use of that word because its meaning was already clear -- wouldn't Moore have had every right to say: "Then Wittgenstein is talking nonsense when he utters the word 'grammar'"?]

Query: geometry, give 3 things which represent plane, line, point.

Plane: flat sheet of paper; line: edge of a ruler (straightedge); point: x marks the spot on a treasure map.

Now what does anyone mean if he says that those three things "represent" planes, lines and points?

Suppose someone says: "I can't show you the thing itself: I can't tell you what the thing itself is directly -- but I can give you a sort of indirect definition of it." And further: "These words are of course names, but I can't show you the things they name. But I can tell you what the things they name are like." [The False grammatical account: How may we account for the persistence of textbook grammar: "a 'noun' is the name of a person, place or thing"?]

Isn't the need for "representations" a confession that the words 'plane', 'line' and 'point' cannot go undefined in geometry -- not if the student is to understand what we are talking about; geometry was not originally a blind calculus, but a visual technique for "measuring the earth": if logically geometry does not require drawings -- and it does not -- so much the worse for logic. "The game not only has rules; it also has a point." (PI § 564) And logic is only about rules.

How can anything "represent" something that doesn't exist -- because doesn't 'represent' mean: 'stand for', or, 'stand in for', 'go proxy'? No, of course, "one-dimensional and two-dimensional objects don't exist" (and a geometric point has no dimensions at all; it is "zero dimensional" -- i.e. a point is in no way it comparable to an object: the word 'point' has a completely different grammar from the grammar of a name-of-object word): everything that exists in our world [in the mind of our form of life -- i.e. in the grammar, or, concepts, of our language] is three-dimensional; but in plane geometry we are only interested in two of those three dimensions, and therefore we make rules to exclude the application of three-dimension words (e.g. 'height' or "thickness") from our definitions of the words 'geometric line' and 'geometric plane'. Of course, in doing this we are redefining those words in a way that makes them not names of objects: what we have done is to radically change their grammars.

If you want the student "to get the right idea", you have to compare "one-dimensional and two-dimensional objects" to three-dimensional objects: e.g. a line is like the edge of a ruler or what we can draw by running a pencil beside a straightedge. -- But that is a comparison, not a representation. What is the difference? If only this, that the word 'representation' suggests one object standing in for another object. But that is not what we have here. Instead: the child [student] must learn three-dimensional grammar before it can learn one- and two-dimensional grammar, because one- and two-dimensional grammar is an additional set of rules added to (or subtracted from) three-dimensional grammar.

[All the queries in this section came from Asia-Pacific. And if one thought like Nietzsche (cf. Human, all too human § 256) one would say: "Teachers who don't think (because that would mean questioning authority, which would be disrespectful) fostering students who don't think (because that would mean questioning authority, which would be disrespectful). A "good student" is one who memorizes and follows rules (set by others)." But first, the situation with respect to the study of geometry is no different in the West than in the East; and second, while it is true that Confucianism values tradition ("The young ones are supposed to listen and not to ask questions"), it is also true that in Confucius there is a profound love of truth. We could therefore not call the schools of Asia Confucian, any more than we would call the schools of the West Socratic.]

Zero-dimensional Nonsense

Note: an earlier version of this page used the expression 'one-dimensional' rather than 'zero-dimensional'. But a line in geometry, having only length (but neither width nor height) is one-dimensional [It is bi-directional, but uni-dimensional]. But because, for the sake of making an analogy, I called location a dimension (and thus said that points are one-dimensional) what I earlier wrote was confusing (and confused). However! I will for the most part let it stand, because the analogy I made does make the grammar of 'point' clearer.

A point doesn't "have position" [is not "an object having position only"] -- a point is a position [a location, an address]. Why does the expression 'one-dimensional' exist then? Because there was a two-dimensional and a three-dimensional and therefore, one thought, there had to be a one-dimensional object as well. Nonsense is created in just this way. Because a point, being like all other noun, the name of an object, had to be an object of some kind: so it was assigned the quality of position only, and having-position-only was dubbed being 'one-dimensional' [See note directly above]. Philosophy was equated with rationalism -- i.e. with "discovering" the real definitions of abstract objects. And in this way self-mystification was created, and in this way mystification of students is perpetuated. It is only if you break with the misunderstanding of the logic of our language that "all nouns are names of objects" that you can de-mystify geometry, because then you will see that the word 'point' in geometry is not the name of an object, not the name of any object: it is not a "one-dimensional object" [nor a "zero-dimensional object"]: rather, a point is not an object at all.

There is of course nothing to stop anyone from calling a location or position a "dimension" [It doesn't "sound English", but it can be done], just as there is nothing to stop anyone from calling time "the fourth dimension". But it is misleading, to say the least, to apply a spatial-object-word (i.e. 'dimension') to non-spatial-object-words (i.e. 'time', 'position'). If we are ever to heal our wounded understanding, if we are ever to begin to understand the logic of our language, we must stop regarding all nouns as if they were names, and when we do that, the expressions 'one-dimensional' [See note at the top of this sub-section] and 'fourth dimension' will lose their charm for us.

Variation. To avoid confusion, the so-called dimensions (of which we have identified five) could be identified by letters rather than numbers, thus: p = position [i.e. location, but we will use 'l' for 'length'], l = length, w = width, h = height, t = time. Note that we would not naturally call p and t dimensions; we might instead speak of "defining characteristics", and thus: points have p as a defining characteristic; lines have p and l; planes have l and w; bodies [or, solid bodies] (to use the Stoic term) have l, w and h; while time has -- t? [Note also that planes, as well as bodies (?), do not have position; because position is relative to something else, and the Euclidean plane is the entire universe: there is no "something else" to give it position; whereas points and lines have position relative to the plane they are locations in.] If we are going to call time a dimension, why not also call location a dimension? Thus: 'point' DEF.= 'an object which has only the dimension of location, nothing else'. -- But is an object with only location an object at all? If we state that an object has no characteristic but location, do we not thereby "define it out of existence", like the hippopotamus in Russell's room (Russell's room has locations, but it does not have hippopotamuses; and no more than "imperceptible hippopotamuses" can you cage locations and take them away to the zoo; so why call them objects [place them in the category 'object'])?

Query: what are the 3 undefined terms in geometry, and what are their properties?

But aren't terms defined if [when] their properties are stated? If you know something's properties, don't you know what it is? What do geometers mean by 'undefined' then?

Query: 3 undefined rules of geometry.

This sea of language suggests countless things [A definition is, after all, a rule, and 'definition' = 'meaning'], above all utter confusion. But it owes us a definition not only of 'undefined' but also of 'rule'.

Which definition of 'definition' does Geometry use?

Query: meaning of definition in geometry.

Of course this is the question we should have begun with. What do we call [are we calling] a 'definition' in geometry? Perhaps in that context it will be clear what it might mean to say that 'point' is an undefined term in geometry. As was said above: if the only type of definitions we want are verbal definitions -- Perhaps we are compiling an unillustrated dictionary e.g. --, then we may in that context say that the color-words as well as sound-words are indefinable. Because this is not to deny that the color-words and sound-words can be defined in other ways -- ostensively or aurally e.g. --, but that those other ways [types of definition] are of no interest to our purpose [or investigation].

Suppose in the context of the Philosophy of Geometry someone said: "There are of course definitions for the words 'point', 'line' and 'plane'; however, because those definitions are never used in geometric proofs, I refer to those words as undefined terms." I would say that someone who defined the expression 'undefined term' that way -- and Note: that would be a jargon definition [a definition belonging to jargon, a stipulative definition contrary to normal usage] -- was courting confusion and contributing to mystification rather than to clarity.

Query: geometry, why are they called "undefined terms" when in fact we can define them?

Why indeed? How can a word be both defined and undefined at the same time? Because there are many things that we call definitions; and also because definitions (which are rules for using words) may play various different roles in games (if we compare using language to playing games). But how may we account for the persistence of the notion of "terms which are undefined and yet not meaningless"? Why does the expression 'geometric point' continue to be defined for students -- i.e. its use explained to them -- as if it were the name of some infinitely small object, when that is not its actual use at all? Because geometry is taught by mathematicians, and mathematicians do not make good philosophers? On the other hand, certainly logicians who speak of "indefinable terms" have perpetuated this fog.

The expression 'undefined but not indefinable'. -- Is this equivocation -- an equivocal use of the word 'define'? Rather than 'equivocal', which suggests that we can say what the two "voices" are, I would use the word 'vague': a vague use of the word 'define' (What is too vague is nonsense). And what of the expression: 'indefinable but not undefined'?

Not only can we define the "undefined terms" of geometry: we also do and must define them if we are going to do anything with them.

Query: describe your room including the three undefined terms of geometry.

Geometry is not anti-rational, but a philosophy of geometry may be.

Provided you are in a box-shaped room: if you look at a corner of your ceiling, you will see three lines lying in three distinct planes [The walls are planes and the seams where they meet are lines] interesting at 90 degree angles. These are Descartes' {x,y,z} axes, and the intersection of these three lines is the "origin" on the Cartesian graph, or, point {0,0,0}.

Question: do you think that the word 'point' is the name of an object at the intersection of the ceiling and the two walls? Do you think that the word 'origin' is the name of an object at that intersection? Then get a ladder and fetch the objects you say are named by 'point' and 'origin' down from that intersection and place them on your desktop.

Mathematicians as Philosophers

Query: meaning of geometry according to mathematicians.

The "meaning of geometry" is a philosophical, not a mathematical, question. Why do mathematicians make bad philosophers? Because mathematicians work by instinct: they rely on intuition when they work, and they can do this because in mathematics your work can be checked at the end of the day against a calculus. But in philosophy there is no such way to check your work: there is no equation that must work out correctly: there is no system of rules to act as a safety net. Philosophical criticism is of an entirely different nature. Someone who works by instinct in philosophy will be satisfied with his convictions, but that is not what philosophy is: someone's convictions.

Or idle conjectures. Philosophy isn't something you do after you have set aside your serious work. It is itself serious work; the most serious.

Query: geometry, what are points made to show?

Locations. Addresses, e.g. on the Cartesian axes: {x,y,z} -- This is a location (cf. a town's latitude and longitude), obviously not the name of an object (although you can self-mystify yourself with fanciful pictures into imagining that it is; even if you do, however, such "pictures" are not the meaning of the word 'point' if 'meaning' = 'use in the language').

A point is a location in space. Or in plane geometry: a point is a location in the plane, an address. This is the best general definition of the word 'point' as it is used in geometry that I can think of (although I still owe you a definition of 'space', for that word is not used in Euclid's system, nor in the Cartesian system). However, that is not by itself an adequate definition of the word 'point' in geometry.

Query: Wittgenstein's own views about the Philosophy of Geometry.

That is what my Philosophy of Geometry does not represent itself as being, because I do not know what Wittgenstein's own views were. In the students' notes titled Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939, Wittgenstein asks the question: "how is it even so much as possible to look for the trisection of an angle?"

It seems that a compass can only be used to bisect an angle -- but what does 'can' mean here: what kind of impossibility is the trisection of an angle? Is it a practical matter: as countless generations of students before you have done, you look but cannot figure out a way to do it? But if it is a matter of logic (i.e. of a system of rules), then aren't you trying to do something illogical if you look for something that the rules do not allow to be done (If you follow the rules, you cannot trisect an angle -- But how do you know that before you look -- And how do you know it after you look)? Do you think that the consequences of the rules you are following already exist before anyone tries to seek them out? If geometry is a calculus, and if by 'calculus' we mean a game played according to the strictest rules (as is chess e.g.), then is what you are doing exploring the consequences of the rules of the game (As in chess: is checkmate possible using only the King and a Bishop?) Can you try to prove that 7 + 5 = 13? (Note: the reason the trisection of an angle is impossible using only a straight-edge and compass belongs to mathematics: no power of 2 is divisible by 3. "So, 2, 4, 8, 16 ... But how do you know that the nth power of 2 will never be divisible by 3?" I don't know. Maybe there is a proof of that; I certainly don't know how to answer that question.)

Query: empiricism, chess.
Query: rationalism, empiricism, chess.

You say: everything [every possibility] is pre-determined by the rules of the game, and yet you still have to discover what the rules make possible: is checkmate possible from this position in five moves? Is that search-discovery to be called empiricism? How is it different from physics? Are there not "laws of nature", then, which predetermine what is possible? We believe so -- i.e. we believe in the uniformity of nature: but unlike in chess, where the laws are conventional [i.e. more or less arbitrary rules, adopted at our discretion], the "laws" of physics attempt to be a reflection of nature [which is independent of human conventions], although the theories of physics in fact consist of a selection of the facts and a point of view; they are always subject to modification, revision, replacement, reconception, in the face of new evidence or anomalies.

Is there anything equivalent to a scientific theory in chess? A principal difference would be that the possibilities of chess are demonstrable once and for all time: unlike in physics, discovery in chess comes to an end. Or does it? In the particular case it may; although questions about chess -- i.e. about what the rules of the game of chess allow --, e.g. whether checkmate is possible from such-and-such position in such-and-such number of moves, would seem to be countless [although I have no idea how this question looks from the point of view of a calculating machine].

If I make a negative discovery in chess -- i.e. if I find that the rules disallow such-and-such a possibility, e.g. that checkmate is not possible in four moves from such-and-such a position, -- how do I know that I have not overlooked possibilities [that my research has been exhaustive? Objective doubt requires objective grounds for doubt, but how do I know that I have not overlooked any grounds]? I want to say: "Chess is like a simple mechanism -- i.e. one whose parts are easily surveyed, like the gears of a clock" [There are only 32 chess pieces deployed on 64 squares, and the rules for their movement could not be more strict (How do I know that if I obey the rules of this game, then I will never find myself in an ambiguous situation with respect to what is or is not allowed?)] -- i.e. that the rules of chess can clearly show you what is possible -- i.e. that as a matter of experience these possibilities are easy to survey, at least if they are shown to us one move at a time [step by step].

Proving a negative, e.g. that the trisection of an angle is impossible: how do I know when to stop looking? What justifies my stopping? What, if anything, justifies my looking in the first place: am I simply acting in the confused way of an irrational person? Suppose someone tries to walk through a wall of the house: he tells us, in effect, that he is simply following an analogy: there is a front door and a back door, so there must also be a side door. Following analogies. -- Is trying to find the trisection of an angle any more irrational than asking for "the location of the mind" -- because both are a matter of rules?

How do you know what is possible [i.e. what the rules allow] before you look? -- And how do you know what is possible after you look? I.e. what does justification look like in the case of chess -- i.e. in the case of rules of a game? And how is it different, if it is different, from justification in physics? We say: chess is not "answerable to experience" -- by which we mean: to any facts external to the game of chess itself; and in this respect it is unlike physics. But on the other hand, that the rules of chess make this-or-that possible is a matter of experience. If this metaphor pleases you, say that chess is a self-contained world, a world whose laws are fully known, and in this respect it is unlike the natural world (God knows all the laws of nature and to Him the natural world is as easily surveyable as is the game of chess for us); but I would avoid this metaphor because of its equivocal use of the word 'law' if we do not introduce the picture of a deity for whom "natural law" = conventional law (i.e. rules of a game).

If such remarks strike us as too vague, then how are such questions to be answered if not by considering specific examples? -- But of course that is exactly the way the philosopher wants to answer such questions: i.e. not by carefully considering examples but by relying on his insight alone. We must cure ourselves of this laziness if we want to do worthwhile -- i.e. deep -- work in philosophy:

A philosopher's work can only be as good as the examples he brings forward.

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