Wittgenstein's Logic of Language (Background) | Bibliography

Abstract Chessmen, Philosophical Jargon, Grammatical Jokes, Geometric Points

These are logic of language ("How is sense distinguished from nonsense in philosophical problems?") remarks, which were suggested to me by the Internet searches that directed (or misdirected) visitors to this site.

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Standards and Definitions

Philosophy and Jargon

Query: is there a way to learn philosophy without getting any jargon involved? Philosophy in easy language.

In general, philosophers invent new concepts or revise concepts that are common currency; these inventions or revisions are necessarily jargon. (Wittgenstein's revision of the concept 'grammar', for example.)

Query: ways to understand the difficulty of philosophy.

If the tool (i.e. a concept) a philosopher needs to do some work does not exist, then he must invent a new one or adapt an old one, as e.g. Wittgenstein's word 'language-game'. Then others must learn to use that tool as well, if they are to understand the philosopher.

As to the difficulty of trying to think philosophically, the difficulty is to stay intensely focused on an amorphous problem, not to let your mind wander off to restful things. Wittgenstein likened it to trying to swim underwater (Malcolm, Memoir 2e, p. 47): you must not let yourself come up for air. It may be that thinking in many other subjects demands intense focusing as well; but philosophy has some peculiarities maybe not found elsewhere -- because philosophy calls everything into question, including the meaning of the question itself, and philosophy itself, and the philosophizer himself.

A philosopher believes that he has managed to break out of the paths or grooves our thinking commonly runs in (Z § 349), to see things in a new way. And sometimes he will give expression to his new view in new language (jargon).

But is jargon necessarily simply an abbreviation for what can be restated in everyday language by means of lengthy explanation? Mustn't it be -- as it takes many pages to "translate" Wittgenstein's jargon-word 'grammar' into familiar concepts, but it can be done? Compare "a metaphor that cannot be restated in prose" -- is not a metaphor; in philosophy, it is, instead nonsense ("absolute sound, without sense"), I think.

Even with an explanation one needs, nonetheless, a long time to learn to think in the new way: what a philosopher has learned is difficult to learn.

Wittgenstein told Malcolm: "Thinking is digesting" (Memoir 2e, p. 37), and a new philosophical idea is an extremely complex carbohydrate -- it needs ruminating, as the effect of not chewing its cud on the health of a cow is the same for the health of the human understanding.

"If a philosopher has seen something new, then maybe it won't be possible to restate what he has seen in jargon free, familiar language." -- But now, however, what is an example that demonstrates that impossibility? The language of Wittgenstein's Philosophical Investigations doesn't translate back into the eccentric language (jargon) of his Tractatus Logico-Philosophicus.

To speak in ordinary comprehensible language

Technical expressions are a danger for every system of philosophy ... For they may become formulae which hinder the natural development of thought in the same way as ruts in a road hinder traffic. So to find out what are its real contents it is reasonable to test a system of thought by setting aside the expressions which it has coined for its own use and compelling it to speak in ordinary comprehensible language. (Albert Schweitzer. Indian Thought and its Development [1935], tr. Russell (1956), Preface, p. ix)

That is what I tried to do in my explanation of the meaning of Wittgenstein's revised or limited concepts 'grammar' and 'logic' and 'meaning', although I don't know if all philosophical jargon can be rendered as "a simple story in words of one syllable", although maybe it can. But a simple story may not always be a short story.

The question isn't who is going to do the rendering of jargon, "compelling it to speak in ordinary comprehensible language", but who is going to judge whether or not the rendering is correct, with no loss or change of meaning?

Reading Philosophy in Translation

Imagine two languages. Language A has only the words {one, two, three, four, five} or {1, 2, 3, 4 ,5}, and Language B only the words {six, seven, eight, nine} or {6, 7, 8, 9}. [Numbers are words.] Suppose a translator wants to translate Language B's '7' into Language A; he may choose '5 + 2' or '3 + 4' or '2 + 2 + 2 + 1' or etc. But does '5 + 2' have the same meaning as '3 + 4'?

Look for what are called 'explanations of meaning'. (PI § 560)

However, suppose that between two languages, e.g. classical Greek and modern English, there are important words ("concepts") that do not have simple equivalents, as is the case with e.g. the words 'beauty', 'wisdom', and 'justice'. Translators use the English word 'justice', but it seems clear that Plato would not, as indeed at a certain point it becomes difficult to determine what he is talking about in the in Book One of his Republic. Whatever the Greek concept that is the subject of that dialog is, it does not seem to exist in English.

Will '5 + 2' always serve just as well as '3 + 4'? To begin a philosophical investigation of that topic we need to imagine [describe] examples of where it may not.

If we compare the many translations of Plato [Five examples of Apology 21a-d], we see that the translations say similar things -- but they seldom say identical things. Somewhat, although not exactly, as there are many ways to translate '7' in the example above, there are also many ways to translate Plato's words from Greek into English. And, thus, I think: when you read philosophy in translation, you get only "a general sense of the original"; you cannot push the particular wording too hard [far]. We say, "We are reading Descartes' Discourse on Method in class", whereas we are only reading Descartes as translated by this or that translator.

"Plato was saying something like or similar to what you read in any given translation; but he was not saying exactly what you read in any translation, so don't try to make too much of the particular words you find in the translation." -- Is that correct, or what is a clearer way to say that? Do we look at philosophers in translation as if we were looking through gauze? When I write about the Philosophical Investigations, ought I to say "Wittgenstein according to Anscombe"?

[Cf. Confucius: "the importance of naming things correctly" if one is to think what is true.]

Is the chess-king a chess-king?

Query: abstract objects, chess.

Here is a false grammatical account: "This piece of wood is not a real chess king (which is an abstract object -- i.e. one that cannot be seen); the piece of wood only stands in for one; it is like a word." -- And therefore chess pieces are not like words -- in every way [all respects] -- i.e. a simile's application is limited; otherwise it would not be a simile, but instead an identity ['A is like B' does not equal 'A is B']. The resemblance -- "the comparison [analogy] Wittgenstein made -- consists in both [the use of] a word and [of] a chess piece being governed by rules. But the word 'king' or ('chess-king') is not a king, whereas the chess piece is a king.

Query: abstract chessman joke; Wittgenstein.

Chess matches are really battles fought in the heavens, in the invisible realm; the chess pieces are mere tokens of this. That is the "abstractness" of mathematics.

Query: numeral, metaphysics.

Exactly. [Why don't we speak of negative numerals?]

Can you play chess without the king?

Query: what would geometry be without points?

What would chess be without the king? Is that the apt comparison in this case: can you play chess without the king? "Chess without the King": objective: to capture [take] as many of your opponents' chess men as possible. But what would geometry be without points? Can you define the word 'line' without reference to points: e.g. "No two lines in the plane have more than one point in common"? Can you do geometry without points?

Are points in geometry like the king in chess or is that not the apt comparison? Is the plane the chess board? Can you play chess if the board is not divided into squares? In chess each square is an address; in geometry the points are addresses in the plane; can you play chess or do geometry without addresses? What does 'can' mean here -- logically possibility of course, but the question here is: are some rules (or "ground rules") essential to the game (making it impossible to play the game in their absence)?

You can describe chess without the king -- i.e. make up rules for playing without that piece [chessman]; but can you make up rules for playing/doing without addresses? And if the only limit is concept-formation (imagination) ... but is that the response: you can play any game that you can invent followable rules for playing, and so you can play chess and do geometry without addresses -- if you specify how (PG i § 82, p. 127). [But that response seems too facile, and I am wary of it because I suspect I'm missing the deeper question (CV p. 48, 62). On the other hand, I have been using the definition 'point' = 'unique address in the plane' for quite a number of years now, which suggests that if I am blind to what is deeper here it seems I shall stay that way until the end.]

Another word for 'address' is 'direction' as in 'directions [to a place]'. Every point in the plane is defined relative to all the other points in the plane, and thus they provide directions to -- where to find -- any individual point. (Even without a city map, a particular street in the city might be found, if one wandered about long enough. But there is no equivalent to this in the geometric plane because no point in the plane is absolute, whereas all addresses in the city are.)

Geometric Points, again and again

Note: these notes continue the main discussion in the Philosophy of Geometry of the conceptual confusion caused by a false grammatical analogy in that subject.

Grammatical Jokes

Query: how many dimensions does a geometric point have?

This apparent question -- i.e. it appears to be a question; it is syntactically in order -- is nonsense, i.e. an undefined combination of words, mere babble. Only objects have dimensions, and the word 'point' in geometry is not the name of an object. And, no, a point is not an object of "zero dimensions", as if the word 'point' were the name of a mysterious intangible object. -- The expression 'zero dimension' is an undefined combination of words; it does not follow that because the combinations of words 'two-dimensional' and 'three-dimensional' have uses (meanings) in geometry that 'zero dimensional' must also have such a use. (Why not also 'minus-two-dimensional'? -- do you think those words must have meaning as well?)

One dimensional = zero dimensional, a position without substance, empty space

We could say -- although it would be better not base the grammar of the word 'point' on this model of thinking about language at all (since that model is the source of all the confusion in the philosophy of geometry) -- that a point is an object of one dimension, but only if we understand that a point's "dimension" is its unique address in the plane (But even using the word 'it' here is misleading). An address indicates a location -- not an object at the location. An object of one dimension would be an object without extension -- i.e. it would not be an object. And thus if points are one dimensional, then the word 'point' is not the name of an object.

With respect to "object-ness", there is no difference between an object of one dimension and an object of no dimensions -- i.e. in neither case are we talking about an object. A point's one dimension is location. A geometric point isn't an atom (not even a metaphysical, i.e. mythical, atom, an ultimately "uncuttable" object, so small that it cannot be divided even in half). A point is "neither a something nor a nothing" (PI § 304), but instead the word 'point' has a use in the language of geometry, although that use is not as the name of an object.

One dimensional = zero dimensional, if a dimension in geometry is a dimension in space, i.e. if the word 'dimension' is not used metaphorically (as e.g. "Time is the fourth dimension").

'The word point names an abstract object' is a classical example of what Wittgenstein means by the expression 'grammatical joke' (PI § 111). Other examples are 'What color is the number 3?' and 'Why is a circle round?' and 'What does a negative number of things look like?'. As well as this Weather report and forecast: "Today's gloomy weather is being caused by clouds and rain. Tomorrow's gloomy weather will be caused by bright sunshine." The second sentence makes clear the grammatical joke that the first sentence is.

Another example is the vague picture "Reality is really only atoms in empty space" (Arthur Eddington). A doctor says, "There are cancerous cells in your body ... But don't worry -- The body's cells aren't really real; and there are no cancerous atoms." A conversation between two non-realities, because, of course, at the atomic level of investigation, human beings don't exist.

Through the Looking-Glass

In Lewis Carroll there are many examples of grammatical jokes, as e.g. when the White King mistakes the word 'nobody' for a proper name ("Who did you pass on the road?" "Nobody," said the Messenger. "Quite right," said the King, "this young lady saw him too"), and the White Queen's "The rule is, jam tomorrow and jam yesterday -- but never jam today" (but the grammars of 'yesterday', 'today', and 'tomorrow' are interconnected: if there is never jam today there can't have been jam yesterday, nor can there be jam tomorrow) .

And, pace the same queen ("Why, sometimes I've believed as many as six impossible things before breakfast"), it is not grammatically possible to believe a proposition that is simply an undefined combination of words, no matter how much, as the Queen suggests, one "try harder" (Indeed, it is not even possible to try). In other words, although it is possible to believe a real impossibility (e.g. "Theaetetus flies" (Plato, Sophist 263a)), it is not possible to believe a logical impossibility (e.g. that an effect comes before its cause, as the Red Queen demands: "No, no! Sentence first -- then verdict", as the rules of syntax allow her to), no more than one can do something logically impossible, as e.g. draw an invisible elephant.

The rules of syntax allow Mearns' poem, "Yesterday, upon the stair, I met a man who wasn't there. He wasn't there again today. I wish, I wish he'd go away."

Query: if philosophy originated from Greece, where is philosophy now?

Is that an example of following a misleading grammatical analogy or syntactic model, and therefore also of a grammatical joke? Philosophizing comes into being in Greece, but if man abandons Greece, moving elsewhere, he does not "take philosophy with him", as if 'philosophy' were a name like 'hat' or 'coat'.

And yet we could say that man does take philosophy with him, e.g. "when he goes for a carriage ride" (John Locke), and indeed wherever man reasons on the model of the Greeks -- that is, by the natural light of reason alone in logic, metaphysics, or ethics -- that is where "philosophy is now". And so whether the query is nonsense or not will be determined by the meaning we assign or try to assign to it.

A 'point' is 'a unique address in the plane'

Query: real life example of a geometric point.

Suppose a teacher gave this command to a student: "Go over to that point on the other side of the room and bring it back to me"? If 'point' were the name of an object, then that combination of words would have a meaning in our language, which of course it does not: No one can bring you back the "point" on the other side of the room (because that is an undefined combination of words). The student could only reply, "I don't understand: I don't know what you want me to do". But the command: "Go over to that point on the other side of the room" is a "real life example of a geometric point" -- i.e. of how we don't use the word 'point' in geometry, because we don't use the word 'point' in geometry the way we use the word 'book' in the command "Walk over to the shelf and fetch the geometry book back to me."

If you want to understand how we use the word 'point' in geometry, examine how we use the words 'address' and 'location'. A point DEF.= a unique address in the plane defined relative to other unique addresses ("points") in the plane.

Maps (Mercator projections) show longitude and latitude lines (rather like a Cartesian grid). If we say, Consider the line through co-ordinates {50° South, 130° West} and {40° North, 40° West}, do you imagine that at those points there are objects of some kind called by the name 'point'? (If there were, they would be quite waterlogged!)

Euclid's rules state, Through any two points in the plane there is one and only line. But those "points" are simply co-ordinates of the plane; they are addresses, not objects, geometric or otherwise.

If we use graph paper, do you imagine that at the intersections of the grids there are objects named 'points'? If you board a train scheduled to stop at "... and all points north", do you imagine there are objects named 'points' at those railroad stops? These questions are absurd -- and that is their point: to make the grammatical joke here explicit.

The difficulty in philosophy is to say no more than we know. (BB p. 45)

And metaphysical hypotheses are not what we know. And indeed, in geometry -- they are not merely superfluous conjecture -- but are instead the expression of the mystified understanding trying to "grasp hold of ghosts" (PI § 36) because it is under the spell of this misleading picture of the logic-grammar of our language: Words are names and the meaning of a name is the thing the name stands for.

Suppose a teacher asks a student to read aloud from the textbook, but interrupts the student with the words "A this point we'll stop". But if you look at textbook's page, you will not see any object named 'point' at that point.

"That's because points have no dimensions." -- Curious it is then that we are able to point to something that cannot be seen (for what "has no dimensions" is invisible).

"But I can use some ink to mark the page to indicate the point where I stopped reading." -- But are you saying by this that: "The ink mark represents an invisible object named 'point', and although I cannot see that object, I somehow know where it is"?

What you know is the address of the place where you stopped reading. Nothing more.

Certainly this is absurd; it has always been absurd -- and that is why countless generations of school teachers have taught children to ... misconstrue the grammar of the word 'point' in geometry, replacing logic of language with belief in points. (But geometry is not a religion.) And thus we have the factitious, hypostatized points of geometry.

Philosophy and the teacher of geometry

Query: what philosophy of mathematics is best for teaching plane geometry?

One that is isn't based on a false description of the logic of our language, i.e. one that does not treat geometry's nouns as if they were the names of "abstract objects". That remark is for the Philosophy of Maths. But in the sense of 'philosophy' = 'policy' or 'principle', then not this one: If the brats don't understand, it's not because my explanations are unclear (or nonsense) but because the stuff I am trying to explain to them is inherently difficult to understand.

Query: undefined mathematical object.

"Certainly I don't understand this. No one has ever understood this." What you can say is that the query suggests a picture to you -- But it does not tell you how to apply that picture. What is a mathematical object (undefined or otherwise) when it's at home? What is the picture -- is it of a triangle, e.g., positioned in a realm on "the other side of the sky", some Platonic Form in the realm of shapes? But is triangle a Form or do all shapes have a common defining quality or nature (Meno 74d, 72c) that is their Form? For if the latter is the case, then there is no picture of any shape for the combination of words 'mathematical object' to suggest, not if by the word 'picture' we mean what we normally mean by that word (i.e. something drawn e.g. on a canvas).

If we use chalk to draw a triangle on the blackboard, that drawing may be called a 'mathematical object' if we like -- but can the lines of that triangle also be called mathematical objects, because they are not the lines of the definition of the word 'line' as that word is used in plane geometry (e.g. the lines on the blackboard have width and color)? And towards what end are all my remarks made? -- Towards breaking the stranglehold of a false account of the grammar of the language of geometry.

Query: witch is defined using the undefined terms 'point' and 'line'.

Given that they are all imaginary, although it seems that geometry's lines do correspond to something not-too imaginary -- (e.g. the edge of a straightedge, whereas neither 'point' nor 'address' does; addresses are found in a telephone directory, but the word 'address' does not mean '27 Lime Lane', which is an example of an address, but has a different use in our language; cf. the demonstrative 'this') -- but witches do not.

"Correspond ..." On the other hand, lines within geometry are not straight or not-straight. The circumference of a circle is neither curved nor not-curved. These words have no visual meaning inside geometry, where there is nothing to see; where there is nothing to see, words have no visual meaning. A strictly one-dimensional object does not correspond to anything, even to something imaginary, as the witches of fairy tales do. Believing that geometry's points, lines and planes have a physical presence "somewhere" is like believing that there really are witches (living beings to drown or burn at the stake or otherwise torment) in that both are beliefs to be disabused of, although only one of the two, namely "geometric objects", is a conceptual muddle.

Query: geometry. What is another name for points?

The best would be 'addresses' or 'unique addresses in the plane'. In geometry the word 'point' = the word 'address' (Those words have the same grammar).

Points invisible

Query: when is a point in geometry visible?

When we die and go to Geometry heaven, where we shall meet Frege and the young (but not the old) Bertrand Russell. However, if we use the word 'point' the way we normally do, or maybe only to invent some meaning for the words of the query: A point in geometry is visible when, instead of making his point in words alone, your instructor makes his point with a drawing on the blackboard. (Cf. Wittgenstein's invention of a use for the combinations of words 'the agent of thought' and 'the location of thinking': The activity of thinking "is performed by the hand, when we think by writing; ... and the locality where thinking takes place ... is the paper on which we write ..." (BB p. 6-7))

I wonder if Lewis Carrol's "Snark" was not a geometric point, for that can also be hunted, apparently, although it can never be found, like a real world image of a geometric point. (I have not read that book but I will imagine that is its topic.)

Words that are not geometry's jargon

Query: the three words in geometry that do not have definitions are?

That is, that do not have definitions specific to geometry. That is a correct grammatical account for the word 'point', but is it also for the words 'line' and 'plane'? We define the word 'plane' as 'a two dimensional surface' (just as we define the word 'line' as 'a one dimensional extension'), do we not?

Note that the surface of a sphere is not two dimensional. Visually: anything two dimensional is flat; however, the word 'flat' is an undefined word in geometry -- i.e. it is a word we apply only to drawings we use to illustrate geometry. (Cf. a one dimensional extension has no width; however, the word 'width' is also undefined in geometry.)

The difference between 'point' and 'line' and 'plane' is the we normally use the ... -- do we normally use any of those words to name an object? Yes, all three, e.g. in a drawing on a sheet of paper (as described in our earlier distinction between a point and a "point marker"; so that we may now speak of "line markers" and "plane markers" as well as "point markers"). However, that is not the use of those three words in geometry: In geometry all three words are merely the addresses of locations in space.

Note that the word 'space' is undefined in Euclidean geometry, because what would its meaning be there other than 'the plane', because the plane is the only location in Euclid: What is the case (reality, as it were) is the existence of addresses in the plane;and outside the plane there are no locations at all. In the Cartesian plane an address is named by three axes {x,y,z}, but, again, the word 'space' is undefined within that system. When we speak of two-dimensional space (Euclid) or three-dimensional space (Descartes) we are talking about ways we can apply geometry in the world of our experience, which is the only place where the word 'space' is defined -- i.e. has its uses. Rather than calling 'point', 'line' and 'plane' by the more than a bit confusing title "undefined terms", they might be called 'contact with experience' terms.)

Query: which undefined geometric term is described as a two dimensional set of points that has no beginning or end?

A "set" that has no limits? The query's "description which is not a definition" (because school teachers say it isn't) is of 'geometric plane'. What would 'set' mean in this case? I think that: 'A plane as a system of unlimited addresses, a "system" because once one point is named all other points in the plane are named based on their relationship to that point'.

Geometry and the "wounded understanding"

Query: difficulty in making definitions for geometric terms.

The difficulty does not lie in geometry but in our confused understanding (Kant's "the wounded understanding") of the philosophical grammar or logic of our language: we define words, not "things"; our failure to understand that is the only "difficulty" here.

Query: mathematical difference between defined and undefined terms in geometry.

If in geometry definitions of words are needed only if the definitions are used in geometric proofs, then there may be many words that are "undefined" in geometry, not only the names of the three basic elements.

Query: undefined definition in math.

Is the word 'definition' defined in geometry: for does it ever enter into a proof. Indeed, is 'undefined term' defined in geometry, for it also is never used in any geometric proofs. And that is the key insight -- that dismissing Euclid's definitions of 'point', 'line' and 'plane' in favor of calling those words "undefined" is done for the sake of orientation only, and consequently it should never have been done at all: for geometry is about rules (grammar); it is not metaphysics (a shadow reality in which abstract objects dwell).

A definition for the word 'point' is needed by geometry so that we know what we are talking about -- or, in other words, which neighborhood and street, if not the specific house, we are discussing.

But, as you can see, our very form of expression betrays us by presenting us with a false picture of how our language works: "what we are talking about", as if that "what" had to be an object of some kind, as in the following false picture of how our language works: "All nouns are names, and the meaning of a name is the object the name stands for."

Query: an undefined term in math is impossible.

What kind of impossibility would that be? Whether possible or impossible is dependent on how the word 'undefined' is being used (i.e. defined) in maths. What we might say is that, it is impossible for there to be undefined words in maths -- if we need to know what we are talking about (in contrast to uttering noise = babbling nonsense) in maths; however, it may not be necessary for us to know what we are talking about. (Cf. What are numbers?.) Blindly following rules contrasts with Understanding what one is doing -- but in either case, one nonetheless follows the rules, because otherwise one is doing maths. That's what mathematics is for most of us (What it is for the mathematicians who invent new rules, new calculi, I don't know).

Query: describe a point in geometry.

Compare: "Describe a the." What we do in is case is to describe the use of the word 'the', and the definite article is not the name of any object or phenomenon. The same is the case with the word 'point' in geometry. (Why can't we draw a point in geometry?)

Query: what is the word 'it' in grammar?

The word 'it' (as in the example "Is your friend at home?" -- "It seems not." -- i.e. in cases where the word 'it' is not a pronoun [not that a pronoun is a common or proper name, either]) is often like the word 'the' -- i.e. although it is not the name of an object or phenomenon, that fact about our language is simply set to one side/disregarded by the account of language meaning "All words are names and the meaning of a name is the object the name stands for". Question: But what if there is no object [or phenomenon] for the word to stand for? And may that not be very often the case?

Query: negative effect on the foundation of geometry of the three undefined terms.

We begin with a geometry that is not 'geometry' = 'measuring the earth', because in geometry 'point', 'line', and 'plane' are not names of objects on earth (Not that they are objects in Heaven either). The three "undefined terms" name rules for using those words within geometry, not outside geometry in our world.

Query: negative consequences of having undefined terms in geometry.

As some philosophers of geometry use the term 'undefined term', the negative part comes at the beginning and is not consequent to the notion of there being "undefined terms" in geometry -- and that negative part is "grammar stripping" by on the one hand claiming that e.g. 'point' is the name of geometric object, but on the other hand claiming that it is an object of no dimensions. Thus the only negative consequence is the intellectual confusion [self-mystification] that comes with all grammar-stripping.

Query: meaning of point, line, plane.

Misleading form of expression: 'a line is made up of points'. "But, then, if a point has no dimensions, then a line must also have no dimensions." That does indeed appear to follow from the expression 'made up of points' -- but if and only if the word 'point' is the name of an object.

Query: invisible points in geometry.

"The meaning of a word is the object the word stands for, if not a visible object, then an invisible one." -- If you do not break free from that picture you will never understand the logic of our language, and therefore you will never see the world aright, but you will always be the victim of language. [Seeing that (and only that) was the achievement of Wittgenstein in philosophy.] Does this combination of words have a use in our language: 'invisible points on a map'? The word 'invisible', like the word 'width', is undefined in geometry, "sound without sense", "air".

Query: do speech sounds have meaning? Plato.

If to 'have meaning' means 'to name an object', then not all "speech sounds" have "meaning". [But this query is about Plato's Cratylus, where the meaning of words is looked for in the syllables that compose them, where 'meaning' = 'etymology' or 'roots'. (Morphemes and phonemes.)]

Query: another way to say 'geometric object'?

No, you should stop saying it altogether, not look for another way to say it. Nonsense by any other name is still nonsense [i.e. an undefined combination of words]. Rather say that a "geometric object" is a unique set of points in the plane (but not as if the word 'point' were the name of an object of some strange, unearthly kind or other, rather than of an address (or, location) in the plane).

That is not another way to say 'geometric object', but it is what you should say instead when you are talking about lines, triangles, circles and other geometric figures. E.g. the triangle ABC in the plane is a unique set of addresses in the plane. "That is its only reality", you might say.

Triangle ABC is a different set of unique addresses from the set of triangle EFG in the plane (unless, of course, we have assigned different names to the same points in the plane such that A=E, B=F, C=G, because of course a single address or location may be given more than one name).

If we laid a Cartesian grid on the plane, then we could use Cartesian co-ordinates (i.e. {x,y} co-ordinates) to fully describe the location of e.g. a triangle in the plane; the co-ordinates would be the address (the set of points) of the geometric figure in the plane. How many co-ordinate pairs would be required? In the case of a triangle, at least three (corresponding to our normal way of identifying a unique triangle in the plane, namely, ABC in the remarks above); beyond that, as many as are required to identify points of intersection with other geometric figures under consideration in the plane (or e.g. the point on the line where the altitude of an isosceles triangle intersects the base of the triangle). [Note, that in the case of the plane in plane geometry, the combination of words 'size of the grid' is undefined -- i.e. in this case, the grid cannot be compared to the different mesh sizes of different types of fish net, some designed with smaller holes to catch smaller fish, others sized with larger holes designed to catch only larger fish; that metaphor has no application here.]

Query: what is the size of a geometric point?

That combination of words should make you laugh, as should 'What is the color of the number 3?' or 'How much does a poem weigh?' It is an example of a "category mistake", or, of a grammar of parts-of-speech mix-up. The persistent misleading picture: "If we are talking about something, and if what we are saying has meaning, then we must be talking about some object (or "process" -- i.e. procession of objects, although they may be "gaseous")."

"What size is a geometric point?" cf. "What size is a concept?" This is an example of trying to apply the grammar to one part of speech to another part of speech. "How big is a point? How big is a concept?"

Query: why a point has no dimension; geometry.

Not "because we don't want it to have"! but because: in geometry the word 'point' is not used to name an object; it has a different use in the language. Or, in other words, 'has no dimension' = 'is not a name'. There is no "why" -- but instead there is a definition of -- or rules for using -- a word.

Query: first day of class geometry: what is a point?

If children are sent to school to be confused by incompetent teachers, this is a good place to start. Don't ask what a point is; ask how we use the word 'point'. Because if you look at that use, you will see that we do not use the word 'point' to name anything.

Query: why is a line is like the edge of a ruler not a good statement?

But it is a good statement. Remember that, as is the case with all comparisons, when we say that A is similar to B, we must indicate in exactly which way/s A is similar to B. If A were like B in every way, then A would be B -- i.e. A and B would be identical, whereas a geometric line and the edge of a ruler are like each other in some ways but not in others. (The only case in which a comparison is not at good statement is where A and B are not similar in any way that is useful to the understanding.) Remember that in Euclid's geometry there are demonstrations using a compass and straight-edge (Q.E.F.).

Query: which undefined geometric term is described as a two dimensional set of points that has no beginning or end?

Note.--At first reading I thought the query was asking for the word 'line' rather than 'plane' , because I passed over its "two dimensional", and the word 'line' is what the following remarks are about.

"... has no end points." Of course the picture we have -- [but this picture doesn't make the geometric term "undefined"; we also picture triangles and circles] -- is of a straight line, but the challenge is to define 'straight' non-visually, i.e. using points = addresses only.

What distinction does the query make with the words 'definition' and 'description'? The description consists of comparisons -- but are only comparisons possible in the case of axiomatic geometry's use of the word 'line'?

"Any two points define a line, which, in contrast to rays and line-segments, has no beginning or end points [This makes it continuous; but what would be an example of an non-continuous set of points? For example, {all points in the line AB plus the point D which is not in AB}], which does not intercept itself: if ABC, then not BAC or BCA [This makes the line straight]." We can begin to define 'line' this way, adding as many explanations of meaning (i.e. rules) as we find necessary, and use this definition in a dis-proof (e.g. if ABC and BAC are assumed, then A, B and C are not points all in the same line).

Did the Greeks make a distinction between axiomatic and visual geometry (Protagoras' criticism of the tangent points to that distinction, although it does not recognize it)? Not if 'line' is -- i.e. must be (according to the rules of the game) -- defined by means of a straightedge; and that there are proofs of construction using straightedge and compass suggests that the distinction was not made, although we would say that the distinction is needed if geometry is to indeed measure the earth (Need a circle be visually round?)

Query: can a point in geometry be anywhere?

It can be anywhere in the plane (which is a grid of unique addresses {x,y}). In geometry: What is the case is the relative position of named points (addresses) in the plane. [Cf. the verbal formulas of the TLP.]

What is Jabberwocky (Through the Looking-Glass, i and vi) without Humpty Dumpty's explanations of language meaning?

'Twas brillig, and the slithy toves
  Did gyre and gimble in the wabe:
All mimsy were the borogoves,
  And the mome raths outgrabe.

A misleading form of expression, for "undefined terms" are defined, although not by geometers

Query: why we need undefined terms.

If those terms are meaningless, absolute sound without sense, then why not just dispense with them -- or replace them with different undefined words? -- This shows you that those terms are defined; we could not replace them with undefined jabberwocky: find tove A on the borogove n in wabe p between toves B and C. The reason we need 'point', 'line' and 'plane' is that those words have defined roles to play in geometry.

[Alice, if she'd been a bit common, might have asked Humpty Dumpty, What's "wabe" when it's at home? (Words and "their original home" in the language.)]

Query: meaning of "not defined" in maths.

Is the distinction between 'undefined' and 'not defined' important here? The proposition 'Geometry does not define these terms' and the proposition 'These terms are undefined in geometry' are quite different. Compare: the proposition 'Philosophy does not define the word 'the' ' is very different from the proposition 'The word 'the' is undefined in philosophy'!

Query: what do they mean by the term "not defined" in maths?

Exactly: 'undefined' in maths = 'not defined by maths' -- but not "absolutely undefined", much less "indefinable", but only: "not defined by maths".

And hast thou slain the Jabberwock?
  Come to my arms, my beamish boy!

Illustration by John Tenniel to 'Through the Looking-Glass' by Lewis Carroll.
He took his vorpal sword in hand ... The vorpal blade went snicker-snack!

This way ... because the other way is mistaken

Wittgenstein: "A philosopher says, "Look at things this way!" ... but first, that is not to say that people will look at things that way ... it's possible, too, that the impulse towards such a change in the way things are looked at must come from another direction" (cf. CV p. 61-62). From which direction I don't know, but apparently not from mine. To learn to see the logic of language -- i.e. what gives language meaning -- aright is hard, but, further, no one will undertake it unless he senses that the model of language meaning that comes as second nature to him -- namely "Words are names and the meaning of a name is the essence of the thing the name stands for, regardless of whether that thing is tangible or abstract" -- is dysfunctional, resulting in the mystification of oneself produced by speculative theories about the essence of some abstract object presumed to be named by an abstract term. (Wittgenstein's old and new ways of thinking contrasted.)

The picture of the meaning of a word, especially of an abstraction (i.e. "the name of an abstract object"), as a nebulous, amorphic quasi-object to the side and just above and outside one's forehead. (It's no more silly a picture than others, as e.g. the picture of word meaning as a saintly halo a word carries about with it. The aim is to get rid of all idle pictures of language meaning, as e.g. the pictures of word meaning as abstracted essences, and Platonic Forms.)

"Look at this another way," the philosopher says. Someone who is unfamiliar with ducks and rabbits would -- indeed, could -- only see the outline of the figure ("the duck-rabbit"), not a rabbit or a duck. It is logically possible for someone to see only one aspect (e.g. a duck) while being blind to the other aspect (the rabbit) until the other aspect is pointed out to him; the philosopher points the other aspect out to him: "Look at the figure this way!"

But someone may never have thought about language meaning, simply presuming that language is the transparent clothing of thought: "We name things and use those names to talk about things; that much is obvious, and we really don't need to know anything more about language than that; what concerns us are things, not their names." Someone who presumes that will see neither the duck nor the rabbit, much less both.

Query: why are they called undefined terms if they have definitions?

Now, how can they be undefined if they have definitions -- i.e. doesn't 'defined' = 'to have a definition'? And this shows that there is nothing so absurd (Pensées vi, 363), nothing so philosophically stupid (cf. George Orwell's double think, for this is a contradiction in sense), that students can't be trained to accept it. (That is the backhanded meaning of 'learned ignorance'.)

In this context recall the sign versus the use of the sign in the language distinction: spoken sounds or marks on paper versus what is done with those sounds or marks that gives them meaning. If a geometric term can at the same time be both "undefined" and "defined", then isn't it evident that the word 'definition' is being used equivocally -- or is nonsense?

The expression -- i.e. combination of sounds, marks -- 'undefined term' belongs to a jargon invented by some philosophers of maths and geometry. (By the word 'jargon' I mean in this context: using a word in a way that is contrary to common usage , i.e. normal practice.)

There are definitions by Euclid, by the Pythagoreans, as well as my own account. But, according to some philosophers of maths, those are not "real definitions" (because either they are too obscure in meaning to be useful-helpful, or they are superfluous, or contribute more obscurity than clarity (Pascal), or because they are never used to justify a step in a proof. A 'defined term' in geometry means: a term that is used in the justification of a step in a proof. "Undefined terms" -- Well, you can talk that way if you want to mystify children with paradoxical jargon. It would, however, be far clearer to simply say, if it is necessary to say anything, that definitions of 'point', 'line, 'plane' in geometry are given so that we know what we are talking about, not in order that we can use those definitions in proofs. What the definitions do is to stop you from stepping off the path onto a false path, e.g. should you forget that no two lines have more than one point in common.

And this is not a mere verbal quibble; it is not a question of "mere semantics" (although it is that too, of course). Because it concerns the master question of "logic of language" -- namely, how is sense to be distinguished from nonsense? (PG i § 81, p. 126-7) That is why which meaning of the word 'definition' we choose to use in our thinking has nothing "mere" about it. Paradoxical jargon cannot "heal the wounded understanding".

Query: early geometers' definitions of undefined terms.

Euclid: a point is that without parts. Pythagoras: a point is a unit -- (a point is that without parts = atom = unit) -- with position. But always the assumption is that the word 'point' is the name of an object, if not visible then invisible, and that is the innocent move in language's conjuring trick (PI § 308), the false picture of the logic of our language.

Question: How does one define the undefined term 'point'? for how can there be "a definition of an undefined term"? Or is there an "undefinition" for an undefined term? A word that even when it is defined, as we normally use the word 'define', remains undefined -- that is a queer customer indeed.

Query: why point, line, plane are meaningless in geometry.

Well, this is it. As we normally use the word 'undefined', 'undefined' = 'meaningless'. Which shows that in maths, 'undefined' is jargon rather than the normal use. And this is why, unless you want to confuse students, you will not use jargon. But, well, then, how shall we put 'undefined' into prose? What title would be both normal and appropriate? It's a nice question. One I don't have an answer for.

Query: Eddington Jabberwocky.

How does Eddington's remark about electrons compare with what I wrote above about replacing 'point' in geometry with 'tove'? How could Eddington be correct -- i.e. if the words 'point', 'line', 'plane' really were undefined (unintelligible noise, sounds which have no more meaning for us than bird song) -- and what else would it mean to say as Eddington does that 'point' might be replaced with 'tove' with "no loss of meaning"? Now, do you want to make that claim?

Query: enumerate and differentiate the undefined terms in geometry.
Query: gave the meaning and examples of three undefined terms of geometry.

What does 'differentiate' mean if not 'define' ("set the limits to", "mark off the limits, the boundary of"). There are indeed assignments that make children stupid (thoughtless, uncritical), although I would like to believe that these assignments are given students to make them think, to make them question what appears to be double-talk.

Query: words with no definition; geometry.

A list of geometry's meaningless "words", i.e. signs without sense. Well, if children are willing to accept religious beliefs without questioning, why not the notion of "undefined terms"? [Why are they called undefined terms when we can in fact define them?] But here is an example of a "sign" with no definition in geometry:

Query: width definition in geometry.

The word 'width' is an example of an "undefined term" in geometry -- i.e. a word that is not used in axiomatic geometry at all, where instead words such as 'congruence' are used. In the context of geometry the word 'width' is a meaningless sound or ink mark, a "sign without a sense".

Query: grammar / how to use the word point.

Well, that is what a grammatical explanation ought to tell you, but schoolbook grammar does not. -- And that is the role of definition of geometry's basic terms, 'point', 'line', 'plane' -- a grammatical explanation tells us how to use those words.

Query: occult geometry on the dot.

Could that "abstract", "occult", "invisible" world of ghostly objects be the object of religious worship, or rather, of a religion? That, based on my observation of life, is hardly likely; which tells us something about religion. (Even Frege did not worship numbers.)

Again and again these questions arise ... Wittgenstein thought that it must be this way "so long as there is a verb 'to be' that looks as if it functioned the way the verb 'to eat' does -- i.e. to name an action, and so long as all nouns seem to us to function the way the words 'cow' and 'desk' do -- i.e. to name objects" (cf. CV p. 15 [MS 111 133: 24.8.1931]; cf. PI II, x, p. 190, para. 11).

Query: why points are considered as undefined terms in geometry?

Why "considered" -- is this a matter of opinion -- or is it instead a matter of selecting a definition? The fundamental mistake in philosophy -- not to distinguish factual from conceptual-grammatical questions; only among the former are found matters of opinion; the latter concern definitions.

Query: difference between dot and point geometry.

A dot is a point-marker, like an "X" on a treasure map. In printing the word 'point' may be defined a 'a dot of ink'. However, that is not the way the word 'point' is used in geometry, where 'a point' simply means 'a location', a unique address in the plane e.g. Or again: in printing, the word 'point' is the name of an object; in geometry the word 'point' is not the name of an object.

If you are given the address "17 North Lane", you do not know what you will find at that location: it may be a house, a park, a bank, but none of those objects is the meaning of the address '17 North Lane' which is simply a point on a map -- on any map that shows that location; '17 North Lane' is not the name of some object. cf. the demonstrative pronoun 'this', which is more aptly called a "pointing-word" because it is not a stand-in for a name, as the word 'pronoun' may suggest, but has a very different use in our language. [To correctly understand the logic of our language it is necessary to see that not all nouns are names of things; when that becomes clear to you, many other things will as well.]

Query: a line is a set of points; Euclid.

In just the way that a neighborhood street is a set of addresses.

Query: what does the following suggest, a point, a line or a plane?

Ever and anon ... But, no, a pin hole does not "suggest" a point, but a pin used to mark a location on a map is an example of a point-marker -- i.e. of how the word 'point' is used in geometry.

Once upon a time, rather than on paper, plane geometry was illustrated by marks make in the sand. Demonstrations required only a straight edge, a pair of compasses and a pointed object (one end of the pair of compasses e.g.) to draw lines in the sand. The sharp end of the compass left a depression in the sand, of course, similar in shape to an ink dot on a sheet of paper, but I wonder if that is a possible origin for the notion that a geometric point is an object. For is there any reason why anyone would suppose that there must be an object -- an invisible, "geometric object", of course -- named 'point' at that depression in the sand where the compass point was positioned?

Points, arrows, locations

The drawing below shows four "points" -- i.e. four locations in a plane, three of which are locations on the line m, and one of which (namely, D) is not a location on line m. In this drawing does the question of "what a point is" arise? Why shouldn't the "three undefined terms" be 'location', 'line' and 'plane' -- would that not be clearer, at the very least with respect to the first term?

Four points not all on the same line, 1 KB

If we remove the arrow labeled 'D' in the drawing, then what remains? Do you think there must remain some unmarked object there called a 'point'? A plane is not composed of points except in the sense that a city is composed of locations; if you open a city-directory of addresses, you will not find a collection of objects therein.

What is geometry about?

Query: point-like objects. Geometry.

"But if geometry isn't about objects, then what is it about?" What is chess about? The proofs of geometry do not require drawings any more than to play chess requires a chess board and chess men. What is geometry about? If it is about anything, it is "about" rules. For if you are you asking what its subject matter is, then the answer is that it has no subject matter. Axiomatic geometry -- i.e. geometry consisting only of definitions, postulates and deductive proofs -- is independent of anything external ("the world") to itself, and, indeed, whether or not it has, in some way or another, application to anything external to itself -- is a question experience must answer.

Query: word for give the meaning of.

The word that comes to mind is 'define' -- a definition gives the meaning of a word, phrase or sentence. [One might say: a sound might have any meaning, and so we must set its limits (cf. 'to confine') if it is to have a use in our language.] A grammatical account, an explanation of meaning, is a definition; even if its form does not suggest that it is, its use shows otherwise.

Query: what does undefined mean?

The words 'defined' and 'undefined' are opposites (antonyms). The first word means 'having been given a meaning', and the second means 'not having been given a meaning'. However, in the jargon of philosophers of mathematics -- and only in their jargon -- 'undefined' does not contrast with 'defined' in the normal way where an undefined term is simply meaningless ("sound without sense"); instead, 'undefined' contrasts with 'a proof depends on its explanation and, therefore, the word is required to be defined'; but if a word's meaning does not need to be explained for the sake of a proof, then it is said to be "undefined", as is the case with 'point', 'line', and 'plane', which however are not meaningless -- otherwise we would have no idea what we were talking about.

"Why do we call them "undefined terms" when in fact we must define them?"

One might think, Well, jargon, so long as its meaning is explained, does no harm (cf. Charmides 163d). However, in the case of philosophers of geometry's jargon-word 'undefined', it is not that way. Because 'undefined' suggests 'mysterious in nature', as if geometric points were occult natural phenomena the nature of which can only be "suggested" while the thing itself cannot be shown. The only mystery here is self-mystification. Euclid's common notions (e.g. "A point is that without parts") are not "obscure in meaning", but they are not a full account of the grammar of 'geometric point' either.

Unless it is our desire to confuse people, we should not use the expression 'undefined terms in geometry'. If it is really necessary to make a distinction, we might e.g. use the expressions 'a definition used in a proof' versus 'an orientation definition'.

Philosophical Grammar is Questions about Logical, not Real, Possibility

Query: what is the name of an object with no dimension that is represented by a small dot?

Should not the response be: "Don't talk nonsense. How can an object have no dimensions!" That is a grammatical remark, a reminder about what we mean by the word 'object' (cf. the word 'ghost' is not the name of an object). The word 'geometric point' has a different use in our language than being the name of an object. The word 'point' is not defined ostensively; it is neither the name of a static object nor is it the name of a fluid phenomenon. It is not a name at all. (Until you see that not all words are names, you will not see the question 'What is a geometric point?' aright.)

"How can an object have no dimensions?" This is not like asking "How can an object be colorless?", for an object may be transparent; but it is like asking "How can an object have no location or shape or size or weight?" These are grammatical questions: they ask for an explanation of meaning. The possibility they enquire about ("How can") is logical, not real possibility (They are not a question about laws of nature, for example).

Query: an undefined term in geometry, it names a location and has no size.
Query: a location in space that has no dimension and is represented by a dot.

Even with the addition of "location in space" the proposed definition, because it includes the provisos "that has no dimension" or "has no size" and "is represented by a dot", is still wedded to the notion ["picture"] that 'geometric point' is the name of an object (How can a dot, which is visible, represent what, having no size (no dimensions), is invisible? The notion of "representation" here is mistaken; instead: a dot is a point- or location-marker, like "X" on a treasure map; the "X" on the map does not represent an "X" on the ground -- There is no "X" on the ground). The proposed definition is still wed to the picture: all words are names of things.

Can a location have dimensions? The word 'location' may be equivocal, and thus we must distinguish between an address and, for example, the building found at that address. The address of a building is not an object, although the building is: the building has dimensions, although the address does not (cf. My name is not six feet tall, although I am, nor does my name have a moustache, although I have, of sorts).

What would it mean to say that a geometric point has no dimensions, other than that the word 'dimension' has no meaning if applied to a geometric point (because 'geometric point' is non-name-of-object word [cf. the word 'no' -- What would it mean to say that "No has no dimensions"? It would mean nothing])? Or in other words, the grammars of 'geometric point' and 'dimension' do not intersect, do not connect up. What cannot, by definition, have any dimensions also cannot have no dimensions -- that is, having no dimensions is not an alternative quality; it is not like not-being-red when something is instead blue.

Query: objects that suggests the idea of a point.

The telephone directory, which is a list of addresses, especially the old-fashioned printed directory, is one example.

Query: what do you call the indefinable term that means a location in space and is represented using a dot?

By the word 'point' in geometry we mean 'any unique address in the plane'; it is not "represented", but indicated: the location of the point in a drawing of the plane is specified by a conventional sign ("X marks the spot"), and it does not matter if a dot is used or any other conventional sign (an arrow e.g.). Comment: but if 'point' were an "indefinable" term, then it would not be logically possible to define it, but yet that is exactly what the query has done: "... that means a location in space and is represented using a dot".

Query: geometric object with no dimensions; it is only a location.

The persistence of these queries -- half mistaken ("object"), half correct ("only a location") -- shows you that Wittgenstein's later view of sense and nonsense is a universal Gestalt shift -- but one that is very hard to make. It is not like switching between the duck and rabbit aspects of the duck-rabbit image, which is fairly easy to do. It is instead more like seeing that there is no rabbit or duck to be seen: that there is only a picture-duck and a picture-rabbit where before you thought there must be a living duck and living rabbit; -- it is like seeing for the first time that the image is only a drawing; it is only ink marks on paper. (And once you see that then you can ask: what gives those ink marks meaning.) I am not good at inventing metaphors, but I think that thinking that 'point' must be the name of an object is like thinking that the duck and rabbit must be real.

"The meaning of a name is an object" -- It is very hard to break with that model [or picture] of how language works; it seems to have its roots in instinct. However, there is no such thing as a "geometric object" [cf. There are no psychological objects -- e.g. 'mind' is not a name-of-an-object word] -- i.e. the grammar of 'point', 'line', 'plane' is not the grammar of name-of-object word, but those words have a different use in geometry. "Let the use teach you the meaning" (Philosophical Investigations II, xi, p. 212)! "Don't think, look!" (ibid. § 66) -- i.e. Don't rationalize, but instead "empiricize"!

Query: I am a location in space; it takes only one letter to name me. Geometry.

We are getting closer, clearer; however, what does 'name' mean here? "The Red House" (In England people liked to give their houses proper names) is an address; however, {x,y,z} with origin at ... would not necessarily be better: the question is: what is our point of reference? A map of an English village where each of the houses has a proper name might serve just as well as Cartesian co-ordinates or degrees latitude and longitude -- provided that we have a known point of reference. Thus, for geometry, better than "location in space" might be "location in such-and-such a [specified] plane". (An address like "The Red House [or a point labeled 'R'] somewhere in space", because it is indeterminate, would not name a geometric point.)

"Words that name the same thing"

The English word 'synonym', it says in the dictionary, comes from the Greek for 'name' [onyma] -- thus, synonyms would be "words that name the same thing". The picture of "word = name of something" is very deep in us; surely it is instinct [and that "surely" is also a picture that seems -- but is 'verification' defined in this case? -- to have a natural, compelling appeal for us; -- and that notion of a "natural, compelling appeal" is also perhaps the product of instinct].

Query: proof of geometric point.

Proof that geometric points exist. -- Now that is an interesting idea: how do you verify Euclid's "common notion"? Protagoras: no geometric tangent can be found in nature (But, then, if "points have no dimensions" by definition, then we ourselves have made the verification of a tangent's existence impossible (Zettel § 259)). If "A point is what has no part", then how do you know whether or not any points exist? For if 'point' is claimed to be the name of an object, shouldn't we ask whether or not any such object exists? (Although, of course, prior to the question of truth or falsity is the question of sense and nonsense: e.g. is any object "absolutely simple"?)

Suppose someone asked: "Does a geometric point have any empirical existence?" This would show that the person was either (1) asking a grammatical question, or (2) unable to free himself of the "physical object" vs. "abstract object" picture. ("Do abstract objects exist?" -- That is a grammatical, not a metaphysical, question: it asks for a verbal definition of 'abstract object' ... if indeed that combination of words is not nonsense.)

Query: a geometric point doesn't really exist.

Cf. the proposition 'fairies don't exist', which is a rule of grammar; but 'Troy doesn't exist' is a thesis (for if that proposition were a grammatical rule, then it would have been impossible for anyone to search for the historical city of Troy; just as it is logically impossible for anyone to search the forest for fairies, for even if someone did invent a criterion for verifying that fairies exist, that criterion would not belong to the way we normally use the word 'fairy'; it would be a redefinition of, a new grammar for, that word; cf. 'Adam and Eve didn't really exist'; fairy tales and religious myths share this aspect of their grammars). Now, which is 'geometric points don't exist' -- a rule of grammar or a statement of fact (thesis)?

Objection: "but Frege believed that geometric points existed in something comparable to the Platonic world of Forms: imperceptible but nonetheless real." But what does that have to do with the way we use the word 'point' in plane geometry? Frege has a "picture", and if that picture states a thesis -- then it is a metaphysical thesis. (Of course Frege's "picture" itself has a grammar, but it is not the grammar of 'geometrical point' in plane geometry. Cf. The Orphic-Platonic picture of "the soul" adopted by Catholic Christianity; that picture does not belong to the grammar of words such as 'psyche' and 'mind', as we normally use those words. For as we normally use those words, they are not names of objects -- of any kind.) Frege might say: "I believe that geometry's points really exist", and someone else might say: "I believe the contrary", but that would be a disagreement belonging to metaphysics; it would not be a disagreement about the grammar of 'geometric points' in plane geometry.

On the other hand, as another account, doesn't the address of the house I live in "really" exist? To say that my address really exists or that geometric points really exist is to say this: that the words 'address' and 'point' are not meaningless combinations of letters or sounds -- i.e. this is to give a grammatical account (Cf. to say that 'geometric points doesn't really exist' is to say that the word 'point' as used in plane geometry is not a name).

Query: mathematicians use numbers like philosophers use words.

Why, do mathematicians talk a lot of nonsense? Do mathematicians write a formula on the blackboard and then ask themselves: "Now, what does that mean?" Do they say when arguing: "I think I mean something by this formula ..... I think this formula makes sense ..."? Can you use numbers ("numerals") to write nonsense; do mathematicians do that very often? Because that is the way philosophers use words.

The Consequences of an Inappropriate Form of Expression

Ce n'est pas un point géométrique, 2 KB

Query: what is a geometric point?

Once you have posed the question that way, you've already presupposed that 'geometric point' is the name of something. Don't ask what a thing is; ask how we use a word. An ill-chosen form of expression may stand in the way of your understanding geometry.

An unsuitable type of expression is a sure means of remaining in a state of confusion. It as it were bars the way out. (PI § 339)

That and instinct is the answer to the following queries.

Query: why do children have misunderstandings in geometry?

Because their teachers (or "mis-teachers") lack a philosophical understanding of geometry, and therefore cause unnecessary confusion for their students; with their talk about "undefined terms" e.g. The Philosophy of Mathematics, at least as I have written about it, is not idle speculation; it is simply an effort to describe in a non-theoretical way what we actually do; e.g. how is language used in geometry?

Query: what is that in mathematics people do not like?

"What are we doing, and why are we doing it?" Step back from your subject matter and answer that question for students. Not approaching mathematics from a philosophical ("I want to understand what I am doing") -- as well as an historical (How many children can say where the expression "the square on the hypotenuse" comes from? [Many teachers use the opaque expression "the square of the hypotenuse" (but where did the notion of multiplying the length of the hypotenuse by itself come from?), thus making the procedure mysterious -- seemingly arbitrary -- to children, because they do not know its origin] or what "5 times 7" means?) -- point of view may be why people do not like mathematics; do we not tend to dislike (or at least shun) what we do not understand? Mathematics can be made a fascinating study rather than simply an uninteresting business of learning to follow a collection of rules that come from who knows where. (Children are not natural enemies of learning, but bad teachers can train them to be.) Above all, slow down. Learning is digesting; it cannot be rushed.

Query: difference between statement and definition; geometry.

In axiomatic geometry, is the distinction between grammar and statement of fact clear? But there are no statements of fact in axiomatic geometry: instead, there is a distinction to be made between rules and the consequences of following rules.

The Socrates of Plato: Seeking a Universal Standard for Ethics

Query: Socrates tells Euthyphro that he is searching for the Form of piety; what does he mean by that?

Just the opposite: instead, it is what Socrates says he is searching for that explains what Plato means by 'Form' in the Euthyphro: a standard of judgment = Form. [Plato, Euthyphro 6d-e | The limits of Know thyself is guidance in the particular case]

Query: Euthyphro and the standard of measurement.

That is the eternal question, and the eternal interest in the Euthyphro: is there a universal [an absolute] standard of measurement in ethics? That standard was what the Platonic Socrates sought. If we wish to measure a length of wood, we have a standard of measurement (a meter stick [ruler] e.g.); but note that we do not have "a standard for our standard" [The standard is what comes at the end of the line; it is used to justify, but it is not itself justified -- i.e. justifiable], but instead we simply agree to make use of it. In a word, axioms cannot be justified (bedrock does not lie on bedrock); that is logic.

However, Socrates' approach to ethics in Xenophon is practical rather than theoretical. The good is the useful, and it is simply an unwillingness to use reason, i.e. it is self-contradictory, to claim that what is harmful to man (either to his body or to his ethical-soul) is the good [beneficial] or useful for man.

Query: definition of quotes made by Socrates.

Here 'definition' clearly means: an explanation of meaning (Philosophical Grammar i §§ 24, 32, p. 60, 68-69), which in Wittgenstein's logic of language is given by stating rules (as in rules of the game or "language-game") for the use of a word or a form of expression -- or, in Wittgenstein's jargon: its "grammar".

Query: Socrates says nothing can be defined.

Plato's dialogs about "the cardinal virtues" among the Greeks (temperance, courage, justice, piety, wisdom) may be unsuccessful at finding a definition (in Plato's sense of that word) -- but his dialogs were not intended to be "the end of the matter"; we mustn't give up, but instead must keep looking: there is a difference between saying that Plato was unable to define those words, quite another to say that it is logically impossible to define those words (although it may by logically impossible to give a Platonic definition of those words -- i.e. Plato's preconception may be a false account of the grammar of those words). As Socrates says at the end of the Euthyphro: "As for me, I will never give up seeking until I know." [Socrates' mission in philosophy.]

Query: Question everything. Socrates.

But maybe it should be: Question everyone. That was Socrates' response to Apollo's oracle at Delphi. He sought to question everyone who had a reputation for being wise [wisdom], and what he discovered from his search -- its conclusion ["working hypothesis"] -- is called "Socratic ignorance".

Old photograph of Athens, showing its acropolis, 30 KB

Source: C.E. Robinson, Zito Hellas [Hellas (1955), Plate I, opp. p. 50]: "The Acropolis at Athens from the west. The view is taken from the Pynx Hill where the [democratic and sovereign] Assembly [of all Athenian citizens] was held. To the left lies modern Athens on the site of the ancient city. Beyond lies the conical hill of Lycabettus, and in the further distance Mt. Hymettus." The mountain is about ten miles from the city's center.

N.G.L. Hammond in his history of Greece wrote that "Socrates drank the cup of hemlock as the last rays of the sun were lingering on Mt. Hymettus" [cf. Plato, Phaedo 116]. I like the indistinctness of this old photograph, because it suggests a world that no longer exists, a world which is "not easily assimilated" to our times. If anyone says that he understands that world, he must say what he means by 'understand'.

Query: theory of Socrates; what the man know is nothing about what he suppose to know.

If "supposed to" = ought to, we appear to have no innate knowledge of what is most important for us to have knowledge of -- namely ethics, or, "no small matter, but how to live our life" (Plato). Man's condition from birth is that -- although like all living things he aims for his perceived good -- he is not born knowing what is the good for him, but instead must seek to discover it (and here most men appear to stumble). The Greek's answer was to obey the Delphic command "Know thyself" -- What does Socrates discover?

But "supposed to know" may mean "think we know (but do not), although we ought to do". But I cannot think but that often when the word 'theory' is invoked in philosophy, we do not stop to ask what we might mean by it. For example, what might 'ethical theory' mean other than 'ethics' -- i.e. what would the word 'theory' make clearer here? By 'Socrates' theory' the query may mean: Socrates' account (or, explanation) of what he believes himself to know with respect to human wisdom (Plato, Apology 23b). For Kant will give a very different account, and that account might be called "Kant's theory in ethics" or simply "Kant's ethics".

Infinitely better to have something to ask or say in broken English than to have nothing to ask or say in the English of educated men. It is not the form but only the sense [meaning, use] of language that is important to Wittgenstein's later philosophy (or, "logic of language" in my jargon). "The sign-post is in order -- if, under normal circumstances, it fulfills its purpose" (PI § 87).

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