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The Philosophy of Geometry

What is a geometric point? Is the word 'point' the name of an object? Wittgenstein's method of describing the use of a word in the language stands in contrast to the theories of Euclid and Pythagoras about what geometry's points, lines and planes are (or "really are" despite appearances to the contrary).

In Wittgenstein's logic of language the word 'point' in geometry may be defined thus: 'a geometric point' is 'a unique address in the plane' or, in other words, by the word 'point' in plane geometry we mean 'any unique address in the plane'. This sets aside all metaphysical theories (fanciful pictures of invisible objects, dimensionless and approximated to by ink dots, compass points in the sand, and pinwheel holes), notions which have their origin in the presumption that: "All nouns are names, and the meaning of a name is the object the name stands for."

An alternative way to "heal the wounded understanding" (Kant) in philosophy would be simply to not use the word 'point' in geometry at all, but to exclude that word from the language of geometry altogether. We might use e.g. the expression 'address in the plane' abbreviated simply to 'address'.

Now, why do I, who am not a mathematician, presume to discuss this topic (Cf. Wittgenstein's Lectures on the Foundations of Mathematics, 1939 p. 13-14)? First, because this is not a topic in mathematics, but rather a topic in the Philosophy of Mathematics. And second, because I only talk about how the word 'point' is used in the two geometries I am basically familiar with from school: Euclidean and Cartesian.

Geometry: 'not defined' versus 'undefined'

Note: in maths, 'not defined' ≠ 'undefined'. That is, 'undefined' in maths = 'not defined by maths', not "absolutely undefined", much less "indefinable", but only: "not defined by maths". Geometry does not define the word 'the', but we do not for that reason say that the word 'the' is undefined in geometry. (Geometry and Jabberwocky)

Although the fundamental view here is sound, because the view it overthrows is based on a false understanding of the logic of our language, the remarks on this page belong to the time they were written. Later some are bound to strike me as philosophically stupid, but the remarks won't have been stupid simply because I am stupid, but because philosophy is that way: it is always possible with time and thought to see more deeply into a philosophical problem than we see now.

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Outline of this page ...

• The False grammatical account: Geometric points are names of objects
  • Half-way house (partial rejection): Geometric points are names of objects whose size doesn't interest us
• Complete rejection of the false grammatical account: Geometric points are not names of objects (nor are points themselves objects)
  • Points versus "point-markers"
  • Does geometry need a definition for the word 'point'?
  • The Role of drawings in geometry
  • "What is a point?" is a question that belongs to the Philosophy of Geometry (Foundations of Mathematics)
    • Reality is number, which is both its stuff and form (Pythagoras)
      • The Forms as non-mathematical-numbers (Plato and Aristotle)
  • We know the word's meaning, until we are asked to explain it
  • Philosophy of Mathematics is the view from outside mathematics: What we imagine about points is of no importance to geometry
    • No theorem necessarily has an application outside geometry
• Life-long perplexity about geometry that begins at school
• Endnotes
  • An adequate definition of 'point'
  • Deafness to the grammatical joke

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Definition for Orientation

In Wittgenstein's logic of language, 'undefined' = 'nonsense' (i.e. "mere sound without meaning", noise). But some philosophers have said that in geometry 'undefined term' means that "a definition of the term has or would have no role in the demonstration of any theorem". But definitions, even in geometry, are often given for the sake of orientation, and to avert misunderstandings; a definition is "an explanation of meaning" (PI § 560) given for just that purpose, to help us know our way about (ibid. § 123).

And, further, without such definitions we could not apply pure geometry to the world of our experience (Geometry's contact-with-sense-experience words, e.g. 'line' and 'plane', but not 'point'), which is of course something we may want to do.

Geometry: "undefined vs. meaningless". Until we say what you are calling a 'definition', then the word 'undefined' is without meaning.

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The False grammatical account: Geometric points are names of invisible objects

Children are told that geometric points are "inventions of the human mind. No one has ever seen a point." -- Statements like this make people stupid. They are not even useful for orientation. Geometry is not a natural science, and geometric points are not theoretical constructs.

A dictionary might define the word 'point' as: an ink dot in printing or writing, e.g. a period or decimal point; an element in geometry having position, but no extension. These two definitions are bound together in our imagination; we should not underestimate the importance of this.

Always the presumption is that: the word 'point' is the name of something -- either of something visible or invisible. (Ever and anon: "If our talk is meaningful, then we must be talking about some object -- if not a visible one, then an invisible one.")

Perhaps we make the mistake of confusing the meaning of the word 'point' with techniques for representing points? But 'represent' suggests 'stands for'; -- something is presumed to be 'stood in for' ("I can't show you the thing itself, but ..."). I want to break completely with that "grammatical" model.

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Half-way house: partial rejection of the false grammatical account: Geometric points are names of objects whose size doesn't interest us

Suppose we said: 'geometric point' means an ink dot whose size does not interest us; we treat the dot as indivisible, i.e. as an "atomic dot". Just as every line has some width, but its width doesn't interest us, anymore than its color. We find geometric points in any geometry book. And if anyone says 'those are merely representations', then either he tells us representations of what, or we take no notice. (These are grammatical remarks -- if they are anything.)

There is nothing essential about using an ink dot; an asterisk or a star would serve just as well. This is not a matter of "approximating to a real point" (a "real point" is not round, for that matter).

'This is an example of a geometric point: * ' -- This would be a grammatical rule for using the ink mark (i.e. "sign"). However, that would not fix the sign's grammar; we would still have to say that we are excluding the question-sign 'What are the dimensions of a geometric point?' from our language.

Or again: it is possible to be interested in the asterisk from various points of view. The sign or its use e.g. Just as we can ask about the indefinite article 'a' as opposed to the shape of the ink mark 'a'.

Then evidently '*' is not a name, because the meaning of a name is explained by pointing to its bearer. Is the letter 'a' a name? '*' no more names anything than 'a' names the letter 'a'; -- 'a' is the letter 'a'. And '*' is a geometric point.

The shape of the ink mark we designate 'a geometric point' is of no more importance than the shape of the piece of wood we designate 'the king' in chess. The ink mark or the piece of wood's meaning is its role in the game. If we designate the ink mark '*' a geometric point, then that ink mark is a geometric point. "By convention?" Well, by what else!

Geometry is no more about points than chess is about pieces of wood. Geometry, like chess, has no subject matter, if by 'subject matter' we mean something independent, as e.g. plants are independent of botany. Geometry can be compared to -- can even be called -- a game played according to fixed rules; we can imagine a people who had no other use for geometry than that of a game. And a chess game can be given the form of a geometric proof, e.g. we can speak of the 'castling postulate' as the justification for the step of castling.

Below I say that 'point' is not the name of an object -- and that a point is not an object. But let us suppose that it is. Can't we give just such an account of its grammar -- i.e. haven't I done that in the above account?

We are talking about the spatial and temporal phenomenon of [the language of geometry], not about some non-spatial, non-temporal phantasm. But we talk about [these signs] as we do about the pieces in chess when we are stating the rules of the game, not describing their physical properties. (PI § 108)

... a phantasm like those ideal, undrawn, geometric straight lines of which the actual lines we draw are mere tracings. (PG ii § 35, p. 427)

The question 'What is a [point] really?' is analogous to 'What is a piece in chess?' (PI § 108)

The dictionary's definition, 'without extension', defines the geometric point out of existence. To return it to existence, we have only to modify the definition: 'whose extension does not interest us'. (Seen in this light, Euclid's "A point is what has no part" cannot be improved upon.)

We make the rules of the game: the piece of wood that has the role of 'the king' may or may not be required to be of a particular shape (its profile may or may not be included in the rule book). We may specify a shape for 'the point' or we may not; -- but what interests geometry about the point is not its shape but the rules for using it.

We know from Aristotle that Protagoras ... used against the geometers the argument that no such straight lines and circles as they assume exist in nature, and that (e.g.) a material circle does not in actual fact touch a ruler at one point only. [Note 1]

Did Protagoras make a fair criticism of the geometers? Does he force us to speak of geometry as a study of "ideal objects"? What he forces us to do is to recognize that "we disregard certain aspects of reality when we work" (see Philosophy of Science). We do not conjure up an "ideal object" if we say that the width of a line or that the dimensions of an ink dot does not interest us.

"You know of course that a mathematical line, a line of thickness nil, has no real existence.... Neither has a mathematical plane. These things are mere abstractions." (H.G. Wells, The Time Machine i)

We "abstract", i.e. we select some qualities (e.g. length for a mathematical line, length and breadth for a mathematical plane) and disregard others (thickness in both examples) -- an abstraction is a selection -- and we call these selections "abstract objects". But these "objects" are our own creations; they have no independent existence, e.g. as intangible objects in a ghost world. (The words 'selection' and 'abstraction' are synonymous.)

If we study colors, we cannot point to the color white except by pointing to white objects -- though the only aspect of the objects that interests us is their color. But we are not, therefore, forced to say that we are studying an "ideal object" named 'white'.

[However, is it clear what Protagoras is claiming? Does he not owe us a definition of 'point'?]

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Complete rejection of the false grammatical account: Geometric points are not names of objects (nor are points objects)

But the word 'point' is not used in geometry to name anything. A point is not a "geometric object" (Hilbert); it is not an "atomic dot". There are no objects in geometry. A point is a location (and a location is a relationship, not an absolute); a 'geometric point' is a point of reference. That is the grammar of the word.

A point is a location in space. Or in plane geometry: a point is a location in the plane. That is the best general definition of the word 'point' as it is used in geometry that I can think of: By the word 'point' we mean 'a location' or 'an address'. However, that definition is not by itself the clearest definition for it does not specify everything it should. (Note that the word 'space' is undefined in Euclidean geometry, not because it is an "undefined term", but simply because it is never used there. And if it were, what would its meaning be?)

A point is an address. In Euclidean geometry, "a point is a relation" -- a relation to what? To other points in the plane. That is, when we say that 'point' = 'an address in the plane', that address look like this: ABC -- i.e. the address of point B is given relative (i.e. relationally) to points A and C. It is possible to begin a proof: "Let A be a point of line n", because that too states a relationship (a line being a set of points determined by the definition of the word 'line' in geometry).

By the word 'point' we mean 'a unique location in the plane'. How is that location determined? Relative to other locations. Is that circular? Only in the sense that every system of assigning addresses (designating locations) begins with an arbitrarily assigned point of reference (e.g. "point A in line n") -- and there must be other points in the system. (If a system had only one point, there could be no addresses in that system.)

In Cartesian geometry, a point is an address on the Cartesian axes: {x,y} -- for there are only two dimensions in plane geometry. [Note 4] An address indicates a location (cf. a town's latitude and longitude), but an address is not itself the location it points out, just as a map is not the territory it maps. (In geometry's plane, unless an origin is arbitrarily chosen, -- (Note that the combination of words 'absolute point of reference' is universally undefined) --, as in order to set up Cartesian co-ordinate map, then the word 'origin' is undefined.)

Although we can mystify ourselves with the fanciful pictures that suggest themselves to us, such "pictures" are not the meaning of words, if 'meaning' DEF.= 'use in the language' (PI § 43), as it does in Wittgenstein's logic of language, for such pictures do not describe our use of words. We are not taught to use the word 'point' by means of such pictures, e.g. the picture of 'point' as the name of an invisible object (An invisible = imperceptible "object" is no object at all: i.e. the meaning of the word 'point' is not the "thing" the word 'point' stands for, because it does not stand for any thing). Although at school we may be given that fanciful picture, we do not learn to use the word 'point' in geometry or anywhere else by means of it. (Points and a comparison of the geometric plane to a paper map, of geometry to map-reading.)

Points versus "point-markers"

We should distinguish between a point (a "point somewhere") and a point-marker (or, location-marker; e.g. "X marks the spot"; the ink mark 'X' is the point-marker on the treasure map); only the latter expression names an object e.g. an ink dot. The sign 'a point on line n' does not name an object sitting on line n (as if it were a bird sitting on a telephone wire); however, 'the point-marker on line n' does name an object, e.g. an ink dot on the line such as we make in geometry drawings or on maps.

Note that we can ask of an "X that marks the spot": At what point is that X found on the treasure map? Again, as is the case with any ink dot used in a geometry drawing, 'X' is merely a point-marker, not a point.

An alternative to using ink dots to mark points on a line would be to use arrows to point out locations on the line. In that alternative the point-markers do not even touch the line they mark [indicate] points of.

'Point' and 'point-marker' -- a distinction that we easily overlook, because we use the word 'point' to mean both a point and a point-marker, i.e. both as a name-of-object word and not as a name-of-object word.

We are also mislead by the forms of expression we use. The proposition 'There is a point named 'P'' appears to have the same grammar as 'There is a book named 'P''. And if we follow that analogy, then 'There is a point P' must have same grammar as 'There is an object named 'P''. But that is a false analogy, and it is an example of the kind of misleading grammatical analogies that get us into trouble in philosophy (PI § 90).)

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Does geometry need a definition for the word 'point'?

But consider further: a definition in geometry should avert every foreseeable misunderstanding.

If we suspect that the student will take 'point' to be the name of an object, then we should e.g. state the rule: Points have no length. Otherwise when students move in their studies from the congruence of line-segments to the length of line-segments (The word 'length' can be given a meaning even in pure geometry by designating one line segment in the plane as the unit of measurement, and thus any segments which are congruent to the unit segment are 'of the same length', and 'addition' consists of stringing line segments congruent to the unit segment together), they may ask if segment PRQ is longer than segment PQ. We want to avert this misunderstanding without instructing them in the logic of language. We want to teach them geometry.

'A line-segment is a set of points. But a segment contains points; it is not composed of points. (Other-wise we would be speaking of the length of what has no length, namely points.)' Once we establish the picture of a point as an object, we require such grammatical rules in order to avert the confusion which we ourselves have fostered.

"No definition of 'point' is ever used in a geometric proof" (Waismann), and therefore 'point' is said to be "undefined" (Note that this also tells us the corollary meaning of 'defined' in geometry: a word is defined in geometry if that word's definition is used as the justification of a step in some geometric proof). Even if that is correct, however, does that make definitions of "undefined terms" superfluous? I do not need to define the word 'bird' in order to tell my friend that a bird was singing on the roof all morning; but if I said 'An x was singing on the roof', my friend would have to ask me what I was talking about. A teacher cannot just substitute 'quasil' for 'point' and expect students to understand. A definition,

an explanation [of meaning,] serves to remove or to avert a misunderstanding -- one, that is, that would occur but for the explanation ... (PI § 87)

That is, the definition offered of the word 'point' in a geometry textbook should be one based on the teacher's experience of which misunderstandings need to be averted for the student. That would be the definition of 'point' geometry needs.

'Points have no dimensions' -- is a rule of grammar, just as are 'Numbers have no dimensions' (e.g. 'How big is the number 3?' is nonsense) and 'Colors have no dimensions' (e.g. 'Is the color blue larger than the color red?' is nonsense). [Note 2] -- But it would be better to avoid the need for such rules: the best way to avert misunderstandings is not to lay the ground for them in the first place.

In this drawing, the asterisks are point-markers, and A, B and C are the points (i.e. locations) on line n indicated by the point-markers.

If we express ourselves this way, does any question of "what a point is" arise? No explanation of meaning needs to be given to the student here, because the word 'point' is used here the way it is used in our language of every day; as e.g. when we say 'This is a good point to stop for a rest' and 'At that point I lost interest'.

Suppose we do not use the word 'point' in the illustration, but instead write '... and A, B and C are the locations'.

Suppose we rewrote Hilbert's Axiom of Incidence I, 2, using the word 'location' in place of 'point'. 'For any two distinct locations A and B, there exists no more than one line that passes both through A and through B.' Would we still be tempted to follow the geometers and call points "objects"? [Note 3]

Suppose we are shown a map from antiquity. We point to a point on the map and ask: what is the modern name of this place? Or this location? Or this point? Does any question of "what a point is" arise?

'This is the point where you made the error in your calculation' = 'This is where you made the error in your calculation'. Both 'point' and 'this' are demonstrative signs.

We use the word 'point' to point out, just as we use an arrow in a drawing. (An arrow is an excellent point-marker, better than a dot or a star or an asterisk). Its grammar is similar to the demonstrative pronoun or pointing-word 'this' (see 'Nonsense' and contradiction). In geometry, the word 'point' is not a noun (or, "substantive").

Consider this use of the word 'point': 'Choose any point between Genoa and Naples.' -- Isn't Rome such a point? But 'point' is not another name for Rome. In this example the word 'point' is no more likely to be mistaken for the name of an object than is the word 'this' in 'This is a chair'.

'This sign points to the city.' A point is a sign-post.

'Point P on line n' designates (but does not name; cf. 'This point of line n') a location on n. One is not tempted to call the word 'location' the name of an object; it is an unusual noun in this respect. We can say: location-grammar applies to the word 'point' but not object-grammar.

A bookmark is a point-marker: it indicates where we stopped reading. But 'point' is not another name for Chapter Six. A point is an index, like the index finger.

'A line is a set of points.' -- This definition shows that 'line' is not a name-of-object word in geometry. Lines are not essential to geometry drawings: the only element these drawings need are point-markers. Such a drawing might not be as easy to read as one with lines -- but it would remove the temptation to suppose that geometry is the study of any kind of object; especially if we used arrows for the point-markers. Geometry is a study, not of objects, but of relationships.

Use 'place' rather than 'point'! For example: 'If a place B lies between a place A and a place C, then A, B and C are three distinct places of a line; and then B also lies between C and A.' (Hilbert's Axiom II, 1) (Of course, this form of expression -- like any other -- may suggest inappropriate pictures; e.g. if we substitute 'on' for 'of', it may suggest railroad stations on a railway line -- i.e. name-of-object grammar.)

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The Role of drawings in geometry

Children at school are taught geometry by means of visual aids such as blackboard drawings and straightedge and compass constructions; indeed hasn't everyone -- including Isaac Newton (who was bored by Euclid's "proofs of the obvious") -- who has learned geometry learned it this way? A philosopher of geometry then points out that drawings are not essential to geometry, just as the board and chess pieces are not essential to the game of chess (PG i, Appendix § 6, p. 223); those are also simply visual aids. All that is essential to either chess or geometry is the respective rules of those games.

Question: would someone who gave no visual sense to the language of geometry mean by 'geometry' what we mean by 'geometry'? But then does a man born blind who has learned geometry by touch mean something different by 'geometry' than the sighted man does?

The person who says 'A definition of the word 'point' is not essential to geometry' is someone who has already learned geometry. 'Not essential' means that such a definition does not justify a step in any proof. However, no proof which contains the word 'point' is intelligible to someone who has not learned to use the word 'point'; from that point of view a definition of the word 'point' is essential to geometry. And so is a definition of the word 'proof', though that definition also justifies no step in a proof.

Does geometry need a definition for the word 'point'? Is the word used differently there than in our life of every day? A doctor does not need to define 'human body' before he can talk to us about medicine. However, our concept 'point' is much more fluid than our concept 'human body'. When we want to use a fluid concept in a fixed way, we need to make a definition to fix that way.

And geometry does give such a definition for the word 'point'. 'Choose a point C in line-segment AB such that ABC', but by definition there is no such point. That is, aren't the Pasch-Hilbert axioms of order ("betweenness postulates") rules for using the word 'point' in geometry? Just as the axioms of incidence define 'line' and 'plane'. Their form may not make it obvious that they are definitions -- but isn't that their use?

Is there any distinction, other than formal, to be made between axioms and definitions? A statement is a 'definition' only if it is used as an explanation of meaning; so a formal definition or a formal axiom may be used as a definition. But a formal axiom may also be used to justify a step in a proof -- as may a formal definition (e.g. that of 'congruent triangles'); in that case neither is used as an explanation of meaning and so neither is a definition. The use made of a sign in a particular case, and not the sign's form, determines what kind of sign it is, what its "part of speech" is.

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We can distinguish between a drawing's having an "ostensive meaning" and a "pictorial-conventional meaning". For example:

has only a pictorial-conventional meaning (It says that AC is congruent to BC), whereas:

has only an ostensive meaning (It says that a line intersects itself). The drawing below has both:

Geometry drawings are restatements, a "putting into other words", of the proof, or of particular parts of the proof. One statement is easier for us to survey than the other. -- But not just as '693 + 479' is a restatement of 'six-hundred and ninety-three plus four-hundred and seventy-nine', or as R-Q3 is a restatement of moving a chessman on the chessboard. -- Because geometry drawings cannot be substituted for the written proofs; because nothing can be inferred from a drawing. Because the apparent relationships shown in the drawing are not justifications -- i.e. they are not definitions, postulates, proven theorems, or rules of logic (Euclid's "Common Notions" e.g.).

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"What is a point?" is a question that belongs to the Philosophy of Geometry

But what right have I to talk about geometry if I have studied no more than plane geometry? -- As if knowing more geometry would throw light on its use of the word 'point'!

The question of "what a point is" is a question, not of geometry, but of philosophy (of logic of language, of sense and nonsense).

We must try to approach the question from all possible points of view, in order to free ourselves from the charm of any particular one of them.

"It is high time we compare these phenomena to something different" -- one may say. -- I am thinking e.g. of mental illnesses. (CV p. 55 [MS 133 18: 29.10.1946])

I am thinking of mathematical signs; we always compare this language to name-of-object language. (But I am also thinking of the comparison of language to games.)

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Reality is number, which is both its stuff and form (Pythagoras)

From one point of view, the stuff of reality is always changing its form. From another point of view, what changes is not the form of reality but its stuff. But from the Pythagorean point of view, this is a distinction without a difference: reality (both stuff and form) is number:

[Source: Note 5]

The Pythagoreans defined 'point' as: a unit with position. But the combination of words 'a unit with position' is nonsense unless there is more than one unit -- i.e. position is relative or, in other words, a relation among points. A 'figure' is a 'constellation of points'. And the word 'unit' indicates that a point's location in the constellation must be unique. This is where geometry begins: A point is that without parts, two points define a line, three points not all in one line define a plane. This is Euclid. [Note 6]

The Pythagoreans say, "Look!" But they do not mean that points and numbers (if there is a difference) are objects to be sense perceived: The Pythagoreans have a picture -- but not a picture that is a physical hypothesis. "Look!" means something like "Look beyond the drawing to the reality reasoning conceives!" That there is a reality corresponding to the points in the drawing is the Pythagorean metaphysics.

... Plato refused to accept the Pythagorean idea of the point-unit and spoke of the point as "the beginning of a line" (Aristotle, Metaphysics 992a20 ff.), so that the point-unit, i.e. the point as having magnitude of its own, would be a fiction of the geometer, "a geometrical fiction" (ibid. 992a20-21), an hypothesis that needs to be "destroyed". (Frederick Copleston, A History of Philosophy: Greece and Rome (1947), I, xix, 4, 2, p. 159-160)

Plato's response to the Pythagoreans is an instance of what Pascal criticized, for both Plato and the Pythagoreans imagine that they are saying what geometry's points really are (i.e. stating "real" definitions), and so Plato says that what the Pythagoreans say is "a geometrical fiction" (i.e. a false hypothesis, rather than "mere sound without sense"), whereas both are only saying how they are using -- (or imagine they are using, for they may both be uttering nonsense) -- the word 'point' (i.e. stating a "nominal", i.e. verbal, definition).

An example of the method of metaphysics: If a thing has size ("magnitude"), then it is divisible into smaller magnitudes. Therefore, there are no point-units (atoms) if points have magnitude as the Pythagoreans claim. But if a thing has no magnitude, then it is nothing; but nothing can be constructed of point-units if point-units are nothing, as Plato seems to say that they are. And what then? And then nothing -- i.e. the thing to see is that we are not talking about points, but about how we use the word 'point' in geometry (in contrast to how the metaphysicians of mathematics imagine ("picture") points really to be).

In my asterisk grammar, the rule for using 'point' is: disregard the shape of the bearer of the name. But we can make this mythological: we can say 'Extension is really only accidental, not essential, to points'. Have I the right to say any more here than: I would discourage that way of thinking (LC p. 63)?

Am I doing any more than insisting on one grammatical account rather than another?

I say that this picture ... stands in the way of our seeing the use of the word as it is. (PI § 305)

And that is what I want to describe: how we use the word 'point' in geometry. I want to replace mythos ("stories", i.e. fanciful pictures of things) with logos ("the quiet weighing of linguistic facts" [Z § 447]); -- that is the program of Wittgenstein's logic of language: to rid our thinking of idle pictures, by showing them to be just that. These pictures are logically idle because they do not describe our actual use of language; they are not alternative grammatical accounts. And they are philosophically idle because they are not investigations of reality; they are fairy tales.

But am I saying any more than: 'Look at it this way! In geometry we make rules for operating with signs, just as in chess we make rules for operating with pieces of wood. Neither chess nor geometry has an independent subject matter'? -- Is that all I have the right to say? Yes, because I am saying no more here than: Use Wittgenstein's logic of language! If we use that logic, then we can speak of "false grammatical accounts", because then that language is defined.

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We know the word's meaning, until we are asked to explain it

We know how to use the word 'point' well enough until we are called to give an account of it (PI § 89). Because we try to give the wrong type of account. Rather than try to describe the use of a word as we know it, we try to explain "what a point is" -- i.e. to theorize about an invisible object. We become like a philosopher gesticulating with the sign 'I know' (OC § 482). We say 'A point is ...' while the meaning of these words "seems to float before our eyes"; and we want to point to that as proof positive that we understand what we are saying. But the logic (i.e. Wittgenstein's logic) of our language does not work that way.

That we now have ways to describe -- i.e. to remind ourselves of -- the use of a word shows the value of Wittgenstein's work.

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But if it is simply grammar that excludes 'name of an ideal object' as a part of speech, then why not make just this rule: to allow it as a part of speech? (cf. PG i § 82, p. 127) But is there any need for that? It already is a part of speech -- in fairy tales.

Suppose we made this rule: that by 'point' we mean an invisible object. (The grammar of 'ideal object' on this account is the same as that of 'invisible object'.) But does just uttering that rule accomplish anything? It gives us a picture with one hand ('object'), and then erases it with the other ('invisible'); this is an example of grammar stripping. Does it give us a picture at all?

What is an artist's drawing of an elf a picture of? The artist's idea of an elf? But then the words 'idea' and 'drawing' are interchangeable. 'Points are ideas. An idea is an invisible object.' This is another "picture" that does not describe how a word, namely 'idea', is used.

But is the grammar of 'point' like the grammar of 'elf' -- is this a helpful comparison? Is this a useful grammatical set: {elf, gryphon, point}?

Suppose a student, echoing Protagoras, says: "You utter the word 'point' but all you show us are chalk marks; -- and yet you say that the chalk marks are not points. And you even take away the chalk marks in the case of the tangent, and yet you say that you are still talking about an object."

"A point is a location." -- But what about the case of the tangent? The student really thinks he can imagine a line touching a circle "at one point only". But he cannot. Because we cannot imagine nonsense. On the supposed grammatical model ideal-object, the word 'point' has no visual meaning; and that means that if we cannot draw it, we cannot see it with our mind's eye either.

But how, then, can undefined language suggest pictures to us? It doesn't, unless we give the language a meaning. What happens in this case? Using e.g. my asterisk grammar for the word 'point' we can draw a tangent. But what we draw using this grammar is a conventional picture: the line and circle touch "at only one point" by definition. And that picture -- and not a picture of an "ideal tangent" -- is what the student imagines.

The student takes 'point' to mean an ink dot, and says: Look! a tangent covers more than one of those. But the geometers were not talking about dots in so far as dots have extension -- i.e. they were not talking about dots at all, whether in ink or sand or water. 'Atomic dot' means: disregard a dot's extension; it has no other meaning. And that is why an "atomic dot" cannot be imagined. Nothing in geometry can be imagined if geometry's language is taken to be about "ideal objects". But that model is a false grammatical account.

Does the combination of words 'ideal object' have any meaning? The statues made by the Greek sculptors could be called 'ideal objects'. Indeed, we do call them that, 'idealized forms' e.g. But they are not invisible.

*

Philosophy of Mathematics is the view from outside mathematics: What we imagine about points is of no importance to geometry

When I began to teach myself geometry -- as any child might do -- I used the form of expression 'Place a point P on line n such that ...' rather than 'Choose a point P on line n such that ...' -- What is remarkable is that this did not matter in the least: it did not affect my proofs.

'Place a point' (object) and 'Choose a point' (location) can be used interchangeably with no harm done. Use any form of expression you like as long as you don't break the rules (axioms, postulates) of the game or ask for nonsense (e.g. 'Choose a segment PP').

This is an important method of proof: 'If p, then q; but not-q. Therefore not-p.' We use this e.g. to prove that two lines have no more than one point in common -- by assuming that they have two points in common. But we do not break the rules of the game to do this; that is the difference between a postulate and a theorem: we can propose any theorem we like without changing the rules of the game. A theorem is proved or disproved, but we do not prove the contrary of a postulate; we simply discard it (e.g. the Parallel Postulate is discarded by non-Euclidean Geometry, by Albert Einstein and other theoretical physicists).

Go ahead! talk about points as though 'point' were the name of an invisible object. It does no harm.

Frege can speak of a "geometric heaven", Wittgenstein of "rules for operating with signs" (cf. BB p. 6, 15-16). -- But all that is external to the "calculus" (the account books) of mathematics. Regardless of which form of expression you use, as long as you use the word 'point' to do its proper work in the proofs, imagine anything you like about points -- as long as you have '2 + 2 = 4', think anything you like about numbers; it does not affect the proof -- your calculation.

Philosophy of Mathematics is the view from outside; it is the understanding of the calculus that is not needed for calculating. "What is mathematics?" is not a mathematical question. And mathematicians can dismiss it as a "matter of opinion", something to chat about.

That is why mathematicians (like scientists) make bad philosophers: they are not gripped by philosophical questions; they are not gripped by philosophy. We do not "chat" about understanding what is important to us.

The definitions, postulates, previously demonstrated theorems, and rules of proof (e.g. Euclid's "Common Notions") that justify the steps in a geometric proof are examples of the mathematical foundations of geometry. The 'Foundations of Mathematics' = 'the Philosophy of Mathematics' -- i.e. what philosophers and mathematicians say about geometry -- are not.

*

No theorem necessarily has an application outside geometry

In his geometry Euclid gave rules for reconstructing our world from as few principles as possible. According to his postulates the ideal lines of geometry are mirrored by lines in the perceptible world. That is because what Euclid wanted or believed his geometry to be was what the etymology of the word 'geometry' suggests, namely 'to measure the earth'. According to Euclid, if the universe were to disappear save for a copy of his geometry and a straightedge and compass, William Blake's Urizen could recreate our world.

Euclid's geometry was designed "to measure the earth" or "survey the land" (cf. Herodotus 2.109; a 'geometer' meant a 'land-surveyor'), and his geometry can be used that way, at least on earth. The relation between ideal geometry and perceptible reality is not inside geometry -- it is an external relation; nonetheless, Euclid was designed with that relation in mind. So maybe it should not cause too much perplexity that we can use geometry to determine, say, the width of a river without having to drag a tape-measure across the river -- or, rather, certainly it should perplex us that we are able to know such things as the width of a river in every particular case using an a priori method!

But it does not follow from the instances of where Euclid can be applied that Euclid must be applicable everywhere there is space. No theorem derived from postulates necessarily has an application in the world. And so it is not necessarily the case that Blake's Urizen could reconstruct every structure of our world using Euclid's geometry and a straightedge and compass.

---

Life-long perplexity about geometry that begins at school

From a geometry textbook used in the New Jersey state schools (ca. 1984) for children of age fourteen: "A location or a pinhole suggests the idea of a point. Points have no dimensions. Points are represented by dots and named by capital letters." That is from page one. On page five: "An undefined term is a word which has a meaning that is readily understood ... The basic terms of geometry -- point, line, and plane -- are undefined." Page fourteen: "Postulates are statements that describe fundamental properties of the basic terms."

No child should ever be subjected to such semantic bombast. Because there is no need for it. It is not in the nature of geometry, but only in the way of thinking of mathematicians.

Given the conceptual confusion in our textbooks -- the pictures of "abstract ideas" and "invisible, underlying entities" --, it is not to be wondered at that most students develop a lifelong perplexity about geometry; that while they may pass the tests that are given them, they never feel at home with geometry and are convinced that they don't really understand the thing at all.

Ironically, there is a statement on page fourteen of this textbook that could have pointed the way out of this confusion, had its authors been willing to take it seriously: "Definitions are explanations of how words are to be used."

(1) "... suggests the idea of a point." But what is the idea of a point? Is that idea the "readily understood" meaning of the "undefined term"?

Let us ask instead for an explanation of how the word 'point' is to be used. -- What is the grammar of the word 'point'?

Consider this statement: 'Through any two points there is one and only one line.' Is it a statement of fact about points, or is it a rule for using the word 'point'? Doesn't this incidence postulate belong to the grammar -- and so to the definition -- of the word point?

Here is a proof that 'Through any two points there is more than one line':

But, of course, this drawing doesn't prove anything. It simply breaks the rules of the game.

The drawing is a redefinition of the word 'point' -- one which, if we use it, we play the game of geometry wrong or not at all (cf. OC § 446). ("Play the game wrong" -- i.e. follow different rules, as in some alternative game.)

(2) What use is made of the sentence 'Points have no dimensions'?

In the drawing: Point A (x,y: 2,3). The sentence is a rule that tells the student not to say: 'Point A stretches from 1.9 to 2.1 on the x-axis.'

The dot A stretches from etc., but not the point A. Dimension-grammar is not to be applied to points; or, in other words, we use the word 'point' differently than we use the word 'dot'. On the other hand, it does not matter in the least if the student says 'Point A stretches ...' because this statement doesn't alter the postulates; the student is simply told: The dimensions of Point A do not interest geometry; they do not enter into this game. Just as the color of the chalk used is of no concern to hopscotch; the color is there, but it does not have a role in the game.

It is as if a child were to move the castle in chess along a row but stop half-way between squares; his strategy is to threaten both columns. We tell him: The rules of chess do not allow this. (At this point the rule is something the child must accept, even if it perplexes him. It would be the exceptional case, however, if we were unable to clarify the situation for the child by directing its attention to other games and their rules. At least here we are talking about something comprehensible to most children, i.e. games and rules; and not about the "properties of undefined terms".)

The mistake is to define the word 'point' before stating the incidence postulates. Clarifications of the meaning of 'point' should come with the explanations of the postulates: 'We are using the word 'point' in a more limited way than we use it in the language of every day.'

(3) There is no reason not to say that "points are named by capital letters". But points are "represented by dots" is confusing. It would be less misleading to say that 'points are represented by points'; -- i.e. there are not two different objects here: dots and points. Dots are points; but in geometry, we use the words 'dot' (or 'point-marker') and 'point' differently.

(4) The expression "property of a term" suggests -- I don't know what. I suppose the number of letters a word has can be called a "property" of that word. But by 'postulate' is meant -- and the use shows what is meant -- a rule of grammar. The postulates belong to the grammar of the "basic terms"; the postulates define the words 'point', 'line', 'plane'. Or again: the grammar of the word 'point' is given when all the postulates concerning points are given; these postulates define the word 'point' -- i.e. they give a grammatical explanation of how that word is to be used in geometry. (That is why it is a mistake to "suggest the idea of a point" before stating the postulates.)

(5) "An undefined term is a word which has a meaning that is readily understood." -- This could be a quote from Pascal or W.E. Johnson ("indefinable signs"); it is a nest of conceptual confusion. Words don't have any meaning other than that which we give them; and a word to which we haven't given a meaning -- i.e. haven't defined -- is meaningless. (BB p. 69, 73) This is most clearly the case when we are using a word in a way that the word is not used in "the language-game that is its original home" (PI § 116). And that is why we must -- and do -- define 'point', 'line', 'plane', in geometry by means of postulates.

"Readily understood". -- If the children do not use words in ways that the teacher has told them not to, then the children may be said to 'understand' (just like Hegel's students). But there are many different things we call 'understanding', and being perplexed about what "a point really is" is not one of them. The textbook and the teacher create this perplexity, and then sweep it under the rug with words like 'readily understood'; they might as well say 'obvious' or 'self-evident'. They do not have a philosophical understanding of what they are doing, and do not understand that a teacher needs this if he is not to mystify his students.

"The word is undefined but its meaning is readily understood." Here we have the picture of meaning as something occult, mysterious -- an essence abstracted by the mind, which is unable to put into words what the essence it has abstracted is.

(6) Why not compare geometry to a game? Why not characterize it as rules for making drawings and talking about them? Because that is what children do in school. The children learn the rules as they play the game.

By 'point' we mean a dot whose dimensions do not interest us. The area of the chalk or ink mark does not enter into the game played with the word 'point'. Indeed, it is explicitly ruled out of the game. Or again: 'Point A has a diameter of 1/8th inch' is not a move in the geometry language-game. Isn't that the way we use the word 'point' in geometry? The business is so straightforward that one might wonder why it was ever thought necessary to confuse children with a lot of idle pictures.

What stands in the way of the adoption of this matter of fact way of looking at language -- of regarding words as tools? ["Why are they called undefined terms when in fact we can define them?"]

---

Endnotes

Note 1: Thomas Heath, A History of Greek Mathematics (Oxford: 1921; reprint New York: 1981), i, p. 179. "For neither are perceptible lines such lines as the geometer speaks of (for no perceptible thing is straight or round in the way in which he defines 'straight' and 'round'; for a hoop touches a straight edge not at a point, but as Protagoras used to say it did, in his refutation of the geometers)" (Aristotle, Metaphysics 997b34-998a3, tr. Ross). If judged by the standard of sense perception, Protagoras' geometer does not describe reality.

Maybe something like this. In order for a tangent to intersect a circle or arc at one point only, a tangent would have to be (i.e. the word 'tangent' would have to be redefined to mean), not a line, but the vertex of an angle (e.g. the apex of a triangle and a circle can interest at one point only). I.e. visually 'a tangent intersects a circle at one point only' is nonsense (an undefined combination of words). (Cf. "Why is a circle round?" for visually a circle might have any shape.) [BACK]

Note 2: The latter two rules of grammar strike us as strange. Why? Because we never refer to them. 'How big is the number 3?' is nonsense, but 'At what dimensions will these numerals print?' is not nonsense: the second combination of words is defined in our language, but the first is not. Likewise 'Is the color blue larger than the color red?' is nonsense, but 'Is this red patch larger than that blue patch?' is not.

Another negative definition of 'point'

As well as saying that a point is that which has no dimensions, we could say: in geometry a 'point' is not a unit of measurement (of length e.g., but not of volume, speed, weight or time either). This is a rule of grammar, but another rule that we never refer to. [BACK]

Note 3: David Hilbert, Foundations of Geometry, 10th ed.; tr. Unger (La Salle: 1971). I taught myself geometry with proofs using this textbook: Brumfiel, Eicholz, Shanks Geometry (Reading: 1960). [BACK]

Note 4: That there are only two dimensions is one of the unspoken (implicit) axioms in Euclid, like the Betweenness Postulates: were it not for this rule a plane, which we want to have the properties of a flat sheet of paper, could be shaped like the letter V, a folded sheet of paper. The form of expression 'points of a plane' may be clearer than the expression 'points on a plane' (or 'points already on a plane'). Their meaning is identical, but the latter form may suggest that points are objects like ink spots on paper or like the "points" of Pointilism. [BACK]

Note 5: W.K.C. Guthrie, The Greek Philosophers (New York: 1950), p. 14. [BACK]

Note 6: If Euclid had replaced "A point is that without parts" with "A point is any location in space (i.e. the plane)", he might have put an end to the confusion from the very beginning -- by preventing it from ever having started. And if the Pythagorean "unit" were replaced with "location" -- i.e. if instead of defining 'point' as 'a unit with position', the Pythagoreans had simply defined 'point' as 'a position', then everyone might escape the fly-bottle (PI § 309). Or so it may seem.

An adequate definition of 'point'

We mustn't overlook an important point here. The term 'a geometric point' cannot simply be defined as 'a location in space' -- because all geometric figures are nothing more than sets of points -- i.e. they are all nothing more than locations in space; there are no objects in plane (axiomatic) geometry. Nor in Euclid can we define 'a geometric point' as 'a particular location in space' (which is in any case worse than vague, because the word 'space' is in fact undefined in geometry), but must instead define 'a geometric point' as 'a unique location in the plane', although rather than the word 'location' it may be clearer to use the word 'address'. But if we want to be as clear as possible, then a definition such as the following is needed: By 'a geometric point' we mean 'a single member of the set of all points in the plane'.

Or, if we picture the plane as a Cartesian grid, then by the word 'point' we mean an 'address in the plane, a particular pair of co-ordinates {x,y} in the plane'. Any particular point -- i.e. any particular location in the plane -- may be identified by a unique {x,y}; the choice of an "origin" in the plane for the Cartesian grid is arbitrary and of no importance beyond for naming -- i.e. assigning {x,y} values -- to the points in the plane.

A 'point' is an 'address in the plane'. Is that an un-definition (like Humpty Dumpty's un-birthdays)?

The Pythagorean 'unit' is an 'atom' or "uncuttable" and therefore "singular", but it is still imagined to be the name of an object of some kind, e.g. numbers as objects.

Deafness to the grammatical joke

"A dot is the best representation of a geometric point because it has the smallest surface area of any shape." But a geometric point has no surface area at all; no object can "approximate nothing". And so why doesn't that combination of words make our common sense cry out: "There is no surface area? Isn't that another way of saying that an object doesn't exist?" It should tell us that 'geometric point' is -- not the name of an "ideal object" in the mind or on the other side of the sky -- but instead not the name of any object at all, that the word 'point' has some other use in geometry. But our ears are calloused by our years at school (PI § 348). By a din of nonsense we are trained to babble without understanding, formulas of words without meaning, until it becomes second nature to us.

Wittgenstein to Drury apropos of a university exam question: "Questions like that make people stupid." Children are trained to be stupid. Stupefied as if hit on the head with a club, numbed by lack of understanding of the philosophy of mathematics -- i.e. of the use of language in geometry. The troubled eyes of students in geometry class is not due to the inherent difficulty of the subject matter but to the "difficulty" (i.e. utter confusion) in their teacher's presentation of it. People who are confused (transitive verb) look confused. (Philosophy for the teacher of geometry)

Someone points up at the night sky, points into open space. "See, just over there is the point where the asteroid is predicted to enter the earth's atmosphere." -- Who would ask in this case what thing the word 'point' is the name of? That question does not even arise. Is it comprehensible that a grammatical joke played with the words 'geometric point' could be sustained across more than 125 human generations? Compare the longevity of astrology. "... in sanity we are in the midst of madness." Does this case show that the human capacity for self-delusion is demonstrably limitless?

... is that alternative

I don't know if it demonstrates that, but what it does demonstrate is the consequence of blindly following a rule -- namely the rule that "the meaning of a word is the thing the word stands for" (regardless of whether that "thing" is perceptible or imperceptible, tangible or "abstract") -- and of being unable to imagine an alternative rule (which is what Wittgenstein offered).

... why do we feel a grammatical joke to be deep? (PI § 111)

And that is the depth of "What is a geometric point?" [BACK]

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