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"I am honestly disgusted with the other way of thinking"

The "other way" is not only the old way of thinking, because for most of humanity it is the present way of thinking as well, so powerful is the hold the old pictures of how language works have. Those pictures are:

  1. "a noun is the name of a person, place, or thing", and
  2. the meaning of a name is the thing the name stands for, as e.g. the essence (common nature) of a group of things named by a common name,

-- regardless of whether that thing is a tangible object, or an "abstract object", i.e. an intangible "something" which can be known but not told to others (which makes 'know' in that case nonsense, because when Plato says rhetorically, "And that which we know we must surely be able to tell?" he is stating a definition from Socrates).

Those pictures (and others also nebulous and non-objective) are the way of thinking Wittgenstein was "honestly disgusted with", and in contrast to them is Wittgenstein's "new" logic of language.

(Source of the page title above: Lectures & Conversations on Aesthetics, Psychology and Religious Belief, ed. Barret (1966), p. 28)

When Wittgenstein says "the other way" he is talking not only about language meaning ("Words are mere spoken sounds, marks on paper, meaningless in themselves -- and so logic's question is, what gives words meaning?") but also about ways of reasoning in philosophy, as e.g. wild -- because in no way verifiable -- speculation about the reality that "must" be behind concept-formation (an example of which is "the theory of abstraction"), trying to guess the essences of things ("analytic propositions"), and confusing where the limits of phenomena and concepts lie.

Wittgenstein struck on his "new way of philosophizing" very early after his post-Tractatus return to thinking about philosophy, although it would still be several years before he fully thought his way out of the the old way:

This method consists essentially in leaving aside the question of truth and asking about sense instead. (CV (1998 rev. ed.) [MS 105 46 c: 1929])

The Introduction to this site describes Wittgenstein's "new way of thinking" about philosophical problems, and the Synopsis contrasts that way with "the other way". The topics below are all discussed using Wittgenstein's "logic of language" -- i.e. the meaning of the word 'meaning' that Wittgenstein's selected in order to preserve an objective distinction between sense and nonsense in philosophy.

If I changed the title of my "A Synopsis of Wittgenstein's Logic of Language" to "The Elements of ..." (cf. The Elements of Euclid, Book One) or "An Introduction to ..." or even "A Grammar of ...", such titles might make its meaning clearer, and they would do away with the confusing word 'synopsis' in the title.

Topics on this page ...

Note: The logic of language remarks on the present page supplement discussions elsewhere on this site. Note that words marked "Query" below are Internet searches that were directed and misdirected to this site.

Is mythology divine historiography?

Preliminary: before discussing a question from Plato's Euthyphro (Is it logically possible for 'holy' to mean 'what the gods love', or in other words 'what is pleasing to the gods'?), it is well to recall what we mean, or at least what I mean, by "Socratic ignorance".

Query: what does Socrates conclude the oracle at Delphi means by saying that there is none wiser than Socrates? Why does the oracle at Delphi say Socrates is the wisest man?

In Plato's Apology the god does not say that Socrates is the wisest man (contrary to Diogenes Laertius and Xenophon). His oracle does not say that Socrates is wise, only that no man is wiser than Socrates. And Socrates believed that Apollo's oracle had meant that no man is wise, not that Socrates is wise. Why did Socrates' believe that? Because he put the oracle's words to the test of reason in order to discover their meaning. Using his method of question and answer (dialectic), Socrates tried to find a wise man in Athens, "the place where speech is most free" (Plato, Gorgias 461e). After the Apology, by 'wisdom' Plato's Socrates, now, I think, the child of Plato's imagination -- for the following standard is not found in Xenophon, although Ethics is Socrates' sole concern in both authors' accounts -- means 'being able to state real definitions of the cardinal virtues -- what is the common-nature (or essence) of courage, of piety, of temperance, of justice, of wisdom?' But Socrates cannot find any man who has such knowledge. In any case, by whichever criterion he had set for 'being wise', Socrates was unable to find a wise man in Athens, and this was why he believed that the god [Apollo was the patron god of philosophy] had meant, not that Socrates was wise, but that no man is wise.

However, if 'wise' and 'ignorant' are opposite in meaning, then Socrates is less ignorant than men who imagine that they know what they do not know. But Socrates does not claim that his being less ignorant than others makes him a wise man. It is rather the case that: all human wisdom is worthless -- i.e. it is not wisdom at all, but only a presumption by man of knowing what he do not know and imagining himself therefore to be wise. But Socrates is not presumptuous in this way, and therefore he is ready for philosophy: "for no one seeks to know what he imagines himself already know" (Plato, Meno 84c), and "philosophy begins in wonder" (Theaetetus 155c-d) -- that is, in being puzzled by something you do not understand but wish to know.

The 'holy' is 'what the gods love' or, in other words, 'piety' is 'whatever is pleasing to the gods'

Query: Euthyphro makes ethics mysterious, because we don't know what the gods love.

Euthyphro thinks he knows what he does not know; he thinks he knows what all the gods love because he is very learned in all the tales told by the poets about the gods. And he believes all of them, even the tales which show the gods doing shameful things. Euthyphro believes that the stories about the gods are historiography: mythology is the history ("what really happened") of the gods. And thus Euthyphro believes that he knows what the gods love, until Socrates' dialect (question and answer, always with step-by-step agreement with his companion) leads him to agree that different gods love different things. And therefore if it is true that the gods do not all love the same things but instead love different things, then piety or holiness cannot be what all the gods love. And therefore what Euthyphro has suggested cannot be the standard of measurement that Plato's Socrates is looking for in the Euthyphro when he attempts to discover what piety or holiness is. For Xenophon's Socrates 'to know' is: to be able to give an account of what you know, for Plato's Socrates 'to know' is: to be able to state a one-thing-in-common definition (Meno 74d, 72c) which serves as a standard of judgment as to what a thing is and is not. Euthyphro's account or definition hides a contradiction in his reasoning, and that is what shows that Euthyphro does not know what he thinks he knows. Socrates' method in logic is to uncover a contradiction in his companion's account of what the companion thinks he knows; that is the Socratic refutation or elenchus.

Euthyphro's ethics are "mysterious" only because his account of what he knows is contradictory. He has no standard of measurement, and yet he believes that he knows what piety or holiness is. Rather than "mysterious", his ethics are self-mystification or self-delusion. In sum he has no ethics, if Ethics is fully rational as for Socrates it is, but at the end of Plato's dialog Euthyphro runs away rather than acknowledge this either to Socrates or to himself.

Euthyphro is religious in a very different way from the way Socrates is, for Socrates does not believe that the poets' crudely anthropomorphic tales are a history of the gods. Like Plato and Xenophon, he believes that the gods must be good, and although we cannot really know much more than that about them, it is clear that they cannot be capricious or shameful. Because they are good the gods neither harm anyone, nor wish to see anyone harmed; they are benevolent towards men. Because, unlike men, the gods are wise.

Because the gods are good, they love what is good. But if by 'piety' we mean 'correct behavior towards the gods' (The question asked of Euthyphro is: how can we know what we must do if we are to behave correctly towards the gods?), then 'to do what is pious' would 'be to do whatever is good'. But of what use is the word 'piety' if piety is not concerned exclusively with correct or good behavior towards the gods? (We use the word 'holy' differently than we use the word 'piety'. For instance, piety is a virtue, but holiness is not a virtue, and whereas all piety is holy, not all holiness is piety. For example, a sacred place -- if it be granted that there is one place more sacred than another -- is holy, but it is not pious; what is pious is to treat a sacred place with reverence. Whether or not it would be natural to also call that reverence 'holiness' I don't know; I seldom see or hear that word used. (Those are of course simply remarks about the English language concepts 'holy' and 'pious' -- i.e. grammatical reminders about how we use the words 'holiness' and 'piety'.)

"What the gods love" does not define the common nature of "holiness"

The word 'piety' or 'holiness' cannot be defined as 'what the gods love' -- neither given Euthyphro's beliefs about the gods nor Socrates' beliefs either. Because, according to Socrates and contrary to Euthyphro, the gods are good, they all love the same things -- i.e. they all love everything that is good, just as all men would do were it not for their ignorance. If the holy is what[ever] the god's love, then what[ever] is holy is simply what[ever] is good, and what[ever] is good is what is holy. But that is only a formula, a tautology, and quite useless.

In other words, 'piety' is 'what pleases the gods' or 'holiness' is 'what is pleasing to the gods' cannot serve as definitions of 'piety' or 'holiness'. Because either different and contrary things please different gods (Euthyphro) or because 'what pleases the gods' = 'whatever is good' (Socrates).

Query: why, even if there were but one god, would Socrates say that we cannot define holiness as that which is loved by that god?

Because whatever is holy is also good, and it is loved by the god because it is good, and not for some other reason. (If a god loves what is evil, then what does a demon love? The question is rhetorical.) If virtue is knowledge, as was Socrates' view, then, because the title 'sophist' ("one who knows") is worthy only of a god (Phaedrus 278c-d), the gods will always choose what is good. Piety may be a good (an excellence proper to) for man -- but that alone would not tell us what is and what is not holy in any particular circumstance (It is not a serviceable standard).

An "other way of thinking" about the meaning of language

Query: confusions about the meaning of philosophy.

There is no essence of philosophy -- i.e. to the concept 'philosophy'; we call a variety of things 'philosophy'. But if you have the preconception about logic or language (a linguistic-meaning world-picture, so to speak) that the meaning of a word is what all applications of that word have in common (a common nature or essence), then it will confuse you if you are unable to figure out what this one-thing-in-common is, which the "theory of abstraction" tells you must be there. (However, that is not a scientific theory because it does not account for all the data -- i.e. the evidence that most words do not have essential meanings; it is a metaphysical theory (about what "really" is, despite any "appearances" to the contrary), not a theory that summarizes our experience.) Or if you suppose that words have true or real meanings ("persuasive definition").

"... with the other way of thinking" in Geometry

It is most useful to define 'a geometric point' as 'a unique location (or, address) in the plane' (The Philosophy of Geometry as reviewed by logic of language). A location can be named (e.g. "point B in the line AC"), but that name is not the name of an object; naming a location solely by Cartesian co-ordinates {x,y} -- and excluding the word 'point' altogether from geometry -- makes this clear (e.g. "the location {x,y : 7,8} in the plane". There may be anything at all at that location, but {7,8} is not the name of whatever that anything is).

In the jargon of some geometers, if the definition of a word is never used as the justification for a step in a geometric proof, then that word is said to be "undefined". This is related to W.E. Johnson's view of language, specifically his notion that there are "indefinable signs". And that is the view with which I am "honestly disgusted". (See on 'indefinable signs': "primitive terms that cannot be defined", and on W.E. Johnson's non-distinction of sense from nonsense.)

Trying to change the fashions of thought by talking about it is as promising as trying to change the fashions of dress by talking about it. (CV p. 62 [MS 134 143: 13.-14.4.1947 § 8]). But the topic of "undefined terms" in geometry illustrates the traditional, persistent and pernicious fashion -- i.e. "way of thinking" -- that "I am honestly disgusted with", and so I am drawn to talk about it again and again. But each time, perhaps I see a bit deeper into the problem.

"The logic of our language is misunderstood" (but not in the TLP's sense). It is a misunderstanding of "the connection between grammar and sense and nonsense" (BB p. 65) that encourages anyone to use the word 'undefined' in philosophy to mean anything other than 'meaningless', unless they are deliberately wanting to invent a misleading jargon-word. But what exactly is that misunderstanding -- i.e. what is the mistaken view of language that encouraged anyone to invent the expression 'undefined term' in geometry? Was it the picture that there are indefinable words -- i.e. words that cannot (logically "cannot") be defined because there are no more readily understood [or, simpler] terms to define them with (as if the only type of definition, the only way of explaining the use of a word, were verbal).

Hence, did the geometers who turned 'undefined' into a jargon-word intend to do so, or had they had thought that they were using that word in the normal way? The normal way in their view, if I have given an account of it, would be that the words 'point', 'line' and 'plane' are indefinable. But they used the word 'undefined' which does not mean 'logically impossible to define'. They said that there was no need to define these words (given that such definitions were not required for any geometric proofs), and therefore that there was no need for geometers to worry about their meaning. Indeed, that their meaning -- their "real meaning", not their conventional meaning (which is of no importance), of course -- was a philosophical question. Was that their view -- that we define things, not mere words: we are not concerned with the conventions for using the word 'point', which are quite arbitrary, but with what a point really is. That is, a definition tell us what kind of thing a point is -- was that their view? It may or it may not have been.

They might also have said that the word 'geometry' is undefined in geometry, meaning that no definition of that word is ever needed for a proof in geometry.

Whatever their view of language -- now the expression 'undefined term' is jargon, as in 'the three undefined terms'. [What are geometry's "no need to define" terms, and in what sense doesn't geometry need to define them? That form of expression would be clearer.] But if we recognize that -- shouldn't that be the end of it? No, because even the humble "sign" -- i.e. the physical part of language, such as sound or ink marks -- is important -- because "same sign" suggests "same meaning" or "same grammar" (in Wittgenstein's jargon, which because he did not explain his extended use of the concept 'grammar' to anyone puzzled even G.E. Moore when he attended Wittgenstein's lectures). We are not so aware of language as to be expecting to find "same sign, different grammars". If we see the word 'undefined', we expect it to have the same meaning here as there, there as here, as if that word had an essential meaning. It is a mistaken view of language meaning, but it is our inherited view, "the other way of thinking".

It is only because 'undefined' has become for us a jargon-word in geometry that the response "Because it is a jargon-word" is the answer to the question "Why are they called undefined terms when we can in fact define them?"

To call a geometric point a "geometric object" is like calling an elf a "fairy tale object". You only do that because you are wedded to the definition of 'noun' as "the name of a person, place or thing", a definition from which it follows that all nouns are names of objects, if not visible objects then invisible ones, for 'thing' may mean anything. But that definition of 'noun' embodies a false account of the logic of our language -- because those who invented that definition believed they were stating a "real definition" (one based on Plato's notion of common natures or Forms). But if we do no more than describe our language, then we find that not ALL but only SOME nouns are such names, and a definition based on that description would embody the correct account of the function of nouns when the word 'noun' is defined by syntax.

As things stand now, if you define the word 'noun' the way the textbooks do, then you will have to say that some nouns (as 'noun' is defined by syntax) are nouns (as 'noun' is defined by the textbooks) even though they are not. To call a geometric point a noun, then, suggests a picture to you that neither has any application to reality (for it must be possible to compare a picture of reality to the reality it is claimed to picture) nor tells you how the word 'point' is used in geometry. In a word 'point' both would be and would not be a noun, because you are using the old way of thinking's metaphysical account of how our language works.

If we are using the word ... as it is normally used (and how else are we to use it?) ... (PI § 246)

The following remarks use the word 'undefined' as we normally use that word. Which is the way geometers imagine that they are also using that word in their Philosophy of Geometry, but are not.

Query: can you define an undefined term in geometry?

Using Bertrand Russell's "Theory of Descriptions", we can rewrite this way: 'There is no x such that x is both without a definition' [For that is what 'undefined' must mean if it means anything: 'without a definition'] 'and has a meaning' [which is the very thing that is stated by a definition].

That is, if that rewriting makes anything clearer to us, not because that is "the true logical form" of that proposition (which is a view belonging to metaphysics, not logic [a discipline which is non-theoretical]).

How can there be such thoughtless queries, such absurd combinations of words, as these? Do these children not think at all; do they never question anything their teachers tell them? Thoughtless teachers make thoughtless students -- and the cycle repeats itself through countless mind-numbed generations, for of course, their teachers when they were students must also have swallowed slogans, repeating the same nonsense. It's Orwellian.

Here the word 'nonsense' means 'nonsense' in logic's sense of that word ['sounds without sense'], not in that word's sense of 'foolishness'. The slogan 'Freedom is slavery' is not mere foolishness; it is meaningless.

Query: can we give the exact definition of the undefined words in geometry?

When does 'undefined' not mean undefined? And if you define an "undefined word", is it still undefined? Rather than 'can', the question is: must you -- if you're going to know what you're talking about and not merely make babbling sounds -- define any undefined term in geometry? (In any case, what kind of "can" would this be -- empirical or logical possibility?) An "exact definition"? Is an "inexact definition" acceptable to you -- "Well, the definition is rather vague and we don't really know what we're talking about, but that's all right"?

If we define 'definition' as 'rules for using a word', then if a word is undefined there are no rules for using it. Now, how do we play a game if we don't know what its rules are? Is that logically possible?

The question that would show understanding here is: In which sense of 'undefined' are geometry's three "undefined terms" undefined?

The objection to the form of expression 'undefined term' is that it suggests a fundamentally thoughtless attitude towards language, and the consequences of that attitude are shown by the next query:

Query: what would you do to understand undefined words since they are undefined?

If you understand how a word is used, you must be able to describe how that word is used -- i.e. "give an account" of its use, that is, define it. That is our standard for saying that anyone 'understands how a word is used', the standard in philosophy that is. However, knowing how to use a word and being prepared to describe how that word is used in our language are not the same thing. When do we say that anyone 'understands how the "undefined words" are used' in geometry; what criterion do we use?

Drawings are not essential to plane geometry (although used as explanations of meaning)

In Euclid, there are not only axiomatic proofs, there are also straight-edge and compass proofs, drawings that show how the words 'point', 'line' and 'plane' are being used in geometry. In the second type of proof, the criterion for understanding is that the student makes geometry drawings correctly (as we all learned to do, or very few of us would ever have learned geometry). It is also important that 'line' (or 'straight line') is a word from our everyday language, as is the word 'point' (and there would be no confusion about the meaning of 'point' in geometry if no one were taught the grammatical myth that 'point' is the name of a "geometric object"). Making correct drawings and correctly following rules like the rule for using the word 'line' below -- That is what we call 'understanding' here.

It is only later, when we are taught that drawings are not essential to axiomatic geometry, that we become perplexed about how the meanings of 'point', 'line' and 'plane' are to be stated in language alone -- i.e. by one linguistic sign being substituted for another linguistic sign (verbal definition); and that is what tempts some philosophers of geometry to call those words "undefined terms". The truth is that we learn to use those words by ostensive definition -- i.e. by someone pointing to drawings -- and it is only after those drawings are taken away from us that we wonder how those words are to be defined now.

Trying to define those words without drawings is like trying to define the word 'pain' without describing how we learn to us that word.

Rule: Through any two points in the plane, there is one and only one line in the plane. "Of course, we don't really know what a point is, and we don't really know what a line or a plane is, and so we don't really know what we're talking about, but ..."

On the other hand, does it matter -- for work in geometry -- whether someone can "give an account" -- i.e. state the rules for using the words 'point', 'line', 'plane'? Isn't it enough that they use those words correctly? We might set either requirement -- (1) passing only one test (following the rules) or (2) passing both tests (following the rules and giving an account of the rules) -- as our criterion for saying whether someone 'understands' geometry or not. We might say that a philosophical understanding is only required in philosophy (I would not say that).

Augustine: "For He willed to make them Christians not mathematicians." But I will to make them, not only geometers, but also philosophers of geometry. For I want students to have a philosophical understanding of what they are studying and why they are studying it. I will to make them, at the earliest age possible, philosophers, men who live the examined life of Socrates.

Earlier I remarked that human beings are by nature learning creatures. And indeed human beings have to learn everything. The bird is not taught, so we believe, how to build its nest, but human beings have to learn -- i.e. have to be taught -- even what it is safe to eat and even how to procreate, things as fundamental as that to "the preservation of the individual and of the species". We acquire much of the language we speak, but we are not born knowing how to read and write or knowing how to "add and take away", but we must be taught. The bird does not need to learn what is the excellence appropriate to the avian form of life, what is the good life for birds. But man must learn about himself, if he is ever to know. Man knows -- i.e. he can give an account of what he knows. But because man can know, he can also not know: man can be ignorant: he can know that he does not know, and he can also believe that he knows when he does not know. A child does not know what is the good or the bad for it -- but it can be taught to ask and be answered (but only if the man is as thoughtful as his child).

"I grow old learning new things" -- a man can say that, but I don't know of how many animals it can be said.

Query: what do you mean by the term undefined?
Query: what do you mean by "undefined term"?
Query: why are the undefined terms called undefined?

That is the intelligent question, if that is a student's question for his teacher and not vice versa. "Why are they called undefined terms?" Why are you speaking in the passive voice -- who exactly calls them 'undefined terms'? And why do they do that?

Query: draw and explain the 3 undefined terms and give examples or representation of each.

But why only the three "undefined terms" -- for mustn't you also be able to draw and explain the "defined terms" such as 'triangle' and 'circle' if you are to understand how those words are used in geometry? Because geometry's definitions of the "defined terms" are hardly easy to understand without drawings. Why do you need to draw a line but not a triangle?

To speak of drawing and explaining "undefined terms" is an abuse of language, an abuse of the excellence that is uniquely appropriate to man -- namely, his reason. How can you draw and explain undefined language -- for is not anything you draw going to be equally correct and incorrect -- i.e. arbitrary? But if geometry really does not need to define those words, then does it matter how you "draw and explain" them?

Query: why line point and plane exist when in fact they are undefined?

But they don't "exist" if they are undefined -- until they are defined! But not as if it were [logically] possible to define anything into existence. What is correct is that some conceptual systems [frames of reference] recognize or take notice of things that other conceptual systems do not; e.g. one system might have only the single word [concept] 'clouds', whereas another has 'cirrus clouds', 'stratus clouds', 'cumulus clouds' and so on -- but not as if the latter system saw what was not there to be seen; a system that sees what is not there to be seen is called fantasy.

Whether or not geometry's valid theorems are applicable to the world [or only to geometry's plane] is a question of experience (although according to the Rationalists -- who confuse conceptual with factual investigations (Z § 458) --, beginning with Plato, it is assumed not to be). cf. Protagoras' objection: in experience a line doesn't touch a circle at only one point, just as in experience a line doesn't have only one dimension [length], but also has width. Which tells you that in geometry we are not talking about the lines and circles of experience. And thus we see that geometry is a game played according to exact rules -- and precisely because of this, rules must be stated for each and every more in its game; 'ambiguous' in geometry = 'meaningless'.

The proposition 'There is an x such that x has such-and-such qualities' is either a rule (definition) or it is a statement of fact -- i.e. it is either a logical proposition or an empirical proposition (and hence verifiable by experience). cf. Russell's "Theory of Descriptions" and 'golden mountain'; thought can be about what doesn't exist -- but that [i.e. thought] doesn't therefore make it exist!

The so-called -- and confusedly called -- geometric objects only "exist" because their names are defined. A definition is a rule for using a word -- and without a definition 'point', 'line' and 'plane' would be nonsense ("sounds without sense", meaningless). But those words are not and cannot be defined ostensively. We do not need, nor indeed have, a verbal definition of e.g. 'fire'. But we do need a verbal definition (a strict set of rules) for the words 'point', 'line', 'plane' -- because outside geometry the "geometric objects" named 'point' [But 'geometric point' is not the name of any type of object], 'line' and 'plane' have no existence, if you like to call those word 'names of objects' and if you like to call existing in a geometry textbook -- or existing in the "heaven" (or shadow worlds) of Frege and of the Platonic "Forms" -- 'existence' in order to confuse everyone, including yourself, with pictures that have no application to the world of experience.

Again, note that it is the words 'line', 'point' and 'plane' that are undefined, if they are undefined, not as it were the things named 'line', 'point' and 'plane', if those words are the names of anything. E.g. in one system the word 'cirrus clouds' is undefined, but the thing cirrus clouds is not as it were undefined. Not, that is, if we use the word 'undefined' as we normally use it; and as it is used also in logic of language studies which exclude "real definition", for logic is not theoretical, nor is our everyday language (Z § 223).

If the word 'point' were the name of an object, then there would be four dimensions of space: (1) one-dimensional (point), (2) two-dimensional (line, which is bi-directional in the plane), (3) three-dimensional (figures, e.g. triangles, which are multi-directional in the plane), and (4) four-dimensional (solids, e.g. pyramids, which demand at least three planes).

Many children -- in Malaysia e.g. (also I think, based on the queries from there, in the Philippines, which is where all the queries in this section came from) -- are forced to study maths and sciences in a language [English] they do not understand and which may only be their teacher's second or third language. That is difficulty enough for them, but in this case the students are also being asked to study philosophy -- i.e. to ask and answer a philosophical question -- in a language they do not understand.

Query: in geometry what are planes that do not have and never will have any points in common called?

With respect to geometric planes in axiomatic geometry "Now is always", and so the addition of the phrase 'never will have' is nonsense. And with respect to empirical geometry (or empirical anything else), the word 'never' is undefined (What is logically possible may happen; only a theory or world-picture says that it can't. What is a real possibility is whatever happens [These are of course grammatical remarks]). This query seeks the word 'parallel', I think.

Query: representations of undefined terms.

"What points, lines and planes really are in themselves, we cannot know; but we can represent them, as it were, as shadows of themselves." -- That confuses geometry with religion. The proposition 'We cannot see the gods, but we can represent them' is a rule of grammar; but there is no analogous rule in geometry.

The title of this page. But "the other way of thinking" is the ancient way of thinking that has now become "the way of not-thinking". I want to rid Philosophy of Geometry of the notion of anything occult and replace that notion with logic of language -- because I want to give a true account of the "grammar" of the language of geometry, not to theorize about "what geometry really is". Because a metaphysical theory does not describe the way we use language in geometry; if such a theory has any relation to grammar, it is as a grammatical myth -- i.e. a false account. And a metaphysical theory's relationship to reality is one of fantasy, of a "picture" that cannot [the impossibility is logical, not "real"] be compared with what it is said to picture; it does not describe reality; it floats free of it.

"What are numbers really?" is a grammatical, not a metaphysical, investigation (PI § 90), a conceptual, not a "real" investigation -- i.e. it is not an investigation of an independent reality [of anything that exists independently of language; like numbers, geometry is either defined into "existence" or simply does not exist], but only of the rules of a game (Z § 458). As is "What is geometry really?"

Query: identify the undefined words in geometry.

Begin by identifying the word 'undefined' as an undefined word in Geometry, until 'undefined' is identified as a jargon-word there. Summary:

If  (1) 'undefined'1 = that word as some geometers use it.

And  (2) 'undefined'2 = that word as we normally use it.

Then  (3) 'undefined'1 ≠ 'undefined'2

Or, in other words, (1) and (2) are the same sign but have different grammars. cf. 'cloud bank' and 'piggy bank'; the sign 'bank' here has two distinct meanings. And although there are resemblances, these are only curiosities (for anything may be compared to anything else, in some way or another), not defining. For example, many things called 'bank' may be compared to walls (as also 'river bank') and some clouds are shaped like pigs and clouds flow and money flows in and out of banks and so on. And if we are going to determine the meaning of words by assembling resemblances -- This will be yet another meaning of 'meaning' -- rather than by describing (by means of rules) the way those words are used, then any two words at all may have the same meaning. But that meaning of 'meaning' will be of no use to logic of language, because it will not make an objective distinction between sense and nonsense.

Now if for an expression to convey a meaning means ..... our expression may have all sorts of meanings, and I don't wish to say anything about them. (cf. BB p. 65)

Again, in both (1) and (2) we might say that 'undefined' means 'without rules for use'. Thus in (1) there are no rules for using a definition of 'point', 'line' or 'plane' in geometry proofs; nonetheless in (2) there are rules for using the words 'point', 'line' and 'plane' in geometry. It may be remarked that there is no contradiction between (1) and (2). However, that (1) requires special explanation if it is to be understood at all, shows that it is an instance of jargon rather than of our normal use of the word 'undefined'. Further, although we might in (1) replace 'undefined terms' with 'terms whose definitions are unnecessary for proofs in geometry', we cannot replace 'undefined terms' in (2) that way. And indeed, there is no reason to use the same word in (1) as in (2) at all. Rather than 'undefined terms', the words 'point', 'line' and 'plane' could be called e.g. 'no-proof terms' with no loss of meaning, substituting one jargon expression for another.

Query: the characteristics of 3 undefined terms.

But they aren't undefined if you can say what their characteristics -- i.e. their defining characteristics -- are, for any "accidental characteristics", if there were any, would be of no interest to geometry (and indeed would be excluded from geometry, for is a rule that has no role in a game a rule of that game at all?).

Query: give the examples of objects that signifies the point.

Of course there can only be such objects if the word 'point' in geometry is itself the name of an object (Otherwise what meaning will 'signify' have here?). You want to make an analogy, but is that analogy valid in this case? Not if the word 'point' in geometry is not the name of an object.

If  (1) 'point'1 = that word as it is used in geometry.

And  (2) 'point'2 = that word used to mean 'dot of ink'.

Then  (3) 'point'1 ≠ 'point'2

Note that (2) is an everyday use of the word 'point' -- but it is neither the only nor even the most frequent everyday use of that word.

Summary: 'a point in a plane' ≠ 'an ink dot on a map', but instead 'the location marked by the ink mark on the map' and a location is not an object; a location is not an object of any kind. (And those are of course grammatical remarks.)

The word 'point' in geometry does not name a "geometric object". [What I have called] Wittgenstein's logic of language breaks with that picture -- i.e. with the picture that all words must be names of objects "of some kind or another", if not visible, then invisible. Because that picture is a misleading grammatical analogy, one that does not describe the way we actually use language. (It is not the case that if we are talking about something then we must be talking some object.)

Query: the undefined terms of geometry and the reason why they are not yet defined.

"Well, we haven't discovered their definitions yet, but someday we shall. After 2,500 years of not knowing what we're talking about, some day we'll know. We'll define them yet!" And will this be the discovery of some new empirical fact about a non-empirical world where the undefined terms exist -- or will some super clever person discover the words of the magic formula (just the right combination of words) for defining those terms which everyone has been searching for (PI § 75) for so long?

A query like that shows the absurd confusion caused when a jargon-word is used without telling anyone that it is a jargon-word!

Query: what is the meaning of the undefined word point?

What does it mean to say that an undefined word has a meaning? Unless it is recognized that the word 'undefined' is a jargon-word in geometry, then responding to this query amounts to responding to a parrot, for as we normally use the words 'undefined' and 'meaning' an undefined word is meaningless. But geometers who use the expression 'undefined term' do not use those words in the normal way.

Question: are the "undefined terms" necessarily (essentially) "undefined" in geometry -- that is, is it logically possible to define the words 'point', 'line', 'plane' in such a way that their definitions can be used to justify a step in a geometric proof? But geometry does not seem to need geometric definitions of its "undefined terms" (and, so far as I can tell, it does not need that type of definition for them). But is what is unnecessary also without meaning? No, a rule of the game that is never employed ought to be excluded from the rule book by Ockham's principle, but that does not make the rule meaningless (despite its being useless -- 'useless', that is, not in Wittgenstein's sense, but in the sense of 'good for nothing').

"Through any two points in the plane, there is one and only one line." Does that definition of 'line' really never serve as the justification for a step in a geometric proof? Well you know I don't know. I thought I had used it that way in a proof that a line does not intersect itself, but that may have involved the Betweenness Axioms, which are not Euclidean (although it was thought by these axioms' inventors that they are implicitly there). But what if we do not use those axioms in our proofs, then is that definition of 'line' -- if it is a definition of 'line' -- never used in a proof?

Back to the query. What is the meaning of the word 'point' in geometry? Here is the word 'meaning' in the context in which Wittgenstein said that it was more helpful to the understanding to ask for the use rather than the meaning of the word. Because standing behind the word 'meaning' is "the other way" of thinking, the old, but apparently immortal picture: "All words are names, and the meaning of a name is the object it stands for."

And so the meaning of the word 'point' in geometry must be the geometric object that word stands for. But although we cannot see that object, we can nonetheless show its corresponding objects in our world, such as ink dots, pin holes, things like this.

And so long as you are unable to imagine an alternative to that account of the "grammar" of our language, which is a false account, then how else can you define words in geometry other than as the names of geometric objects? It is that alternative account that Wittgenstein's "new way of thinking" offers.

The word 'use' suggests a tool (of which there are many kinds) whereas the word 'meaning' (in the minds of many) suggests only "name and the object it stand for".

If we did not draw the triangles of this world to explain what we meant by the word 'triangle' in geometry, no ordinary person would know what axiomatic geometry was talking about. If we excluded such drawings from our explanation of meaning -- it would be like trying to learn chess without ever being shown a board and chess pieces but only signs like 'P-K4'. But the triangles of this world are no more the triangles of geometry than the lines of this world are the lines of geometry. And thus why is not 'triangle' an "undefined term"? Because it has a definition that is used as a justification for steps in geometric proofs. But any word that does not have a definition that is so-used is dubbed 'undefined'. And that is why some geometers call them 'undefined terms' even though we can -- and do -- in fact define them (as we normally use the word 'define' = 'explain the meaning of'), for otherwise we would have no idea what we were talking about.

Query: why are the three terms considered undefined?

Are they only "considered" to be undefined -- i.e. is it a matter of opinion that the three terms are undefined? But the difference here is not a difference in opinions -- but a differences in frames of reference. A difference in ways of thinking.

Difference in ways of life, points of reference

What is not a matter of opinion is the difference between the two ways of thinking -- i.e. the two different ways of looking at our language, either the account of language in which all nouns are names of objects ("the other way of thinking"), or the account of language as the relation between "grammar and sense and nonsense" (Wittgenstein). The older account can be shown to be false -- i.e. can be shown not to be a description of the way we use our language -- but only if the non-theoretical account of language, or in other words, Wittgenstein's selected meaning of 'meaning', is accepted. And there is nothing to force anyone to adopt his new frame of reference (point of view). Different frames of reference are not a difference in opinions, but in ways of life (PI § 167), as is the difference between the examined and the unexamined life.

Nonetheless, it is unphilosophical not to make clear to yourself which -- if any -- logic of language you are making use of when you philosophize -- i.e. how you are objectively distinguishing between sense and nonsense -- if you are making that distinction. And if you are not making that distinction, then what are you doing when you philosophize?

The absence of a logic of language in Descartes' method, when he attempted to apply the method of mathematical deduction to philosophy [This was not original; it had been Plato's method in philosophy as well] in order to discover the truth about reality (the "really real" reality as opposed to the "only apparent" reality of experience), took the language Descartes had learned in his childhood and youth for granted -- as if he had neither learned nor needed to learn the language he speaks.

He says that there are "innate ideas" such as the idea of the God of the philosophers which he finds in his own mind, without the need for this idea to have been learned -- and, from which it follows, without the need to have learned the language in which to talk about this god. No, Descartes does not say that -- however, he does assume it, for if he is setting aside everything he has learned by means of his five senses (regarding that knowledge as, unlike maths, uncertain and therefore not "really" knowledge at all), then mustn't he also set aside the language he learned to speak. And if he said that he did not need that language in order to identify his idea of the God of the philosophers, then we would no more understand him than we understand William James' Mr. Ballard, who said that he had had thoughts about God before he had learned any language (PI §§ 342-349).

Is what Descartes and Mr. Ballard claim logically possible -- i.e. can it be described? How does Descartes distinguish sense from nonsense? If he says that this distinction is an innate idea, then his distinction is subjective -- i.e. it is no distinction at all: for whatever seems correct to Descartes is going to be correct, which only means that we can't talk about correct and incorrect in his case (ibid. § 258).

One later development of Descartes' method of introspective reasoning is: "We are now doing Cartesian linguistics. -- How does a thought get from one mind to another?" That picture of language is found today in the social science of linguistics.

That is, or seems to be, the nature of philosophy -- that there will always be Aristotelian, Cartesian, Hegelian, and so on, thinkers. The differences between such thinkers in not a difference in opinions, for a difference in opinions within a single frame of reference my be resolvable, whereas a difference in frames of reference is not. A frame of reference can only be criticized from outside it, making use of course of a different frame of reference.

Query: importance of undefined terms in the map.

Exactly. Look at how the word 'point' is used when we talk about maps, which should show you that 'point' is not the name of any object ... if your mind is open to that possibility (that a noun need not be a name, but may be used in countless other ways). The "grammar" of the word 'point' in map-reading is not name-of-object but of pointing-out-word; cf. the word 'this' in 'This is a chair'; the word 'this' is not the name of this particular chair, nor of any other chair, nor of all chairs. It is what grammarians call a "demonstrative pronoun" [although the word 'pronoun' -- or 'a word that is used in place of a noun' -- is misleading here, for 'noun' suggests 'name of a person, place or thing'; the clearer form of expression, from the point of view of meaning (not necessarily of syntax), would be "demonstrative word"], and that is the grammar (in Wittgenstein's sense of 'grammar') that the word 'point' has in geometry, just as it has in map-reading.

Query: is it possible to define all the terms in geometry including the three terms which are points lines and planes?

If we use the word 'define' as we normally use it, then not only can all of geometry's terms be defined -- but, also, they must be defined, for otherwise they would be nonsense. All explanations of meaning come to an end, but that does not mean that there are cases where no type of explanation of meaning is possible. For to say that there are such cases is to say that there are words that are undefined as we normally use the word 'undefined' -- i.e. that there are meaningless words which nonetheless play a role in our language. -- But if they have a role to play, then that role can be described, and when we describe that role, we define the word; so that any word that has a role in our language can be defined.

If there is impossibility here, it is neither logical nor empirical impossibility, but only a failure of imagination. If a child, who is capable of learning, does not understand one explanation, then you must invent others until it does understand; otherwise, do not say that the child has failed, but that you as a teacher have failed the child.

Query: give examples of objects that would illustrate undefined terms.

Always it is assumed that a word must be the name of an object -- that the word 'point' just has to be the name of an object. But what if 'point' is not the name of an object? Then how can you so much as even look for x if the x you are looking for has no value ("value of a variable"). "Give examples of objects that illustrate x." But if the value of the variable x is not an object, then you have been ordered to perform a contradiction. Give examples of objects that illustrate the word 'the' e.g.

If you were hunting for a golden-fleeced sheep, although none may exist, at least you would know if you found one. But if you were seeking "an object with no dimension" ... you would be seeking something that does not exist and which therefore you cannot find -- nor indeed even seek. And that is what an "undefined term" in geometry is -- when no one has told you that by 'undefined' geometers do not mean what we normally mean by 'undefined'. And so you are looking for the meaning of an "undefined term" using the word 'undefined' as we normally use it. But as we normally use that word, 'undefined' = 'nonsense', so that you are looking for the meaning of a meaningless word -- that is, for a variable x that has no value.

Query: who is the philosopher of geometry?

The philosopher Pythagoras, although there is a difference between the philosophy of geometry and geometry itself, did important work in geometry as well. I don't know whether Euclid would have been called a philosopher at the time in which he lived; the word 'geometer' (originally, 'one who measures or demarcates the earth') did exist then. The Stoics did have a category 'mathematics', however they meant by the word what we mean by 'physics'. Was Euclid a philosopher in our sense of the word 'philosopher'? No, but in the days when 'philosophy' = 'learning', certainly. Plato did distinguish between geometry and philosophy.

Freedom through logic of language (the examined life)

Ever and anon I return to this topic. Elsewhere I wrote of being freed from unreason in religion by Albert Schweitzer, as I have also written of being freed from unreason in ethics by Socrates.

Query: how Aristotle defines psychology?

That, precisely that is "the other way" of thinking that I am "honestly disgusted with". That is being surrounded by vagueness and confusion -- as if every question were factual [every definition a "real definition" (i.e. hypothesis about the nature of some thing)], none conceptual. That fundamental confusion and the consequent vagueness is, among many other things, what Wittgenstein's logic of language is salvation [freedom] from.

What is this "psychology" when it's at home? What is it that Aristotle defines? What type of definition would this be -- is it a theory about what some natural phenomenon, maybe some concept, named 'psychology' really is? "Aristotle gives a definition" -- a definition of what? What is this "psychology", this "abstract object" that is pictured in the query to have an independent existence of only God knows what kind and only God knows where? Except that even God doesn't know -- because even God cannot know nonsense. I want this exotic thing "psychology" tied down with ropes; I want more than the fleeting glimpse the early explorers to the New World had when they claimed to have seen or rather almost seen ...

Of course the query may simply ask: "How did Aristotle use the word 'psychology'?" But there is a Gestalt shift world of difference between asking that and asking "How did Aristotle define psychology?"

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