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Geometrically-defined circles versus visually-defined circles. Definitions of 'round', 'equidistant', 'congruent', and 'circle'.

If it cannot be put into words, and the Tractatus Logico-Philosophicus says that it cannot, then how can it be that Wittgenstein does put it into words?

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Is there a general definition of the word 'chair'?

We do not define the word 'chair' as "something to sit on" (Wittgenstein), for one may sit on a table and one may sit on the floor and those are not examples of what we call by the common name 'chair'. We do not define the word 'chair' verbally at all, but instead by pointing to examples of chairs: 'This is the type of thing we call a chair'. But note that there is no archetype (Parmenides 133c), no absolute archetypal chair (no Form "chairness" or "chairhood" independent of Plato's "metaphysical picture"). We define the word 'chair' ostensively: 'This and they like ...', but not 'This and only what is identical' (And so Wittgenstein speaks of "family likenesses", but that is not a "theory of meaning", merely a characterization of how we in public fact use words, and it has no predictive power; it is a simile of doubtful use.)

The combination of words 'something to sit on', which is an apparent stab at giving a general definition of 'chair', is not "the normal definition" of the word 'chair'. There is no general definition of 'chair' at all.


"... if they [philosophers] do ask and they want a definition, they do not want the most natural definition, e.g. of 'chair' they do not want the definition 'something to sit on'. Why are they not satisfied with the normal definition of chair, or, to put the question in another way, why do they wish to ask for the definition of a physical object?"

Source: "From the Minutes of the Moral Science Club, 23.2.1939" in Wittgenstein in Cambridge (2008), Document 246, p. 295.

I can give no sense to the expressions 'definition of chair' or 'definition of a physical object' -- (Would Wittgenstein have spoken this way in 1939? I am puzzled if he did) -- and the fact that form of expression is commonly uttered tells me nothing about whether that combination of words is defined language or nonsense.

Query: Socrates. Define a chair.

What use would that be to Socrates' inquiries in ethics? "A chair is a piece of furniture designed or adapted to be sat in or on." Why would Socrates want such a definition? (It is a nominal definition, an account of how we use the common name 'chair' in our language. But there is "no real definition of chair", although there is an ostensive definition -- i.e. pointing at examples of chairs.) To serve as a model of the type of account Socrates is seeking, namely, universal (absolute or defining common quality) definitions? Plato gives the examples of 'quickness' (Laches) and 'clay' (Theaetetus); I gave the example of 'simile'.

[Question: That many things may be used to serve as chairs or that some places resemble chairs, e.g. indentations in stone -- do these facts belong essentially to the definition of 'chair'? Do they belong to an explanation of the meaning [i.e. a description of the use we make] of the word 'chair'? Only if we need them to be to make our meaning clear, to avoid a misunderstanding? Is what does or does not belong to a definition determined by what we want the definition for?]

Query: defining a chair as something that people sit on is correct. Because it is a factual statement.

While it may appear the definition "something that people sit on" identifies one thing that all chairs have in common, it does not say what differentiates chairs from all other things people sit on, or in other words it does not set the limits of the class {chair} and therefore it fails the second requirement [test] of Socratic definition.

As remarked above, people sit on rocks, tables, the ground, using these as [as if they were?] chairs. And further, there are chairs that people do not sit on, e.g. chairs that are purely decorative, not intended to be sat on and which would not support the weight if they were; and others that are imaginative designs that because of their shape could not be sat on or in. There are also holographic chairs.

"... using these as if they were chairs", as if to say those are not really chairs. But if 'chair' is defined as 'something that people sit on', then rocks, tables, and the ground really are chairs, but the decorative and illusory shapes of chairs are not. By our own definition.

And so the query's thesis is incorrect: it is not a factual statement: it is not a description of how we normally use the word 'chair', and it should not satisfy the lexicographer.

Query: does the chair exist? Philosophy.

"Philosophy" -- well, this is it (Wittgenstein: on overhearing a conversation between two philosophers: "They are not crazy, just doing philosophy" (OC § 467)). At school a professor asked us whether we thought we knew we were sitting on a chair. Most people thought so (I don't know what the rest thought). Why? "Because otherwise I would be lying on the floor." To which the professor's reply was "If the chair isn't there, the floor isn't there either." (But then neither is the professor, nor possibly anyone else for that matter.) Samuel Johnson thought he did know, but for the wrong reason (as did G.E. Moore).

As we normally use language, the professor's question is nonsense. If you speak English you know -- no, not that you are sitting in a chair -- but that this is what we call a 'chair', this 'sitting', this 'being in a condition to know'. Notwithstanding that, however, is there a metaphysical use of language -- is there a "deeper level" of language use? No, but there is a clearer one (Berkeley: Perception (and reality))!

Query: it still has chairness. Philosophy.

If said object, if the query is talking about an object, is a chair, how much [i.e. what] can you take away from the chair before we can say that it no longer "has chairness", i.e. that "it" is not a chair "anymore"? Cf. "essence reduced"; if x is essentially a chair, then it cannot cease to be a chair; but then we are not talking about an object ["physical object"], but about an idea ["abstraction"]. An object can always be changed, but not an idea.

Query: nominal definition of philosophy.

Why -- is there another kind? [Are these real definitions of philosophy?] But, then, is there even a nominal [rather than a real definition] of philosophy? Maybe, 'philosophy' is 'the rational study of logic, ethics and metaphysics'? Is that a general definition of the word 'philosophy'? Is that an example of a "real definition of philosophy"? But, then, do we know "what philosophy really is", i.e. what "the true meaning of the word 'philosophy' is?

If it cannot be said, then how can it be that you are saying it?

About the words 'nonsense' and 'meaning' and 'nonsense with meaning' (Count Eberhard's Hawthorn).

The main point [of my book, apparently] is the theory of what can be expressed (gesagt) by props [i.e. propositions] -- i.e. by language -- ... and what can not be expressed by props, but only shown (gezeigt); which, I believe, is the cardinal problem of philosophy. (Letter to Bertrand Russell dated 19.8.1919, R.37, or, as renumbered in Wittgenstein in Cambridge, 63)

It is significant [Why, what does it signify? that he would not have used that word to describe his later philosophy? anything else? That he recognized that it was a (metaphysical, or, philosophical) theory], I think, that Wittgenstein uses the word 'theory' here. And it is a theory of the type: despite any [all] appearances to the contrary, in reality [such-and-such is the case, namely, that despite our saying x and thinking x, x can neither be said nor thought] ... The obvious question, of course, to ask Wittgenstein would have been: if it cannot be said, then how is it that you are saying it? If it cannot be thought, then how is it that you are thinking it? [You assert that it cannot be expressed, but as a matter of fact you express it -- now, how can that be?] Which shows the absurdity of this "theory of meaning" [The theory is not nonsense, but it belongs to grammar -- i.e. what it does is to redefine the word 'nonsense'; it does not, as Wittgenstein imagined it did at that time, state "what nonsense really is"], for nonsense that can convey meaning is, after all, not nonsense. (Not that Wittgenstein would have accepted this criticism at the time he wrote the TLP, which is a work of thoroughgoing metaphysics).

Query: ineffability of logic. Wittgenstein.

Exactly ... because, notwithstanding, Wittgenstein does say the unsayable (ineffable). Now how can that be?

"Why is a circle round?" Yes, why is a circle round, for might it not be square or triangular instead?

Suppose one took the view that every proposition -- i.e. every combination of signs that has the form 'proposition' -- were [in very deed] a statements of fact, that even definitions were statements of fact, and not mere statements of fact about language [usage] either, but statements of fact about the things that language names -- and thus that 'A circle is round' is a statement of fact?

The Attitude of Philosophy

Some nice quotes (what Schweitzer called "plums" or rather "raisins", as in a cake) were taken from my page about the Philosophy of Psychology. But it wasn't that I had a brain at the time that I wrote that page, but that I had a good grasp of Wittgenstein's ideas then, ideas which have now in many ways faded from my memory [About slogans in philosophy, about why they are often useful to us: as reminders, e.g. of past investigations we once made but have since forgotten: "Slogans are easy, and stick in the memory" (M. Sc. Cl. 23.2.1939)]. But I doubt that anyone who hasn't studied (thought about Wittgenstein's ideas to the point of making them, with whatever limited understanding one is able, one's own) Wittgenstein's work will understand those plum quotes, however, if only because, but this isn't the only because, the word 'grammar' is jargon of course (What does 'This is a grammatical joke' mean to someone who has not learned that jargon). But why did I ever have that grasp? It was not because my interest in Wittgenstein's work was academic -- but because it was driven by a thirst for understanding (for clearing away "the vagueness and confusion") and for the truth. (I have never done scholarly work in the History of Philosophy. Indeed, as Wittgenstein said about the background of his own work: "I think I have never invented a line of thinking but that it was always provided for me by someone else & and I have done no more than passionately take it up for my work of clarification." (CV [MS 154 15v: 1931 § 2]) And that is what I have done with Wittgenstein and Albert Schweitzer, although my aim in philosophy has ever been not only for clarity, as vital as that has been to me, but always also for the truth.)

Another Internet writer cursorily referred to me, apropos of my page about the Philosophy of Geometry, as a "Wittgenstein commentator", and in a some places -- but not that page -- (e.g. when I write about the TLP or try to give an account of Wittgenstein's later ideas, as I do in the Introduction and Synopsis of Wittgenstein's Logic of Language) that is -- or may be -- correct (What does 'commentator' mean here? A dictionary of synonyms lists: 'critic', 'interpreter', 'expositor', 'annotator' [Dictionary authors are given no credit for their work, other than the highest of all honors: that their work is helpful to countless dictionary users]); but what is correct is that almost all of my tools for thinking in philosophy come from Wittgenstein (The Synopsis is my exposition-explanation (of meaning) of Wittgenstein's tools for thinking in philosophy), but most of my pages are not attempts to "exposit" or "interpret" -- i.e. make the meaning clear of -- Wittgenstein's thought and tools, but to make use of those tools for my own logic of language studies.

And that is the attitude of philosophy, based on what I do, according to me.

Wittgenstein's ideas versus my account of Wittgenstein's ideas

In the Introduction to my Synopsis of Wittgenstein's Logic of Language I wrote:

Why "ask for the use rather than for the meaning"? Because 'meaning of a word' suggests an object [cf. "Studying the grammar of the expression 'explanation of meaning' will teach you something about the grammar of the word 'meaning' and will cure you of the temptation to look about you for some object which you might call 'the meaning'" (BB p. 1)] (whether visible or invisible), whereas 'use of a word' suggests a tool that we use to do some work in our life.

But that is not what Wittgenstein said, or rather is reported to have said, at the meeting of the Moral Science Club (Cambridge, 23 February 1939, op. cit. p. 296; I have not altered the original punctuation). He gave a very different, and more than a bit puzzling, reason:

Why is it in a great number of cases useful to ask for the use and not the meaning? Because meaning suggests an object, whereas use a number of objects spread out in time.

However, the context of Wittgenstein's statement shows that he was responding to someone else's answer to his question "In a vast number of cases it is possible to replace "the meaning of a word" by the use of a word. In what way is it useful.... Suppose I ask: What is a zebra?" Maybe I point to a picture in a book. But that particular picture is not the meaning of the common name 'zebra'; we do not use the word 'zebra' that way. And thus Wittgenstein's statement, or reported statement, above about "a number of objects spread out in time".

Do I understand that statement? No. In the Philosophical Investigations § 43 Wittgenstein says, or certainly seems to say, that we do not define common names of objects by asking for their use rather than their meaning; instead: "the meaning of a name is sometimes explained by pointing to its bearer". A common name is of course a type of tool, as are all other words; the question here is if the tool comparison is helpful for explaining the meaning of a word when we are talking about the part of speech 'common-name'. If someone asks what a Zebra is, he are not asking for a general definition of 'common name'. And, indeed, if he is going to understand the ostensive definition we give him, he must already understand what we mean by 'common name' -- namely, "This and the like are zebras". (What I have just written does make anything clearer here.)

Query: Wittgenstein: do not ask what the meaning of a word is; ask what its use is.

Because 'meaning' suggests one [a single] thing whereas 'use' suggests a number of things? But does 'use' suggest that? Maybe 'uses' would -- but then wouldn't 'meanings' also: "What are the meanings of this word?" If you say 'its use', doesn't that suggest one [a single] thing? But the "thing" suggest is no kind of object, whereas the word 'meaning' may suggest an "abstract object" -- for, after all, what is an essence when it's at home?! The second understanding of "ask for the use" is the serviceable one, I think: because 'use' suggests a tool [conceptual tool] rather than "an abstract object" [Hypostatization occurs when you can't imagine an alternative].

Query: the connection between language and logic.
Query: a clear distinction between logic and language.

The connection between language and logic is the connection between a game and its rules (That is not a metaphor; it is a definition of 'logic', but not the only possible one, just as that is not the only possible comparison on which to base a definition of 'logic').

Query: why is a circle round?

And so what shall we say? Putting our hand under our chin and pondering ..... Before coming to an understanding (i.e. my view) of "Wittgenstein's logic of language", I would not have known what else to do. Analytical philosophers claim mysteriously to abstract the essence (occult abstraction) of circles: If we ponder the concept 'circle' we find that it is the essence of circles to be round. The mystical "must": Analysis of the concept 'rain' reveals that rain must fall downwards rather than upwards. (Query: "philosophical and logical meaning of down?")

At first blush it appears obvious that a circle is round "by definition". But what would be the source of that definition? Natural law, God as divine legislator? Suppose instead we call a word's 'definition' what we give when we describe how we use that word in our language. Then, as the following discussion shows, describing our use of the word 'circle' isn't simple.

In pure geometry (whether Euclidean or later non-Euclidean, or in a word: axiomatic geometry), in contrast to applied (to the world of sense experience) geometry, a circle is not round (nor not-round either). The figures of pure geometry have no shape. And that is, of course, a grammatical remark (i.e. an account the of "rules of grammar" (in Wittgenstein's jargon) in the particular case of pure geometry, a mere reminder of the facts in plain view). Because 'shape' is defined visually or tactilely, and the figures (circles, triangles, polygons) of pure geometry are not.

Query: Why a circle is round.

The Geometric circle versus the visually circular

There are no shapes in axiomatic geometry: a word such as 'round' is undefined there (i.e. it is a sound or mark without meaning). Axiomatic geometry is not visual (but not as if "non-visual" meant: concerned with "ghost-like objects"; someone who plays chess without using a board and chess pieces but instead using a different notation (cf. PG; i, Appendix § 6, p. 223) -- because a board with chess pieces is a notation -- is not moving ghosts about when he writes B-K3). In applied geometry, a circle may or may not be visually round -- and which it is an empirical question. For instance, a circle in 3-dimensional space (e.g. in a mountainous landscape) may not be visually round, but more or less amoeba-shaped (if 'circle' is defined as 'all points equidistant from a given point of reference').

Cf. Projecting (mapping) a three-dimensional surface onto a two-dimensional space, e.g. a globe onto a flat sheet of paper.

To assert dogmatically that a circle must be round is to be held captive by a grammatical picture, e.g. a drawing in two-dimensional space, such as an instructor at school draws on a flat blackboard using a compass and chalk (That drawing is a notation for single-plane geometry); the assertion mistakes a rule of grammar (definition) for a statement of fact. Why must a circle be round? Because someone has chosen to define the word 'circle' that way (The only "must" in that case is logical necessity ).

Query: a general definition is a fact.
Query: a circle is round because that's what a circle is. Logic.

For Socrates, according to Aristotle's account, and as found in Plato's early dialogs -- Yes, a general definition is a statement of fact about the essential nature of some thing (where essence belongs to reality rather than to the conventions of our language).

Holding that all propositions are questions of fact, none questions of "grammar" -- would that not be a peculiar state of mind, namely, madness? -- For that view would make it impossible to clear up the most fundamental confusion -- namely, the confusion over the distinction between sense and nonsense (defined and undefined words of combinations of words). The combination of words 'Why is a circle round?' is an excellent example!

Is it nonsense to say that a circle is round because it is the essential nature of a circle to be round, and that therefore any given circle must in fact be round despite any appearances to the contrary? What is the essence of a circle? The notion that "general definitions are statements of fact rather than of convention" ...

And here -- Wittgenstein's decisive decision about his method in philosophy -- the remark: "I want to say: I must begin with the distinction between sense and nonsense. Nothing is possible prior to that; I can't give it a foundation" (PG i § 81, p. 126-7). And then the question is: how do we make an objective distinction between sense and nonsense? And Wittgenstein replies -- at least in my simplistic account ("simplistic" because Wittgenstein had far more reservations about the universal applicability of the comparison of language to a collection of games, where what characterizes a game is its rules, than is apparent from my account) he replies -- by using the concept 'grammar' (as Wittgenstein will define the word 'grammar'). And thus: not all propositions are statements of fact, but instead some propositions state rules of grammar; and confusing those two type of propositions is the fundamental mistake made by philosophers.

"Why is a circle round?" This a grammatical joke. Or, other forms: 'Why can't a circle be square?', 'Why are all circles round?', 'Why is a circle round rather than square?'. (Note: the expression 'squaring the circle' is defined in Greek geometry where it means 'constructing a square with the same area as a given circle'; that, however, is not apropos of our discussion here.)

Even in natural language, words are tools; they do not "control our minds"; we are not their hapless, helpless victims. Not in "Wittgenstein's logic of language" we aren't, and that belongs to the Gestalt shift in Philosophy.

Compare: cases where we say that the meaning of 'x' is determined by the method by which x is measured or verified.

Query: why are metaphysical statements not empirical?

Because they cannot be put to the test of experience. That belongs to the definitions of both 'metaphysical' and 'empirical'. What may be claimed is that metaphysical statements are statements about non-perceptible reality -- i.e. statements about things invisible rather than visible, and they may be invisible despite any mere appearances to the contrary ("the reality behind reality"). Whether that is a true account of the grammar of such statements is another matter. This query is not like asking 'Why is the sky blue rather than green?' (for the sky is not blue by definition), but is instead like asking 'Why is water wet rather than dry?' or 'Why is a circle round rather than square?' Why are metaphysical statements not empirical? Because if they could be put to the tests of experience, we would not call them 'metaphysical statements': we would call them 'empirical statements' -- i.e. the query is equivalent to 'Why is a statement non-empirical rather than empirical?', i.e. it asks for a definition.

Query: language as a cage, Wittgenstein.

But imagine that picture, not as a metaphysical theory (as in the TLP), but as a way of looking at language, versus the picture of language as a poorly fenced (or even unfenced) pasture. What are the limits of sense and nonsense? It varies from one particular case (or particular type of case) to another which picture is applicable: a game played according to strict rules or not such a game (to the extreme case where there is nothing we normally call a 'rule').

Query: what do you understand by logic? explain in your own words.
Query: logical problems for language as use theory.
Query: every word has a meaning which is the object for which it stands.

... and for which one may have a "theory" about the object's nature. Thus: the question "What do you yourself understand by logic?" and not "How do we use -- Note that our words are compared to tools; a comparison is not a theory of meaning -- the word 'logic'?" ["Impressionistic meaning"] And there it is, in three queries, man's wounded understanding crying out to be healed.

The true meaning of words

Query: every word has a meaning, namely the phenomenon for which it stands.
Query: words we don't know the true meaning of.

I often wonder who this "we" is, all mankind? If we compare a word to a tool, then what would it mean to say that, although we had invented this tool, we did not know its true use? Would that mean the we might be using it wrongly or that it might have a potential use we do not know of? If we don't know a words "true meaning", then do we know its meaning at all? Contrast that with: the "meaning" of a tool is the actual work we use that tool to do. (The notion of a word's having a "true" meaning, namely, the nature of the "thing it names", belongs to metaphysics, not to logic of language. [Essence belongs to grammar, not to "reality in itself" ... or does it always? | "The meaning of a word is the essence of the thing the word names"])

We don't know what love is, and so we don't know the true meaning of the word 'love'. We do not know if there is an afterlife, and therefore we don't know the true meaning of the word 'man' [or 'death']. Can't the word 'meaning' be defined that way? Yes, that is one meaning of 'meaning', but it does not yield a "logic of language", i.e. an objective distinction between sense and nonsense, does it? And we are not going to get any work done in philosophy without making that distinction. We get lost in the vagueness of the phenomena, or the Socratic inadequacy of the definitions of the names of the phenomena.

Is black darker than gray?

"First the Professor considered various common sentences into which the idea of comparison entered. Some of these, such as those comparing different human beings, are experiential statements. Others, for instance 'Black is darker than gray', are more properly regarded as grammatical. The distinction is important when the ideas of length and measurement are discussed; one of the chief difficulties in mathematical philosophy is that arising from definitions masquerading as experiential statements."

Source: "From the Minutes of the Trinity Mathematical Society, 19.2.1940" in Wittgenstein in Cambridge: Letters and Documents 1911-1951 (2008), Document 268, p. 316. This is a report of what Wittgenstein said, not apparently or necessarily the actual words he spoke (which is why I have enclosed the quotation in double quotes). Wittgenstein's lecture was titled "The Descent of Mathematics".

For 'grammatical' we can always substitute 'definitional' (and vice versa) in Wittgenstein's jargon. Is 'Black is darker than gray' a rule of grammar? You know I don't know. Is it an experiential proposition -- might it be false that black is, or always is, darker than gray? Cf. 'Black is darker than white', 'Red is darker than pink' ... The question is: does 'Black is darker than gray' belong to the common definitions of the words 'black' and 'gray' and 'darker than' such that no one could say: "It never occurred to me to ask myself that question; how would I know that without comparing a color sample of black and a color sample of gray?" (Although, even if someone did compare color samples, how would he know which color was darker than the other? Or would this simply be a case of "whatever seems correct is going to be correct" (cf. PI § 258)? -- But Wittgenstein does not say here that 'Black is darker than gray' has the grammar of a proposition of experience (either of an objective or a subjective type) -- but says that 'Black is darker than gray' is itself a rule of grammar.)

Is black darker than yellow? I reply: I don't know. But that only means: What is the word 'darker' to mean in this context -- does that word have a normal use in this context? Could someone say to me, "Look here, Robs, you don't seem to know your own language. For it should be obvious to you that 'Black is darker than gray' is a grammatical rule and not an experiential proposition"?

Is the way out of this fly-bottle (PI § 309) -- i.e. out of my perplexity over 'darker than' -- to be found in the next paragraph?

"The lecturer went on to discuss other statements which he described as being "well within the region of the obvious", as "the whole is greater than the part". A proposition of this nature is independent of time, but involves certain restrictions on the method of measurement." (Wittgenstein in Cambridge op. cit.)

The clue may be in the statement "... involves certain restrictions on the method of measurement", which is a rule of grammar. So that if you are going to say that one color is darker than another color, the question is: How do you measure this? Without a method of measurement specified, the expression 'darker than' doesn't necessarily have any meaning at all. (And if we look at the proposition 'The whole is greater than the part' I think that, without too much imagination, we can describe cases where we would say that the whole is not greater than the part. So that Wittgenstein gives a true grammatical account when he says that there are restrictions as to measurement on that proposition [or principle (for some say it is an axiom of nature)].) But we do not normally define 'darker than' by a method of measurement. Am I saying, then, that 'Black is darker than gray' is an undefined combination of words rather than a rule of grammar? Is not 'Gray is darker than white' then a grammatical rule? I don't know what to say (except that Wittgenstein's proposition looks like something that one might find in "Linguistic Analysis" where "they let the words talk to them"; see the word 'obvious' in the quoted passage above).

"Questions were asked about the relationship of physics to mathematics. After many unsuccessful attempts ... in the [M.H.A.] Newman-Wittgenstein controversy ..." (ibid.)

The relationship, or an aspect of the relationship, may be summed up this way, I think, that mathematics can be a tool of physics, but it cannot be its master. Mathematics cannot be the master of physics because as in the case of geometry there is no necessary relationship between pure geometry [axiomatic geometry] and applied geometry: they may coincide or they may not; they are not necessarily identical (or what would it mean to say that they were).

That was not the Rationalist view of geometry or of physics, of course. E.g. Descartes' "There can be no extension which is extension of nothing" and therefore no vacuums, despite any apparent experience to the contrary.

All that Descartes did, of course, was to define the word 'extension'. But that was just what he wanted to do (although he imagines that his definitions are real rather than verbal), because Rationalism's project in philosophy is to seek a priori knowledge, which it believes it can do just as geometry does by making deductions from its axioms, although the Rationalists will hold their own axioms to be indubitable (i.e. impossible to doubt the truth of), and therefore that their deductions are also indubitable, escaping the need to subject themselves to any test of experience. And thus inside the vacuum tube must be a ghost (cf. the notion of an imperceptible ether, which some physicists fall back on in order to "save the appearances"), for if there were nothing inside a container, then its sides are (ipso facto) in contact with one another -- i.e. they must be in contact with one another (but this is only by definition).

I think, what a long-winded bore Descartes is, and how well written are Pascal's scientific papers by contrast. And how under appreciated as a philosopher Pascal is: Pascal's criticism of Rationalism in "On the Geometrical Mind" is devastating, as are his experiments proving the existence of something Descartes' definitions and deductions proved could not exist, namely, the vacuum.

The reason Pascal is ignored by those who lecture on the topic of Rationalism (Descartes, Spinoza, Leibniz) is because Pascal's criticism shows the whole "early modern philosophy" project to be rather: pointlessness.

Query: Euclid's geometry and its relation with philosophy.

Philosophy is the view from outside the calculus: it is blah, blah, blah (allez, allez, allez) to the calculus (i.e. without effect).

"Theology as grammar"

Query: the meaning is the use we make of the word.

Amended to "... the use we make of the word in the language", that is a good grammatical summary of the relationship between the concepts 'meaning' and 'use'. But 'meaning' = 'use' is only one of many uses we make of the word 'meaning'; it is not as it were the essence of language meaning.

Query: gods cannot tell lies.
Query: Apollo cannot lie.

That "the gods do not tell lies" (because "Virtue is knowledge" and because the gods are thoroughgoingly rational, they know what is good (This part of the "because" I do not see) and therefore always do what is good) is a rule of grammar, a clear example of "theology as grammar" (PI § 373), for it belongs to a definition of the word 'gods' as that word was used, not popularly (where all manner of scandalous stories were told about the gods, the popular myths that Plato decried), but by thoughtful persons (It is the result of critical thought; the concept 'God' was created by man for man's use, and the philosopher has no use for the pictures of clowns and ethical monsters as "higher beings", but rather: the gods are rational without also being animals is a picture philosophy has uses for). Note: these are remarks about the Philosophy of Theology (The word 'theology' is not correctly rendered "talk about the gods"; it is rather "talk about the meaning of talk about the gods").

Apollo cannot lie, but can Apollo be mistaken? Can his oracle make a false statement? Being all-rational does not mean being all-knowing. But on the other hand, being fully rational means not thinking you know what you don't know. And so Apollo may not know, but if so, then Apollo would not claim to know.

Dietrich Bonhoeffer took some "Notes" from "W.F. Otto, the classics man at Königsberg, The Gods of Greece" (Letters and Papers from Prison (1971), p. 333; cf. p. 365n55) dated by Bethge as end of June 1944:

"God" -- is not a demonstrable entity (ibid. p. 331).

No self-revelation of the gods. Apollo reveals "the right", but not himself. (p. 332)

That is "theology as grammar". Those notes from Otto are historiographical theology, for Otto is not simply trying to report "what really happened" (Thucydides 2.48.3), but to interpret it -- because many, maybe most Greeks did believe, as Plato's Euthyphro did, that the personalities of the individual gods had been revealed.

Would Aquinas' Five Ways of Knowing that God exists (Summa Theologiae 1A, 2, 3) belong to the grammar of 'God', e.g. first cause -- and this is what everyone means by the word 'God'? Theology, in this case, as explanation of meaning? That is one way of looking at this. But remember that is not what everyone means by the word 'God'; for cf. Bertrand Russell's Theory of Descriptions: the word 'God' may be defined by various descriptions, no one of which is necessarily defining of the word 'God' -- and not every possible definition by descriptions would be acknowledged by everyone, and most certainly not every description will have a place in the life of every religious believer. Maybe you could say that Aquinas' Five Ways belong to the grammar of 'Catholic Christianity's God'.

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