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Undefined in Philosophy and in Mathematics

How could the questions 'What are the definitions of the three undefined terms in geometry?' and 'In geometry what is the undefined object of no dimensions?' ever have been asked? As if an innate madness (PI § 36), these absurd forms of expression have been repeated endlessly through the generations. (The philosophy of geometry and "grammatical jokes".)

Topics on this page ...

It is only the attempt to write down your ideas that enables them to develop. (Recollections p. 109)

That is what these logic of language remarks are. And therefore some are rough (very rough) and may be wrong (indeed, certainly are wrong or have been changed by later thinking), although others are my "considered opinions" (ibid. p. 141). The queries to which I have responded are Internet searches that directed or misdirected visitors to this site.

The Meaning of 'Undefined' and Mathematics

Note: This continues the discussions why are they called 'undefined terms' when we can in fact define them? and the persistence of "undefined terms" (Summary).

Query: what is the mathematical meaning of undefined?
Query: why are they called undefined terms? geometry.

These are very strange queries. What is the meaning of 'defined' in mathematics? Is it different from the meaning of 'defined' in natural language? [What is the difference between mathematics and natural language?]

If we state rules for using a sign (i.e. a word, a phrase or sentence, considered only as a physical object such as a sound or ink mark), haven't we defined that sign? And if we do not state rules for using a sign, then isn't that sign undefined -- i.e. meaningless? How would this be different in mathematics than elsewhere?

If a sign is undefined (i.e. if we are given no rules for using it), then we can do nothing with it in the language, because an undefined sign is nothing more than marks on paper or spoken sounds; in other words an undefined word is "sound without sense" -- i.e. nonsense.

Would we ask: what is the mathematical meaning of 'meaningless'? or Why are they called 'meaningless terms' (geometry)? ["Query: (1) what are the three meaningless terms in geometry, and (2) what is their meaning?" Is this not ridiculous?]

Euclid's definitions are without clear meanings, and they serve no purpose within his system

According to the older view the axioms describe facts of immediate intuition. They deal with "ideal" points, lines, planes and those relations, which are characterized by the words 'incident', 'congruent', 'between', 'parallel'.

Accordingly Euclid begins with the definition of the basic concepts; as he puts it ... point is that which has no parts. These definitions formed a stumbling block from the very beginning, for their meaning is extraordinarily obscure....

However, we must say, above all, that even if such a definition were understood[,] strictly speaking it would have no value for Euclid's system. Not a single proof depends on its explanation; it is never used; it stands entirely outside the remaining system of propositions. (Friedrich Waismann, Introduction to Mathematical Thinking, tr. Benac (1951), p. 73)

That is what, I believe, 'undefined' is intended to mean in the context of mathematics by philosophers of mathematics. -- Note that Waismann is in effect saying that this is a philosophical, not a mathematical, topic -- either (1) that everyone already knows what the basic terms mean [they are "facts of immediate intuition"] and therefore that there is no need to define them [They are called 'basic' because they are used to define other terms, but there is nothing more basic than the basic terms with which to define the basic terms], or (2) that the definition of a basic term, e.g. 'point', has no role to play in any calculus: e.g. the definition "that which has no parts" is not the justification for any step in a proof, and therefore "strictly speaking" it has "no value for Euclid's system".

My own views on those uses of the word 'undefined' -- and note that those uses belong to the jargon of some philosophers of mathematics, not to our language of everyday -- are stated in my paper Philosophy of Geometry. On the other hand, it might be said that I am merely making a fuss over arbitrary signs [that is, in Wittgenstein's jargon: ink marks on paper, spoken sounds, the purely physical part of language] -- that it is simply the case that the queries above approach the problem from the wrong end. However, language matters: to call a sign arbitrary does not mean that conflicting uses of it can't cause the greatest confusion in philosophy; the words 'meaning', 'definition', 'undefined', happen to be examples of just such signs.

Query: plane geometry, meaningless terms.

Quite. Because if 'undefined' does not = 'meaningless', then what does it = ?" (This query is very much an example of "... out of the mouth of babes" (Matthew 12.3) and "The King's New Clothes".)

Query: undefined equals undefined in math.

Or more clearly: 'undefined' in mathematics = 'undefined in mathematics'. (That notation is a minor reason to make the sign versus meaning of a sign distinction.) But is that correct? No, it isn't. However, this is: 'undefined in mathematics' = 'not defined in order to justify any step in mathematical proofs (but instead defined for the sake of orientation)'.

What justification is there for turning a common word, e.g. 'undefined', into a jargon word? In science, maybe a theory will justify this (Wittgenstein wrote (CV p. 44)) (I am thinking of the way 'pain' is defined in medical science), but mathematics is not theoretical: in itself it cannot come into conflict with experience and therefore cannot be falsified by experience. In any case, the question of whether there are -- or can be -- "undefined terms" in mathematics is a question that belongs to the Philosophy of Mathematics, not to mathematics itself. Nonetheless, if we use these words the way we normally use them, what we can say is that: if any sign that is used in a calculus is undefined, then the calculus itself is undefined [e.g. './2' cannot be introduced into a calculus without making the calculus nonsense, because './2' is itself nonsense (i.e. an undefined sign)].

[I have always disliked the very word 'intuitive', because it suggests a mode of knowing. -- But there is no such mode. All 'to intuit' means [amounts to] is: 'to guess'. [That remark is not a theory of knowledge (whatever that is when it's at home), but belongs to "grammar".] 'Intuitive' suggests an occult, a mysterious way of knowing. But it is not a form of knowledge or a way of knowing: 'to know' means: to have compelling grounds, sufficient reasons; what has the logical status ['logic' = 'grammar' in Wittgenstein's jargon and philosophy] of a guess is not knowledge.]

Query: knowledge is definition.

If to be able to give an account of what you know = to give a Platonic-Socratic definition or simply being able to say what a thing is, then yes, "knowledge is definition" -- i.e. 'knowing x' = 'being able to define x'. But then, according to "the older view", do we know what a point in geometry is?

Waismann in the quotation above writes "the older view", but of course all he was justified in saying was: a different view or an alternative view to the one Waismann was setting forth in his book. [The Philosophy of Mathematics is the view from outside.]

Query: what is the three indefinite terms of geometry?

Well, 'indefinite' [without limits or clear boundaries] is one sense of 'undefined', but is that the sense that concerns geometry; -- i.e. is that what geometers do with the terms 'point', 'line', and 'plane' -- talk in vague terms?

Query: why is there a need for undefined terms in any mathematical system?

Are you asking or telling with this query? One possible response is given by Blaise Pascal and W.E. Johnson: that there are not "undefined" but instead "indefinable" simple terms that lie at the foundations of maths. But the query is not a question to be responded to: it is instead a thesis in search of adductions to support it -- i.e. it is not philosophy. Alternative thesis-queries for which support might be sought are: Why is there not a need for undefined words in maths? or, Why must any undefined words be prohibited in/from any systematical maths?

Query: another name for 'undefined' in math.

Well, what might you call "undefined terms" that isn't misleading? The thing is: if you don't begin that way (i.e. with the vague notion of "undefined term"), then the only question that arises is whether a definition is used in geometry proofs or not -- and this you indicate, not by what you exclude, but by what you include.

Question: Is it not this way, that in maths, 'undefined' ≠ 'not defined' and 'not defined' ≠ 'undefined'? That is, 'undefined' in maths only means "not defined by maths". Geometry does not define the word 'the', but we do not for that reason say the word 'the' is undefined in geometry. (Geometry and Jabberwocky)

Query: the undefined object in geometry with no dimensions.

Is that an "undefinition" of the word 'geometric point'? The query is a grammatical joke (PI § 111); it is Russell's hippopotamus. Why doesn't our common sense rebel against this obvious absurdity; why don't we respond: "Nonsense: if "it" is undefined then it is without meaning, and if "it" has no dimensions then it is not an object"? Because we are in school and must conform or fail. In this way our ear for language becomes calloused long before we reach the age of taking up philosophy (ibid. § 348).

[Other examples of grammatical jokes.]

Another "negative definition" of 'point'

This continues the discussion of the philosophy of geometry: the grammar of the word 'point' -- is 'point' a name-of-object word?

Rather than, or as well as, saying that a point is that which has no dimension, you could also say: in geometry a 'point' [definition of the word] is not a unit of measurement (of length e.g.), or: a geometric point is not a unit in any system of measurement.

And a "positive definition" of 'point'

A point is DEF.= a unique address in the plane defined relative to other unique addresses ("points") in the plane. At least three points are required for there to be any points in the plane -- i.e. for there to be an addressing system in the plane ("Three points define a plane" -- but the three points also define one another).

"Is Zero a Number?"

Query: philosopher who said zero divided by zero is undefined.

As I recall, the sign '0' replaced the sign '.' The latter sign was simply a null placeholder: 'zero' = 'no number' or 'no quantity'; e.g. '703' could be read: 7 hundreds, no tens, 3 ones. Thus what would it mean 'to divide a number by no-number' or 'to divide no-number by no-number'? What would './.' mean? It might mean anything we cared to define it to mean; as it stands it is nonsense -- i.e. an undefined combination of signs.

If the query's "undefined" does not allude to "real definition" (in which case, what on earth might it mean [By a 'real definition' is meant a "definition of a thing", often or usually in philosophy of an "abstract thing"]), then what does it allude to? Someone says: '[such-and-such-sign]' is undefined; this may be an important insight, as it was in the case of Einstein and 'simultaneity' (but not all nonsense is disguised nonsense [(PI § 464)]).

Question: is zero a number? The sign '0' suggests to us that it is, because it looks like '1', '2', '3' etc. "The power language has to make everything look the same ... is most glaringly evident in the dictionary ..." (CV p. 22). And that appearance in this case fosters self-mystification. If we connect the grammar of 'number' with the grammar of 'quantity', then because zero is not a quantity, zero is not a number. But we might define the word 'number' many other ways.

The Origin of Vagueness -- and of Clarity

Rules and the consequence of their absence. We must not let the sharp distinction between a sign and its meaning [use, grammar] dissolve. I would say: if we lose that, we lose everything. We mustn't babble about "concepts" as if these had an existence independent of signs and their definitions.

"A concept is like a cloud hovering above and beyond us" is a picture that tells us nothing about our use of the word 'concept'; it is like the pictures suggested by the words 'mind' and 'time'. But what does tell us something about the way we use the word 'concept' is that we have uses for the expressions 'I have a vague notion', 'I have a vague idea'; the question is: what is the source of that vagueness? The answer is that this vagueness is due to our trying to use undefined or ill-defined language.

Philosophers say "I am turning this concept over in my mind, trying to make it clear to me what the concept really is", but that is not the technique we use to define words. "But can't one invent a new meaning for a word, and can't that new meaning emerge from vagueness?" Of course one can invent a new meaning -- and that is what one does: it is not as if one discovered the "true meaning" of the concept, the "true meaning" of the word, as if the concept, the meaning of word had an independent existence --, but the question is: what is that new meaning? And the answer is: it is a convention; it is a definition; -- i.e. it is rules for using a sign. That is how we remove vagueness: we make a rule.

An "unformulated definition"

The notion of an "unformulated definition" that we would recognize as correct if only we were told it (PI § 75), is like the notion that a word "must" have an essential meaning that we can "somehow" abstract. That essential meaning would be the "concept" we turn over in our mind, like the rationalists "letting the words speak to them", trying to discover [elicit] its true meaning.

That picture of the "process of abstraction" is what makes it impossible to establish an objective distinction between sense and nonsense (meaning and meaningless); because it places meaning "in the mind" (or wherever Plato's Forms dwell) -- i.e. it offers us a picture of the meaning of language as something both occult and subjective (W.E. Johnson's "If I say that a sentence has meaning for me, no one has the right to say it is senseless."). Wittgenstein's example of Johnson's notion:

[The] phrase 'I think I mean something by it', or 'I'm sure I mean something by it', which we so often hear in philosophical discussions to justify the use of an expression is for us no justification at all. We ask ... 'How do you use this expression?' (BB p. 65)

And the answer we require -- i.e. the definition ("explanation of meaning") -- must come in the form of public rules that anyone can follow. That is the only meaning of 'meaning' that interests [what I have called] Wittgenstein's logic of language.

An example of how vagueness is overcome is the definition I offered for the word 'concept' as 'rules for using a word'. What makes a concept vague is the absence of a rule that clarifies the use of a word; and where there is no rule in common usage, then if we want to put an end to vagueness, then we must make a rule to end the vagueness. And what we do is to invent -- not discover a pre-existent -- rule. I.e. we cannot grasp [or, abstract] the essence of what, in the absence of a rule, has no essence. Words, concepts, do not have meaning in themselves. We do often ponder a vague notion [idea, concept] in the hope of getting clear about it; the only question is: what is the origin of our clarity, if we do make a concept [idea, notion] clear? It is the rule of "grammar" we have made.

"We believe in the Uniformity of Nature"

Query: riddles, bird in the sky.

Here is one riddle: At this moment, how many birds are there in the sky? One wants to answer: "The earth is huge and there are estimated to be millions upon millions of birds ... No one knows, but of course the number must be determinate [There must be an exact number]." And now the further riddle: what does 'must' mean here? Why "must" there be an exact number, even if there is no method for measuring that number? [I am not asking for a metaphysical-theoretical method, a thought-experiment; my question is: as a matter of what we actually do [how we actually live], and might have been asked at any time in mankind's history.] "There must be a determinate number of birds in flight at this moment"; -- why "must" there be?

[Yes, a child could draw a picture of a globe with birds fluttering around it; yes, we can count the number of birds. Yes, the picture is there -- but what is its meaning?]

This is similar to Russell's "It must be the case that either it snowed or it did not snow [on a particular day long before records of the weather were kept]". If there is no defined way to verify how many birds are in flight, then what does it mean say that there is an exact number? What time did I get up this morning? I did not look at the clock; nonetheless one thinks: I must have gotten up at an exact time, although what that time might be is unverifiable. But if there is no way to verify an exact time -- then is there an exact time?

What possesses us here is a picture. [We are held captive by a rule.] We believe in the uniformity of nature, and in that picture, regardless of whether anyone takes the measurement or not, there is a definite number.-- It is as though we believed that there are absolute co-ordinates: you must have gotten up at an exact time, whether anyone noted [or was even in a position to note] that time or not. The word 'must' indicates a rule, and that is what the "uniformity of nature" is: a rule for constructing statements about events.

"We believe in the uniformity of nature" sounds as if it could be the first proposition of a religious creed. What is the difference? What we call "uniformity" or "laws of nature" are of a variety of forms and these forms are derived from our experience of nature [life]. Would it be correct to say that "We believe in Newton's Third Rule for Reasoning in Natural Philosophy"? What would 'believe in' mean here? Would it not mean that we hold this picture [world-picture, even Weltanschauung] before us: it is the frame of reference through which we look at the natural world? The generalization of this picture might be called a frame of reference, but it is not as if this frame of reference were independent of our experience of many particular regularities in the world: it is an induced frame of reference, although it is not justified (nor justifiable) in its generality [universal application: it does not follow from our observance of particular regularities that "all events have a cause", which is what the general statement of the uniformity of nature says. Experience only says that some regularly occurring events have identifiable causes -- not all events.]

The Philosophical Mind

The open-mindedness of the ancient Athenians as reported in Acts of the Apostles. The native Athenians and foreigners living there, despite the Macedonian and later the Roman conquests, still four hundred years after the death of Socrates and the Sophists -- the Athenians were curious to learn new things, to be presented with new ideas. (Acts 17.21)

But also: they lived in a society that tolerated the discussion of new ideas -- i.e. one that was not afraid of new ideas. Free speech, which was something the Greeks invented, was practiced and welcome even from visitors without fear that they would undermine the established order of society. What if the Apostle Paul had tried to speak that way -- "The god whom you worship in ignorance, I have come to make known to you" -- anywhere else.

But at the same time, the Athenians were a skeptical audience (Acts 17.32), open-minded rather than dogmatic, and therefore inclined more to thought than to deed, to philosophy rather than rioting. Which is perhaps why their various rulers allowed them free speech.

Schweitzer's idea that Jesus does not demand of his hearers that they think dogmatically, if it is accepted, makes it possible to reconcile faith with reason. But it is not possible to reconcile the acceptance of dogma with the way of life of reason. Wittgenstein would not, I think, have agreed, but would have kept everything in its place by means of distinctions; I, on the other hand, believe that philosophy must pass judgments on "forms of life".

"Religious Hypotheses - Subjective Hypotheses"

However, will someone who "believes in" Christianity (as in a set of propositions that must be accepted "on faith") be satisfied with Schweitzer's account of the Christian religion in which "dogmatic belief" is of no importance? On Schweitzer's account all that is needed to be a Christian is to have the spirit of Christ: doctrines divide, but the spirit unites. (The point of the double-quotes in this paragraph is: don't regard the meaning of these forms of expression as simple or straightforward.)

[I don't know whether one can really say in philosophy, "My life is my argument". -- But couldn't Socrates have said that "It is possible to live the way that I have lived -- that is, the life of a philosopher"?]

Certainly Graham Greene's Monsignor Quixote belongs to an entirely different frame of reference from Schweitzer's -- but also from Wittgenstein's, because although, unlike Schweitzer, Wittgenstein did talk about the same "dogmas" as Greene, Wittgenstein treated them in a very different way, as life-guiding pictures rather than as hypotheses-manqué. Greene's view, on the other hand, is the traditional view of religion: where faith and doubt are at war with one another. I want to say that: Greene treats religious belief as if it were a matter of hypotheses -- but of hypotheses of a very strange nature --, because all judgments about the truth of "religious hypotheses" are subjective -- as if an hypothesis could be tested, not objectively, but subjectively, in the context of religion. The "unobjective test" is: what seems right to me, what feels right to me.

The notion of religious belief as subjective hypotheses is a "form of life". It is certainly not a "form" of reason, however.

[I find myself unable to think my way back into the "faith-doubt" -- or, "belief-in" and "doubt-in" -- world-picture of Monsignor Quixote. Is this because of the logic of language Gestalt shift or because of my application of thoroughgoing reason to every aspect of my thinking -- i.e. my unwillingness to look at religion that way? The latter, because I am not saying that those forms of expressions are nonsense in Green's book.]

In Greene's world "faith" and "belief" are not identical: faith may perdure despite the loss of belief in defined dogmas (Schweitzer's own example, if the word 'faith' should be applied to it).

Why does the concept 'God' exist?

God was made for man, not man for God; as God was made by man, not man by God. (Those are of course grammatical remarks about the word 'God', which like all words, is a human tool, invented by man for man's use.)

Graham Greene's Doctor Fischer of Geneva. You can invent any picture you like and give it the name 'God', such as Doctor Fischer does with his God of humiliations and little presents. But go back to first principles. Why are we interested in God? What do we want to say with that concept 'God'? If we cannot identify God with the good, then we cannot worship God, and we cannot believe that the world has a sense, or, a deeper meaning, that is good. What Doctor Fischer does is to invent a picture, but it is a picture that has been divorced from our original interest. It is not a picture that responds to the question of how the God of Nature is to be identified with the Good -- unless one regards it as a negative response (and then its specific content is no more than a portrait of an Olympian as the god of nature).

As if to say, "If God is known by the things he has made, then what account [picture] of God is consistent with what he has made?" Doctor Fischer can be seen as responding to that question.

When we talk about God, the concept 'God', we mustn't lose sight of why we talk about God, the concept 'God'. No one worships the laws of physics; no one loves Newton's three laws. And I would say that was precisely the problem with pictures such as Greene's of the God of Nature -- namely, that God must be an answer, not merely one more phenomenon.

[Related page: Graham Greene's Monstrous Gods.]

The question of metaphysics [and this includes natural theology] is this: are there real definitions of concepts [of "abstract objects", of "abstract objects of thought"]? Because if there are not, then metaphysics is either jargon-making or nonsense. Is there a real definition of the word 'God'?

If you draw out the grammar of that word as it is taught to children, you find rules of the form 'God is a person, but God is not a person' again and again. How we are able to live with those grammatical contradictions (i.e. nonsense), I do not know, but we are. This is a form of life, and it shows that the language used in religion may even be nonsensical -- i.e. that language is not always -- if it ever is -- the key to understanding it.

Nothing can preclude the invention of a conception of God (i.e. the invention of a concept 'God') that, in the words of Saint Hilary, would be "worthy of the Creator of all that is". "Nothing can preclude ..." -- That is another way of saying: "The limit of religious thought -- is concept formation", which is to say: imagination.

[Is there really nothing new under the sun: were all philosophical ideas -- "eternal questions" -- already known to the Greeks? That depends on the precision with which one defines those ideas. It would, on the one hand, seem strange to say that the Greeks had exhausted all logical possibilities; but on the other hand, maybe there really aren't many such possibilities: "... it shows how little has been done when these problems have been solved" (TLP "Preface"), or as Wittgenstein would later say: dissolved ... which is not the same thing as to say: made to go away. "Why is there anything rather than nothing? Is the whole of reality perceptible to the senses? does human life have a meaning ("Providence")? and Are good and evil absolute and universal?" Etc. The riddle of existence, of eternal questions without answers, is there -- like our life.]

Heaven and Hell

Neither heaven nor hell is very far away from any of us. Because heaven is the presence of truth, goodness, love, and hope (Heaven: their complete presence), and hell is the absence of truth, goodness, love, and hope (Hell: their complete absence). And of course, goodness and love is what we mean by 'God' (cf. the "five ways" of Thomas Aquinas: "... and this is what everyone calls 'God'" (Summa Theologiae 1A, 2, 3)). Hell is the absence of God.

"Above all remember this -- heaven is here" (Greene, The Power and the Glory ii, 1), which is related -- but not identical -- to "The kingdom of God is within you". What you can say is that: the intimation of heaven is here in our midst. If we did not know love, if we did not know goodness, we would have nothing out of which to construct the concept 'kingdom of God'. We make an analogy from the love and goodness we experience in this world to ... well, to nothing: it is a single-ended analogy, a notion (picture).

"Do you think you are more worthy that Christ should have died for you than for some other person?" (The Power and the Glory ii, 1) That is what despising other human beings says about us. (A remark of that kind says more to anyone who wants to know what "being a follower of Jesus Christ" means than "the articles of faith" can. Imagine if on the way to Golgotha Jesus had only been able to call to mind the lowest, most debased forms of human life -- would he not them have been willing to die for rapists, torturers, for those who harm children.)

Christ had seen into the brothels of cities as He hung upon the Cross, and He had not come down. He had known everything .... (Marshall, This Sorry Scheme (1925), xv)

"... it was for the sake of sinners that I was handed over to death, that they might return to the truth and sin no more ..." (Mark, following 16.14, variant ending known to St. Jerome)

"For I come to call not the righteous but sinners to repent." (Luke 5.32)

"How we would expect God to be"

As to Nature and Nature's God, the how-ness of the world is far more perplexing than its that-ness (the aspect Wittgenstein called "the mystical"). -- This is because the God of Nature is so different from what we would expect that God to be.

How different our expectations are than from what we find -- is that the origin of that notion that we live in a "fallen world", that the result of Adam's disobedience was not only the fall of mankind but also of the world itself? (Religious thought: there is evil in the world not because God is capricious but because man is sinful.) Plato invented the identification of the concept 'God' with the concept 'the Good'. And this distortion of concepts -- if that's what it is (because the Greek universe was uncreated, and thus God is not Nature's master?) -- has taken root in Western thought (where the universe is created, and God is Nature's master).

The world does not match our idea of what it should be -- that mismatch is the wonder of the world's how-ness (If the world were otherwise, its that-ness might not astonish us). But isn't that a dysfunctional way of thinking, of looking at things? How shall we account for dysfunction? Well, we can't. "Evolution has created a creature that is at odds with Nature." The other animals appear at peace with Nature (but only if 'at war with Nature' has any application to animals). "Life presents us with nothing but riddles" -- but are not such riddles instances of dysfunction?

Wittgenstein, at least in his early stage, thought that asking about the "riddle of existence" was dysfunctional (The dysfunction is to misunderstand the logic of our language). But later he wrote: Live in such a way that the riddle is not a murky background to your life but instead a bright halo (CV p. 27). And further, he wrote (and late in life, this): "Life can force the concept 'God' on us." (ibid. p. 86) (So I don't suppose that Wittgenstein himself ever solved the riddle that he once said did not exist.)

For me the concept 'God' makes nothing clearer, and as such life had at one time forced that concept away from me, because I could not help but demand that the concept 'God' be an explanation (That picture held me captive (cf. PI § 115) for many years, although it no longer does). Wittgenstein was a far deeper human being than I am [although I have not specified a standard of measurement for 'depth'], and he said: One can only speak about these things from one's own level.

The impulses of our hearts wax and wane -- but so does our admittance of philosophical questions. If someone is convinced by a grammatical investigation that the concept 'monotheistic God' is nonsense, he will stop seeking that God; he will not echo St. Augustine's "our hearts are not at rest until they rest in You" if he believes that "You" to be without meaning, a sign without a sense.

That is, if someone believes that God is what Falstaff wished honor to be -- a word, air -- a name that names nothing.

How does thought come to such a meaningless proceeding as making man enter into a spiritual relation with an unreal creation of thought? (Albert Schweitzer, Civilization and Ethics, 2nd ed. (London, 1929), tr. C.T. Campion, Chapter 20, p. 241)

What is belief (worship, prayer) in God, then, if not belief in (i.e. a relationship with) an invention of your own mind (i.e. fantasy)? Schweitzer is not talking about "God" here per se; however, what he is talking about are simply other "names for God" (e.g. "the Absolute").

Three Forms of Captivity

Physical. The body is locked up in a cage. Locke: "When my body goes for a carriage ride, my mind goes with it." [If I am awake and alert to my sense perceptions, I may want to say that. But if I fall asleep in the carriage and dream, then where goes my mind?]

Existential. And that image makes me think: We are all God's captives: captives inside the bodies of animals; captives in the natural world; captives in our worldly ignorance; and captives above all in our ultimate ignorance of God and His aims for us, if there is any such thing.

Conceptual. God the Creator [God as the intelligence, the explanation behind non-sentient [thoughtless and dead as a stone] Nature]. The picture is there; it too may hold us "in its grasp" (CV p. 79) as it was fixed there in childhood -- and to the child's mind it seems there is no getting away from it. It may not guide our deeds [Indeed, how could this picture do that?], but it does guide our thinking about "what is really real".

There is no "the question of ultimate reality" -- i.e. that combination of words is meaningless (undefined); but we are nonetheless inexorably drawn to ask ourselves about it: "Why is there anything? What is life's meaning?" we cannot help but ask ourselves.] Wittgenstein recognized the [logically possible] relationship between "wondering at the existence of the world" and "God created the world", although he told Drury that he did not believe in a being named 'God' [a "Personal God"] (Recollections p. 107-108). Albert Schweitzer wrote that he had never believed in a "God-personality directing the world" (Letter to Erwin R. Jacobi in 1962).

Meaning is also shown by a statement's consequences

There are many techniques for getting clear about the meaning of a word, [a sentence, a sign]. One is to ask how we learned to use that word; this gives us an account [description] of a primitive [a child's] use of language [of a primitive language-game] (LC p. 1-2), which helps us to know our way about [It serves as orientation] (PI § 123; cf. § 151). Another method is to ask how we would teach [instruct, give directions to] someone else to use the word. In both these cases we describe the use of word as if it were a move in a game: "If this word were a game piece, like a chessman [or a ball], what would the rules for moving it?"

Another way of knowing [finding out, learning; another method for discovering] what a person [someone] means by a statement is to ask what consequences he draws from it [i.e. what its implications are]. For example, if someone uses the expression "the eye of God", what consequences does he draw, e.g. does he go on to talk about eyebrows in this connection? (LC p. 71). Of course this is a consequence that someone might or might not draw, e.g. he may go on to talk about the eyebrows of Zeus. This helps us to understand the meaning of a statement: it helps us know our way around, e.g. what more we should expect, what picture, if any, we can draw to illustrate his statement.

Grammar tells what kind of object anything is. (Theology as grammar.) (PI § 373. Always?)

A similar technique for understanding the meaning of a statement is to ask how it is verified -- or, how something is measured (e.g. "potential energy").

"May Statements" in Psychology

Psychology. The willingness of human beings to accept "may statements" -- i.e. 'It may be that ...' -- not merely to consider, but to believe in statements that are not only unverified, but in many cases unverifiable. "It may be", but then it may (just as well) not be. [cf. What only seems correct also only seems incorrect -- and that means that the word 'correct' is without meaning here, because there is no criterion for its being correctly applied (PI § 129)].

It is not only disturbing that human beings upon hearing these "may statements" accept them as if they were statements of fact; it is also disturbing that these statements, which are nothing more than conjecture -- i.e. wild speculation (They are not even hypotheses that anyone defines how to or intends to put to the test) -- readily are turned into catchwords, "explanations" that become a substitute for thinking [critical thinking; thoughtful examination of the particular case].

The willingness to believe in the truth of such conjectures as if they were the insights of seers into the really real is very much like superstition, if indeed it isn't superstition. 'Plausible', 'believable' -- should not be made to equal 'accepted', 'believed'. [cf. Using the words 'probably' or 'likely' ('It seems likely that ...') without explaining exactly why it is likely. If we mean 'I guess' or 'I am inclined to say' (and therefore also 'disinclined to say'), then we should say just that.]

The essence of psychological analysis is persuasion, "the replacement of one myth with another", in the absence of verification. (LC p. 27, 51)

That is, you cannot simply say "The limit of psychology -- is concept formation". Because psychologists are not metaphysicians. They claim to be making statements of fact. It would be truer to say "The limit of psychology -- is verification".

"I'm amazed that soothsayers don't break out laughing when they meet one another" (I don't remember which ancient said that). One might say the same about psychologists.

Fallacy -- that because there is a science of abnormal psychology, there must also be a science of normal psychology. But there is no such science. 'Mentally healthy' means no more than 'not mentally ill'. The "scientists" -- psychologists -- of normal psychology are in reality peddlers of value-laden ideologies, philosophers-manqué.

"Unanswerable Questions" - "Unquestionable Answers"

Query: answers without questions.

If we can have questions without answers, then why not also answers without questions -- i.e. is the one notion any less absurd than the other? If the combination of words 'a question essentially ["essence belongs to grammar" (PI § 371)] without an answer' is not nonsense, then why should 'an answer without a question' also not be nonsense? Because aren't both as if to say: 'Here is an object that has no connection whatever to sitting down, but it is nonetheless what we commonly call a chair.' 'The word 'point' in geometry is not the name of a "physical object" but it is nonetheless the name of an "abstract object", like the word .' [But isn't 'physical object' a pleonasm, and 'non-physical object' an oxymoron?] 'It is logically impossible -- i.e. impossible by definition -- to answer this question, nonetheless it is a question.' 'There is no question for this statement to be the answer to, nonetheless it is an answer.'

We want to reject the last sentence. As if to say: a question is "logically prior" to an answer, and that is why one can have a question without an answer, but not vice versa. (Surely, that ought to be the easiest thing in the world to explain, but then why do I find it difficult?)

We cannot know that we cannot know; agnosticism [i.e. the claim not to know] is a dogma [i.e. because we cannot even know that we do not know]. (Sextus Empiricus, Hyp. Pyrr. i, 7, quoted or paraphrased by Will Durant Sextus "flourished" [floruit] at the end of 2nd Century A.D.)

Maybe we can only say that we do not know something at present, never that anything is "essentially unknowable" (if only because, the limit of knowledge is -- concept-formation). Sextus: It is logically impossible to know that we "cannot" know -- but what kind of impossibility is this? (Logical possibility belongs to grammar; but is it clear whether Sextus is talking about logical or real possibility here?). -- But isn't this like the question "Is the roundness of the earth a theory that has become a fact?" -- i.e. if you recast the question, change the ground rules for answering it, do you still want to call it the same question? There is a logical-grammatical difference between an unanswered question and an unanswerable -- i.e. an essentially unanswerable (i.e. defined in such a way as to be unanswerable) -- question. But if we redefine the question in order to make it answerable ...

Is "an unanswerable question" any less absurd than "a meaningful word that is indefinable"? Are they not both grammatically self-contradictory -- i.e. contradictions in sense?

Pictures of abstract objects (Queries and Remarks)

Query: abstract concept, picture, dictionary.

This is another very strange query. It reminds me of the old book See it & Say it in Italian, where line drawings of common objects and activities accompanied Italian language words and sentences. Does the query ask for "concrete pictures of abstract objects"? This is similar to a query for pictures of the 4th Dimension (as if time were a dimension in the sense that space is).

Query: ritual, language-game, Wittgenstein.

Certainly we can compare rituals to games (as indeed we can compare most human activities to games). -- But does this comparison show us either the meaning of the language spoken in rituals or the "meaning" of the ritual itself? Is it a comparison that is helpful if we want to understand the nature of ritual? ["Religion and language-games".] On another hand, by 'ritual' the query may simply mean a "custom, usage, institution" (PI § 199) and Wittgenstein certainly called a language-game that. See The Fable of the Born-Blind People for an example of that.


Let silence be your general rule; or say only what is necessary and in a few words. Talk, but rarely, if occasion calls you, but do not talk of ordinary things. (Epictetus, Manual 33, tr. Crossley)

Pascal wrote that if you want people to think well of you, then don't speak (Pensées i, 44). But Epictetus was speaking on an entirely different level: not about life in this world, but about life from the point of view of eternity. Life is serious. It is not l'esprit.


Philosophy in Readable Sentences would be a title but more importantly a motto for a good book. "A simple story in words of one syllable" -- but even a complicated story can be told using such words. -- If I really want to be read internationally, and that is the origin of my site's visitors, I need to rewrite Wittgenstein's Logic of Language in a much simpler style of English. Above all, I think, I need to shorten my sentences and get rid of dependent clauses.


Protagoras. There is a difference between saying that "Mankind is the measure of all things" and "Each individual man is the measure of all things" although both can be given a compelling sense in the context of "forms of life" (life forms) and "a deeper skepticism".


A human being's body is half of what he is. But is the body really that much? In the case of some human beings perhaps it is more: I once characterized, whether fairly or not, most young men as having "no heart, no soul, no brains, only appetites"; Solzhenitsyn: "bronzed bodies and petrified souls". But some human beings are at peace with their bodies, while others are not.

In the animals we see a unity of mind and body not seen in man.


Query: multiplication hand signs.

Is one less tempted to say that the hand signs of deaf people are about abstract objects? Less tempted -- or perhaps more tempted. Do hand signs seem to make language more, or less, concrete than do spoken sounds and ink marks?

What defines a game?

Query: do rules define a game?

Preface: "What you must not do with a philosophical question is to try to answer it" (Waismann). The method instead must be: to ask whether or not the question has any clear meaning. The fundamental mistake we make in philosophy is [on this account]: to try to answer questions before we have asked ourselves: what exactly is being asked?

Philosophical questions tend to lack any definite sense, and thus in trying to answer them we wander off into a labyrinth of confusions of our own making, from which there appears to be no way out. "A philosophical problem has the form: 'I don't know my way around'" (PI § 123); and, of course, how can anyone know his way around nonsense (i.e. bare signs with no defined meaning).

The following is a good example of that, of my mistakenly assuming that I know the meaning of the word 'define' in the search query.

In Wittgenstein's logic of language, using language is compared to playing games where what defines a game is its rules, or absence thereof. But definitions are given [made] for specific purposes. A game might be "defined" -- i.e. classified or categorized -- by a particular feature of the game, e.g. by its equipment (boards, balls, nets) or by its physical space requirements (a court or playing field of such and such dimensions); e.g. we might want to know: which games will children be able to play at a particular location? Such features belong to the rules of the game; they are internal to games.

But we might define games by external criteria, e.g. by the profitability of the game for the professional player or by the expense of staging it; in that case we are not concerned with the game's rules. That a game can be played professionally, and the cost of nets and rackets, are not determined by the rules of the game. But we may choose to define a game that way.

Suppose one said however, "No, what I want is to know about the things we all agree are called by the common-name 'game'. In that context, is a game defined by its rules?" If someone asks us for the meaning of the word 'chess' or what chess is, is it necessary to answer that person's question by describing the rules of chess? Suppose we did that, but the person then replied, "No, no, I am not interested in the rules of play; what I want to know about is the shape of the chess pieces." There is a rule that requires there to be a chessman designated as 'the king', but there is no rule requiring the king to have a particular shape. But would we call a description of the possible shapes of chess pieces 'a definition of the game of chess'?

The query owes us a definition of 'to define'. It may mean: tell us what the essential nature of a game is, in the Greek sense of 'to define' -- i.e. to set the limits of and set apart from: what does a game include and what does it exclude -- i.e. how is it unique ... if it is unique -- i.e. if there is an essential or general definition of 'game'. From that point of view rules alone do not define a game, because many things that are not games have rules (e.g. law courts, banks).

Is there a real definition of games? Does it make sense to ask: what is the essential nature of games; what are games in themselves? The question is puzzling, like Augustine's question about time. "What is a game really?"

"This and similar things are called 'games'". And do we know any more about it ourselves? Is it only other people whom we cannot tell exactly what a game is? (PI § 69)

If anyone asks us what a game is, we point to examples of games. But if we are asked, "Is a game defined by its rules?" how should we reply? Is there any reply except: what are you calling a 'definition' here? We define the word 'game' by the technique we use to define most common names: we give examples of games; -- and giving examples is where definition comes to an end in these cases. But if we use the word 'define' to mean 'characterize', then there are countless ways to characterize games: There are many ways we might choose to cut [divide] reality up into bits, just as there are many ways to cut a pie. The only limit of this cutting up is concept-formation (human imagination).

What defines a particular game? Where are the limits to be set? In chess there are rules defining 'checkmate', but there is no rule that says that a player must play with the intention of checkmating his opponent. One may from the start of the game choose instead to play with the intention of reaching a stalemate ("draw") or even of losing. Does the reason for playing the game belong to its definition?

Rather than "where what defines a game is its rules", we could say: where what interests us about a game is its rules -- i.e. this is how we are choosing to look at games. We might have chosen to look at them in some other way; just as Wittgenstein chose one sense of the word 'meaning' for his logical philosophy (logic of language), in order to make an objective distinction between sense and nonsense; but he could have chosen another, if he had had some other purpose in mind ("it is possible to be interested in a phenomenon from many points of view"). Wittgenstein said: Let us look at language this way, because if we do, we have a method for getting clear about its meaning: Let us compare using language to playing games, where what defines a game is its rules. (The rules for using language, including rules of sense and nonsense [conventional definitions], are called 'grammar', a word which in Wittgenstein's usage [i.e. jargon], means the same a 'logic'.)

The question this query suggests is a conceptual one -- i.e. it concerns the use of words --, although is presented as if it were an empirical question: "What is a game really?" rather than "How do we wish to define the word 'game'?" [If the question asks for a report about how we use the word 'game' -- i.e. for the facts about our common practice -- then as was said about: we define the word 'game' by means of examples: we do not define it by pointing to the essence of games (which is what "Is a game defined by its rules?" presupposes).]

The passive voice is not suited to philosophy [not suitable for logic] -- i.e. "Is a game defined by its rules?" -- By whom, and for what purpose?

Postscript: And so this investigation ends where it began -- in its self-constructed maze. Because the question "Do rules define a game?" has no clear meaning. In fact, it is nonsense.

Was I trying to "define games" or to define the word 'game'? This confusion is what Wittgenstein called "the essential thing about metaphysics" -- that it "obliterates the distinction between a conceptual and a factual investigation" (Z § 458) The question "What is essential to a game?" ["What defines a game?"] is not a question of fact, unless it is a question about the facts of how we actually use the word 'game' [e.g. about the practices that a dictionary tries to report -- or better (because dictionary compiles very often incorporate their misunderstanding about the logic of our language into their reports): unless it is about the type of grammatical account Wittgenstein gave of the word 'game' (partly quoted above)]. When dealing with fundamental [foundational] problems, which in philosophy are as "hard as granite", it is easy to lose one's way about: as in Wittgenstein's "old city" metaphor: if you approach the same street from different sides you may no longer know your way around.


For I would have you think, not jump through hoops -- (the hoops are the established points of view and doctrines held up and passed on to you by your school instructors) -- as if you were a trained animal rather than a thoughtful human being.

Query: why we cannot define philosophy?

Compare: "undefined term" in plane geometry. This is rote-learning: the answers are already known; just repeat the traditional "rote". The philosophical question is: can we define philosophy?

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