INDEFINABLE TERMS
In mathematics, logic, philosophy and formal
systems, a primitive [indefinable] notion is a concept that is not defined in terms of previously defined concepts.
Wikipedia
Indefinable means Unanalyzable. We cannot describe the term with other terms. The term cannot be “taken apart” or broken down into simpler parts.
*******
In
the following pages I will call the non-explicable manifestations of
‘somethings’ Indefinables, and state that only our mental
registrations of the above, that is
Abstracts, lend themselves
to definitions. p3
[…] there are
no words available in our terminology which could adequately convey
their meaning ‒ Indefinability, to use my designation.
p94
The essence of these seemingly contradictory
data of reality lies in their unconquerable indefinability.
Consequently, anything we might say about them would not be semantically
meaningful but, at best, a lucky
label going no further than ‒ if
you
like ‒ a superficial indication of something of which ‘we do not know
what it is, and only know what it does’. And this is exactly why I call
these ‘somethings’ Indefinables.
Josephine Pasternak;
Indefinability; 1999; p123
*******
§4- […]
the names
of simple ideas are not capable of any definitions; the names of
all complex ideas are.
§5- I will not here trouble myself, to prove that
all terms are not definable from that progress,
in infinitum, which
it will visibly lead us into, if we should allow, that all names could be
defined. For if the terms of one definition, were still to be defined by
another, where at last should we stop? But I shall from the nature of our
ideas, and the signification of our words show,
why some names can, and others cannot be defined, and which
they are.
§6- I think, it is agreed, that a
definition is nothing else, but
the showing the meaning of one word by several other not synonymous
terms. […]
§7- This being premised, I say, that
the names of simple ideas, and those only,
are incapable of being defined.
The reason whereof is this, that the several terms of a definition,
signifying several ideas, they can altogether by no means represent an idea,
which has no composition at all: and therefore a definition which is
properly nothing but the showing the meaning of one word by several others
not signifying each the same thing, can in the names of simple ideas have no
place.
John Locke; An Essay Concerning Human Understanding; 1689/1997; bk3; ch4
ALL INSTANCES OF “LOVE IS” IN SCRIPTURE (ESV)
The word LOVE is not defined in scripture.
Psalms 26:3
For
your steadfast love is before my eyes,
and I walk in your faithfulness.
>> The IS of finding oneself
Psalms 57:10
For
your steadfast love is great
to the heavens, your faithfulness to the clouds.
>> The IS of predication
Psalms 63:3
Because
your steadfast love is better than life,
my lips will praise you.
>> The IS of predication
Psalms 69:16
Answer me, O LORD, for
your steadfast love is good;
according to your abundant mercy, turn to me.
>> The IS of predication
Psalms 108:4
For
your steadfast love is great
above the heavens; your faithfulness reaches to the clouds.
>> The IS of predication
Psalms 109:21
But you, O GOD my Lord, deal on my behalf for your name’s sake; because
your steadfast love is good,
deliver me!
>> The IS of predication
Proverbs 15:17
Better is a dinner of herbs
where love is
than a fattened ox and hatred with it.
>> The IS of finding oneself
Song of Solomon 1:2
Let him kiss me with the kisses of his mouth! For
your love is better than wine;
>> The IS of predication
Song of Solomon 8:6
Set me as a seal upon your heart, as a seal upon your arm, for
love is strong as death,
jealousy is fierce as the grave. Its flashes are flashes of fire, the very
flame of the LORD.
>> The IS of predication
Hosea 6:4
What shall I do with you, O Ephraim? What shall I do with you, O Judah?
Your love is like a morning cloud,
like the dew that goes early away.
>> The IS of a simile
John 11:3
So the sisters sent to him, saying, “Lord,
he whom you love is ill.”
>> Not a claim about Love
Romans 13:10
Love does no wrong to a neighbor; therefore
love is the fulfilling of the law.
>> The IS of consequence
1 Corinthians 13:4
Love is patient and kind; love does not envy or boast; it is not arrogant
>> The IS of predication
1 John 4:7
Beloved, let us love one another, for
love is from God,
and whoever loves has been born of God and knows God.
>> The IS of finding oneself
1 John 4:12
No one has ever seen God; if we love one another, God abides in us and
his love is perfected in us.
>> The IS of consequence
5/15/26
In addition to its symbolic uses ‘Beauty’ has also its emotive uses. These
have often been responsible for the view that Beauty is indefinable, since
as an emotive term it allows of no satisfactory verbal substitute.
Ogden & Richards; The Meaning of Meaning; 1956; p247
(1/14/25)
Primitive terms can be both of existing and imaginary entities.
Numbers are the primitives of the imaginary world of math.
Point and Line are the primitives of the imaginary world of geometry.
Being a primitive, indefinable term, does not mean that a term is meaningless.
Primitive terms are understood intuitively.
| INDEFINABLES |
EXPERIENCED |
NOT EXPERIENCED |
| EXISTS |
> LOVE
> LIFE
> COLORS:
RED, BLUE, etc.
> THINKING
> LIGHT
> MATTER
> ENERGY
> MOTION
> SPACE
> GOOD |
> TIME |
DOES NOT
EXIST |
X
[7/23] |
> NUMBERS-
ZERO, ONE
> POINT
> LINE
> REST
> IDENTITY
(Colin McGinn; Logical Properties) |
Set theory: The concept of the
set
is an example of a primitive notion. As Mary Tiles writes: The ‘definition’ of ‘set’ is less a definition than an attempt at explication of
something which is being given the status of a primitive, undefined, term. As evidence, she quotes Felix Hausdorff: “A set is formed by the
grouping together of single objects into a whole. A set is a plurality, thought of as a unit.”
[Set Theory; 1957; p11]
Mary Tiles;
The Philosophy of Set Theory; 2017; p99
§9 - Nor have the modern philosophers, who have endeavored to throw off the jargon of the Schools, and speak intelligibly, much better succeeded in defining simple ideas, whether by explaining their causes, or any otherwise. The atomists, who define
motion
to be a
passage
from one place to another, what do they more than put one synonymous word for another? For what is
passage
other than
motion? And if they were asked what
passage
was, how would they better define it than by
motion? For is it not at least as proper and significant, to say,
‘passage is a motion from one place to another’, as to say,
‘motion is a passage’, etc.
This is to translate, and not to define, when we change two words of the same signification one for another; which when one is better understood
than the other, may serve to discover what idea the unknown stands for; but is very far from a
definition,
unless we will say, every English word in the dictionary, is the definition of the Latin word it answers, and that motion is a definition of
motus.
Nor will successive application of the
parts of the superficies of one body, to those of another,
which the Cartesians give us, prove a much better definition of motion, when well examined.
John Locke; An Essay Concerning Human Understanding; 1689/1997; bk3; ch4
The thing, then, to be fully realized at this stage is,― that in every
simple idea or ultimate notion we have something that must be started from,
something that must be taken for granted, something to be referred to
individual experience, and that― except in the case of correlatives, where
together with a thing we can produce its opposite― nothing can be done in
the way of making the meaning clearer or more intelligible. When we have
said of Light that it is the opposite of Darkness, with a partially opposed
correlative in Shade,― we have said all on the subject that words can say.
We need never hope to make plain the nature of Life to a man that does not
understand it when put in opposition to Death. Cold and Heat are mutually
implicated, and their names may be brought together with effect, using
luke-warmness as the intermediate form; but beyond that we cannot go. Pain
and Pleasure are self-explaining, or else not to be explained at all. So
with seeing, hearing, touching, tasting, smelling, fatigue, repose,
sensations of the viscera or systemic sense― in a word, all those primary
sensations that form the basis and the raw material of individual
experience. Who will venture to define Toothache to one who has never felt
it? or what hope can we have of being understood when we try to explain
Vision to a man born blind? These experiences, one and all, are ultimate,
and as ultimate, they must be accepted; and the words that stand for them
must be allowed to stand, without mark or comment.
William Leslie Davidson; The Logic of Definition; 1885; p74-75
Naive Set Theory: The
Empty Set is a
primitive notion. To assert that it exists would be an implicit axiom. Wikipedia
Euclidean Geometry: Under Hilbert’s axiom system, the primitive notions are point, line, plane, congruence, betweenness and incidence.
Wikipedia
Euclidean Geometry: Under Peano’s axiom system,
the primitive notions are point, segment and motion.
Wikipedia
Peano Arithmetic: The
successor
function and the number Zero are primitive notions.
Wikipedia
BETWEEN
THINKING
RELATION
TRINITY
OMNIPOTENCE
OMNISCIENCE
OMNI-ANY
*******
Even in the domain of logic, we need what Peano has called “primitive ideas” and “primitive propositions.” In fact,
since all definitions of terms are effected by means of other terms,
every system of definitions which is not circular, must start from a
certain apparatus of undefined terms. It is to some extent optional what
ideas we take as undefined.
Deduction, then, can only begin by a
stating and assuming the truth of the necessary primitive propositions,
“explaining” them if necessary.
Ernst Schröder called the
collection of the primitive ideas and primitive propositions of logic
the “faith of the logician.”
[…] all reasoning without exception
assumes some at least of the primitive premises.
These sciences
have now grown up to the stage of frank admission that they rest on a
basis of indefinables and indemonstrables.
Philip E. B. Jourdain; Indefinables and
Indemonstrables in Mathematics and Theology; The Monist; 1919; p547,
548, 549,
[…] it is logically necessary that some things should be indefinable […]
A. C.
Ewing; Review: The Nature of Thought by Brand Blanshard; Mind; 1944; p79
Just as we can’t define or explain “Red” to a blind person, we can’t define
Motion or Time to a sighted person.
[8/30/23]
§10.
‘The act of perspicuous, as far forth as
perspicuous’, is another Peripatetic definition of a simple idea;
which though not more absurd than the former of
motion, yet betrays its
uselessness and insignificancy more plainly, because experience will
easily convince anyone, that it cannot make the meaning of the word
light (which it pretends to define) at all understood by a blind man: but the definition of
motion appears not at
first sight so useless, because it escapes this way of trial. For
this simple idea, entering by the touch as well as sight; ‘tis
impossible to show an example of anyone, who has no other way to get
the idea of motion, but barely by the definition of that name. Those who tell
us, that light is a great number of little globules, striking briskly on the bottom of the
eye, speak more intelligibly than the Schools: but yet these words
never so well understood, would make the idea, the word
light stands for, no more
known to a man that understands it not before, than if one should
tell him, that light was nothing but a company of little tennis-balls, which fairies all day
long struck with rackets against some men’s foreheads, whilst they
passed by others. For granting this explication of the thing to be
true; yet the idea of the cause of
light, if we had it never
so exact, would no more give us the idea of
light itself, as it is
such a particular perception in us, than the idea of tile figure and
motion of a sharp piece of steel, would give us the idea of that
pain, which it is able to cause in us. For the cause of any
sensation, and the sensation itself, in all the simple ideas of one
sense, are two ideas; and two ideas so different, and distant one
from another, that no two can be more so. And therefore should
Descartes’s globules strike never so long on the retina of a man,
who was blind by a gutta serena*, he would thereby never have any idea of
light, or anything approaching to it, though he understood what little globules were,
and what striking on another body was, never so well. And therefore
the Cartesians very well distinguish between that light which is the
cause of that sensation in us, and the idea which is produced in us
by it, and is that which is properly light.
§ 11.
Simple ideas, as has been shown, are only to be got by those
impressions, objects themselves make on our minds, by the proper
inlets appointed to each sort. If they are not received this way,
all the words in the world, made use of to explain, or define any of their names,
will never be able
to produce in us the idea it stands for. For words being sounds,
can produce in us no other simple ideas, than of those very sounds;
nor excite any in us, but by that voluntary connexion, which is
known to be between them, and those simple ideas, which common use
has made them signs of. He that thinks otherwise, let him try if any
words can give him the taste of a pineapple, and make him have the
true idea of the relish of that celebrated delicious fruit. So far
as he is told it has a resemblance with any tastes, whereof he has
the ideas already in his memory, imprinted there by sensible objects
not strangers to his palate, so far may he approach that resemblance
in his mind. But this is not giving us that idea by a
definition, but exciting
in us other simple ideas, by their known names; which will be still
very different from the true taste of that fruit itself. In light
and colours, and all other simple ideas, it is the same thing: for
the signification of sounds, is not natural, but only imposed and
arbitrary. And no definition of light or redness, is more fitted,
or able to produce either of those ideas in us, than the sound
light, or red, by itself.
For to hope to produce an idea of light, or colour, by a sound,
however formed, is to expect that sounds should be visible, or
colours audible; and to make the ears do the office of all the other
senses. Which is all one as to say, that we might taste, smell, and
see by the ears: a sort of philosophy worthy only of Sanco Panca*,
who had the faculty to see Dulcinea by hearsay. And therefore he
that has not before received into his mind, by the proper inlet, the
simple idea which any word stands for, can never come to know the
signification of that word, by any other words, or sounds,
whatsoever put together, according to any rules of definition. The
only way is, by applying to his senses the proper object; and so
producing that idea in him, for which he has learned the name already. A
studious blind man, who had mightily beat his head about visible
objects, and made use of the explication of his books and friends,
to understand those names of light, and colours, which often came in
his way; bragged one day, that he now understood what
scarlet signified. Upon which his friend demanding, what
scarlet was? the blind man
answered, it was like the sound of a trumpet. Just such an
understanding of the name of any other simple idea will he have, who
hopes to get it only from a definition, or other words made use of
to explain it.
§12.
The case is quite otherwise in
complex ideas; which
consisting of several simple ones, it is in the power of words,
standing for the several ideas, that make that composition, to
imprint complex ideas in the mind, which were never there before,
and so make their names be understood. In such collections of ideas,
passing under one name, definitions, or the teaching the signification of one word, by
several others, has place, and
may make us understand the names of things, which never came
within the reach of our senses; and frame ideas suitable to those in
other men’s minds, when they use those names: provided that none of
the terms of the definition stand for any such simple ideas, which
he to whom the explication is made, has never yet had in his
thought. Thus the word statue
may be explained to a blind man by other words, when
picture cannot, his senses having given him the idea of figure, but
not of colours, which therefore words cannot excite in him. This
gained the prize to the painter, against the statuary; each of which
contending for the excellency of his art, and the statuary bragging,
that his was to be preferred, because it reached further, and even
those who had lost their eyes, could yet perceive the excellency of
it. The painter agreed to refer himself to the judgement of a blind
man; who being brought where there was a statue made by the one, and
a picture drawn by the other; he was first led to the statue, in
which he traced with his hands, all the lineaments of the face and
body; and with great admiration, applauded the skill of the workman.
But being led to the picture, and having his hands laid upon it, was
told, that now he touched the head, and then the forehead, eyes,
nose, etc. as his hand moved over the parts of the picture on the
cloth, without finding any the least distinction: whereupon he cried
out, that certainly that must needs be a very admirable and divine
piece of workmanship, which could represent to them all those parts,
where he could neither feel nor perceive anything.
§13.
He
that should use the word
rainbow, to one who knew
all those colours, but yet had never seen that phenomenon, would, by
enumerating the figure, largeness, position, and order of the
colours, so well define that word, that it might be perfectly
understood. But yet that definition, how exact and perfect soever, would never make a blind
man understand it; because several of the simple ideas that make
that complex one, being such as he never received by sensation and
experience, no words are able to excite them in his mind.
§14.
Simple ideas, as has been showed, can only be got by experience,
from those objects, which are proper to produce in us those
perceptions.
When by this means we have
our minds stored with them, and know the names for them, then
we are in a condition to
define, and by definition
to understand the names of complex ideas, that are made up of them.
But when any term stands for a simple idea, that a man has never yet
had in his mind, it is impossible, by any words, to make known its
meaning to him. When any term stands for an idea a man is acquainted
with, but is ignorant, that that term is the sign of it, there
another name, of the same idea which he has been accustomed to, may
make him understand its meaning. But in no case whatsoever, is any
name, of any simple idea, capable of a definition.
§15.
Fourthly, but though the names of
simple ideas, have not the help of
definition to determine their signification; yet that hinders not
but that they are generally
less doubtful and uncertain, than those of mixed modes and
substances. Because they standing only for one simple
perception, men, for the most part, easily and perfectly agree in
their signification: and there is little room for mistake and
wrangling about their meaning. He that knows once, that whiteness is
the name of that colour he has observed in snow, or milk, will not
be apt to misapply that word, as long as he retains that idea; which
when he has quite lost, he is not apt to mistake the meaning of it,
but perceives he understands it not. There is neither a multiplicity
of simple ideas to be put together, which makes the doubtfulness in
the names of mixed modes: nor a supposed, but an unknown real
essence, with properties depending thereon, the precise number
whereof are also unknown, which makes the difficulty in the names of
substances. But on the contrary, in simple ideas the whole
signification of the name is known at once, and consists not of
parts, whereof more or less being put in, the idea may be varied,
and so the signification of its name, be obscure, or uncertain.
§16.
Fifthly, this further may be observed, concerning simple ideas, and
their names, that they have
but few ascents in linea prædicamentali* (as they call it,)
from the lowest species, to the summum genus*. The reason whereof
is, that the lowest species being but one single idea, nothing can
be left out of it, that so the difference being taken away, it may
agree with some other thing in one idea common to them both; which
having one name, is the genus of the other two:
v.g. there is nothing can
be left out of the idea of
white and red, to make
them agree in one common appearance, and so have one general name;
as rationality being left out of the complex idea of
man, makes it agree with
brute, in the more general idea and name of
animal. And therefore when
to avoid unpleasant enumerations, men would comprehend both
white and
red, and several other such simple ideas, under one general name;
they have been fain to do it by a word, which denotes only the way
they get into the mind. For when
white,
red, and
yellow, are all comprehended under the genus or name
colour, it signifies no
more, but such ideas, as are produced in the mind only by the sight,
and have entrance only through the eyes. And when they would frame
yet a more general term, to comprehend both
colours and
sounds, and the like
simple ideas, they do it by a word, that signifies all such as come
into the mind only by one sense: and so the general term
quality, in its ordinary
acception, comprehends colours, sounds, tastes, smells, and tangible
qualities, with distinction from extension, number, motion,
pleasure, and pain, which make impressions on the mind, and
introduce their ideas by more senses than one.
§17.
Sixthly, the names of simple ideas, substances, and mixed modes,
have also this difference; that those
of mixed modes stand for
ideas perfectly arbitrary:
those of substances, are
not perfectly so; but refer to
a pattern, though with some latitude: and those of simple ideas are
perfectly taken from the existence of things, and are
not arbitrary at all.
Which what difference it makes in the significations of their names,
we shall see in the following chapters.
The names of simple modes, differ little from those of simple ideas.
*
gutta serena = a condition of partial or
complete blindness with a transparent, clear pupil
Sanco Panca = Fictional character in the novel Don Quixote, Sancho Panza.
linea prædicamentali = a predicamental line
summum genus =
The highest, most all-inclusive genus
John Locke; An Essay Concerning Human Understanding; bk3; ch4
Since all terms that are defined by means of other terms, it
is clear that human knowledge must always be content to accept some
terms as intelligible without definition, in order to have a starting
point for its definitions.
Bertrand Russell
What has been said about undemonstrated
propositions is equally true for undefined terms. […]
That there must be undefined terms is clear from any attempt to define a term.
Cohen and Nagel; An Introduction to Logic and
Scientific Method; 1939; p142
When we set out to construct a given discipline, we distinguish, first of
all, a certain group of expressions of this discipline that seem to us to be
immediately understandable; the expressions of this group we call
PRIMITIVE TERMS or
UNDEFINED
TERMS,
and we employ them without explaining their meanings. At the same time we
adopt the principle: not to employ any of the other expressions of the
discipline under consideration, unless its meaning has first been
determined with the help of primitive terms and of such expressions of the
discipline whose meanings have been explained previously. The sentence which
determines the meaning of a term in this way is called a
DEFINITION, and the expressions themselves
whose meanings have thereby been determined are accordingly known as
DEFINED
TERMS.
We proceed similarly with respect to the asserted statements of the
discipline under consideration. Some of these statements which to us have
the appearance of evidence are chosen as the so-called
PRIMITIVE STATEMENTS or
AXIOMS (also often referred to as
POSTULATES, but we shall not use that term in this technical meaning here);
we accept them as true without in any way
establishing their validity. On the other hand, we agree to accept
any other statement as true only if we have succeeded in establishing its
validity, and to use, while doing so, nothing but axioms, definitions and
such statements of the discipline the validity of which has been established
previously. As is well known, statements established in this way are called
PROVED STATEMENTS or THEOREMS, and the process of establishing them is
called a PROOF.
Alfred Tarski; Introduction to Logic; 1995; p118
Definition, as is evident, is only possible in
respect of complex ideas. It consists, broadly speaking, in the analysis
of complex ideas into their simple constituents. Since one idea can only
be defined by another, we should incur a vicious circle if we did not
admit some indefinable ideas. This obvious truth is fully recognized by
Leibnitz, and the search for the simple ideas, which form the
presuppositions of all definition, constitutes the chief part of his
studies for the Universal Characteristic. Thus Leibnitz says (Monadology,
§§ 33,35): “When a truth is necessary, its reason can be found by
analysis, resolving it into more simple ideas and truths, until we come
to those which are primary. […] In short, there are simple ideas,
of which no definition can be given […]”
Bertrand Russell; A Critical Exposition of the Philosophy of Leibniz;
1900; Ch 2; Sec 11
Euclid’s initial assumptions for geometry consisted of a number of
definitions, followed by a set of postulates and axioms. We shall follow the
modern procedure and arrange that the number of undefined elements and
relations shall be as small as possible. In terms of these undefined
elements and relations we shall state certain unprovable propositions, which
we shall call axioms. These axioms will be chosen so that no one is a
consequence of the others: they will be independent; they must also be
consistent.
[…]
To a non-mathematician it often comes as a surprise
that it is impossible to define explicitly all the terms which we used. This
is not a superficial difficulty but lies at the root of all knowledge; it is
necessary to begin somewhere, and to make progress one must clearly state
those elements and relations which are undefined and those properties which
are taken for granted.
Gilbert de B. Robinson;
The Foundations of Geometry; 1940; p7,8
If every word in the English language were
definable, there would be a circularity within the group of definitions. One
would, for instance, end up attempting to define ‘bent’ as ‘not straight’
and ‘straight’ as ‘not bent’ or something similar. To avoid such circularity
we must concede that some words are indefinable.
Irwin Goldstein; Must there be Indefinable Words?; Metaphilosophy; v17; n1; 1986; p91
If I am asked “What is good?” my answer is that
good is good, and that is the end of the matter. Or if I am asked “How
is good to be defined?” my answer is that it cannot be defined, and that
is all I have to say about it. … My point is that “good” is a simple
notion, just as “yellow” is a simple notion; that, just as you cannot,
by any manner of means, explain to anyone who does not already know it,
what yellow is, so you cannot explain what good is. Definitions … which
describe the real nature of the object or notion denoted by a word … are
only possible when the object or notion in question is something
complex. … But yellow and good, we say, are not complex: they are
notions of that simple kind, out of which definitions are composed and
with which the power of further defining ceases.
G. E. Moore;
Principia Ethica; cited in
Richard Cartwright; Philosophical Essays; 1987; p71
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Russell, Bertrand >
A Critical Exposition of the Philosophy of Leibniz
Silberger, D. M. >
Are Primitive Words Universal for Infinite Symmetric Groups?
Solomon, J. >
Is the Conception of “Good” Undefinable?
Tarski, Alfred >
Introduction to Logic
