Categories From Within

1.417-520      G-c.1896-2

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    THE LOGIC OF MATHEMATICS;
AN ATTEMPT TO DEVELOP MY CATEGORIES
           FROM WITHIN*

     # 1. THE THREE CATEGORIES

  417. Although the present paper deals with mathematics.
yet its problems are not mere mathematical problems. It is
not proposed to inquire into the methods of reasoning of
mathematics particularly, although this subject will incidentally be
touched upon. But mathematics performs its reasonings by
a logica utens which it develops for itself, and has no need of
any appeal to a logica docens; for no disputes about reasoning
arise in mathematics which need to be submitted to the
principles of the philosophy of thought for decision. The questions
which are here to be examined are, what are the different
systems of hypotheses from which mathematical deduction can
set out, what are their general characters, why are not other
hypotheses possible, and the like. These are not problems
which, like those of mathematics, repose upon clear and
definite assumptions recognized at the outset; and yet, like
mathematical problems, they are questions of possibility and
necessity. What the nature of this necessity can be is one of the
very matters to be discovered. This much, however, is
indisputable: if there are really any such necessary characteristics
of mathematical hypotheses as I have just declared in advance
that we shall find that there [are], this necessity must spring
from some truth so broad as to hold not only for the universe
we know but for every world that poet could create. And this
truth like every truth must come to us by the way of
experience. No apriorist ever denied that. The first matters which
it is pertinent to examine are the most universal categories of
elements of all experience, natural or poetical.

  * c. 1896. The first four pages of the manuscript are missing.
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  418. We remark among phenomena three categories of
elements.
  The first comprises the qualities of phenomena, such as red,
bitter, tedious, hard, heartrending, noble; and there are
doubtless manifold varieties utterly unknown to us. Beginners in
philosophy may object that these are not qualities of things
and are not in the world at all, but are mere sensations.
Certainly, we only know such as the senses we are furnished with
are adapted to reveal; and it can hardly be doubted that the
specializing effect of the evolutionary process which has made
us what we are has been to blot the greater part of the senses
and sensations which were once dimly felt, and to render
bright, clear, and separate the rest. But whether we ought to
say that it is the senses that make the sense-qualities or the
sense-qualities to which the senses are adapted, need not be
determined in haste. It is sufficient that wherever there is a
phenomenon there is a quality; so that it might almost seem
that there is nothing else in phenomena. The qualities merge
into one another. They have no perfect identities, but only
likenesses, or partial identities. Some of them, as the colors
and the musical sounds, form well-understood systems.
Probably, were our experience of them not so fragmentary, there
would be no abrupt demarcations between them, at all.* Still,
each one is what it is in itself without help from the others.
They are single but partial determinations.
  419. The second category of elements of phenomena
comprises the actual facts. The qualities, in so far as they are
general, are somewhat vague and potential. But an occurrence
is perfectly individual. It happens here and now. A
permanent fact is less purely individual; yet so far as it is actual, its
permanence and generality only consist in its being there at
every individual instant. Qualities are concerned in facts but
they do not make up facts. Facts also concern subjects which
are material substances. We do not see them as we see
qualities, that is, they are not in the very potentiality and essence
of sense. But we feel facts resist our will. That is why facts
are proverbially called brutal. Now mere qualities do not
resist. It is the matter that resists. Even in actual sensation
there is a reaction. Now mere qualities, unmaterialized, cannot

 * Cf. 313; also vol. 6, bk. I, ch. 5.
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actually react. So that, rightly understood, it is correct to say
that we immediately, that is, directly perceive matter. To say
that we only infer matter from its qualities is to say that we
only know the actual through the potential. It would be a
little less erroneous to say that we only know the potential
through the actual, and only infer qualities by generalization
from what we perceive in matter. All that I here insist upon
is that quality is one element of phenomena, and fact, action,
actuality is another. We shall undertake the analysis of their
natures below.
  420. The third category of elements of phenomena consists
of what we call laws when we contemplate them from the
outside only, but which when we see both sides of the shield we
call thoughts. Thoughts are neither qualities nor facts. They
are not qualities because they can be produced and grow, while
a quality is etemal, independent of time and of any realization.
Besides, thoughts may have reasons, and indeed, must have
some reasons, good or bad. But to ask why a quality is as it
is, why red is red and not green, would be lunacy. If red were
green it would not be red; that is all. And any semblance of
sanity the question may have is due to its being not exactly
a question about quality, but about the relation between two
qualities, though even this is absurd. A thought then is not
a quality. No more is it a fact. For a thought is general. I had
it. I imparted it to you. It is general on that side. It is also
general in referring to all possible things, and not merely to
those which happen to exist. No collection of facts can
constitute a law; for the law goes beyond any accomplished facts
and determines how facts that may be, but all of which never
can have happened, shall be characterized. There is no
objection to saying that a law is a general fact, provided it be
understood that the general has an admixture of potentiality in it,
so that no congeries of actions here and now can ever make a
general fact. As general, the law, or general fact, concerns the
potential world of quality, while as fact, it concems the actual
world of actuality. Just as action requires a peculiar kind of
subject,matter,which is foreign to mere quality, so law requires
a peculiar kind of subject, the thought, or, as the phrase in this
connection is, the mind, as a peculiar kind of subject foreign
to mere individual action. Law, then, is something as remote
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from both quality and action as these are remote from one
another.
  421. Having thus by observation satisfied ourselves that
there are these three categories of elements of phenomena, let
us endeavor to analyze the nature of each, and try to find out
why there should be these three categories and no others. This
reason, when we find it, ought to be interesting to
mathematicians; for it will be found to coincide with the most
fundamental characteristic of the most universal of the mathematical
hypotheses, I mean that of number.

              # 2. QUALITY

  422. What, then, is a quality?
  Before answering this, it will be well to say what it is not.
It is not anything which is dependent, in its being, upon mind,
whether in the form of sense or in that of thought. Nor is it
dependent, in its being, upon the fact that some material
thing possesses it. That quality is dependent upon sense is the
great error of the conceptualists. That it is dependent upon
the subject in which it is realized is the great error of all the
nominalistic schools. A quality is a mere abstract potentiality;
and the error of those schools lies in holding that the potential,
or possible, is nothing but what the actual makes it to be. It
is the error of maintaining that the whole alone is something,
and its components, however essential to it, are nothing. The
refutation of the position consists in showing that nobody does,
or can, in the light of good sense, consistently retain it. The
moment the fusillade of controversy ceases they repose on other
conceptions. First, that the quality of red depends on anybody
actually seeing it, so that red things are no longer red in the
dark, is a denial of common sense. I ask the conceptualist, do
you really mean to say that in the dark it is no longer true
that red bodies are capable of transmitting the light at the
lower end of the spectrum? Do you mean to say that a piece
of iron not actually under pressure has lost its power of
resisting pressure? If so, you must either hold that those bodies
under the circumstances supposed assume the opposite
properties, or you must hold that they become indeterminate in those
respects. If you hold that the red body in the dark acquires
a power of absorbing the long waves of the spectrum, and that

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the iron acquires a power of condensation under small pressure,
then, while you adopt an opinion without any facts to support
it, you still admit that qualities exist while they are not
actually perceived -- only you transfer this belief to qualities
which there is no ground for believing in. If, however, you
hold that the bodies become indeterminate in regard to the
qualities they are not actually perceived to possess, then, since
this is the case at any moment in regard to the vast majority
of the qualities of all bodies, you must hold that generals exist.
In other words, it is concrete things you do not believe in;
qualities, that is, generals -- which is another word for the
same thing -- you not only believe in but believe that they
alone compose the universe. Consistency, therefore, obliges you
to say that the red body is red (or has some color) in the dark,
and that the hard body has some degree of hardness when
nothing is pressing upon it. If you attempt to escape the refutation
by a distinction between qualities that are real, namely the
mechanical qualities, and qualities that are not real, sensible
qualities, you may be left there, because you have granted the
essential point. At the same time, every modern psychologist
will pronounce your distinction untenable. You forget perhaps
that a realist fully admits that a sense-quality is only a
possibility of sensation; but he thinks a possibility remains possible
when it is not actual. The sensation is requisite for its
apprehension; but no sensation nor sense-faculty is requisite for the
possibility which is the being of the quality. Let us not put the
cart before the horse, nor the evolved actuality before the
possibility as if the latter involved what it only evolves. A
similar answer may be made to the other nominalists. It is
impossible to hold consistently that a quality only exists when
it actually inheres in a body. If that were so, nothing but
individual facts would be true. Laws would be fictions; and,
in fact, the nominalist does object to the word " law, " and
prefers "uniformity" to express his conviction that so far as the
law expresses what only might happen, but does not, it is
nugatory. If, however, no law subsists other than an expression of
actual facts, the future is entirely indeterminate and so is
general to the highest degree. Indeed, nothing would exist but the
instantaneous state; whereas it is easy to show that if we are
going to be so free in calling elements fictions an instant is the
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first thing to be called fictitious. But I confess I do not take
pains accurately to answer a doctrine so monstrous, and just
at present out of vogue.
  423. So much for what quality is not. Now what is it?
We do not care what meaning the usages of language may
attach to the word. We have already seen clearly that the
elements of phenomena are of three categories, quality, fact,
and thought. The question we have to consider is how quality
shall be defined so as to preserve the truth of that division.
In order to ascertain this, we must consider how qualities are
apprehended and from what point of view they become
emphatic in thought, and note what it is that will and must
be revealed in that mode of apprehension.
  424. There is a point of view from which the whole universe
of phenomena appears to be made up of nothing but sensible
qualities. What is that point of view? It is that in which we
attend to each part as it appears in itself, in its own suchness,
while we disregard the connections. Red, sour, toothache are
each sui generis and indescribable. In themselves, that is all
there is to be said about them. Imagine at once a toothache,
a splitting headache, a jammed finger, a corn on the foot, a
burn, and a colic, not necessarily as existing at once -- leave
that vague -- and attend not to the parts of the imagination
but to the resultant impression. That will give an idea of a
general quality of pain. We see that the idea of a quality is
the idea of a phenomenon or partial phenomenon considered
as a monad, without reference to its parts or components and
without reference to anything else. We must not consider
whether it exists, or is only imaginary, because existence
depends on its subject having a place in the general system of
the universe. An element separated from everything else and
in no world but itself, may be said, when we come to refiect
upon its isolation, to be merely potential. But we must not
even attend to any determinate absence of other things; we
are to consider the total as a unit. We may term this aspect
of a phenomenon the monadic aspect of it. The quality is what
presents itself in the monadic aspect.
  425. The phenomenon may be ever so complex and
heterogeneous. That circumstance will make no particular difference
in the quality. It will make it more general. But one quality
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is in itself, in its monadic aspect, no more general than another.
The resultant effect has no parts. The quality in itself is
indecomposable and sui generis. When we say that qualities are
general, are partial determinations, are mere potentialities,
etc., all that is true of qualities refiected upon; but these things
do not belong to the quality-element of experience.
  426. Experience is the course of life. The world is that
which experience inculcates. Quality is the monadic element
of the world. Anything whatever, however complex and
heterogeneous, has its quality sui generis, its possibility of
sensation, would our senses only respond to it. But in saying this,
we are straying from the domain of the monad into that of the
dyad; and such truths are best postponed until we come to
discuss the dyad.

               # 3. FACT

  427. Next, what is fact?
  As before, it is not the usage of language which we seek to
learn, but what must be the description of fact in order that our
division of the elements of phenomena into the categories of
quality, fact, and law may not only be true, but also have the
utmost possible value, being governed by those same
characteristics which really dominate the phenomenal world. It
is first requisite to point out something which must be excluded
from the category of fact. This is the general, and with it the
permanent or eternal (for permanence is a species of generality),
and the conditional (which equally involves generality).
Generality is either of that negative sort which belongs to the
merely potential, as such, and this is peculiar to the category
of quality; or it is of that positive kind which belongs to
conditional necessity, and this is peculiar to the category of law.
These exclusions leave for the category of fact, first, that
which the logicians call the contingent, that is, the accidentally
actual, and second, whatever involves an unconditional
necessity, that is, force without law or reason, brute force.
  428. It may be said that there is no such phenomenon in
the universe as brute force, or freedom of will, and nothing
accidental. I do not assent to either opinion; but granting
that both are correct, it still remains true that considering a
single action by itself, apart from all others and, therefore,
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apart from the governing uniformity, it is in itself brute,
whether it show brute force or not. I shall presently point out
a sense in which it does display force. That it is possible for
a phenomenon in some sense to present force to our notice
without emphasizing any element of law, is familiar to everybody.
We often regard our own exertions of will in that way. In like
manner, if we consider any state of an individual thing, putting
aside other things, we have a phenomenon which is actual, but
in itself is not necessitated. It is not pretended that what is
here termed fact is the whole phenomenon, but only an
element of the phenomenon -- so much as belongs to a
particular place and time. That when more is taken into account,
the observer finds himself in the realm of law in every case,
I fully admit. (Nor does that conflict with tychism. * )
  429. On the other hand, if the view be limited to any part
of the phenomenal world, however great, and this be looked
upon as a monad, entirely regardless of its parts, nothing is
presented to the observer but a quality. How much, then,
must we attend to, in order to perceive the pure element of
fact? There are certain occurrences which, when they come
to our notice, we set down as "accidental." Now, although
there is really no more of the factual element in these than in
other facts, yet the circumstance that we call them par
excellence contingent, or "accidental," would lead us to expect that
which distinguishes the realm of fact from the realms of quality
and of law, to be particularly prominent in them. We call such
facts " coincidences," a name which implies that our attention
is called in them to the coming together of two things. Two
phenomena, and but two, are required to constitute a
coincidence; and if there are more than two no new form of
relationship appears further than a complication of pairs. Two
phenomena, whose parts are not attended to, cannot display
any law, or regularity. Three dots may be placed in a straight
line, which is a kind of regularity; or they may be placed at
the vertices of an equilateral triangle, which is another kind of
regularity. But two dots cannot be placed in any particularly
regular way, since there is but one way in which they can be
placed, unless they were set together, when they would cease
to be two. It is true that on the earth two dots may be placed

 * See vol. 6, bk. I, ch. 2, and 6.102.
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antipodally. But that is only one of the exceptions that prove
the rule, because the earth is a third object there taken into
account. So two straight lines in a plane can be set at right
angles, which is a sort of regularity. But this is another rule
proving exception, since