THE MATHEMATICAL PRINCIPLES
As has been pointed out,[1] the aim of the second part of the
_Analytic of Principles_ is to determine the _a priori_ principles
involved in the use of the categories under the necessary sensuous
conditions. These principles Kant divides into four classes,
corresponding to the four groups of categories, and he calls
them respectively 'axioms of perception', 'anticipations of
sense-perception', 'analogies of experience', and 'postulates of
empirical thought'. The first two and the last two classes are grouped
together as 'mathematical' and 'dynamical' respectively, on the ground
that the former group concerns the perception of objects, i. e. their
nature apprehended in perception, while the latter group concerns
their existence, and that consequently, since assertions concerning
the existence of objects presuppose the realization of empirical
conditions which assertions concerning their nature do not, only the
former possesses an absolute necessity and an immediate evidence such
as is found in mathematics.[2]
[1] p. 246.
[2] The assertion that all perceptions (i. e. all objects of
perception) are extensive quantities relates, according to
Kant, to the nature of objects, while the assertion that an
event must have a necessary antecedent affirms that such an
antecedent must exist, but gives no clue to its specific
nature. Compare "But the existence of phenomena cannot be
known _a priori_, and although we could be led in this way to
infer the fact of some existence, we should not know this
existence determinately, i. e. we could not anticipate the
respect in which the empirical perception of it differed from
that of other existences". (B. 221, M. 134). Kant seems to
think that the fact that the dynamical principles relate to
the existence of objects is a sufficient justification of
their name.
It needs but little reflection to see that the distinctions
which Kant draws between the mathematical and the dynamical
principles must break down.
These two groups of principles are not, as their names might suggest,
principles within mathematics and physics, but presuppositions of
mathematics and physics respectively. Kant also claims appropriateness
for the special terms used of each minor group to indicate the kind of
principles in question, viz. 'axioms', 'anticipations', 'analogies',
'postulates'. But it may be noted as an indication of the
artificiality of the scheme that each of the first two groups contains
only one principle, although Kant refers to them in the plural as
axioms and anticipations respectively, and although the existence of
three categories corresponding to each group would suggest the
existence of three principles.
The axiom of perception is that 'All perceptions are extensive
quantities'. The proof of it runs thus:
"An extensive quantity I call that in which the representation of the
parts renders possible the representation of the whole (and therefore
necessarily precedes it). I cannot represent to myself any line,
however small it may be, without drawing it in thought, that is,
without generating from a point all its parts one after another, and
thereby first drawing this perception. Precisely the same is the case
with every, even the smallest, time.... Since the pure perception in
all phenomena is either time or space, every phenomenon as a
perception is an extensive quantity, because it can be known in
apprehension only by a successive synthesis (of part with part). All
phenomena, therefore, are already perceived as aggregates (groups of
previously given parts), which is not the case with quantities of
every kind, but only with those which are represented and apprehended
by us as _extensive_."[3]
[3] B. 203-4, M. 123.
Kant opposes an extensive quantity to an intensive quantity or a
quantity which has a degree. "That quantity which is apprehended
only as unity and in which plurality can be represented only by
approximation to negation = 0, I call _intensive quantity_."[4] The
aspect of this ultimate distinction which underlies Kant's mode of
stating it is that only an extensive quantity is a whole, i. e.
something made up of parts. Thus a mile can be said to be made up
of two half-miles, but a velocity of one foot per second, though
comparable with a velocity of half a foot per second, cannot be said
to be made up of two such velocities; it is essentially one and
indivisible. Hence, from Kant's point of view, it follows that it is
only an extensive magnitude which can, and indeed must, be apprehended
through a successive synthesis of the parts. The proof of the axiom
seems to be simply this: 'All phenomena as objects of perception are
subject to the forms of perception, space and time. Space and time are
[homogeneous manifolds, and therefore] extensive quantities, only
to be apprehended by a successive synthesis of the parts. Hence
phenomena, or objects of experience, must also be extensive
quantities, to be similarly apprehended.' And Kant goes on to add that
it is for this reason that geometry and pure mathematics generally
apply to objects of experience.
[4] B. 210, M. 127.
We need only draw attention to three points. Firstly, no justification
is given of the term 'axiom'. Secondly, the argument does not really
appeal to the doctrine of the categories, but only to the character
of space and time as forms of perception. Thirdly, it need not appeal
to space and time as forms of perception in the proper sense of ways
in which we apprehend objects, but only in the sense of ways in which
objects are related[5]; in other words, it need not appeal to Kant's
theory of knowledge. The conclusion follows simply from the nature of
objects as spatially and temporally related, whether they are
phenomena or not. It may be objected that Kant's thesis is that _all_
objects of perception are extensive quantities, and that unless space
and time are allowed to be ways in which _we must perceive_ objects,
we cannot say that all objects will be spatially and temporally
related, and so extensive quantities. But to this it may be replied
that it is only true that all objects of perception are extensive
quantities if the term 'object of perception' be restricted to parts
of the physical world, i. e. to just those realities which Kant is
thinking of as spatially and temporally related,[6] and that this
restriction is not justified, since a sensation or a pain which has
only intensive quantity is just as much entitled to be called an
object of perception.
[5] Cf. pp. 37-9.
[6] The context shows that Kant is thinking only of such
temporal relations as belong to the physical world, and not
of those which belong to us as apprehending it. Cf. p. 139.
The anticipation of sense-perception consists in the principle that
'In all phenomena, the real, which is an object of sensation, has
intensive magnitude, i. e. a degree'. The proof is stated thus:
"Apprehension merely by means of sensation fills only one moment
(that is, if I do not take into consideration the succession of many
sensations). Sensation, therefore, as that in the phenomenon the
apprehension of which is not a successive synthesis advancing from
parts to a complete representation, has no extensive quantity; the
lack of sensation in one and the same moment would represent it as
empty, consequently = 0. Now that which in the empirical perception
corresponds to sensation is reality (_realitas phaenomenon_); that
which corresponds to the lack of it is negation = 0. But every
sensation is capable of a diminution, so that it can decrease and thus
gradually vanish. Therefore, between reality in the phenomenon and
negation there exists a continuous connexion of many possible
intermediate sensations, the difference of which from each other is
always smaller than that between the given sensation and zero, or
complete negation. That is to say, the real in the phenomenon has
always a quantity, which, however, is not found in apprehension, since
apprehension takes place by means of mere sensation in one moment and
not by a successive synthesis of many sensations, and therefore does
not proceed from parts to the whole. Consequently, it has a quantity,
but not an extensive quantity."
"Now that quantity which is apprehended only as unity, and in which
plurality can be represented only by approximation to negation = 0,
I call an _intensive quantity_. Every reality, therefore, in a
phenomenon has intensive quantity, that is, a degree."[7]
[7] B. 209-10, M. 127.
In other words, 'We can lay down _a priori_ that all sensations have a
certain degree of intensity, and that between a sensation of a given
intensity and the total absence of sensation there is possible an
infinite number of sensations varying in intensity from nothing to
that degree of intensity. Therefore the real, which corresponds to
sensation, can also be said _a priori_ to admit of an infinite variety
of degree.'
Though the principle established is of little intrinsic importance,
the account of it is noticeable for two reasons. In the first place,
although Kant clearly means by the 'real corresponding to sensation' a
body in space, and regards it as a phenomenon, it is impossible to see
how he can avoid the charge that he in fact treats it as a thing in
itself.[8] For the correspondence must consist in the fact that the
real causes or excites sensation in us, and therefore the real, i. e.
a body in space, is implied to be a thing in itself. In fact, Kant
himself speaks of considering the real in the phenomenon as the cause
of sensation,[9] and, in a passage added in the second edition, after
proving that sensation must have an intensive quantity, he says that,
corresponding to the intensive quantity of sensation, an intensive
quantity, i. e. _a degree of influence on sense_, must be attributed
to all objects of sense-perception.[10] The difficulty of consistently
maintaining that the real, which corresponds to sensation, is a
phenomenon is, of course, due to the impossibility of distinguishing
between reality and appearance within phenomena.[11]
[8] Cf. p. 257 note.
[9] B. 210, M. 128.
[10] B. 208, M. 126. The italics are mine. Cf. from the same
passage, "Phenomena contain, over and above perception, the
materials for some object (through which is represented
something existing in space and time), i. e. they contain the
real of sensation as a merely subjective representation of
which we can only become conscious that _the subject is
affected_, and which we relate _to an object in general_."
(The italics are mine.)
[11] Cf. pp. 94-100.
In the second place, Kant expressly allows that in this anticipation
we succeed in discovering _a priori_ a characteristic of sensation,
although sensation constitutes that empirical element in phenomena,
which on Kant's general view cannot be apprehended _a priori_.
"Nevertheless, this anticipation of sense-perception must always be
somewhat surprising to an inquirer who is used to transcendental
reflection, and is thereby rendered cautious. It leads us to feel some
misgiving as to whether the understanding can anticipate such a
synthetic proposition as that respecting the degree of all that is
real in phenomena, and consequently respecting the possibility of the
internal distinction of sensation itself, if we abstract from its
empirical quality. There remains, therefore, a problem not unworthy of
solution, viz. 'How can the understanding pronounce synthetically and
_a priori_ upon phenomena in this respect, and thus anticipate
phenomena even in that which is specially and merely empirical, viz.
that which concerns sensations?'"[12] But although Kant recognizes
that the anticipation is surprising, he is not led to revise his
general theory, as being inconsistent with the existence of the
anticipation. He indeed makes an attempt[13] to deal with the
difficulty; but his solution consists not in showing that the
anticipation is consistent with his general theory--as he should have
done, if the theory was to be retained--but in showing that, in the
case of the degree of sensation, we do apprehend the nature of
sensation _a priori_.
[12] B. 217, M. 131; cf. B. 209, M. 127.
[13] B. 217-18, M. 132.
Strangely enough, Hume finds himself face to face with what is in
principle the same difficulty, and treats it in a not dissimilar way.
"There is, however, one contradictory phenomenon, which may prove,
that 'tis not absolutely impossible for ideas to go before their
correspondent impressions. I believe it will readily be allow'd, that
the several distinct ideas of colours, which enter by the eyes, or
those of sounds, which are convey'd by the hearing, are really
different from each other, tho' at the same time resembling. Now if
this be true of different colours, it must be no less so of the
different shades of the same colour, that each of them produces a
distinct idea, independent of the rest. For if this shou'd be deny'd,
'tis possible, by the continual gradation of shades, to run a colour
insensibly into what is most remote from it; and if you will not allow
any of the means to be different, you cannot without absurdity deny
the extremes to be the same. Suppose therefore a person to have
enjoyed his sight for thirty years, and to have become perfectly well
acquainted with colours of all kinds, excepting one particular shade
of blue, for instance, which it never has been his fortune to meet
with. Let all the different shades of that colour, except that single
one, be plac'd before him, descending gradually from the deepest to
the lightest; 'tis plain that he will perceive a blank, where that
shade is wanting, and will be sensible, that there is a greater
distance in that place betwixt the contiguous colours, than in any
other. Now I ask, whether 'tis possible for him, from his own
imagination, to supply this deficiency, and raise up to himself the
idea of that particular shade, tho' it had never been conveyed to him
by his senses? I believe there are few but will be of opinion that he
can; and this may serve as a proof, that the simple ideas are not
always derived from the correspondent impressions; tho' the instance
is so particular and singular, that 'tis scarce worth our observing,
and does not merit that for it alone we should alter our general
maxim."[14]
[14] Hume, _Treatise_, Bk. I, Part 1, Sec. 1.