Hume offers against infinite divisibility ‘A surface terminates
a solid; a line terminates a surface; a point terminates a line:
but I assert that if the _ideas_ of a point, line, or surface were
not indivisible, ’tis impossible we should ever conceive these
terminations. For let these ideas be supposed infinitely divisible,
and then let the fancy endeavour to fix itself on the idea of the
last surface, line, or point, it immediately finds this idea to
break into parts; and upon its seizing the last of these parts it
loses its hold by a new division, and so on _ad infinitum_, without
any possibility of its arriving at a concluding idea’. [1] If
‘point,’ ‘line,’ or ‘surface’ were really names for ‘ideas’ either in
Hume’s sense, as feelings grown fainter, or in Locke’s, as definite
imprints made by outward things, this passage would be perplexing.
In truth they represent objects determined by certain conceived
relations, and the relation under which the object is considered
may vary without a corresponding variation in the name. When a
‘point’ is considered simply as the ‘termination of a line,’ it is
not considered as a quantum. It represents the abstraction of the
relation of externality, as existing between _two lines_. It is these
lines, not the point, that in this case are the constituents of the
relation, and thus it is they alone that are for the time considered
as extended, therefore as quanta, therefore as divisible. So when the
line in turn is considered as the ‘termination of a surface.’ It then
represents the relation of externality _as between surfaces_, and
for the time it is the surfaces, not the line, that are considered
to have extension and its consequences. The same applies to the view
of a surface as the termination of a solid. Just as the line, though
not a quantum when considered simply as a relation between surfaces,
becomes so when considered in relation to another line, so the point,
though it ‘has no magnitude’ when considered as the termination of
a line, yet acquires parts, or becomes divisible, so soon as it is
considered in relation to other points as a constituent of extended
matter; and it is thus that Hume considers it, ἑκὼν ἢ ἄκων [2], when
he talks of extension as ‘made up of coloured points.’
[1] P. 345. [Book I, part II., sec. IV.]
[2] [Greek ἑκὼν ἢ ἄκων (hekon e akon) = like it or not. Tr.]
What becomes of the exactness of mathematics according to Hume?
273. It is the necessity then, according to his theory, of making
space an impression that throughout underlies Hume’s argument against
its infinite divisibility; and, as we have seen, the same theory
which excludes its infinite divisibility logically extinguishes it as
a quantity, divisible and measurable, altogether. He of course does
not recognize this consequence. He is obliged indeed to admit that
in regard to the proportions of ‘greater, equal and less,’ and the
relations of different parts of space to each other, no judgments
of universality or exactness are possible. We may judge of them,
however, he holds, with various approximations to exactness, whereas
upon the supposition of infinite divisibility, as he ingeniously
makes out, we could not judge of them at all. He ‘asks the
mathematicians, what they mean when they say that one line or surface
is equal to, or greater or less than, another.’ If they ‘maintain the
composition of extension by indivisible points,’ their answer, he
supposes, will be that ‘lines or surfaces are equal when the numbers
of points in each are equal.’ This answer he reckons ‘just,’ but the
standard of equality given is entirely useless. ‘For as the points
which enter into the composition of any line or surface, whether
perceived by the sight or touch, are so minute and so confounded with
each other that ’tis utterly impossible for the mind to compute their
number, such a computation will never afford us a standard by which
we may judge of proportions.’ The opposite sect of mathematicians,
however, are in worse case, having no standard of equality whatever
to assign. ‘For since, according to their hypothesis, the least as
well as greatest figures contain an infinite number of parts, and
since infinite numbers, properly speaking, can neither be equal
nor unequal with respect to each other, the equality or inequality
of any portion of space can never depend on any proportion in the
number of their parts.’ His own doctrine is ‘that the only useful
notion of equality or inequality is derived from the whole united
appearance, and the comparison of, particular objects.’ The judgments
thus derived are in many cases certain and infallible. ‘When the
measure of a yard and that of a foot are presented, the mind can no
more question that the first is longer than the second than it can
doubt of those principles which are most clear and self-evident.’
Such judgments, however, though ‘sometimes infallible, are not always
so.’ Upon a ‘review and reflection’ we often ‘pronounce those objects
equal which at first we esteemed unequal,’ and vice versâ. Often
also ‘we discover our error by a juxtaposition of the objects; or,
where that is impracticable, by the use of some common and invariable
measure which, being successively applied to each, informs us of
their different proportions. And even this correction is susceptible
of a new correction, and of different degrees of exactness, according
to the nature of the instrument by which we measure the bodies, and
the care which we employ in the comparison.’ [1]
[1] Pp. 351-53. [Book I, part II., sec. IV.]
The universal propositions of geometry either untrue or unmeaning.
274. Such indefinite approach to exactness is all that Hume can allow
to the mathematician. But it is undoubtedly another and an absolute
sort of exactness that the mathematician himself supposes when he
pronounces all right angles equal. Such perfect equality ‘beyond what
we have instruments and art’ to ascertain, Hume boldly calls a ‘mere
fiction of the mind, useless as well as incomprehensible’. [1] Thus
when the mathematician talks of certain angles as always equal, of
certain lines as never meeting, he is either making statements that
are untrue or speaking of nonentities. If his ‘lines’ and ‘angles’
mean ideas that we can possibly have, his universal propositions are
untrue; if they do not, according to Hume they can mean nothing.
He says, for instance, that ‘two right lines cannot have a common
segment;’ but of such ideas of right lines as we can possibly have
this is only true ‘where the right lines incline upon each other with
a sensible angle.’ [2] It is not true when they ‘approach at the rate
of an inch in 20 leagues.’ According to the ‘original standard of a
right line,’ which is ‘nothing but a certain general appearance, ’tis
evident right lines may be made to concur with each other’. [3] Any
other standard is a ‘useless and incomprehensible fiction.’ Strictly
speaking, according to Hume, we have it not, but only a tendency to
suppose that we have it arising from the progressive correction of
our actual measurements. [4]
[1] P. 353. [Book I, part II., sec. IV.]
[2] Cf. Aristotle, Metaph. 998a, on a corresponding view ascribed to
Protagoras.
[3] P. 356. [Book I, part II., sec. IV.]
[4] P. 354. [Book I, part II., sec. IV.]
Distinction between Hume’s doctrine and that of the hypothetical
nature of mathematics.
275. Now it is obvious that what Hume accounts for by means of
this tendency to feign, even if the tendency did not presuppose
conditions incompatible with his theory, is not mathematical
science as it exists. It has even less appearance of being so
than (to anticipate) has that which is accounted for by those
propensities to feign, which he substitutes for the ideas of cause
and substance, of being natural science as it exists. In the latter
case, when the idea of necessary connexion has been disposed of,
an impression of reflection can with some plausibility be made
to do duty instead; but there is no impression of reflection in
Hume’s sense of the word, no ‘propensity,’ that can be the subject
of mathematical reasoning. He speaks, indeed, of our _supposing_
some imaginary standard--of our having ‘an obscure and implicit
notion’--of perfect equality, but such language is only a way of
saving appearances; for according to him, a ‘supposition’ or ‘notion’
which is neither impression nor idea, cannot be anything. A hasty
reader, catching at the term ‘supposition,’ may find his statement
plausible with all the plausibility of the modern doctrine, which
accounts for the universality and exactness of mathematical truths
as ‘hypothetical’--the doctrine that we suppose figures exactly
corresponding to our definitions, though such do not really exist.
With those who take this view, however, it is always understood that
the definitions represent ideas, though not ideas to which real
objects can be found exactly answering. Perhaps, if pressed about
their distinction between idea and reality, they might find it hard
consistently to maintain it, but it is by this practically that they
keep their theory afloat. Hume can admit no such distinction. The
real with him is the impression, and the idea the fainter impression.
There can be no idea of a straight line, a curve, a circle, a right
angle, a plane, other than the impression, other than the ‘appearance
to the eye,’ and there are no appearances exactly answering to the
mathematical definitions. If they do not _exactly_ answer, they might
as well for the purposes of mathematical demonstration not answer
at all. The Geometrician, having found that the angles at the base
of _this_ isosceles triangle are equal to each other, at once takes
the equality to be true of all isosceles triangles, as being exactly
like the original one, and on the strength of this establishes many
other propositions. But, according to Hume, no idea that we could
have would be one of which the sides were precisely equal. The Fifth
Proposition of Euclid then is not precisely true of the particular
idea that we have before us when we follow the demonstration. Much
less can it be true of the ideas, _i.e._ the several appearances
of colour, indefinitely varying from this, which we have before us
when we follow the other demonstrations in which the equality of the
angles at the base of an isosceles is taken for granted.
The admission that no relations of quantity are data of sense removes
difficulty as to general propositions about them.
276. Here, as elsewhere, what we have to lament is not that Hume
‘pushed his doctrine too far,’ so far as to exclude ideas of those
exact proportions in space with which geometry purports to deal,
but that he did not carry it far enough to see that it excluded all
ideas of quantitative relations whatever. He thus pays the penalty
for his equivocation between a feeling of colour and a disposition
of coloured points. Even alongside of his admission that ‘relations
of space and time’ are independent of the nature of the ideas so
related, which amounts to the admission that of space and time there
are no ideas at all in his sense of the word, he allows himself to
treat ‘proportions between spaces’ as depending entirely on our
ideas of the spaces--depending on ideas which in the context he
by implication admits that we have not. [1] If, instead of thus
equivocating, he had asked himself how sensations of colour and touch
could be added or divided, how one could serve as a measure of the
size of another, he might have seen that only in virtue of that in
the ‘general appearance’ of objects which, in his own language, is
‘independent of the nature of the ideas themselves’--_i.e._ which
does not belong to them as feelings, but is added by the comparing
and combining thought--are the proportions of greater, less, and
equal predicable of them at all; that what thought has thus added,
viz. limitation by mutual externality, it can abstract; and that by
such abstraction of the limit it obtains those several terminations,
as Hume well calls them--the surface terminating bodies, the line
terminating surfaces, the point terminating lines--from which it
constructs the world of pure space: that thus the same action of
thought in sense, which alone renders appearances measurable, gives
an object matter which, because the pure construction of thought, we
can measure exactly and with the certainty that the judgment based on
a comparison of magnitudes in a single case is true of all possible
cases, because in none of these can any other conditions be present
than those which we have consciously put there.
[1] Part III. § 1, sub init.
Hume does virtually admit this in regard to numbers.
277. To have arrived at this conclusion Hume had only to extend to
proportions in space the principle upon which the impossibility of
sensualizing arithmetic compels him to deal with proportions in
number. ‘We are possessed,’ he says, ‘of a precise standard by which
we can judge of the equality and proportion of numbers; and according
as they correspond or not to that standard we determine their
relations without any possibility of error. When two numbers are so
combined, as that the one has always an unite answering to every
unite of the other, we pronounce them equal’. [1] Now what are the
unites here spoken of? If they were those single impressions which he
elsewhere [2] seems to regard as alone properly unites, the point of
the passage would be gone, for combinations of such unites could at
any rate only yield those ‘general appearances’ of whose proportions
we have been previously told there can be no precise standard. They
can be no other than those unites which, not being impressions, he
has to call ‘fictitious denominations’--unites which are nothing
except in relation to each other and of which each, being in turn
divisible, is itself a true number. We can easily retort upon Hume,
then, when he argues that the supposition of infinite divisibility
is incompatible with any comparison of quantities because with any
unite of measurement, that, according to his own virtual admission,
in the only case where such comparison is exact the ultimate unite
of measurement is still itself divisible; which, indeed, is no
more than saying that whatever measures quantity must itself be a
quantity, and that therefore quantity is infinitely divisible. If
Hume, instead of slurring over this characteristic of the science of
number, had set himself to explain it, he would have found that the
only possible explanation of it was one equally applicable to the
science of space--that what is true of the unite, as the abstraction
of distinctness, is true also of the abstraction of externality. As
the unite, because constituted by relation to other unites, so soon
as considered breaks into multiplicity, and only for that reason is
a quantity by which other quantities can be measured; so is it also
with the limit in whatever form abstracted, whether as point, line,
or surface. If the fact that number can have no least part since each
part is itself a number or nothing, so far from being incompatible
with the finiteness of number, is the consequence of that finiteness,
neither can the like attribute in spaces be incompatible with their
being definite magnitudes, that can be compared with and measured
by each other. The real difference, which is also the rationale of
Hume’s different procedure in the two cases, is that the conception
of space is more easily confused than that of number with the
feelings to which it is applied, and which through such application
become sensible spaces. Hence the liability to the supposition,
which is at bottom Hume’s, that the last feeling in the process of
diminution before such sensible space disappears (being the ‘minimum
visibile’) is the least possible portion of space.
[1] P. 374. [Book I, part III., sec. I.]
[2] Above, par. 258.
With Hume idea of vacuum impossible, but logically not more so than
that of space.
278. Just as that reduction of consciousness to feeling, which
really excludes the idea of quantity altogether, is by Hume only
recognised as incompatible with its infinite divisibility, so it is
not recognised as extinguishing space altogether, but only space as
a vacuum. If it be true, he says, ‘that the idea of space is nothing
but the idea of visible or tangible points distributed in a certain
order, it follows that we can form no idea of vacuum, or space where
there is nothing visible or tangible’. [1] Here as elsewhere the
acceptability of his statement lies in its being taken in a sense
which according to his principles cannot properly belong to it. It
is one doctrine that the ideas of space and body are essentially
correlative, and quite another that the idea of space is equivalent
to a feeling of sight or touch. It is of the latter doctrine that
Hume’s denial of a vacuum is the corollary; but it is the former
that gains acceptance for this denial in the mind of his reader.
Space we have already spoken of as the relation of externality. If,
abstracting this relation from the world of which it is the uniform
but most elementary determination, we regard it as a relation between
objects having no other determination, these become spaces and
nothing but spaces--space pure and simple, _vacuum_. But we have
known the world in confused fulness before we detach its constituent
relations in the clearness of unreal abstraction. We have known
bodies συγκεχυμένος [2], before we think their limits apart and out
of these construct a world of pure space. It is thus in a sense true
that in the development of our consciousness an idea of body precedes
that of space, though the _abstraction_ of space--the detachment of
the relation so-called from the real complex of relations--precedes
that of body; and it is this fact that, in the face of geometry,
strengthens common sense in its position that an idea of vacuum is
impossible. It is not, however, the inseparability of space from body
whether in reality or for our consciousness, but its identity with a
certain sort of feeling, that is implied in Hume’s exclusion of the
idea of vacuum. ‘Body,’ as other than feeling, is with him as much a
fiction as vacuum. That there can be no idea of vacuum, is thus in
fact merely his negative way of putting that proposition of which the
positive form is, that space is a compound impression of sight and
touch. Having examined that proposition in the positive, we need not
examine it again in the negative form. It will be more to the purpose
to enquire whether the ‘tendency to suppose’ or ‘propensity to feign’
by which, in the absence of any such idea, our language about ‘pure
space’ has to be accounted for, does not according to Hume’s own
showing presuppose such an idea.
[1] P. 358. [Book I, part II., sec. V.]
[2] [Greek συγκεχυμένος (synkechymenos) = confused or jumbled-up. Tr.]
How it is that we talk as if we had idea of vacuum according to Hume.
279. By vacuum he understands invisible and intangible extension.
If an idea of vacuum, then, is possible at all, he argues, it must
be possible for darkness and mere motion to convey it. That they
cannot do so _alone_ is clear from the consideration that darkness
is ‘no positive idea’ and that an ‘invariable motion,’ such as that
of a ‘man supported in the air and softly conveyed along by some
invisible power,’ gives no idea at all. Neither can they do so when
‘attended with visible and tangible objects.’ ‘When two bodies
present themselves where there was formerly an entire darkness, the
only change that is discoverable is in the appearance of these two
objects: all the rest continues to be, as before, a perfect negation
of light and of every coloured or tangible object’. [1] ‘Such dark
and indistinguishable distance between two bodies can never produce
the idea of extension,’ any more than blindness can. Neither can a
like ‘imaginary distance between tangible and solid bodies.’ ‘Suppose
two cases, viz. that of a man supported in the air, and moving his
limbs to and fro without meeting anything tangible; and that of a
man who, feeling something tangible, leaves it, and after a motion
of which he is sensible perceives another tangible object. Wherein
consists the difference between these two cases? No one will scruple
to affirm that it consists merely in the perceiving those objects,
and that the sensation which arises from the motion is in both cases
the same; and as that sensation is not capable of conveying to us an
idea of extension, when unaccompanied with some other perception,
it can no more give us that idea, when mixed with the impressions
of tangible objects, since that mixture produces no alteration upon
it’. [2] But though a ‘distance not filled with any coloured or
solid object’ cannot give us an idea of vacuum, it is the cause why
we falsely imagine that we can form such an idea. There are ‘three
relations’--_natural_ relations according to Hume’s phraseology
[3]--between it and that distance which really ‘conveys the idea
of extension.’ ‘The distant objects affect the senses in the same
manner, whether separated by the one distance or the other; the
former species of distance is found capable of receiving the latter;
and they both equally diminish the force of every quality. These
relations betwixt the two kinds of distance will afford us an easy
reason why the one has so often been taken for the other, and why we
imagine we have an idea of extension without the idea of any object
either of the sight or feeling’. [4]
[1] P. 362. [Book I, part II., sec. V.]
[2] P. 363. [Book I, part II., sec. V.]
[3] Above, § 206.
[4] P. 364. [Book I, part II., sec. V.]
His explanation implies that we have an idea virtually the same.
280. It appears then that we have an idea of ‘distance unfilled
with any coloured or solid object.’ To speak of this distance as
‘imaginary’ or fictitious can according to Hume’s principles make no
difference, so long as he admits, which he is obliged to do, that we
actually have an idea of it; for every idea, being derived from an
impression, is as much or as little imaginary as every other. And
not only have we such an idea, but Hume’s account of the ‘relations’
between it and the idea of extension implies that, _as ideas of
distance_, they do not differ at all. But the idea of ‘distance
unfilled with any coloured or solid object’ _is_ the idea of vacuum.
It follows that the idea of extension does not differ from that of
vacuum, except so far as it is other than the idea of distance. But
it is from the consideration of distance that Hume himself expressly
derives it; [1] and so derived, it can no more differ from distance
than an idea from a corresponding impression. Thus, after all, he has
to all intents and purposes to admit the idea of vacuum, but saves
appearances by refusing to call it extension--the sole reason for
such refusal being the supposition that every idea, and therefore the
idea of extension, must be a datum of sense, which the admission of
an idea of ‘invisible and intangible distance’ already contradicts.
[1] Part II. § 3, sub. inst.
By a like device that he is able to explain the appearance of our
having such ideas as Causation and Identity.