Kant, as is well known, demonstrates, in the Transcendental Aesthetic, that space and time are the pure forms of our external and internal sensible intuition, respectively. But we must not forget that this demonstration occurs as part of the “exposition” (Erörterung[1]Kant gives expositio as the Latin equivalent (KrV B38). Note, however, that it contains Ort (= locus) rather than Satz (= positio). A more proper calque would be Aussetzung, but, although Grimm says that that is equivalent to expositio in all its senses, Kant apparently already felt it wasn’t appropriate here (just as it would not be appropriate today).) of certain concepts, namely, the concepts of space and time. It is unclear, to begin with, what concepts these are. But Kant does say more elsewhere (A87/B119–20) about the concept of space: he calls it the “fundamental concept” (Grundbegriff ) of geometry, and further implies that the “exposition” carried out in the Aesthetic is equivalent to, or includes, what he later on calls a “deduction”: namely, a demonstration of the objective reality (reference to some possible object) of the concept in question.

My intention therefore was, and still is, to write a post explaining what this concept is, and how its exposition, or deduction, is related to conclusion that space is the form of our external sense. To understand what this concept is, however, it helps to begin with Locke, whose doctrine in this respect is very close to Kant’s. Eventually I realized that, leaving Kant aside, I had more to say about Locke than would fit, even, in one reasonably long post. So my plan is put up a series of posts about Locke’s theory of space, and then eventually get back to Kant.

According to Locke there is, first of all, a simple idea of space. And then, secondly, there are simple modes of that idea, among which simple modes are our ideas of various distances.[2]See Essay 2.13.2–4, to be quoted and discussed in detail in a future post. As a reminder: a simple mode, somewhat confusingly, is never a simple idea: rather, modes in general are a species of complex idea. A complex idea, generally speaking, is an idea formed by the operation of the mind that Locke calls “composition” (the Latin equivalent of Greek σύνθεσιςsunthesis), “whereby it puts together several of those simple ones it has received from Sensation and Reflection, and combines them into complex ones” (2.11.6).[3]It is evident from other places that Locke doesn’t intend to exclude decompound ideas formed by the composition of various ideas that are themselves already complex. Whether, according to him, there can also be simple modes of a single complex idea, I’m not sure. That is: a mode, as a complex idea, is an idea that attains its unity (the idea of unity that must accompany every idea: see 2.7.7) due to an act of our mind.[4]I will not discuss here the differences between modes and other complex ideas. Suffice it to say that modes are the easy case: the other kinds of complex ideas (ideas of substances, relations) are formed using additional operations beyond mere composition, and therefore raise additional difficulties. Modes in general are then divided into, on the one hand, “mixed” modes, which we form by composing different ideas one with another, and, on the other hand, simple modes, “which are only variations, or different combinations of the same simple Idea, without the mixture of any other” (2.12.5).

What it means to compose the same simple idea with itself can be gathered from Locke’s description of this kind of composition as “enlarging”:

Under this [operation] of Composition may be recokn’d also that of ENLARGING; wherein though the Composition does not so much appear as in more complex ones, yet it is nevertheless a putting several Ideas together, though of the same kind. Thus by adding several Unites together, we make the Idea of a Dozen; and by putt[i]ng together the repeated Ideas of several Perches, we frame that of a Furlong. (2.11.6)

Thus he slips in the somewhat surprising thesis that an idea gains a larger object — becomes, for example, the idea of a larger number or a larger space — by being a larger idea. The object of the idea of three is larger than the object of the idea of two because the idea of three contains the idea of a unit one more time than does the idea of two; the idea of a furlong is the idea of a longer space than is the idea of a perch because it is a longer idea: to be precise, forty time longer than the idea of a perch.[5]In England, in Locke’s time, a furlong was exactly forty perches, also known as “rods.” (A rod, or perch, was five and a half yards. Hence a furlong was, and in theory still is, 660 feet, or one eighth of a mile.)

Now, it is evident that a simple idea, not being divisible, cannot be larger than any other idea in the above sense. Hence it appears we should say that a simple idea is both the smallest possible idea of its kind (that is: smaller than any of its simple modes) and the idea of the object which is smallest in that respect: for example, the smallest (natural) number or, most significantly for us, the smallest possible space. Hume, in an argument he attributes to “Mons. Malesieu,” draws exactly this conclusion:

’Tis evident, that existence in itself belongs only to unity, and is never applicable to number, but on account of the unites, of which the number is compos’d. Twenty men may be said to exist; but ’tis only because one, two, three, four, &c. are existent; and if you deny the existence of the latter, that of the former falls of course. ’Tis therefore utterly absurd to suppose any number to exist, and yet deny the existence of unites; and as extension is always a number, according to the common sentiment of metaphysicians, and never resolves itself into any unite or indivisible quantity, it follows, that extension can never at all exist. ’Tis in vain to reply, that any determinate quantity of extension is an unite; but such-a-one as admits of an infinite number of fractions, and is inexhaustible in its sub-divisions. For by the same rule these twenty men may be consider’d as an unite. The whole globe of the earth, nay the whole universe may be consider’d as an unite. That term of unity is merely a fictitious denomination, which the mind may apply to any quantity of objects it collects together; nor can such an unity any more exist alone than number can, as being in reality a true number. But the unity, which can exist alone, and whose existence is necessary to that of all number, is of another kind, and must be perfectly indivisible, and incapable of being resolved into any lesser unity. (Treatise 1.2.2.3)[6]See, similarly, Leibniz to Arnauld, 30 April, 1687, G2:96–7. See also Nouveaux essais 2.12.7.

But Locke cannot agree to this: he holds both that all bodies are divisible (2.17.12), and that every body occupies a space (2.4.2), from which it follows, on fairly reasonable assumptions,[7]We need to assume, at least, that the whole is greater than the part, as well as Archimedes’ axiom. that no space can be the smallest possible. Hence the simple idea of space, whatever it is, cannot be the idea of the smallest possible space.

The difference between Locke and Hume here actually goes pretty deep. Locke agrees, of course, that what is simple is indivisible. But he does not agree that the simple and therefore indivisible is the site of “true unity,” and that all other unity is therefore “fictitious.” On the contrary: what is simple and therefore indivisible is always merely an idea, according to Locke, whereas unity in the object of an idea — that is, the unity of any unit — is always a unity of composition (i.e., synthesis). It is important to remember that Locke is an empiricist in a full sense in which Hume is not (but Kant is!), namely: that he thinks[8]There is some distortion in reporting this as a disagreement in which Locke thinks one thing, while Hume thinks the contrary. Hume actually fails to find any rational, coherent, and stably credible view on the subject. Hume, or at least the fictionalized “Hume” character of Treatise 1.4.7, when in a sanguine, philosophical temper, “thinks” (more or less) what Locke does about this; but Hume the narrator has just finished showing that this opinion will not hold up to full rational scrutiny. all our ideas arise due to the effect on us of some (external or internal) thing, that thing being thereby their object, that is, that to which they refer. But the effect of such a thing on us is never a single simple idea. This is, in part, because, even if the object were to contain a simple, indivisible power, that power would affect us in different ways, yielding more than one simple idea. Our body is not simple, but divisible, and our sense organs, in particular, are parts designed to be discriminate between the powers of other bodies in very specific respects:

Though the Qualities that affect our Senses, are, in the things themselves, so united and blended, that there is no separation, no distance between them; yet ’tis plain, the Ideas they produce in the Mind, enter by the Senses simple and unmixed. (2.2.1)

More importantly for our present purposes, it is also because we can be affected only by some thing that exists, “for, who is it that sees not that Powers belong only to Agents, and are Attributes only of Substances?” (2.21.16). But “all Things, that exist, [are] Particulars” (3.3.1); “the principium Individuationis … is Existence it self” (2.27.3), and so that thing whose power it is to cause some simple idea in us must have some additional characteristic which makes it different from anything else of its kind, i.e., from any other possible subject of an exactly similar power. This characteristic, according to Locke (and according to many others), is its being in a certain place at a certain time: existence individuates in that it “determines a Being of any sort to a particular time and place incommunicable to two Beings of the same kind” (ibid.). Simple ideas, when we get them clear and “precise,” and consider them “as they are in the Mind,” are always abstract, that is, general ideas, “separate from all other Existences, and the circumstances of real Existence, as Time, Place, or any other concomitant Ideas” (2.11.9). Hence, on the other hand, our idea of the thing that affects us, if we consider it precisely as it is in the mind, will always be complex, which is to say, its unity always depends on us:

As simple Ideas are observed to exist in several Combinations united together; so the Mind has a power to consider several of them united together. . . .  Ideas thus made up of several simple ones put together, I call Complex … which though complicated of various simple Ideas, or complex Ideas made up of simple ones, yet are, when the Mind pleases, considered each by it self, as one entire thing. (2.12.1)

With respect, at least, to the first transcendental predicate, namely unum, Locke is thus, already, a transcendental idealist.

So the correct analogy between number and space, for Locke, is as follows. The idea of unity, for him, in not Hume’s “true unity,” i.e. the unity that must be attributed to some thing thanks to its indivisibility, but is rather “fictitious,” or rather ideal, unity, i.e., the unity, and hence indivisibility, that a thing has insofar as we are pleased to consider it as one. And so, too, the simple idea of space, for Locke, is not the idea of some part of space which, thanks to its indivisibility, we cannot but take as simple, but rather the idea of the simplicity, and hence indivisibility that space has just insofar as we consider it in abstraction from all place, which is to say — given the way Locke defines place (2.13.7) — in abstraction from the size and situation of the bodies in it. What Locke takes to be indivisible is not the smallest space, but rather space as such. And this is why, although he holds that body, which occupies space, is infinitely divisible, he says that space as such is not divisible at all (see 2.13.13, quoted and further discussed below). Or, to put it differently: the simple idea of space is not the idea of a maximally small, and therefore indivisible, space because it isn’t the idea of a space, at all, large or small. The simple idea of space is the idea of space as such; every idea of a space — whether one, two, or three dimensional — is a (simple) mode, and hence a complex idea.

Beginning with any one such simple mode, for example that of a foot or a rod, we can form many other simple modes by enlargement. In fact, Locke even adds — though this raises certain problems, to be mentioned at the end of this post, and discussed further in a future post — that we can form, not only larger modes of the same kind, but also modes of different shapes (and higher dimensionality):

For the Mind having a Power to repeat the Idea of any Length directly stretched out, and join it to another in the same Direction, which is to double the Length of that straight Line, or else join it to another with what Inclination it thinks fit, and so make what sort of Angle it pleases … it can make an Angle of any Bigness: So also the Lines that are its Sides, of what Length it pleases, which joining again to other Lines of different Lengths, and at different Angles, ’till it has wholly inclosed any Space, it is evident, that it can multiply Figures both in their Shape, and Capacity, in infinitum; all which are but so many different simple Modes of Space. (2.13.6)[9]The part I have elided discusses the mind’s power of making shorter modes of length by division, but I will also return to that below.

The obvious question is, however: where do we get that initial simple mode whose combination with itself yields all these lengths, figures, etc.? Hume is able to answer that we do not start with any simple mode at all, but rather with the simple idea of the smallest possible space (or rather, of the smallest possible colored thing). But what can Locke say about this?

The answer to this question must lie in the nature of the idea which is added to the simple idea of space in order to form the idea of a space, namely, the idea of limit, or, in other words, of finitude. Its nature must be rather unusual, since adding a single simple idea[10]I’m not aware that Locke says anywhere explicitly that the idea of limit is simple. However, his explanation of how we come by it (to be quoted directly below) shows that he is classifying it as such. to another one would normally yield only a single mixed mode, whereas in this case we seem to get an infinitude of simple modes.

So, first of all: if the idea of a finite space results from a combination of the simple idea of space with the simple idea of limit, why is it not, by definition, a mixed mode? But now we might as well ask this about the idea of unity. I do not obtain the idea of a furlong, for example, merely by perceiving the idea of a rod forty times in a row: rather, like any complex idea, I form it only insofar as my mind is pleased to consider these forty rods “as one entire thing” (see again 2.12.1). In other words: the idea of a furlong, in a certain sense at least, contains not only the idea of a rod (repeated), but also the simple idea of unity. And the idea of a furlong, similarly, also involves the idea of limit: forty rods, taken together, are not a furlong unless what is thus taken together is only forty rods, no more and no less. This calls attention to a distinction between simple and complex ideas which is more than merely quantitative. A simple idea is the product of complete abstraction, in which everything that has occurred together in the effect of some power — even including transcendental predicates like unity, being, power, and limit — is stripped away to leave something absolutely precise and “naked.” A complex idea results when the mind, as it pleases, then replaces that old, passively received co-occurrence, now dissolved by abstraction, with a new, voluntary unity. Hence a complex idea is never purely abstract, in the way a simple idea must be. And this applies, in particular, to simple modes: they can be “variations, or different combinations” of one simple idea only insofar as they already involve such transcendental ideas as unity and limit.

Now, this also, in a way, answers the further question, why combining the idea of space with the idea of limit doesn’t yield a single complex idea, the idea of finite space in general, but rather an infinitude of simple modes. The addition of ideas like unity and limit to a simple idea is, in itself, nothing but the process of making from it any complex idea whatsoever: it never suffices to individuate a single complex idea. Still, the problem can be put in a different way, which requires a much more complicated answer: given, namely, that a finite space is never barely a finite space, but always one particular finite space out of many possible — in particular, always in one particular position (and orientation), and always of one particular size and shape — what is the source of this specific manifoldness? It can’t be a matter of adding various other ideas onto the original composition of space and limit. Finite space, that is, can’t simply be a genus from which the many possible finite spaces are derived by adding various differentiae: in that case, the ideas of specific finite spaces would be mixed modes. So, somehow, this one simple idea of space, when combined with the simple idea of limit, must yield a very specific series of possible limitations.

To see how this is possible, note, to begin with, how Locke says we get the idea of a limit of space — of course, like all simple ideas, from our senses:

As for the Idea of Finite, there is no great Difficulty. The obvious Portions of Extension, that affect our Senses, carry with them into the Mind the Idea of Finite. (2.17.2)

Now, these “obvious portions of extension” cannot, simply as such, affect our senses, since an empty space, lacking solidity and hence (2.4.5) any power of resistance, impulse, or protrusion, cannot affect anything. But if this idea of limit could, in principle, be received only from the sensation of a space filled with body, then it would best be described as the idea of finite body, rather than as the idea of finite space — or, to put it differently: we would be talking, if anything, about simple modes of solidity. So the obvious portions of extension must be capable of affecting us by means of a body which, though it may fill them now, does not necessarily fill them. Sure enough, this is how Locke explains that we get the idea of a finite pure space:

For a Man may conceive two Bodies at a distance, so as they may approach one another, without touching or displacing any solid thing, till their Superficies come to meet: Whereby, I think, we have the clear Idea of Space without Solidity. For (not to go so far as annihilation of any particular Body) I ask, whether a Man cannot have the Idea of the motion of one single Body alone, without any other succeeding immediately into its Place? I think, ’tis evident he can: The Idea of Motion in one Body, no more including the Idea of Motion in another, than the Idea of square Figure in one Body includes the Idea of a square Figure in another. . . .  When the Sucker in a Pump is drawn, the space it filled in the Tube is certainly the same, whether any other Body follows the motion of the Sucker or no: Nor does it imply a contradiction, that upon the motion of one Body, another, that is only contiguous to it, should not follow it. (2.4.3)

But the motions in question, by which surfaces of bodies may unite or come apart, are exactly those that are involved in Locke’s analysis of divisibility — of the divisibility, that is, that all body has, and that pure space lacks:

To divide and separate actually, is, as I think, by removing the Parts one from another, to make two Superficies, where before there was a Continuity: And to divide mentally, is to make in the Mind two Superficies, where before there was a Continuity, and consider them as removed one from the other; … But neither of these ways of Separation, whether real or mental, is, as I think, compatible to pure Space. (2.13.13)

Hence we may say: the obvious portions of extension can affect us with the idea of a finite empty space because of the divisibility of body.

This perspective helps immediately to understand the quantitative manifoldness inherent in the idea of a finite space. When we contemplate space (the simple idea) together with limit, we are contemplating the idea of a space that might be vacated as some piece of some body comes off another. The diversity of finite spaces is not a diversity of species under the genus, finite space, but, rather, a diversity in the ways bodies can divide. The infinite divisibility of body means, in particular, that the space vacated could always be larger or smaller. That is to say: the simple modes finite space, can always be made to represent a larger or smaller obvious portion of extension than the one by which we happen to have received it. There remains a problem, on the other hand, about the qualitative diversity of the modes of space, that is, their diversity in figure. Recall Locke’s explanation: in joining one line to another to make a figure, the mind “can make an Angle of any Bigness: So also the Lines that are its Sides, of what Length it pleases, which joining again to other Lines of different Lengths, and at different Angles, ’till it has wholly inclosed any Space, etc.” (again, 2.13.6). But if this were taken to mean that the mind can make a closed figure by joining any number of lines of any length at any arbitrary angles, it would, of course, be quite false. On the contrary, if a closed figure, suitable to be the boundary of a flat surface, is to result, the number and lengths of the sides and bigness of the angles must be carefully chosen. Locke’s own favorite geometrical proposition, namely the so-called triangle postulate (that the interior angles of a triangle are equal to two right angles), is a perfect example of this, but there are many others: infinitely many others, perhaps, depending on how you count, but at least one fundamental and independent other, namely the exterior angle theorem[11]The exterior angle theorem follows from the triangle postulate (since it follows from the triangle postulate that the exterior angle is equal to the sum of the two opposite angles), but it remains true in hyperbolic space, even when the triangle postulate fails. On the other hand, both the exterior angle theorem and the triangle postulate are false in elliptic or spherical space. — which, as we will see in a future post, Locke also introduces as an example in one key place. Even given the connection to body and its divisibility, it may seem hard to explain how all of these constraints can follow from a few simple ideas. Stay tuned!