This past spring I engaged in an independent study course on Walden with our excellent undergraduate student, Laurel Mentor (since graduated, and possibly applying to graduate programs this year). This post is based on our conversation during our final meeting, about the 18th chapter (“Conclusion”). I post it here with her permission. Given my generally blabbermouthed nature, I likely contributed more of the conversation by volume, but Laurel’s part was decisive. For one thing, she brought up the problem that started everything off, about the following passage:

I fear chiefly lest my expression may not be extra- vagant enough, may not wander far enough beyond the narrow limits of my daily experience, so as to be adequate to the truth of which I have been convinced. . . .  I desire to speak somewhere without bounds; like a man in a waking moment, to men in their waking moments; for I am convinced that I cannot exaggerate enough even to lay the foundation of a true expression. (18.6)

Isn’t this an odd thing for Thoreau to say? It is indeed, though it had never occurred to me before. It is odd, as Laurel pointed out, in view of, for example, this:

In most books, the I, or first person, is omitted; in this it will be retained; that, in respect to egotism, is the main difference. We commonly do not remember that it is, after all, always the first person that is speaking. I should not talk so much about myself if there were any body else whom I knew as well. Unfortunately, I am confined to this theme by the narrowness of my experience. (1.2)

The narrow limits of his daily experience, in other words, are exactly what Thoreau says, to begin with, he will remain within. Moreover: “The day is an epitome of the year” (17.2). The everyday, the diurnal,[1]With this word, “diurnal,” I allude to Stanley Cavell (although the term does not occur in The Senses of Walden — which, incidentally, I have not re-read at all recently). I don’t know how, beyond that, to give him specific credit for what follows — which would amount, really, to giving myself credit, no doubt falsely, for having understood him. Many of the passages I discuss (especially those that were brought up by me, rather than by Laurel) are ones I remember him reading with special emphasis, either in person or in print. And yet I doubt he would approve of the overall drift of interpretation here. of Thoreau’s life in the woods contains the whole of his experience there. How could he desire to speak without those bounds?

Considering the above quote from ch. 1 (“Economy”), we asked ourselves: well, of which books is it true that, in most of them, the I or first person is omitted? This is actually something of a puzzle. Emerson’s “Self-Reliance” begins “I read the other day.” But, lest we think that this merely shows the similarity between Thoreau and his friends, Thomas Brown’s Lectures on the Philosophy of the Human Mind begins:

Gentlemen,

The subject on which we are about to enter, and which is to engage, I trust, a considerable portion of your attention for many months, is the Philosophy of the Human Mind. (vol. 1 [Boston, 1826], p. 2)

But maybe that is different from “most books” because it is a collection of lectures. Then how about Brown’s Inquiry into the Relation of Cause and Effect? There the I is, indeed, somewhat delayed, but, on the second page of the introduction, we find:

To remove, in some degree, this darkness, is the object of the following pages; in which I shall endeavour, in the first place, to fix, what it is which truly constitutes the relation of cause and effect; — in the second place, etc. ([Andover, 1822] p. 14)

Locke’s Essay concerning Human Understanding begins (not counting, as perhaps we really should, either the Epistle Dedicatory or the Epistle to the Reader[2]The Epistle to the Reader begins: “Reader, I Here put into thy Hands, what has been the Diversion of some of my idle and heavy Hours: If it has the good Luc[k] to prove so of any of thine, and thou hast but half so much Pleasure in reading, as I had in writing it, thou wilt as little think thy Money, as I do my Pains, ill bestowed.” Which could just as well have been written by Thoreau.):

Since it is the Understanding that sets Man above the rest of sensible Beings, and gives him all the Advantage and Dominion, which he has over them; it is certainly a Subject, even for its Nobleness, worth our Labour to enquire into. The Understanding, like the Eye, whilst it makes us see, and perceive all other Things, takes no notice of it self: And it requires Art and Pains to set it at a Distance, and make it its own Object. But whatever be the Difficulties, that lye in the way of this Enquiry; … sure I am, that all the Light we can let in upon our own Minds … will not only be very Pleasant, but bring us great Advantage. (1.1.1; the second emphasis is mine)

And we all know what line Descartes is famous for, and of what it is he claims, first of all, to be sure. So what are these books, in most of which the I is omitted? To what genre do they belong?

The answer that occurred to me is that Thoreau might be thinking of scientific literature, although I feared that might be anachronistic. And so it proves to be. Picking up, more or less at random, vol. 130 (1840) of the Transactions of the Royal Philosophical Society, I find an article by John Herschel, “On the Chemical Action of the Rays of the Solar Spectrum on Preparations of Silver and Other Substances, both Metallic and Non-metallic, and on some Photographic Processes” (pp. 1–51) in which the first paragraph ends:

The facts themselves, in the present state of our knowledge, will, I believe, be found by no means devoid of interest, and may lead, in the hands of others more favourably situated for such researches, and, I may add, in a better climate than ours [lol], to inquiries of the utmost interest. (p. 1)

After that, an article by Michael Faraday, “Experimental Researches in Electricity — Sixteenth Series” (pp. 61–91), in which the third paragraph reads (in part):

Examining this question by the results of definite electro-chemical action, I felt constrained to take part with those who believed the origin of voltaic power to consist in chemical action alone. . . .  I wished not merely to escape from error, but was anxious to convince myself of the truth of the contact theory; for it was evident that if contact electromotive force had any existence, it must be a power not merely unlike every other natural power as to the phenomena it could produce, but also in the far higher points of limitation, definite force, and finite production. (p. 62)

Looking ahead to the 1880’s, I did start to see papers that really lacked the first person, but of course Thoreau couldn’t be alluding to them. And even (or again?) today, the first person, in an admittedly very impersonal use, will turn up in papers of this kind (“we present here the results …”).

All of this I determined after the fact. At the time, once the suggestion was on the table that “most books” might refer to scientific literature, Laurel recalled her training in how to write lab reports: they were indeed told to avoid the first person and phrase things impersonally, using the passive voice. No one typically gives any reason for this rule. But we both agreed that it must be a kind of inverse of the rule by which manuals of style tell us to avoid the passive like the plague, because (as they say) the active construction is more concrete and vivid, and in that sense more interesting. A lab report, then, or more generally the report of an experiment, is not supposed to be interesting: at least, not in that sense (for there is more than one sense).[3]Some more definite explanation is due here as to the relevant sense, and as to the reasons we want to avoid it in lab reports. In correspondence, Laurel has quoted the following from Thomas Nagel’s Mind and Cosmos (Oxford 2012, ISBN 978-0199919758): “It was essential [following the scientific revolution of the 17th century] to leave out or subtract subjective appearances and the human mind — as well as human intentions and purposes — from the physical world in order to permit this powerful but austere spatiotemporal conception of objective physical reality to develop” (p. 36, if Amazon’s “Search Inside this Book” function does not deceive me). I haven’t read this book, and I am not a huge fan of Thomas Nagel in general, but no doubt something like this is correct. Only, to say it really correctly, even according to me, let alone according to Thoreau, would require some delicate work on the senses and history of almost every word in that quote (at least: conception, essential, human, intention, mind, objective, physical, purpose, reality, spatiotemporal, subjective, world; but maybe also austere, develop, leave, permit, powerful, subtract). Going back to the Faraday paper, moreover, we find, after a very long, vivid, concrete, interesting introduction, which discusses, not only the recent literature on the topic, but also, as we have seen, Faraday’s own motivations, hopes, and anxieties, he finally gets down to his experiments in a section labeled §24 i., “Exciting electrolytes, &c., being conductors of thermo and feeble currents,” which begins as follows:

Sulphuret of potassium. — This substance and its solution were prepared as follows. Equal weights of caustic potash (potassa fusa) and sulphur were mixed and heated gradually in a Florence flask, till the whole had fuzed and united, and the sulphur in excess began to sublime. It was then cooled and dissolved in water, so as to form a strong solution, which by standing became quite clear. (p. 66)

All in the passive voice, as you can see. If there were a genre of books that were just extended lab reports, containing the procedure and results of experiments, it is reasonable to suppose that, even in the 1840’s, the I or first person would be omitted in most of them.[4]This would not continue to hold up if we looked still further back in the Transactions, say to the 1660’s. I don’t know when the transition occurs.

As an answer to the question about “most books,” this is not very satisfying. For there is no such genre. I conclude that I don’t really understand what Thoreau is talking about here. Nevertheless, there is something to this answer. When our conversation took another tack, which I will come to in a moment, I ended up searching for this quote, which I found in ch. 5 (“Solitude”):

With thinking we may be beside ourselves in a sane sense. By a conscious effort of the mind we can stand aloof from actions and their consequences; and all things, good and bad, go by us like a torrent. We are not wholly involved in Nature. I may be either the driftwood in the stream, or Indra in the sky looking down on it. (5.11)

But right above that I saw a short paragraph beginning with this sentence: “We are the subjects of an experiment which is not a little interesting to me” (5.10). Thoreau has plenty of fun with the words “experiment” and “experience,” which in some sense ought to mean the same thing (and did mean the same thing at some point), so that “the subject of an experiment” is, no doubt, supposed to be readable as a piece of German idealist technicality. But this works both ways, so that “the limits of my daily experience” or “the narrowness of my experience” might equally make us think of lab reports. And there are other passages that make this clearer. There is the beginning, so to speak, of the Results section:

I learned this, at least, by my experiment: that if one advances confidently in the direction of his dreams, and endeavors to live the life which he has imagined, he will meet with a success unexpected in common hours. (18.5)

and there is the end, so to speak, of the Introduction section: “But to make haste to my own experiment,” followed by the beginning of the Procedure section:

Near the end of March, 1845, I borrowed an axe and went down to the woods by Walden Pond, nearest to where I intended to build my house, and began to cut down some tall, arrowy white pines, still in their youth, for timber. (1.59–60)

In “most books,” this would read: “A house was prepared as follows. An axe was borrowed … Some tall, arrowy white pines, still in their youth, were cut down …”. Only, the “narrowness” of the experiment here carried out has made it interesting, to Thoreau, at least, in the sense in which a lab report is not supposed to be interesting. We are the subjects (and the objects) of an experiment of great interest.

Perhaps this all seems a digression from the original line of questioning. It is that, in part: any conversation I am involved with is sure to contain digressions. But there was, at least, a good reason for me to look up that passage in “Solitude.” In the continuation there, Thoreau says:

I only know myself as a human entity; the scene, so to speak, of thoughts and affections; and am sensible of a certain doubleness by which I can stand as remote from myself as from another. However intense my experience, I am conscious of the presence and criticism of a part of me, which, as it were, is not a part of me, but spectator, sharing no experience, but taking note of it; and that is no more I than it is you. When the play, it may be the tragedy, of life is over, the spectator goes his way. It was a kind of fiction, a work of the imagination only, so far as he was concerned. This doubleness may easily make us poor neighbors and friends sometimes. (5.11)

This doubling, note, is the same as the doubling or splitting of the ego that has crossed the Atlantic with Schelling and Coleridge,[5]When Thoreau says, “I had withdrawn so far within the great ocean of solitude, into which the rivers of society empty, that for the most part, so far as my needs were concerned, only the finest sediment was deposited around me. Beside, there were wafted to me evidences of unexplored and uncultivated continents on the other side” (6.7), the reference in context is to his visit from that “true Homeric or Paphlagonian man,” the Canadian woodchopper Alek Therien. But Thoreau is no doubt thinking also of the literal oceans: elsewhere, he calls this ocean of solitude “the Atlantic and Pacific Ocean of one’s being alone” (18.2). which I previously discussed here. In transcendental reflection, the I or first person, that is, the ego (das Ich), the subject of our experiment (das Subjekt der Erfahrung), splits into an infinite, subjective component — an ego that speaks somewhere without bounds — and a finite, objective one, confined to narrowness. Thus (as is signified also by Emerson’s phrase, “the other day”), the diurnal, or (via epitome) the annual, is doubled:

The present was my next experiment of this kind, which I purpose to describe more at length; for convenience, putting the experience of two years into one. As I have said, I do not propose to write an ode to dejection, but to brag as lustily as chanticleer in the morning, standing on his roost, if only to wake my neighbors up. (2.7)

What this says about Coleridge (who literally did write an ode to dejection), I don’t know. I can say that to get up and crow lustily early in the morning (“earlier and earlier every successive day” [4.22]) would definitely make us poor neighbors. Or we might apply the phrase another way: think of our poor neighbors! But, whoever these neighbors are, and whether or not they appreciate the wake-up call, the method of awakening should be familiar from Schelling: the two years, or days — that is, the two egos — are put together (synthesized) “for convenience,” where I take convenientia, in this case, to translate συμβολήsumbolē. Thoreau’s “Good Genius” (10.6[6]This is a passage that Laurel and I discussed at length in a previous meeting. She brought this good genius together with the “evil genius” who, as it were, directs Thoreau’s axe (that same axe that he began by borrowing?) to a hole in the ice (9.6), which naturally led us to discuss the end of the First Meditation. But I digress — or do I? (as Cavell would add). ) has created the symbol in which his neighbors can (as Emerson put it) “read the other day”:

I do not say that John or Jonathan [i.e., the two shores of Thoreau’s Atlantic] will realize all this; but such is the character of that morrow which mere lapse of time can never make to dawn. The light which puts out our eyes is darkness to us. Only that day dawns to which we are awake. (18.19)

The light of that other day “puts our eyes out” in the way a snail, for example, puts out eyes: that is, our eyes extend from it, it sees through our eyes. “This seeing light, this enlightening eye, is Reflection” (Coleridge, Aids to Reflection [ed. Marsh, 1829], Introductory Aphorism IX, p. 3).[7]But Locke already says this: see again the quote above from the beginning of the Essay.

At this point Laurel’s original question has, from one point of view, been answered: we understand how Thoreau, putting the experiment of two years into one, can speak both with and without bounds. But the speaking, or crowing — at what poor neighbor can this all be directed? Just to remind you, we started with: “I desire to speak somewhere without bounds; like a man in a waking moment, to men in their waking moments” (18.6). So the audience — at least, the desired audience — consists of those same poor neighbors Thoreau hopes (desires) to wake up. “Desire” is so important a term to Cavell that I wish I could say more about it here. (I note only the absence of women, in their waking moments or otherwise, among the objects of Thoreau’s desire: an absence that has been noticed even before our own sensitive age.) Another path we could follow from here (also found in Cavell, if I recall correctly) is to note that, by the doubling of the experiment, Thoreau becomes his own poor neighbor (“Next to us is not the workman whom we have hired, with whom we love so well to talk, but the workman whose work we are” [5.9]; “Some show their kindness to the poor by employing them in their kitchens. Would they not be kinder if they employed themselves there?” [1.103]; “I never knew, and never shall know, a worse man than myself” [1.105]). But what Laurel and I ended up talking about was more the nature of the audience desired that the nature of that desire itself, and more the audience that includes us than the audience that consists only of Thoreau.

As it happens, both the “narrowness of my experience” passage and the “without bounds” passage are in neighborhoods where Thoreau takes up this theme of audience. Soon after the former we find this:

Perhaps these pages are more particularly addressed to poor students. As for the rest of my readers, they will accept such portions as apply to them. I trust that none will stretch the seams in putting on the coat, for it may do good service to him whom it fits. (1.2)

whereas the latter comes immediately after this:

It is a ridiculous demand which England and America make, that you shall speak so that they can understand you. Neither men nor toad-stools grow so. As if that were important, and there were not enough to understand you without them. As if Nature could support but one order of understandings, could not sustain birds as well as quadrupeds, flying as well as creeping things, and hush and who, which Bright can understand, were the best English. As if there were safety in stupidity alone. (18.6)

What he means about toadstools, I frankly have no idea. But, that aside: the passages agree, at least, that this book might be read differently by those with different “orders of understanding,” although the second one suggests that those to whom it is not “more particularly addressed,” however much they can “accept such portions as apply to them,” will not understand it at all. This impression is strengthened, but also made more complicated, by what Thoreau says in the following paragraph:

“They pretend,” as I hear, “that the verses of Kabir have four different senses; illusion, spirit, intellect, and the exoteric doctrine of the Vedas”; but in this part of the world it is considered a ground for complaint if a man’s writings admit of more than one interpretation. (18.7[8]The quote, as I learn from my Norton Critical Edition (ed. W. Rossi [2nd ed., 1992], ISBN 0-393-95905-8), is from J. Garcin de Tassy, Histoire de la littérature hindoui et hindoustani (1839), p. 279: “On prétend qué les vers de Kabîr ont quatre sens différents : l’illusion (mâyá), l’esprit (âtma), l’intellect (man), et la doctrine exotérique des Védas.” Garcin de Tassy, in turn, cites H.H. Wilson, “A Sketch of the Religious Sects of the Hindus,” Asiatic Researches 16 (1828):62: “It may be sufficient to observe, that the doctrines of Kabír, are said to be conveyed in four-fold language, or that of Máyá, Atmá, Man or intellect, and the Védas.” Wilson does not cite any source. Note, for whatever it’s worth, that the word “exoteric” is due to Garcin de Tassy.)

This complicates things, first of all, because it introduces four “orders,” rather than just two (flying and creeping). But it also raises questions about the direction of ordering. This actually is a place where I remember distinctly what Cavell says, and can even find it in writing:

This is characteristic in its orientalizing of the mundane. There is just one text in the culture for which he writes that is known to require interpretation on four distinct levels. (The Senses of Walden [University of Chicago Press, 1972, ISBN 0-226-09813-3], p. 15)

I’m not sure whether he is thinking of the Christian doctrine or the Jewish one or both, nor how he can be so sure which culture Thoreau is writing for (after all, Thoreau doesn’t take scriptures generally to be written only “for” the cultures in which they were composed — the older scriptures probably contain, according to him, “words addressed to our condition exactly” [3.11]). Be that as it may, the hierarchy of senses attributed to Kabīr, especially in Garcin de Tassy’s version, is ambiguous in a way those hierarchies of biblical exegesis are not. Apparently “illusion” (māyā) is the lowest level, which sounds like a much less stable foundation than the either the littera or the פשטpeshaṭ. But if illusion is the lowest, shouldn’t it, so to speak, be the outermost, the least hidden, most accessible? Why then is the fourth order called “exoteric”?

Laurel and I confronted here a serious problem, and one than must have been felt by anyone who studies Walden intensively, to the point where it starts to look like the most difficult work of philosophy ever written, far more difficult than, say, Schelling’s System des transzendentalen Idealismus. Thoreau has provided something for readers such as us to find. You can, admittedly, always doubt whether some particular, twisted trail we’ve followed through the words was intentionally made or not, but on the whole it’s impossible not to think that Thoreau has been there before us. “The future inhabitants of this region, wherever they may place their houses, may be sure that they have been anticipated” (2.1). And yet, wherever he may have gone, we don’t know that he especially wants to be followed. A passage Laurel pointed out, from slightly earlier in the Conclusion:

It is remarkable how easily and insensibly we fall into a particular route, and make a beaten track for ourselves. I had not lived there a week before my feet wore a path from my door to the pond-side; and though it is five or six years since I trod it, it is still quite distinct. It is true, I fear, that others may have fallen into it, and so helped to keep it open. The surface of the earth is soft and impressible by the feet of men; and so with the paths which the mind travels. How worn and dusty, then, must be the highways of the world, how deep the ruts of tradition and conformity! (18.4)

The book is particularly addressed to poor students, and the neighbors for whose sake the experiment and the crowing of its results are of most interest are poor neighbors (“Farmers are respectable and interesting to me in proportion as they are poor — poor farmers” [17.29]). Does that mean that the outermost is the highest? Or that the lowest is innermost? They say, as I hear, that, according to Kabīr, God is Antar, the Inner, and Māyā, Illusion, is his daughter and his bride.

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!