This is the third in a series of posts about Locke’s theory of space.

Having raised, in the first post, a question about the qualitative constraints on the modes of space, I pointed out, in the second post, that such constraints can in principle be derived from constraints on the possible behavior of bodies in motion. In particular, I examined one of Thābit ibn Qurra’s proofs of the parallel postulate — a proof based on the principle that, when a solid body moves in a single direction, every point in the body moves along a straight line. Here I intend to show that Locke considers the principle in question (which I called “Thābit’s axiom”) to be an intuitively certain truth about bodies: considers it, that is, to be a fundamental truth of rational physics. I will also make a start at showing how Locke can use this principle to derive geometrical results.

This principle: but I ought to say, rather, instances of this principle. Highly abstract universal axioms such as this — in Locke’s terminology, “maxims” — don’t, according to him, themselves play any useful role in our demonstrations. To see why, we need to remember (1) what Locke thinks is the end of demonstration and (2) when and how he thinks demonstration is the right means to that end.

As to (1), the answer is deceptively obvious: the end of demonstration is knowledge. This seems so unobjectionable as to be not worth saying, until you realize just how seriously it is meant. Locke himself (like Spinoza) often introduces such deceptively obvious propositions — the most pertinent example here would be, that knowledge is conversant about ideas (4.1.1) — and he relies implicitly on this one, as well. Such propositions may even be “trifling”: that every demonstration aims to produce knowledge is probably supposed to follow from the definition Locke would give of the word “demonstration.” When we realize, to our surprise, what follows from taking them seriously, they are serving the proper purpose of trifling propositions, namely “to shew the Disingenuity of one, who will go from the Definitions of his own Terms, by reminding him sometimes of it” (4.8.5). Our surprise exposes our covert disingenuity: the definition we were prepared to accept was out of line with the consequences we were prepared to draw. It is always possible, indeed, that our error was in accepting the definition, and this technique, whether in Locke’s hands or in others’ — for example, in Socrates’ — may even have the aim of making us reject what seemed trifling as after all false. With some hesitation, however, I judge that in this case, and in many others, Locke’s aim is more direct. We are supposed to accept, or rather to realize that we have already accepted, the surprising consequences, which here include: that a demonstration is useless to anyone who already knows the conclusion, that is, perceives with certainty that the conclusion is true. This is why Locke thinks it clear that, if there are propositions such that everyone, upon entertaining them at all, instantly and certainly perceives their truth, then there can never be any use in demonstrating them:

He would be thought void of common Sense, who asked on the one side, or on the other side, went to give a Reason, Why it is impossible for the same thing to be, and not to be. It carries its own Light and Evidence with it, and needs no other Proof: He that understands the Terms, assents to it for its own sake, or else nothing will ever be able to prevail with him to do it. (1.3.4)[1]×Hamilton (Dissertations on Reid, Note A, p. 784b) takes Locke to be admitting here that “Common Sense or intellect, as the source, is the guarantee of the principle of contradiction.” This is an example of the unfortunate carelessness in interpretation that makes Hamilton’s formidable erudition so much less useful that one might hope. For, first of all, what common sense guarantees here is not the principle of contradiction, but rather the proposition I have been discussing, namely that demonstration presupposes doubt — which proposition is, as I have noted, probably “trifling,” according to Locke. That is: “common sense” in this place reveals no theoretical content. Furthermore, Locke does not regard this kind of maxim (maximally general proposition: see Hamilton’s own discussion of this term, pp. 766b–768a, misunderstanding Locke’s use of it at 767b) as a principle, that is, as a beginning point of knowledge. However, it is true that Locke thinks this maxim and all its instances (e.g. “Sweet is not bitter” and “A rod is not a cherry”) are not trifling and that they are intuitive, that is, perceived certainly to be true once proposed. Only, they cannot be proposed until we acquire, from experience, the ideas therein related. So, in Kant’s terminology: they are all synthetic a priori, but not pure a priori.

A demonstration, therefore, always occurs in the context of willing to know something which we do not, as yet, know, which is to say, in the context of doubt: “before the Demonstration there was a doubt” (4.2.5).

As to (2), then: given that knowledge is certain perception of the agreement or disagreement of ideas, the context of demonstration must be one in which ideas can be perceived without a perception of their agreement or disagreement: a context, that is, in which we do not have intuitive knowledge of the agreement or disagreement of the ideas in question, “where the Ideas themselves, by an immediate View, discover their Agreement or Disagreement one with another” (4.1.9). For without a perception of the ideas themselves, I could not will to perceive their agreement or disagreement; but if I already perceived that agreement or disagreement, there would be no doubt which a demonstration might resolve. But how is it possible to perceive two ideas clearly and yet not perceive whether they agree or disagree? For this Locke turns to just such a geometrical example as we have been discussing:

Thus the Mind being willing to know the Agreement or Disagreement in bigness, between the three Angles of a Triangle, and two right ones, cannot by an immediate View and comparing them, do it: Because the three Angles of a Triangle cannot be brought at once, and be compared with any one, or two Angles; and so of this the Mind has no immediate, no intuitive Knowledge. (4.2.2)

What allows doubt in this case — being willing to know, but not knowing — is that the two ideas in question can’t be directly compared. If they could be directly compared, on the other hand, then the agreement or disagreement would already be known, so that no doubt could intervene and there would be no use for a demonstration. If some general maxim is immediately certain, then, i.e. is a general truth about a sort of ideas that can, in each instance, be immediately compared, there can be no occasion for using the maxim, as such, in a demonstration. Whatever we might use it to demonstrate is already known.

When, at the end of my previous post, I spoke of showing that this axiom is intuitively certain, according to Locke, what I really should have said, then, is that every instance of the axiom is an intuitively certain truth. These “instances” themselves, granted, are typically only lower generalities, not fully concrete particulars. Angelic intellects, perhaps, can carry out and retain the results of an infinite number of particular demonstrations; but we can not, and so, “in this imperfect state,” are forced to deal instead with general ideas — for example, the idea of a triangle, “neither Oblique, nor Rectangle, neither Equilaterial, Equicrural, nor Scalenon” (4.7.9). Still, the instance of the maxim involved in any given demonstration is always specialized in some way or other. Such instances are not demonstrated, according to Locke, by showing that they are instances of the axiom: they are not demonstrated at all. Therefore the axiom, as a general principle, also is not used to demonstrate anything else. It does not appear as a premise in any demonstration. In fact, a demonstration doesn’t really have premises, according to Locke. A demonstration doesn’t proceed from a list of propositions (the premises), via various steps that allow us to write down other propositions (according to rules of inference), to a final proposition (the conclusion). Rather, a demonstration proceeds from one idea (the subject of the conclusion) to another idea (the predicate of the conclusion) via a series of intermediate ideas, such that, at each step, the adjacent ideas can be directly compared, and such that, in consequence, the agreement or disagreement of the adjacent ideas is immediately, i.e., intuitively, perceived.

Now, if I, unlike Locke, take Thābit’s axiom and other such general maxims to be worth our consideration, it is not so much that I disagree with him about (2) (although there are well known problems in reducing everything we call a “demonstration” to this form) as rather that I disagree with him about (1). I deny, namely, that the only or even the main purpose of demonstration is to move ourselves or others from doubt to certainty. Mathematicians, at any rate, seem to value demonstration mostly because they want to understand something about the structure of what we know: how one part depends on another. (This is why, among other things, even after a demonstration of some proposition has been discovered, they may continue to search for a more perspicuous demonstration, or one that proceeds on different premises.) In the context of geometry, in particular, one of the main questions is, so to speak, how much of the Euclideanness of space is required to make a given theorem come out true. This question may be asked, and has been historically been asked, even by people who regard it as immediately and self-evidently true that space is fully Euclidean, i.e. that every instance of every Euclidean axiom is true. Whether Locke has simply ignored the possibility of this kind of question, or whether he on purpose rejects it, I can’t say for sure.

Returning to Locke, then: note first that, in the way he describes the need for a demonstration of the triangle postulate, he implicitly supposes that instances of something like Thābit’s axiom are intuitively certain. The reason the three angles of a triangle cannot be directly compared to two right angles is, that the three angles can’t be “brought” to coincide with the two. This is because “two right angles” are, by definition, two equal, adjacent angles formed by one straight line and another (see Elements 1, def. 10, ed. Clavius [1574], fol. 5\(^{\rm v}\)). To compare some three angles directly to two right angles, they must be “brought” into adjacency — which, of course, can never be done to the three angles of a triangle, no matter how the triangle is moved. If, on the other hand, Locke implies, some angle can be “brought” to some other, then the two can be directly compared. This is the standard for geometric equality, which is to say, for agreement or disagreement of the second kind (qualitative agreement or disagreement) between geometric ideas. But Thābit’s axiom concerns specifically the way in which one thing can be “brought” to another for the purpose of direct comparison. If the axiom is true at all, therefore, then its instances must be intuitively certain. The standard of equality which is to be used at every step in a geometrical demonstration cannot itself depend on any demonstration.

What is the source of this intuitive certainty? That the axiom involves motion implies, as I pointed out in my previous post, that we are outside the realm of pure geometry: the parts of space as such are immovable. When we talk about a solid body, \(S\), which is “moved in its entirety to one side by a simple direct motion,” and a point, \(A\), fixed on \(S\), we must, according to Locke, be using the term “solid” in the physical, not the mathematical, acceptation. This idea of solidity is essential to that sort of substance that we call “body”: a body is something extended and solid. Notice that the idea of body is, therefore, quite different from the idea of almost any other sort (species) of substance. The complex idea of a substance differs from a mixed mode, according to Locke, in that the former, but not the latter, is supposed to reflect the qualities which, taken together, constitute a distinct sort of external thing. This means, on the one hand, that we must know that those qualities possibly coexist (which, in general, we can tell only by experiencing that they actually coexist). But it also means, on the other hand, that the coexistence of those qualities implies something about the thing in which they coexist. That is to say: it is supposed to follow, from the coexistence of those qualities in the external object of our idea, that it will also have certain further qualities. Our ideas of substance are, for this very reason, in general inadequate ideas. We do not, in general, perceive any necessary connection (nexus) between distinct ideas, and thus in general do not know (perceive with certainty) that the coexistence of certain qualities implies anything further. With one small, and yet important, exception:

In vain … shall we endeavour to discover by our Ideas … what other Ideas are to be found constantly joined with that of our complex Idea of any Substance: since we neither know the real Constitution of minute Parts, on which their Qualities do depend; nor, did we know them, could we discover any necessary Connexion between them, and any of the secondary Qualities: which is necessary to be done, before we can certainly know their necessary Co-existence. So that let our complex Idea of any Species of Substances, be what it will, we can hardly, from the simple Ideas contained in it, certainly determine the necessary Co-existence of any other Quality whatsoever. . . . Indeed, some few of the primary Qualities have a necessary Dependence, and visible Connexion one with another, as Figure necessarily supposes Extension, receiving or communicating Motion by impulse, supposes Solidity. But though these, and perhaps some other of our Ideas have: yet there are so few of them, that have a visible Connexion one with another, that we can by Intuition or Demonstration, discover the Co-existence of very few of the Qualities are to be found united in Substances. (4.3.14)

Because our idea of body is made up of primary qualities, namely extension and solidity, body is a sort of substance of which we know the real, not merely the nominal, essence, and the known properties of body then follow from this essence, with either intuitive or demonstrative certainty. Among these properties is the property of divisibility, which I discussed at length in the previous two posts. Divisibility follows from solidity, first of all, because solidity, as Locke says here, is a necessary condition for the reception or communication of motion by impulse. The extreme parts of a given body, considered simply as such, cannot, in general, move towards one another, but they can always move apart from one another: what could resist such motion would have to be another body, and our original body — again, considered simply as such — in no way makes it necessary that any other body exist beyond its own limits. But this motion requires division — that is, the creation of new superficies — because “upon the Solidity of Bodies also depends their mutual impulse, Resistance, and Protrusion” (2.4.5; my emphasis). Solidity is a necessary and sufficient condition, not only for resistance to the motion of other bodies as a whole, but also for any force that would cause them to distort as they moved. We understand, in common terms, what this means: motion through empty space does not tend to make a body stretch, flatten, bulge out, etc. One part of this can be expressed more precisely by saying that a body, in the absence of other bodies, is free to move without any change in the distance between its parts, and this is already enough to imply that bodies are divisible: if the extremities of a body move away from one another, any other parts can maintain a constant distance from one extremity or the other, but not from both. Another part of it, however, is expressed by Thābit’s axiom: a body is free to move such that its parts remain at constant distances from each other, in such a way that every point of the body, and every line in the direction of motion, moves along a straight line. This, then, is why every instance of Thābit’s axiom is intuitively certain.

It is easy to see, in general terms, why a principle like this can constrain space to be Euclidean. In a space of variable curvature, rigid motion is not, in general, possible at all: a figure which occupies one place simply will not fit into other places. But even in a space of constant (nonzero) curvature, where rigid motion is possible, rigid motion is not free of resistance, and it is not free of resistance because Thābit’s axiom is violated. Think of an airplane flying west to east along the equator as a model of motion along a straight line (geodesic) in a curved space. A second airplane flying just to its north cannot both fly straight ahead and maintain a constant distance from the first. No matter what direction the second airplane heads, it will, traveling along some great circle, eventually cross paths with the first. To maintain formation, the second airplane must continually swerve off from a “straight” (that is, great circle) course, and some force must be applied to make this happen. If we imagine that the two airplanes are connected by a solid metal bar, there will be constant stress on the bar.[2]×If we look at the two airplanes as traveling on curved paths in three-dimensional space, rather than as traveling through a curved two-dimensional space, then this force can be seen to arise from the difference between the centrifugal force on the two airplanes, or, in other words, the difference between the forces that are needed to hold each onto the surface of the Earth. From here we could go on to a more general discussion of the relationship between gravitation and geometry. And so, indeed, with the parts of a single airplane, or of any rigid shape: to move rigidly, it must, so to speak, constantly push against the shape of the space it is moving through, because its various points can never both maintain constant distance from one another and all move in straight lines (i.e., along geodesic curves). Thābit’s axiom says that space is flat because it says that rigid bodies — “solids” in the physical sense — can move through space without any such pushing. But just this, the fact that resistance requires solidity, i.e. that a body only needs to push when it is moving against another body, is the necessary relation between ideas that Locke thinks we perceive with certainty. With intuitive certainty, that is: meaning that the ideas themselves are perceived to have this relation in every instance, without any need for demonstration, and in particular without needing to appeal to Thābit’s axiom, or to any general principle or axiom, to make the connection.

To go from this insight about rigid motion and curvature to the proof of a specific theorem like the triangle postulate — to use instances of Thābit’s axiom, in other words, for that kind of specific constraint on the shape of a space — is, however, more complicated than one might at first expect. Indeed, it is more complicated than I at first expected, which is one reason there has been such a long interval since the previous installment in this series of posts. Locke himself perhaps underestimated the difficulty. All he says about how the proof will work is that “in this Case the Mind is fain to find out some other Angles, to which the three Angles of a Triangle have an Equality; and finding those equal to two right ones, comes to know their Equality to two right ones” (4.2.2). Given the way Euclid proves this, however (Elements 1.32, fol. 52\(^{\rm v}\)), we can guess which intermediate angles Locke has in mind.

Figure 1: Euclid’s proof of the triangle postulate (Elements 1.32).

Consider a triangle \(ABC\) whose base lies on the line \(BD\) (fig. 1). The proof proceeds by drawing a parallel \(CE\) to the side \(BA\) (that at least one such a parallel exists can be proved without relying on the parallel postulate). By the the alternate angle theorem (Elements 1.29, fol. 50\(^{\rm r}\)), \(\angle \,ABC = \angle \,ECD\) and \(\angle \,BAC = \angle \,ACE\). Therefore, etc. Here Locke’s intermediate angles are \(\angle \,ACB\), \(\angle \,ACE\), and \(\angle \,ECD\). These angles can be “brought,” for comparison, onto two right angles, and we have shown, on the other hand, that they are equal to the interior angles of the triangle. Or rather: Euclid has shown that, to his satisfaction, using the alternate angle theorem. The question is, however, how Locke thinks we can show it.

Figure 2: A simple proof of the triangle postulate using Thābit’s axiom, of the kind Locke may have contemplated.

Now, the alternate angle theorem is a consequence of the parallel postulate: in the Elements, in fact, the parallel postulate is first used in the proof of 1.29. If Thābit’s axiom can be used to prove the parallel postulate, therefore, it can be used — in combination with the other principles invoked in Euclid’s proof — to prove the alternate angle theorem, and hence to prove the triangle postulate, as well. Such a proof would be needlessly complex, however, since a version of the alternate angle theorem actually turns up as a lemma in Thābit’s overall demonstration of the parallel postulate (his Proposition 6). There must therefore be a more direct proof of the triangle postulate. Given, indeed, the resemblance between the current fig. 1 and fig. 4 in the previous post in this series, we might suspect that this more direct proof would be fairly simple. Thābit’s axiom implies (see fig. 2) that, if we imagine the triangle \(ABC\) as affixed to a solid body, then we can move that body rigidly left until \(\triangle \,ABC\) coincides with \(\triangle \,ECD\), in such a way that \(A\), \(C\), and \(D\) will lie on a single straight line. \(\angle \,ACB\), \(\angle \,ACE\), and \(\angle \,ECD\) will, once again, be the intermediate angles in Locke’s demonstration. It is perhaps intuitive, on the one hand, or, if not, then easily demonstrable, that their sum is equal to two right angles.[3]×See Elements 1.13, fol. 35\(^{\rm r}\), and see also 4.17.14, where Locke gives, as an example of an immediate and hence intuitive comparison of ideas, “that an Arch of a Circle is less than the whole Circle.” And it is, on the other hand, intuitively certain both that \(\angle \,ACB\) is equal to itself and (by rigid motion) that \(\angle \,ECD\) is equal to \(\angle \,ABC\). The question then is only, how to show that \(\angle \,ECA = \angle \,BAC\). It’s impossible to say for sure how Locke would have carried out that final step, but it seems natural to imagine a further application of Thābit’s axiom, this time sliding our solid body diagonally down until \(\triangle \,ABC\) coincides with \(\triangle \,CFG\). Now if we assume that the vertical angle theorem (Elements 1.15, fol. 36\(^{\rm r}\)) is either intuitively or demonstratively certain,[4]×I hope also to discuss in the following post how or whether Locke might be entitled to this assumption. then \(\angle \,ECA = \angle \,FCG = \angle \,BAC\).

My conjecture that Locke had a demonstration like this in mind is only a conjecture. It is an uncharitable conjecture, at that, because if Locke did contemplate a proof like this, he was neglecting something important. How do we know that \(F\), \(C\), and \(E\) lie along a single straight line? This must be true if the plane is Euclidean: it follows from Elements 1.27, fol. 48\(^{\rm r}\) that both \(EC\) and \(CF\) are parallel to \(AB\), and, by Playfair’s axiom, this means that \(EC\) and \(CF\) must be segments of the same line. But the proof of Elements 1.27 uses the exterior angle theorem, while Playfair’s axiom is equivalent (in the presence of the other axioms) to the parallel postulate. In a demonstration that space is Euclidean, then, one cannot take either of these for granted: if Thābit’s axiom is sufficient to demonstrate, via the procedure of fig. 2, that space is Euclidean, we will need to produce a demonstration that \(F\), \(C\), and \(E\) are collinear which does not rely on the parallel postulate or the external angle theorem. But the figure doesn’t make clear, at least, how that might be done.

The moral is not that Thābit’s axiom cannot be used to prove the triangle postulate: on the contrary, we still know that it can, since it can be used to prove the parallel postulate, which in turn can be used to prove the alternate angle theorem, which in turn can be used to prove the triangle postulate by Euclid’s procedure (fig. 1). We may still suspect, even, that there is a way to circumvent some of that complexity, since, as I pointed out above, Thābit proves a version of the alternate angle theorem before he proves the parallel postulate. It remains possible, indeed, that Locke had such a proof in mind, rather than the simpler but invalid one I just attributed to him. In the next post in this series I will show in detail how this would proceed, essentially just following Thābit’s tracks. When this is finished, it will show how the supposedly intuitive truths concerning the divisibility of bodies can be used to limit, a priori, the possible shapes of spaces in this one respect. At the same time, however, it will reveal a certain limit to Locke’s approach, because the proof, while it does not take for granted either the parallel postulate or the exterior angle theorem — does not, that is, take for granted that space is Euclidean as opposed to hyperbolic or elliptical — does take for granted something more basic, about the possibility of reflection. And it will be unclear whether that more basic assumption could be intuitive, according to Locke.