The Apple and the Marble: A Puzzle about Hume’s Principle of Seperability

 

Abstract:

            The principle of Separability says that if two ideas are different you can ‘separate them in imagination’. The expository puzzle is that Hume says you can distinguish the color of an apple from its taste but you can’t distinguish the color of a marble sphere from its shape and it’s not clear what the difference is supposed to be. My proposed answer is (very quickly) that Hume’s atomic theory of ideas gives an objective decomposition of ideas into parts, and that the apple’s color and smell correspond to different parts of the complex idea of the apple, while the marble sphere’s distinct properties correspond to the same part.

In the paper I also describe how this interpretation makes sense of the various appeals to the principle of Separability in the Treatise, and consider an epistemological problem for it. How do we know when the shared properties of two ideas correspond to a shared simple or complex idea part? I suggest an empirical answer to this question.

 

Introduction

            Hume’s Principle of Separability says that, “Where ever the imagination perceives a difference among ideas, it can easily produce a separation” (1:1:3)[1], in other words, if it is possible to ‘distinguish’ two ideas A and B it is also possible to ‘separate them in imagination’ by conceiving of A without B or B without A. This a plausible, if somewhat underspecified, principle and Hume appeals to it at the beginning of some very important arguments: to get the result that we can conceive of the future failing to resemble the past and of a cause unaccompanied by its usual effect. However when we look at Hume’s exposition of what it takes for two ideas to be ‘distinguishable’ and ‘separable’ an apparent contradiction arises.

In the first part of this paper I will present this interpretive puzzle. In the second and third parts I will suggest a way of drawing on Hume’s atomic theory of ideas to answer it. Finally, in the fourth part I will consider some problems with this style of solution. In describing this puzzle I hope at least to reveal a kind of constraint on our interpretations of the principle of Seperability and perhaps to motivate a more positive thesis about what Hume meant by terms like ‘distinguish’ and ‘separate in imagination’.

 

The Apple and the Marble Sphere

            The apparent contradiction that I want to focus on concerns two pairs of ideas: one which Hume takes to be distinguishable (and hence separable) and the other which he takes to be neither distinguishable nor separable.

            Just after introducing his distinction between simple and complex ideas (the latter have components which are ‘distinguishable’ from one another, the former do not) Hume gives the following example of what it is for a pair of ideas to be distinguishable: “Though a particular color, taste and smell are qualities all united together in this apple it is easy to perceive that they are not the same but are at least distinguishable from one another”(1:1:1). So it seems clear that Hume thinks the particular color of an apple and its particular smell are ideas that can be distinguished from one another. And if they are distinguishable, then given his principle of Separability Hume presumably thinks they are separable as well.

In contrast, we see in chapter 7 of the first section of Book I that Hume does not think that the whiteness and the roundness of a marble sphere can be distinguished from one another or separated in imagination: “When a globe of white marble is presented, we receive only the impression of a white color disposed in a certain form, nor are we able to separate and distinguish the color from the form”(1:1:7) All we can do with this pair of ideas is draw what Hume calls a “distinction of reason”- a technical term whose significance we can see in the next quote. Hume writes:

Observing afterwards a globe of black marble and a cube of white, and comparing them with our former object, we find two separate resemblances, in what formerly seemed, and really is, perfectly inseparable. After a little more practice of this kind, we begin to distinguish the figure from the color by a distinction of reason; that is, we consider the figure and color together, since they are in effect the same and undistinguishable; but still view them in different aspects, according to the resemblances, of which they are susceptible. (1:1:7)

That is, we can distinguish the whiteness of the marble sphere from its sphericalness by considering other examples of white things and spherical things and then contrasting the way in which the idea of a white marble globe resembles the white things from the way it resembles the spherical things. But, Hume says, such a distinction of reason “implies neither a difference nor a separation”(1:1:7).

            So here we have a puzzle: why is it that the taste and color of an apple are distinguishable and separable but the color and shape of a block of marble are not? What is the difference between these two pairs of ideas supposed to be?

            The question is interesting on its own and should shed some light, or at least put some restrictions on our answers to the larger question of what distinguishability and separability are supposed to be. For one thing, we can already see from the discussion of the marble sphere that one apparently natural and appealing interpretation of ‘separability’ won’t work. You might think that separating the apple’s taste from its color is just a matter of imaging something which instantiates one of the relevant ideas but not the other. But this can’t be quite right, for it’s clear in the quote above that Hume thinks you can also have the idea of whiteness without having that of sphericity by imagining a white cube, or sphericity without whiteness by imagining a black sphere.

 

Part-Whole Relationships Between Ideas

The closest thing I can find to a solution to the puzzle just mentioned arises from Hume’s atomic theory of ideas. Recall that that Hume takes our immediate perceptual experiences to be complexes of either of simple ideas (in the case of thoughts) or simple impressions (in the case of feelings) and holds that various complex ideas can be composed to form another complex in which both ideas are combined (e.g. you might combine ideas of a horse and of a pair of wings to get an idea of a Pegasus or the ideas of gold and a city to get a city of gold). And general ideas are something like sets of complex ideas instances which we are disposed to call to mind as required (so for example to have the general idea <triangle> is to have a particular complex idea of a triangle together with the disposition to call to mind the other complex ideas of particular triangles). This body of complex ideas is sometimes called the recognition set assocated with a general idea.

There are two things I want to note here. First, this chemical theory of ideas gives us certain objective part-whole relations between simple and complex ideas: there are facts about whether a particular simple idea is a constituent in a complex idea, how many simple ideas two complex ideas share, whether one complex idea is a part of another and the like.  Most strikingly there is an objectively correct finest way of dividing up your present experience (a complex idea) into (simple idea) parts.

Secondly Hume seems to occasionally refer to simple and complex ideas by way of general ideas. For if we interpret expressions like ‘the color of the sphere’ as referring to the general idea (whiteness) which this particular complex idea falls under then we get the unacceptable conclusion that when Hume says that “the figure and color” of the sphere are “in effect indistinguishable” he means that one can’t distinguish between the general ideas of whiteness and sphericity. Instead, I want to suggest that ‘the color of the sphere’ or ‘the whiteness of the sphere’ refers to the simple or complex part of the idea of the sphere which is responsible for its color: the component of the whole complex idea in virtue of it belongs to the associational set of complex ideas belonging to the general idea of whiteness. And more generally, whenever we can identify one component of a complex idea C as being responsible for that idea’s belonging to the recognition set for an general idea A (because, for example, changing other components gives a new complex which is still in the set, but changing this component leads to a complex which no longer belongs to the set) it would seem that we can refer to this component by taking about the Aness of C.

Now, if you buy these two points then I think we can say why Hume thinks the whiteness and sphericity of the marble are not distinguishable and separable while the greenness and apple-y taste of apple are. Specifically I want to suggest 1) that Hume means one can distinguish two simple or complex ideas within a complex when one can see that they form different parts of that complex, and that one can separate them in imagination if you can form another complex idea which has one of these ideas as a part but not the other and 2) that Hume takes the whiteness and roundness of a marble sphere and the taste and color of an apple to differ as follows. When you imagine an apple different components of your experience represent it as tasting a certain way and looking a certain way whereas the same components of your experience of the ball correspond to its whiteness as to its roundness. Here, note that by ‘different components’ I don’t mean spatial parts (it’s hard to say, for example, how such a division would apply to some aspects of imagination and experience which Hume clearly has in mind like taste and smell) but rather different combinations of the basic building blocks of experience which the atomic theory of ideas provides us with. You might call this strategy a mereological interpretation since it involves interpreting distinguishability and separability in terms of divisions within and part-whole relationships between ideas.

Intuitively the idea is that by shutting off your mind’s-tongue you can prune away parts of your experience of the apple in such a way as to leave the color aspect of your idea of the apple but not the taste aspect. But no such act of imaginative pruning can take away the whiteness of the marble without taking away its roundness as well, since the same ‘component’ (it may be a simple idea or a complex idea) - a certain pattern of color on the visual field - corresponds to both the whiteness and the roundness of the imagined marble. Since the same part of the complex impression of the sphere accounts for both its whiteness and its sphericity we can’t distinguish the whiteness of the sphere from it’s sphericity and neither can we cook up another complex which contains the component responsible for one but not that responsible for the other. In contrast the ‘greenness of the apple’ and ‘the apple-y taste of the apple’ pick out different parts of the whole complex idea of the apple. Because of this, one can distinguish these ideas by noting that they belong to different parts and one can produce a new complex which contains one of these ideas but not the other. Thus Hume’s chemical theory of ideas (specifically, the division of experience into parts which it gives us) provides us with an interpretation of ‘distinguishable’ and ‘separable’ which can account for the difference between the apple and the marble.

 

The Principle of Separability in Action

      Now I will try to give a little additional evidence for this resolution of the puzzle by showing how the mereological interpretation of the Principle of Seperability can explain some other applications of the Principle in the Treatise. I will start with the argument against Lockean general ideas. In 1:1:7 Hume considers the question of how general ideas like <man>, <triangle> and <line> can represent men, triangles or lines “of all different sizes and qualities. He argues against the view (which he takes Locke to hold) that these general ideas “represent… no particular [idea] at all”(1:1:7) – that the general idea of a triangle is an idea of a triangle which has no angle measures, side-lengths or any other features which belong to only some triangles. In contrast, on his and Berkeley’s view general ideas “are nothing but particular ones, annexed to a certain term, which gives them a more extensive signification, and makes them recall upon occasion other individuals, which are similar to them”(1:1:7). Thus a person who has a thought involving some general idea like <line> has the (simple or complex idea) of some particular line together with a disposition to think of various other instances of line should the occasion arise. We might put this idea in the form of a slogan that, ‘General ideas are not ideas of general things, but general ideas of particular things’

Now, the principle of Seprability comes into this debate in a slightly complicated way. Hume wants to argue that we don’t have the kind of abstract ideas of a line which is no particular length or a man of no particular height or color which the Lockean view needs. He does this by saying the following two things and then invoking the reverse of the principle of Separability to say that things which aren’t distinguishable can’t be separated: 1) if you could have general ideas you could separate general ideas like line and triangle from “the circumstances, which we abstract from in our general ideas”, and 2) it’s “evident at first sight that the precise length of a line is not different nor distinguishable from the line itself”. These comments are at a little bit mysterious: what does it mean to distinguish the length of a line from the line itself and how is it ‘evident’ that this can’t be done?

But if we plug in the theories of distinguishability and Separability suggested above, they say something rather plausible or at least contiguous with what we’ve already said. You can’t distinguish a line from its length in the sense that within the idea of a particular line e.g. a black line three centimeters long, there are not separate components such that one of them represents the object as a line and others represent it as having a certain color and length. We get 2) because, just as in the case of the whiteness and roundness of the marble sphere, the same part of our idea of the line accounts for its being a line of a particular length as accounts for its being a line. And looking at the same issue the other way around, if the Lockean theory were right there would be some basic featureless idea of a line, a man or a triangle which you could have on its own without combining any other ideas to make it represent a line, man or triangle with certain particular features. So we get 1) as well: if Lockean general ideas existed, in having them we would have managed to separate the part of an idea in virtue of which it is an idea of a line from that in virtue of which it is a particular line.  

And this interpretation makes sense of another use of Seperability later in the Treatise as well. Hume argues that there is no object which “implies the existence” of any other object, “if we consider these objects in themselves, and never look beyond the ideas which we form of them” (1:3:6). So, in particular, the connection between cause and effect can’t be that a cause implies the existence of its effect or vice versa. He uses Seperability to argue for this point, saying that if one object did imply the existence of the other then it would be impossible to conceive of the one without the other, but that “as all distinct ideas are separable, it is evident there can be no impossibility of that kind.”(1:3:6)

The qualification ‘if we consider these objects in themselves, and never look beyond the ideas which we form of them’ is clearly important and I think the theory of Seperability presented so far gives us a way of understanding it. Hume surely does not mean to say that there are no descriptions or general ideas such that the existence of an object falling under the one implies the existence of something falling under the other. For example it’s clear that he doesn’t think that we can conceive of there being an effect without there being anything which is the cause of it if we look at what he says in a later section about the argument that everything must have a cause because having a cause is implied in the notion of an effect. He acknowledges that “Every effect necessarily pre-supposes a cause; effect being a relative term, of which cause is the correlative”(1:3:3) so  every effect is caused. But he denies that every event is an effect, saying,  “this does not prove, that every being must be preceded by a cause; no more than it follows, because every husband must have a wife, that therefore every man must be marryed”(1:3:3)

So what is the sense in which cause and effect are supposed to be separable? Looking at the example about husbands and wives Hume seems to be claiming something like this. Husband and wife are separable not in that you could have a husband without any wife, but in the sense that for any particular person who is a husband you can conceive of that person not being married and hence not having a wife, (though of course if this were the case we wouldn’t still call them a husband). Similarly if you have a cause and its effect, like the lighting of a fire and the appearance of smoke, he wants to say that we can conceive of the cause without its effect in that we can conceive of the one event occurring without the other e.g. of the fire being lit without the smoke appearing. But what do we mean by saying that we can conceive of that man or that event being unyoked from their respective spouse and cause? One way of answering this question would be to invoke a notion of ‘intrinsic’ properties: a cause and effect pair are separable in that you can always have something which shares all the intrinsic properties of the cause inhabiting a world without the effect or vice versa[2]. But how could Hume’s metaphysics support such a distinction?

This is (on the interpretation of Separability here argued for) just where the caveat “if we consider these objects in themselves, and never look beyond the ideas which we form of them” comes in. For recall that on this interpretation the principle of Seperability only directly applies to simple and complex ideas (i.e. not to abstract ones) and guarantees the possibility of pruning down a complex in which two distinct such ideas are contained in such a way as to leave only one of them. That is, the principle justifies the claims about the conceivability of A without B or B without A for distinct A and B exactly when A and B are simple or complex ideas – and not when they are abstract ones. So an effect is independent from its cause and more generally each object is independent from every other in the sense that you can always have the simple or complex ideas corresponding to one of these without the other.

Thus, this interpretation of the principle of Seperability squares with the other uses of the principle of in the Treatise, as well as solving the puzzle of the apple and the marble.

 

But where do the Part-Whole Relationships Come From?

            Despite the arguments just given I am not convinced that the mereological approach to the problem of the apple and the marble advocated here will work. This is because I have doubts about whether it is possible to give an account of how we know when two parts of a complex idea are distinct (and hence when they can be distinguished). You may have already noticed that I haven’t said anything very specific about the sense in which e.g. the taste of the apple and its color correspond to distinct components of one’s experience of an apple and how we can learn facts like this. Hume’s view that complex ideas are composed of component simple ideas certainly seems to require that there are objective facts about how a complex idea divides up into parts. But what is this division and how do we know about it? What justifies our earlier claim that a single component of our idea of a marble sphere represents its whiteness and roundness but distinct components of our idea of the apple represent its taste and smell? In this section I will try my best to spin out Hume’s comments that a) complex ideas are composed of simple ideas as the idea of a golden city is composed of the ideas of gold and city and b) simple ideas are those which cannot be further divided in to parts into a theory of what justifies claims about the distinguishability of ideas.

            I’ll start however, on a somewhat ominous note, with the claim that one of the examples we have so far considered is plausibly among the easiest cases for a theory of the division of ideas. So if there’s a sense in which one can intuitively see that the color of an apple and its taste constitute different parts of our experience of an apple this fact may not justify optimism about one’s general ability to detect when two ideas are distinguishable. This is because color and taste belong to different sensory modalities. Even if we only have a very sketchy idea of what the ‘building blocks of experience’ might turn out to be it seems clear that these blocks won’t span different sensory modalities – that it’s always possible to separate a sight from a sound associated with it or a taste from a texture. Thus, there won’t be a simple idea which e.g. combines the taste and smell of bread since it would always be possible to split these into taste and texture components. And on the basis of this fact alone we can see that the apple’s smell and taste must be composed of entirely distinct ideas. So only a very rudimentary intuition about sensory modalities is needed to give us an answer to questions about distinguishability in the case of the apple. The problem of justifying claims about distinguishability of ideas associated with the same modality may be very much harder.

            It’s not obviously impossible though, as we can see from comparing the epistemic problem of finding out whether two ideas are distinguishable to that in the following fanciful example. You’re in a room and in an adjoining room there is an artist working with a set of stamps. The artist periodically comes out with pictures made by pressing some combination of stamps on a piece of paper, and he takes requests –though rather whimsically and with some amount of artistic license. Noticing certain similarities running through the pictures (e.g. you see circles and half circles but you never see 1/3 circles) you guess that they are made using combinations of some fixed stock of stamps. Now we can imagine that you might proceed to make some empirical hypotheses about which stamps the artist had at as his disposal and which pictures he could produce as follows. First you might look for the patterns which are always found either entirely or not at all. These patterns, one would have reason to suspect, corresponded to individual stamps – so that the artist couldn’t make a picture which contained only part of the image. You might, for example, have empirical reason to believe that the stamps in the room were: &_, ** and _>>_.  Next, given this knowledge you might make statements about whether certain aspects of the picture corresponded to distinct combinations of stamps or we accounted for by the same components. For example, suppose you saw a picture containing the following image and are asked whether the column of &s corresponds to a different combination of stamps from the column of _s:

&_

&_

&_

If we know that the artist has only the three stamps hypothesized above then the answer will be that these aspects of the picture are both accomplished by the same combination of stamps – namely by three applications of the &_ stamp going down the page. (Note, by the way that this is different from the question of whether the artist could produce an image which contained the column of _s without a column of &s. For, using the _>>_ they indeed could produce such a column. This does not show that the column of &s is distinct from that of _s in the sense that they are composed of different combinations of stamps. The contrast between these two cases is, on the mereological theory, analogous to the contrast between the way we can conceive of a sphere that isn’t white, and the more robust sense in which we can conceive of a apple without its characteristic taste.) In contrast if we though that the artist had separate stamps for & and _ or for a row of &s and of _s the right answer would be that these two features of the picture are indeed accounted for by different combinations of stamps.

In addition to making judgments about this kind of question, one might form the further empirical hypothesis that the artist doesn’t have any prohibitions about using certain distinct stamps only in certain combinations e.g. that they haven’t sworn not to put the _>>_ next to the ** unless the &_ occurs on the line above. Given the truth of this additional hypothesis there will be a connection between the facts about which features of a picture are accounted for by distinct stamps and the facts about whether the artist could/would produce a pruned down version of a given picture which contained only the combination of stamps responsible for one feature and not those responsible for the other. For when two features are accounted for by different combinations of stamps the artist could always – given that they don’t have any special prohibitions of the kind mentioned above – use only the stamps responsible for one of the features to create such a pruned down picture.

Claims about the distinguishability of ideas might be justified in an analogous way. First one hypothesizes certain indivisible units – ideas which one never has only part of. These are the simple ideas, and play a role analogous to the artist’s stamps. Then given such a body of simple ideas, one can answer questions about whether certain aspects of a complex idea (the green color in the idea of an apple, the smell etc) are accounted for by different combinations of simple ideas or by the same one. This is (according to the principle of Separability) what it means to say that two aspects of a complex idea are distinct or distinguishable.

In addition, if this analogy, is apt it also clarifies the relationship between distinguishability and separability and hence the status of the principle of Separability which connects these two. For if distinguishable ideas are analogous to patterns made using different combinations of stamps, separable ideas are analogous to patterns such that one might actually find one pattern without the other. It doesn’t follow merely from the fact that complex A is composed of partly simple ideas which do not form part of complex B that one could have idea A without B. For, as far as we know in principle there may be some kind of particular psychological law analogous to the artist’s scruples which rules out combining certain simple ideas unless other simple ideas are also added. In order to get this result one needs the further observation that there are no such particular psychological laws which prevent one from having any proper part of a complex idea on its own. Thus the principle of Separability turns out to express an empirical hypothesis about the freedom of the imagination to recombine simple ideas.

Thus we can give an empirical foundation to the knowledge about which ideas are distinguishable and the principle of Separability itself, provided that people are able to notice common features of distinct complex ideas and note the variations in which the features can arise so as to empirically determine which features of a given complex correspond to a simple idea.  

 

Conclusion

            In this paper I have described what I think is a very attractive and interesting problem for the interpreter of the Treatise. What is the difference between the properties of the apple in 1:1:1 and the marble sphere in 1:1:7 supposed to be? I have also proposed a much less attractive and interesting answer to this problem, which grounds the difference between the two impressions on an objective individuation of experience based on Hume’s chemistry of ideas. I have then tried to show how this interpretation accounts for various key uses of the principle of Seperability in the Treatise and answered some special problems about the ontology and epistemology of this individuation of experience.

I want to conclude however by noting that even though this interpretation of Hume seems to be textually adequate, and the resources for defending it seem to be available to the Humean I sincerely hope it is not correct. Surely the riches of the Treatise can be put on a firmer foundation then vague phenomenal intuitions about which ideas have common features, and perhaps the true solution to the problem of the apple and the marble will show us how to do just that.