On Meaning, Mathematics, and Space

On Meaning, Mathematics, and Space

It has been held that a good amount of philosophy revolves around a clash or competition between subjective and objective conceptions of things.[1] For present purposes we can understand this contrast as consisting in an opposition between conceptions of things from a personal (first-person) point of view and conceptions of things from an impersonal (third-person) point of view, or no point of view. I want to explore this question in relation to the philosophy of language, particularly the nature of meaning. I will focus on Frege. Frege’s theory of meaning splits meaning into two parts—sense and reference. Reference comprises public objects existing in the external world, not points of view on objects—actual tables, tulips, and tapirs. These are objective entities that would exist irrespective of human minds (not counting objects of reference in minds). So far, the theory is rigorously objective—nothing subjective or psychological at all. But then Frege adds the level of sense, glossed as “mode of presentation”. The visual allusion is clear and reinforced by his analogy with the telescope: a sense is a perspective or point of view on a reference. It may not be officially regarded as something mental in itself, but it functions in the theory as something grasped by the mind—how the mind “sees” the world. There can be many senses corresponding to the same reference, as there can be many points of view on the same object. Sense is separate from the reference and can exist without it. It is subjective; it contains a subject-relative point of view. In this respect it is like the classical notion of an idea. So, according to Frege, meaning has two parts, joined together: an objective part and a subjective part. It isn’t only objective and it isn’t only subjective, but both. We could call this a double aspect theory. We might even say there is something it is like to grasp a particular sense, but there is nothing it is like to pick out a particular reference, or be a particular reference.

This two-tiered theory raises thorny questions. Do we really need two layers of meaning? Why not stick with sense alone—isn’t that enough to get meaning going? On the other hand, can’t we get away with reference alone (save for a couple of peripheral puzzles)? Generally speaking, reference seems to carry the main burden of the semantic work. Thus, we get “direct reference” theories. That is, we can try to get by with purely subjective theories or with purely objective theories (shades of the mind-body problem]. The point I am making is that the subjective-objective dialectic plays out in this case also: meaning might be all subjective (sense) or all objective (reference) or a combination of both (sense and reference). We feel the pull of the subjective and we feel the pull of the objective, so we try to accommodate both pulls. Frege is a kind of semantic dualist while others are hardline semantic subjectivists or objectivists. The terrain has much the same topography as in other areas of philosophy. It is an instance of the same general problem. One can take an “inside” view or an “outside” view, or try to combine the two. The objectivist will discount the internal in favor of the external, while the subjectivist will dispense with the external in favor of the internal. Meaning is subject to these familiar tensions. Image theories represent one side of this divide and stimulus-response theories the other. The result is a kind of endless oscillation.

With that case under our belt, we can turn to mathematics, where a structurally similar dialectic plays out. What are numbers (or geometric figures)? We can view them objectively or subjectively, as in Platonism and intuitionism (or formalism). Either they exist independently of the human subject and don’t vary with variation in the subject, or they are creatures of the human mind and reflect its nature; or possibly they are some sort of combination (there are subjective numbers and objective numbers existing in some sort of correlation). Both types of view run into well-known difficulties: numbers become too mind-dependent and subject-relative (psychologism), or they become too distant from the human mind to be knowable (Platonism). If we locate mathematics in the human mind, we make it too subjective; if we locate it in platonic heaven, we make it too objective, i.e., divorced from human faculties. That, at any rate, is the classical dilemma. Space also displays this dialectic: we can make it subjective or we can make it objective or we can multiply spaces. Either we make it a category of the senses (“visual space”) or we think of it abstractly, thus cutting it off from human knowledge (“absolute space”). Or we posit two spaces—mental space and physical space. We have the counterpart to Frege’s mixed subjective-objective view meaning. Or we have an attempt to assimilate space to the subjective or the objective: there is nothing more to space than perceived space or it is removed from conscious awareness, an I-know-not-what. Thus, meaning, mathematics and space exhibit a similar pattern defined by the subjective-objective contrast. This meta-philosophy includes these three areas.[2]

There is something in common to those areas: they are all curiously impalpable. They are not concrete particulars. Accordingly, they tend to attract subjectivist interpretation; they are regarded as “immaterial”. You can’t put your finger on meaning; you can’t throw a number across the room; you can’t kick space around. Nor can you introspect these things as you can sensations. We therefore tend to recoil at objective accounts of meaning, mathematics and space, because they don’t strike us as like physical substances. Nor are they purely ideational. So, we can apply the subjective-objective distinction to two kinds of reality: the concrete and the not-so-concrete. We have subjective-objective quandaries about mind and matter, but we also have them about things that don’t fall neatly into these categories. Still, the cases are united with respect to subjectivism and objectivism. I would also put ethics and aesthetics on the impalpable-but-not-mental side. Are goodness and beauty in the eye of the beholder (subjective) or do they exist separately from the mind (objective)? The question runs deep in philosophical issues and crops up surprisingly frequently. It appears to be the fly-bottle about which Wittgenstein so disparagingly spoke. What depends on me and what does not depend on me? That is the fundamental question. What exists because points of view exist, and what exists whether or not points of view exist? Is the world the totality of points of view or do points of view carry no ontological weight?[3]

[1] This position is associated with Thomas Nagel (The View from Nowhere), rightly so, but hints of it exist in the philosophical tradition, stemming from the rise of empiricism and idealism, as well as materialism. It is another way of talking about familiar issues.

[2] I mean to be including areas not discussed (but alluded to) in Nagel’s The View from Nowhere, thus enlarging the scope of his meta-philosophy.

[3] When Wittgenstein said that the world is the totality of facts, he presumably meant objective facts not points of view on facts (a fact is not a point of view on itself). Berkeley would appear to be saying that the world is nothing but points of view. Quine wouldn’t quantify over points of view.