Mathematics, 3D puzzles, and Logical Interpretations

In his paper, "The Pragmatic Element in Knowledge," C.I. Lewis makes the claim that knowledge is necessarily dependent upon the uses that we intend for it, that "there is a pragmatic element in knowledge" (245). So, essentially, there is no meaning for abstract terms until they are pragmatically related to reality. Lewis' theory asserts that this relation is by means of interpretation and that such an interpretation inevitably conforms to our needs and interests. In this paper I intend to show that Lewis' view is valid by first describing his position and then by offering further support with my own examples.

Lewis' paper begins by stating aspects of knowledge which are necessary for an understanding of reality. He says that "there are three elements in knowledge: the given or immediate data of sense, the concept, and the act which interprets the one by means of the other" (234). He uses "the given" as a label for actual experience of the world around us, the qualia we gain from it, and what it is like for the subjective individual to exist there. With respect to sense data, there subsist individual differences such that where I might see the colour red, you, as a result of slightly different chemical makeup in your eye, might see, what I would consider, the colour blue. This is an extreme example of the kind of subjectivity which we all experience simply because we are not the equivalent people/minds. Next, Lewis specifies, what he calls, "the concept," or pure logic. This aspect of knowledge is such that it can be deduced and verified without any need of connection to the real world. It exists in abstracto as he puts it. An example of this is pure mathematics. Lewis demonstrates in his paper how, with only a few, basic assumptions, anyone with a logical mind can deduce complex mathematics from a limited starting point. In its pure form, it has no association to the world and is but a set of logical patterns which follow regardless of any values we substitute for its X's and Y's. "Thus we discover that the content of pure mathematics is simply the deductive or logical order of purely logical entities, a sort of elaborate logical pattern of abstract terms without any denotation at all" (236). Lewis also points out that "...mathematical truth is a little more certain than almost any other kind of knowledge that we have" (237). He then remarks that we do not require any additional experience to verify it, because its entirety follows logically from its primitive principles (e.g., Euclid's geometry leads to complex linear algebra); it is a priori. The last and most important element of knowledge, in Lewis' opinion, is the pragmatic interpretation of the given by use of the concept in order to arrive at an understanding of the world. He says that this interpretations is the grounding of our logical truths in the real world; defining the variables and making them mean something in reality.

Lewis states that our purely logical truths are the elements of knowledge "...which two minds must have in common" (238). That is, only logical elements may be communicated and shared between two people while still being understood in an equivalent fashion by them both. For example, take the case of the two people who behold different colours for the identical physical phenomenon. Where one experiences what I would label red, the other perceives what I would label blue, yet both identify the colour as "red". In this manner, sense experience is subjective and different for each individual mind. No one can know how "true" red appears (because it doesn't exist; colour exists only as an interpretation of light waves), but only what his individual visual apparatus allows him to perceive. And so two minds cannot share the same understanding with each other with respect to the given. But the concept can be understood in the same way by two minds. Even though one sees blue and the other red, on the abstract level of pure logic, both can speak of things "red" without reference to the material qualia of the experience each of them receives from the world. The word "red" is here used as a label or variable not referring to any specific quality of the visual spectrum. We cause the word to mean something when we assign it the value of that individually, subjective colour in the world which arises from longer wavelengths of light. Thus both people will stop their cars at a "red" light (but not necessarily a red light ;), even though they are seeing very deviant qualities of colour, and both can operate and move about in the world discussing "red" things as if this qualia were the same for both of them. This is the distinction that Lewis makes in his paper; that pure logic can be considered independent of real world experience.

The term "red" is comparable to a variable X in mathematics. It can be talked about (e.g., X > Y) and used in more complex deductions (e.g., X + Y + Y = X + 2Y). These logical relations are equivalent for both the people described above. But it is the individual's qualia that supplies the values for these terms, so that while person A might "see" X = 2 and person B "sees" X = 3, the logical relations are still valid for both interpretations. The concept is, so, crucial to the sharing and advancing of knowledge.

The interpretations is what makes the abstract concepts relevant and practical to the current reality, whereas without any interpretation, the concept is merely a sort of parameter set, awaiting sense data to fill its slots and provide it with meaning. "Order, or logical pattern, is the essence of understanding. Knowledge arises when some conceptual pattern of relationships is imposed upon the given by the interpretation" (240). So the difference between the concept and the interpretation is similar to the difference between pure mathematics and applied mathematics; one is abstract and pragmatically meaningless whereas the other is grounded in the real world and thus embodies practical significance.

Lewis then goes on to explain how we are constantly reviewing and expanding our domain of knowledge through scientific, mathematical, and logical discovery. We are always appending to the complex set of logical relations which we posses and thereby expanding the realm of experience within which we may converse to one another. For example we may talk about black holes and quasars today, whereas in Plato's time such concepts were not yet deduced and thus outside the conversable domain of truth. Lewis expresses that sometimes we discover different explanations for the same thing, but that the "truer" one is the theory which explains the most in the simplest terms and which is useful to the current state of reality. Lewis gives the example of Ptolemyan astronomy versus Copernican astronomy (the heavens move around a stationary earth or the earth moves around the sun and the sun throughout the heavens). The Copernican view was adopted because it made the relative motion of the earth easier to understand and allowed for further astronomical explanations to be developed where they could not have been within the more complex, difficult, and thus less pragmatic, former theory. Hence we adopt the explanation or truth which is more practical and relevant to our given experience. Therefore, interpretation is pragmatic.

But, one might say, interpretation is not needed for knowledge. The concept and the given lead directly to knowledge because they are linked together forming a single set of truths and are inseparable, contrary to what Lewis maintains in his paper. The concept depends on a few simple definitions from which the rest of the abstract is constructed. These simple definitions are "given" from experience and thus the concept is but an extension of the given. No further "interpretation" is then needed because what is known is simply what one has experienced directly. Any "deductions" of logic do not constitute knowledge but mere theories or hypotheses. On might predict that a mass X falling in the Earth's atmosphere will eventually impact the ground because of gravitic forces Y acting upon the mass and velocity Z with which the mass is moving. This does not mean that you know that such a mass will impact the ground, but only that you are highly certain that it might happen based on past and similar occurrences of falling masses. When you throw a ball into the air and it plummets back to the Earth and strikes the ground, you now know only that this specific ball thrown at such and such an angle at such and such a velocity at such and such a time in the Earth's rotation around the sun will return to the Earth at such and such a point on the ground. So abstract concepts are only verifiable through given experiences at which point they are no longer abstract, but specific concepts. Therefore the only true knowledge is that which is experienced in the given verifying only very specific concepts of inductive logic.

This objection to Lewis's theory is fallacious because the idea of the concept is based on those explanations which do not depend on reality to take their form. They can be thought of independently of initial definitions (given experiences), but once these definitions are made, naturally it is grounded and made practical for use in reality. For example we can think of the concept "A + B = A & B". It does not matter what A or B are, the concept is still true. The initial definitions do not matter, but certain definitions are chosen only for practical reasons.

The making of definitions for substitutions of abstract variables is the act of interpretation. The given without logical interpretation is, alone, a "buzzing, blooming confusion" (240) of sensations. Order and understanding are placed upon these sensations by means of logical interpretation. Take computer generated 3D paintings for instance (see for example, the works posted at www.vision3d.com as of 1999). Gazing into one of these images at first gives one the impression of disordered and chaotic colours splashed upon the canvas. But by peering closely and allowing one's cognitive faculties to interpret what is being presented by visual perception, 3D images appear to "pop out" of the chaos. The person who looks at the painting but does not see the 3D image still perceives the same seeming random array of colours (the same information/sensation) as the person who does see the 3D image, but only one of them has knowledge of the painting. In this way, you can see how the concept and the given are separate sets of truths and that logical interpretation is needed to arrive at knowledge.

Therefore, Lewis's theory, that there is a pragmatic element in knowledge, is valid. Logical truths can be understood in purely abstract terms and verified thus without any reference to reality. Given, sense data from the world around us is but a "buzzing, blooming confusion" when passively receiving it. Only with logical interpretation can this confusion be alleviated providing order and understanding, leading to knowledge. Interpretations change over time, changing as the need for better interpretations arise so that we may still use similar words or variables, but the meanings of these labels changes. Where at one point "life" meant "what has a soul", it has now been revised to mean "what can reproduce, adapt, grow, etc." The meaning of the word has been altered "by the degree to which our most vital needs and interests are satisfied" (243). So the interpretations we use are pragmatic in nature, used only to meet out needs and interests. Consequently, there is a pragmatic element in knowledge.

Works Cited

Lewis, C.I., "The Pragmatic Element in Knowledge" in Knowledge: Classical and Contemporary Approaches. Oxford: Oxford University Press, 1995. 234-245.