Belief and Knowledge - Part Six
2 + 2 = 4 Versus 92 x 16 = 1472
In pursuing his strong/weak sense of "know" thesis, Malcolm provides another example to complement his triangle discussion:
"Now consider propositions like 2 + 2 = 4 and 7 + 5 = 12. It is hard to think of circumstances in which it would be natural for me to say that I know that 2 + 2 = 4, be-cause no one ever questions it. Let us try to suppose, however, that someone whose intelligence I respect argues that certain developments in arithmetic have shown that 2 + 2 does not equal 4. He writes out a proof of this in which I can find no flaw. Suppose that his demeanor showed me that he was in earnest. Suppose that several persons of normal intelligence became persuaded that his proof was correct and that 2 + 2 does not equal 4. What would be my reaction? I should say "I can't see what is wrong with your proof; but it is wrong, because I know that 2 + 2 = 4". Here I should be using "know" in its strongest sense. I should not admit that any argument or any future development in mathematics could show that it is false that 2 + 2 = 4.
"The propositions 2 + 2 = 4 and 92 x 16 = 1472 do not have the same status. There can be a demonstration that 2 + 2 = 4. But a demonstration would be for me (and for any average person) only a curious exercise, a sort of game. We have no serious interest in proving that proposition. It does not need a proof. It stands without one, and would not fall if a proof went against it. The case is different with the proposition that 92 x 16 = 1472. We take an interest in the demonstration (calculation) be-cause that proposition depends upon its demonstration.
A calculation may lead me to reject it as false. But 2 + 2 = 4 does not depend on its demonstration. It does not depend on anything!" (pp. 63-64)
Malcolm goes on to argue that cases like 92 x 16 = 1472 represent instances of "know" in the "weak" sense because one is prepared, according to Malcolm, to allow for the possibility of error in calculation. In the case of 2 + 2 = 4, however, one is confronted with a case of "know" in the "strong" sense because one is not prepared to admit, according to Malcolm, any possibility but what one considers to be true (in this case, that 2 + 2 = 4). Moreover, this would remain so, regardless of future developments in mathematics and irrespective of any proofs which may be brought forth that show 2 + 2 cannot equal 4.
Malcolm's arithmetic example (like his hypothetical response to Prichard's triangle example) seems somewhat strained. His asking one to assume that someone came up with a flawless-looking proof that 2 + 2 does not equal 4 is like asking one to imagine someone showed one a flawless-looking proof that what is blue is really white. Malcolm is constructing his examples in such a way that no matter how unlikely they may sound or appear, one must treat them as "evidence" for Malcolm's point of view or thesis.
There is nothing wrong in anyone using the foregoing sort of examples as a way of indicating what one means by a particular idea or term (in this case, "know"). However, the value of such a process will be related directly to the extent to which that characterizing process reflects something accurate about the phenomenon or object or issue to which the 'meaning context' serves as a way of making identifying reference. Unfortunately, Malcolm's 'supposals' tend to overlook a number of possibilities which must be considered in order to arrive at a tenable conclusion about the value of entertaining Malcolm's examples in relation to the problem of coming to grips with the character of knowing.
For instance, how reasonable is one to suppose that someone could not only show one a proof that 2 + 2 does equal 4, but that one also could find nothing wrong with this proof? This is not so much a problem of trying to imagine what the future of mathematics might bring. Instead, it is a case of attempting to imagine how one could take two stones, and then, take two more stones, and then, proceed to come up with something other than a collection of four stones (assuming, of course, nothing further is done to, or with, the stones except to keep them together in a collection).
Even if someone could show that 2 + 2 does not equal 4, and even if one could not find anything wrong with the given proof, how many people, automatically, would reject this proof for no reason and continue to cling to the original idea that 2 + 2 = 4, despite possessing no basis for legitimately holding that 2 + 2 = 4? In addition, even if a person were to cling unreasonably to the idea that 2 + 2 = 4 in the face of a flawless-looking proof which contradicted the idea or belief that 2 + 2 = 4, in what way could one tenably maintain that the claim concerning 2 + 2 = 4 was a matter of knowledge and not of belief? In other words, on what basis would one choose between the two alternatives without having to ask some fundamental questions about what 2 + 2 = 4 meant when evidence supposedly existed which indicated what we formerly took to be the case (i.e., that 2 + 2 = 4) may, in fact, not be the case?
Without answers to these questions, the idea of knowing loses all sense of meaning as far as its being an idea which establishes a defensible relationship of congruency between the character of the would-be knower's understanding and the character of that which would be known. Without answers to these questions, the idea of 'knowing' becomes reducible to a question of whatever one chooses to believe - irrespective of evidence, reason, proof, demonstration, argumentation or reality.
Intuitively speaking, knowledge seems to be, at a minimum, a matter of being able to express something of the character of the reality of some aspect of the phenomenology of the experiential field. This expression of character would be in terms of the available experiential data which concerns the aspect to which one is attending. Without this dimension of an understanding whose character accurately reflects the character of that to which one is attempting to make identifying reference, there seems to be nothing which remains to be said about the idea of knowledge that lends any substantive character to this idea.
One has extreme difficulty trying to see how Malcolm can defensibly contend that 92 x 16 = 1472 depends on its demonstration when 2 + 2 = 4 does not depend on any demonstration. If one can't demonstrate 2 + 2 + 4, one can't possibly demonstrate, in a convincing fashion, that 92 x 16 = 1472. As indicated previously, for the average person, 2 + 2 = 4 rests on the fact one can take any two objects and, then, take any other two objects and place them all together in one collectivity or set and proceed to challenge anyone to demonstrate how putting these objects together will come to more or less than four individual objects taken together.
One may define words in any number of ways such that the signs "plus (+)", "equals(=)", "2" and "4" represent functions and entities other than what is normally the case. Yet, when one considers the meaning of what addition is about in the course of "normal events" and when "+", "=", "2" and "4" all have their usual properties within the perspective from which the operation of addition is normally considered, then, seemingly, 2 + 2 = 4 is every bit as demonstrable as is 92 x 16 = 1472.
In fact, one could easily argue that 92 x 16 = 1472 depends on, among other things, 2 + 2 equaling 4, and if the latter is not demonstrable, then, neither is the former capable of being demonstrated, no matter how many calculations one makes. This is the case because the very idea of calculation is rooted in demonstrating, among other things, that 2 + 2 = 4.
If one opens up the latter statement to doubt, one also opens up the former calculation to doubt. After all, how could one argue that 92 x 16 = 1472 (as opposed to being equal to, say, 1372 or 1561 or 986 or an indefinite number of other possibilities) if one could not show that 2 + 2 = 4?
Alternatively, if one can know 92 x 16 = 1472, this is only because one can know, among other things, that 2 + 2 = 4. Because one can understand, through the emergence of an appropriate connecting insight, how 2 + 2 = 4 expresses an accurate reflection of the character of the experiential context in which one takes four objects, two at a time, and creates a set encompassing all four objects, nothing more and nothing less, one has the beginning of an understanding of how 92 x 16 could be, and is, equal to 1472.
One does not cling to 2 + 2 = 4 unthinkingly and blindly such that one will reject, out of hand, any and all conflicting or contradictory possibilities. The understanding underlying 2 + 2 = 4, which is rooted in everyday concrete experience, represents a willingness to challenge any claim to the contrary.
However, the sense of "challenge" intended here is not that of an unexamined rejection. Rather, it is intended to be construed in the sense of examining the very basis of meaning itself as meaning attempts to reflect, express, represent or establish congruence with the character of that which makes such meaning-experiences possible.
If one is more inclined to check the accuracy of 92 x 16 = 1472 than to check the accuracy of 2 + 2 = 4, this is not because the latter is necessarily an instance of "know" in the "strong" sense while the former is a case of "know" in the "weak" sense. Such an inclination may be because, on the one hand, 92 x 16 = 1472 is somewhat more complicated and involved than 2 + 2 = 4, and, as a result, there are more opportunities for errors to be made. Furthermore, the idea of 2 + 2 = 4 is much more rooted in everyday experience through which it is confirmed or reinforced again and again, whereas 92 x 16 = 1472 is much less a matter of everyday experience, and, therefore, more subject to uncertainty and a subsequent need for confirmation.
Right after the previously cited quote (see page 56 of this chapter), Malcolm states: " ... in the calculation that proves that 92 x 16 = 1472, there are steps that do not depend on any calculation (e.g., 2 x 6 = 12; 5 + 2 = 7; 5 + 9 = 14)" (p. 83). To be sure, certain calculations like 2 x 6 = 12 may be "automatic" in that they are done without conscious awareness or because they are simply pulled from memory, intact (i.e., as a memorized 'fact').
Nevertheless, at some point or another, these calculations were demonstrated to an individual (with apples and oranges, perhaps). As a result, he or she knows how to proceed in making use of these calculations in subsequent arithmetical processes.
An individual's knowledge is rooted in having developed a series of connecting insights concerning certain kinds of past experiences. These insights and experiences could be drawn upon, if required, to work out the actual steps of what was involved in some given calculation of either a simple or more complex nature. If an individual couldn't produce these steps when required to do so, then, one seriously might question if he or she really knew how to proceed arithmetically in examples like 92 x 16 = 1472.
Conceivably, someone could be told to memorize certain arithmetical relationships without being given any concomitant understanding of what the character of the conceptual foundations are which tie these relation-ships together. However, the foregoing exception notwithstanding, the general rule in virtually all cases is that where no arithmetic calculation is directly evident, it is usually implicitly involved. Moreover, quite frequently, when one questions or checks the correctness of an in-stance like 92 x 16 = 1472, one attempts to make explicit, to some extent, that which previously had been treated in an implicit or horizonal fashion.
In any event, there are not two senses of knowing involved in the foregoing examples. There is only one sense. This is the sense in which one says: because one can demonstrate that 2 + 2 = 4, one can also show that 92 x 16 = 1472. To whatever extent one is skeptical of 2 + 2 = 4, one will be skeptical of 92 x 16 = 1472.
In addition, contrary to what Malcolm contends in the previously cited quote, a legitimate proof against the idea that 2 + 2 = 4 also would have ramifications for one's commitment to the belief (in the case one did not know one knew) that one's understanding concerning 2 + 2 = 4 was true or accurate. If one's commitment to belief in the truth of 2 + 2 = 4 would not fall in the face of a flawless-appearing proof to the contrary, then, one might begin to wonder about just what was meant by "truth", "knowledge", "understanding" and "belief".
Malcolm's Ink Bottle
As many of the foregoing pages of discussion have indicated, Malcolm's distinction between two senses of "know" (i.e., "strong" and "weak"), as well as his tendency to argue one cannot distinguish between belief and knowledge by reflecting on the character of the experiential contexts in which belief and knowledge claims are made, both have the effect of pushing one to adopt positions which entail much more scepticism than may be necessary or warranted. As one last demonstration in support of this contention, let us examine Malcolm's "ink-bottle" example.
"Suppose that as I write this paper someone in the next room were to call out to me "I can't find an ink-bottle; is there one in the house?" I should reply "Here is an ink-bottle." If he said in a doubtful tone "Are you sure? I looked there before," I should reply "Yes, I know there is; come and get it." Now could it turn out to be false that there is an ink-bottle directly in front of me on this desk? Many philosophers have thought so. They would say that many things could happen of such a nature that if they did happen it would be proved that I am deceived ... It could happen that when I next reach for this ink-bottle my hand should seem to pass through it and I should not feel the contact of any object. It could happen that in the next moment the ink-bottle will suddenly vanish from sight; or that I should find myself under a tree in the garden with no ink-bottle about; or that one or more persons should enter this room and declare with apparent sincerity that they see no ink-bottle on the desk ... Having admitted that these things could happen, am I compelled to admit that if they did happen then, it would be proved that there is no ink-bottle here now? Not at all! I could say that when my hand seemed to pass through the ink-bottle I should then, be suffering from hallucination; that if the ink-bottle suddenly vanished it would have miraculously ceased to exist; that the other persons were conspiring to drive me mad, or were themselves victims of remarkable concurrent hallucinations....
"Not only do I not have to admit that those extraordinary occurrences would be evidence that there is no ink-bottle here; the fact is that I do not admit it. There is nothing whatever that could happen in the next moment or the next year that would by me be called evidence that there is not an ink-bottle here now. No future experience or investigation could prove to me that I am mis-taken. (pp. 66-68).
A few pages later in his article Malcolm adds:
I wish to make it clear that my statement "Here is an ink bottle" is strictly about physical things and not about ‘sensations’, ‘sense-data’, or ‘appearances’."(p. 71)
This latter additional quote represents part of Malcolm's attempt to respond to those philosophers who maintain that a distinction must be made between statements about sense perception and statements about physical things, and that the former kind of statement should not be confused with the latter kind of statement. Some philosophers might go so far as to say that one really only can know one's sense data, and the physical world, as such, is colored by, or filtered through, this sense data which is at least one step removed from the "real" world.
Malcolm wishes to emphasize that in his "strong" sense of "know" he is not talking about sense data or appearances, but about physical reality itself. He is not talking about what "appears" to be an ink-bottle. Moreover, he is talking about, or so he claims, what "is" an ink-bottle, and if he were talking about "appearances" or "sense-data", he would not be using the "strong" sense of "know".
Even though on any given occasion one's statements actually may be about real physical things, these statements are simultaneously about appearances and sensations. Furthermore, there is not necessarily any thing inherently contradictory about maintaining that statements can, at one and the same time, express truth about, on the one hand, sensations and appearances while, on the other hand, also accurately reflect various aspects of the character of the reality for which the sensations serve as mediating reference points. After all, it is only when one's hermeneutic of experience (which sensations, along with emotions, imagination, and rational faculties help shape and structure) correctly reflects the nature of some aspect of reality's character that there is congruency of sorts between the character of what one understands to be true and the character of some aspect of the phenomenology of the experiential field or of some aspect of that (i.e., reality) which makes a phenomenological field of such character possible.
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