[8-12-05]

Objectivity 1.0

First, I’m interested in the way in which we might approach “A = A.” The most obvious answer to the initial question is rather simple: The two A’s are different; they are different because, looking at the bolded equation [above], we are really seeing pixels, and insofar as we can distinguish between pixels of one color and pixels of no color or of a different color (or, really, what is called whitespace), we can discern two separate instances of pixel-groupings which are different in their location on the screen although they may be quite similar in many respects, and which therefore cannot be called “identical” in the strictest sense. So far in this consideration, beginning with the subheading “Objectivity,” three instances of similar pixel groupings, representing the capital letter in question, have occurred. When I mention “A”, a fourth has occurred.

An approach such as the one outlined in the paragraph above really depends upon (among other things) a concept of identity in which any real thing, or object, has a unique identity, or a unique set of qualities and/or constituent parts. Without going into great detail about such a consideration of “identity” (including what we mean by “qualities” or “constituent parts”), we might consider the possibility that a rational concept of two distinct, real entities would require that each possesses some quality or constituent part which the other does not possess. The two instances of pixel grouping in question may be said to have the cardinal difference of 1) being located in different places in the Universe; or 2) being located in different places on the computer screen, thus having different spacial relations to the other things on the computer screen (a much more localized concept of space/location, although it introduces relativity or relationshipping as well); or even 3) having different sets of photons if we are referring to the pixels which are actually viewed when viewing each. Other distinctions might be made.

Without knowing or even caring what “A” represents, whichever pixel-grouping is in question, we may look at the objective reality of the occurrence — insofar as we are able to view objective reality. If we decide that our senses are able to detect real things (my general assumption for this consideration of objectivity), we can easily detect two separate pixel groupings which are not identical in the strictest sense, thus making the equation “A = A” false, but only if we do not view the pixel-grouping between them in a similarly objective way. The pixel-grouping which looks something like “=”, when considered differently than the other two pixel groupings, may have a conceptual existence as well as an objective existence; we are comparing two sets of pixels which are not identical via a conceptualized pixel-grouping that stands for “equality;” and therefore the entire group of pixels (the three groupings) may represent a false statement if we view the whole from a conceptual basis while viewing some of the parts from an objective basis. If we view the whole entirely objectively — assuming that we are able to do so — the whole is neither true nor false, because the constituent parts are merely pixel-groupings or even merely individual pixels.

Actuality 1.0

The above considerations on objectivity, as they relate to the initial question for this thought experiment, introduce the idea of actuality. Being an idea, the idea of actuality is conceptual, but the concept may actually refer to a reality. [Heh, there’s something of a circular semantic argument in that last sentence — or a tautology?] The differences between our experiences of life seem to depend on our understanding of actuality; and discerning between 1) objective reality, 2) subjective impressions and/or evaluations of objective reality, and 3) sense-experiences of objective reality, seems crucial to our understanding of the concept of actuality — and even our concept of reality.

I imagine that many people, seeing the initial question posed for this thought experiment, would immediately disregard the question because it seems silly or “irrational.” From a mathematical point of view or a common philosophical point of view, the equation is difficult to refute. Some might say that “A = A” is merely a tautology, or an axiom, or an axiomatic tautology, or a tautological axiom, and is therefore neither true nor false, but an assumption or a “frame” upon which we may build a sturdy structure. Others might simply say the equation is true. These are interesting viewpoints, and I expect to be contemplating them further as I continue my thought experiment. At the moment, any of the above standpoints seems to me to be a highly conceptual point of view, might be called rationalist in the conventional sense that the point of view depends on a mental feat moreso than an experiential (or empirical) feat.

To clarify the paragraph immediately above: by use of the word “feat,” I mean an action or act of viewing. The American Heritage Dictionary definition for the adjective “empirical” includes the description “relying on or derived from observation or experiment,” and I take issue with the definition, since it seems to clearly refer to concepts or ideas which have as their basis some perception of actual things, and it therefore suggests an endpoint not dissimilar to the endpoint of a purely rationalist approach: that is, concepts. The adjective is useful, a distinction might still be found between empirical and rationalist approaches, the value and/or virtue of concepts might still be distinguished by reference to the method that has led to them, but either approach will leave us far removed from actuality.

In using “actuality,” I am referring to those pixel-groupings, and any thing which exists despite what we make think of it or even whether we think of it. An empirical feat I would have to be the instance of observation of actuality via our senses [ed. — of which we are aware—?]

My concern with actuality springs from my consideration of how we may be aware of identity: how we discern an identity. When we think of “A = A”, not only are the terms vague from a mathematical or philosophical conceptualization, our tendency to view that collection of pixels as a “concept” while ignoring the actuality of that collection seems to lead us away from identity. If “A” represents any thing, it represents no thing; to say that every thing is self-identical (should we wish to express the idea this way instead) tells us very little about any actual thing. Most of all, the habit of interacting with reality from such a conceptual basis seems to shut the door on actual observations of reality: why would we see “A = A” as an axiom before we see it as a collection of pixels? [ed. — or focus on the concept rather than focusing on the observation of pixels, or while forgetting the actual pixels…]

[8-16-05]

Obviously, if “A = A” can be viewed with a focus on the actual pixel-groupings to either side of the “=” grouping, a dedication to viewing any concrete object will produce similar results. For instance, rather than worrying about comparing apples to oranges, we might worry about comparing apples to apples.

Thinking of the problem in terms of leaves leads me to wonder if any two identical leaves can be found. Ignoring for a moment the spatial-temporal differences between any two objects we call “leaves,” we are nonetheless likely to find other differences between the two objects. Size and shape — the results of more or fewer atoms, plus their distribution, plus their kind — might be found to differ between the two objects.

Any foundation for an epistemology which would require a comprehensive understanding of actual objects runs into the problem of such a differentiation comparison. When I say, “I know for a fact that I held a leaf today,” I may very well know that I have done so, but with these provisos:

1. The term, “leaf” cannot stand for the exact object I held, but only for an approximate reference to that actual object. It does not include whatever features — down to the atom, to the actual size and shape, the entire material reality — which I did not observe. (If I did not turn the leaf over, for instance, I haven’t viewed the entire leaf. If I have turned the leaf over, I have seen a backside while not seeing the front side, although I have seen both sides at different times: it is my mind which combines the two views, but not the part of my mind involved directly in the empirical feat of observation.)
2. When I say “leaf,” I am referring to a set of properties which any object must have in order to deserve that name; but the set is vague, because the term is vague. The set is incomplete, insofar as I haven’t observed the total reality of that object. Even if I have seen the complete set of atoms without seeing the individual atoms (the whole set), I have seen a set which differs from every other actual object to which I may give the term “leaf.”
3. Should I tell another that I have held a leaf today, that other person may not have that actual leaf for reference; so I depend on a common understanding of basic, limited properties for “leaf.” In effect, I am trading in concepts, describing concepts, rather than any actual object. If the other person is present at the time of the holding, we may both have reference to an actual object, in our discussion, although we may each have a limited set of observations (perspectives) of that actual object.
4. Even if I and my interlocutor have at separate times developed the concept of “leaf” by viewing multiple instances of actual objects which have the common properties required for that designation, unless we both have exact prior experiences — an exact viewing of all instances of the actual objects — I cannot depend on a shared concept of “leaf” which would go beyond the vague concept of general properties held in common by such actual objects. I cannot even know that my interlocutor will have the same vague concept of “leaf,” because my interlocutor may have witnessed actual objects which all have a shared property which I haven’t witnessed in all members of my set of actually observed objects.
5. The fact that we trade in concepts of “leaf” rather than actual objects — that we trade in concepts of properties and hope that each of us has in common an equal set of concepts for “leaf” — allows us the metaphorical usage: leaves of a table or leaves of a book. The fact that our concept is vague allows such usage. Question: How, then, can I know that when a person talks about a “leaf,” even a leaf on a tree, that the person is not using the term metaphorically there? Why should I assume that the referent is some actual object, or actual objects, rather than a conceptualization of sets of properties (and not a conceptualization of actual objects)?
6. Without recourse to theories of language, or a dependence on contexts for distinguishing between a “leaf on a tree” and a “leaf in a book,” we can still be fairly certain that sometimes when a person says “leaf,” that person is only considering a limited set of properties of “leaf.” If an epistemological theory has as its foundation the ability to view reality objectively, how does such a foundation separate metaphorical usages (or conceptual formations) from usages which reference actual objects? Can any such epistemology explain why knowledge should be dependable when even actual observations are limited, such that concepts formed from actual observation are limited concepts, or vague concepts? Can higher-level concepts which have as their ultimate foundation limited observations of the actual world be considered infallible?