The Nine Muses The Philosophy of the GOOD


Actuality, Mathematics and Consistency

In this essay I will delve further into the way mathematics helps to broaden the reality created by the individual.

We said in the essay on Intelligibility that: “As to infinity: it is a possibility that can never be reached. It is by definition never ending. It is always approachable but never reachable. It has no actuality so an infinite number is meaningless since number is actuality. As a limit it is approachable, with the proviso of unreachability. As a symbol, it symbolizes a limit, or a meaningless non-entity. Its meaning is created, not given. As an attribute it has reference only to the imaginable not the actual.”

This “actuality” we mention is a very important concept. Is it the same as existence? 1 Not really, since existence is boarder in scope. Things like infinity “exist”, if only in our mind, but as we say above don’t have “actuality”. All existents have actuality, but so do numbers which really only have existence in our minds. Actuality seems to encompass the things or objects we create to represent existents or thoughts. They are definite objects, although they may not correspond to sensual existents. We see that infinity can never be a definite object since it can never be pinned down. This “pinning down” is really what we infer with the term “actuality”. Objects as such can always be pinned down; they are actual. 2

The limit that infinity implies can never be actual; so in essence it does not exist. Let’s examine this. If we take a circle and expand its diameter or circumference toward infinity, it will, according to some geometries become a line as defined in Euclidean geometry. But is this possible? It is conceivable that a segment of a very large circle can act as a line, in all intents and purposes. But this Euclidean geometry is but an approximation. Lines have no beginning or end. Beginning and end in relation to circles are meaningless terms since a circle exhibits neither. Even segments are different things. A line segment is straight while a circle arc is curved. One can only approach the other in an approximation. These approximations can be very precise and very useful, but they still retain the reality, of a fiction.

So what does this tell us about mathematics? It is a system for creating useful approximations; where one object may slide into the guise of another at extremes which can never really ever be reached. Because of the arbitrariness of primary axioms, and their lack of actuality they may be force fitted into the guise of other things within limited precision. Because number is a concept which is itself both actual, yet defined as unlimited in creation; approximation can be brought to a precision or fit which is as precise or demanding as we please.

Therefore mathematics enables us to go that one step beyond, into the realm of the unreal, or even past the imaginable, into the transcendent, but only as a close approximation not the actuality.

We see here again that objects are creations of the mind to allow us to better manage a world where they do not exist as such. Mathematics can cause the metamorphosis of objects from one into another, as shown above with the transformation of a circle into a line, through the application of abstraction and approximation. But if this is true, how can we ever get a handle on consistency? 3 If we are ultimately dealing with approximations then true consistency does not really exist as such. Consistency is itself a creature that is relative to the resolution or approximation we define for the system we are evaluating. Predictability itself is also within the confines of this resolution. At the extremes there is no consistency. The entire system is a fabrication built up on the definitions we impose, and the intelligibility that the representational system constrains us to use. By abstraction and approximation we may get around the inconsistencies and create the actuality of a consistency within the constraints proposed.

So rationality is always concerned with limits as defined within the original axioms of the system. Even the undefined terms create, or impose limits that we may not be at first aware of, but which will later impose a restraint on what is deducible within the system.

Now if we venture to the limits which rationality imposes as I have mentioned in the footnote #5 in the essay on Intelligibility; in analogy to the above line, circle transformation, but concerning the human will and the Divine Will, we see that we might consider the human will an approximation of or part of the Divine Will. Seen in this way, using analogy, we can see that the consistency of purpose in the individual will and Divine Will can only be approached as a purpose containing the entirety of humanity. And only through the concerted harmony and solidity of purpose of the individual wills, can the Divine purpose be finally achieved.

The creation and rational utilization of the individual’s reality is merely a stepping stone toward the final achievement of the Divine Purpose. So all individual lives throughout time are the objects through which the Divine Will works. They are themselves the creations of their own rationality, but are throughout history the creators and implementers of the Divine Rationality.

In this essay I have tried to point out the ultimate self-definition of the individual’s reality. It shows in the completely arbitrary way mathematics develops, and the relativity of consistency shown in the individual’s rational structures. Consistency is a parameter relative to the resolution the system is allowed as contained in the primary axioms adopted. As such mathematics merely reproduces the same structure as the perceptual system, but uses the arbitrary approximations allowed in the use of number and quantity, to allow consistency within the precision (or resolution) allowed.




To return to note's origin click the footnote number at left

1 The existence mentioned here is not the existence of Dasein, but the dependent existence represented within the present-perception.

2 We can think of things which are non-actual as conceivable, but not ever perceivable, probably imaginable though (i.e. they can’t be pinned down although they can be thought of, or contemplated).

3 As I said in the essay on Intelligibility consistency seems to lie at a level where intelligibility lies. It is contained in the axioms of a system, and all the implications these axioms hold, either demonstrable or hidden. So the proof of consistency seems to be within these implications and their complete elucidation. Further the resolution created through these also limit the consistency of the whole system. Therefore a proof of consistency would seem to be very difficult to obtain.

Originally Published:

May 29, 2008


June 22, 2014