Attribute Theory – a proposition
Am I white or black? In reality, I'm dark-skinned caucasian. So to focus on the 'truth validity' of logical statements is superficial and somewhat arbitrary.. Is not not white identically equal to white?* I contend: not exactly and so the deeper notion is set membership or set theory. I am most certainly a member of the caucasian sub-race of human beings.. I belong to that proper subset of human beings; I'm an element of that set. Again, the deeper notion is set membership compared to logical statements which are arbitrary in meaning.. The real question is: is there anything deeper than set membership?
Human beings are traditionally thought of as hybrid spiritual-physical creatures; we have spiritual aspects and physical aspects. The 'problem' is: those characteristics appear to be contradictory. From my standpoint, the spiritual realm needs the physical plane to investigate and explore spiritual concepts just as the physical realm needs the spiritual to associate meaning to human life.. The realms are interdependent.. I specifically avoided the term 'attribute' because of its simplicity but I reinstate it here: attributes are a deeper notion than sets and therefore more core to mathematics and science.
Let's restate the argument before we progress.. We can describe 'reality' in terms of logical statements:
Something is true or not true but this denies 'grey area' which probably accounts for most of human activity.
A deeper notion is set membership; it belongs or does not belong to a set depending on its attributes.
Therefore, 'the science of attributes' is the core of all science and mathematics.
Attributes define set membership and so how we define attributes is absolutely critical to sets, membership, math, and science. From our discussion above, we can see attributes belong to different categories: exclusive to a particular set, shared among sets, and global / common to all sets .. A good related question is: is the attribute of spiritual a global attribute of all living things? Or is it exclusive to human beings? I contend it's a global attribute.. But we sidetrack.. Back to attributes.. There are three types of attributes:
An attribute is exclusive when only one set possesses it.
An attribute is shared when more than one set possesses it.
An attribute is global when all sets possess it.
Examples are: living is a global attribute of living things, sexually reproducing is a shared attribute of living things, and (as far as we're aware) technology producing individuals are exclusive to human beings. Mathematical examples: number is shared between (but not limited to) real, complex, and matrices, symbol is global for mathematical objects/operations, and empty (the set with no elements) is exclusive to the null-set .. At this point, the real question becomes: is there anything more (meaningful) we can say about attributes? Is there anything deeper than attributes? ..In the history of math, the most profound result discovered is this (by Gödel): logic cannot assert its own consistency. In more colloquial terms: the language of science cannot be used to determine the validity of science. In other words, we cannot 'prove' science with science. We must (by fact) appeal to 'other things'.. Modal logic? Meta-math? Spirituality? In a very real sense, there is no 'solution' within science that can solve this conundrum. Math/science is 'fundamentally incomplete'.. The theory of attributes is the only rational / reasonable approach to sets, math, and science. Let's develop it.
*U = W + Wb + Wc, the universe of objects equals all white objects plus their boundary plus their complement. Wb may or may not be part of W (a closed or open set); not white may or may not include the boundary; not not white is at most W + Wb but may be W so it's unclear.