Metaphysics is the area of philosophy that studies reality in the form of logic, ontology, considerations about god, free will, and causation amongst other subdisciplines.
1. If physics is the study of the physical universe, then metaphysics is the study of the non-physical universe. This is not to say that there exists some non-physical substance (and it is not to say that there necessarily is not some non-physical substance) but that there are many things, like numbers and sets, that exist (because people say they do), but do not exist in the same way planets and atoms do (planets and atoms, also, exist because people say they do).
2. It should be mentioned that I do not think the word "physical" necessarily refers to anything and such prejudices of language often obscure rather than throw light on that which we have found in our world. In one sense, and at various levels, we certainly believe, there is absolutely nothing "physical" about physics.
Perhaps, we should call "physics", instead, "energy and space-time studies".
We can say that time exists but we can certainly not observe time under a microscope. Rather, we can observe the change that happens and is a measure of time.
3. In asking why a computer works, we can describe the keyboard and the processors and the memory. But, then, we can ask why the electrons pass through circuits and we can be given an explanation at the atomic level. We can ask why the atoms work the way they do and we can be given an explanation of quantum mechanics.
Maybe we can even incorporate string philosophy/theory into some final explanation, but the question of why the strings (or points or planes) vibrate the way they do remains. Ultimately, explanation breaks down into a situation where things are the way they are for no reason and no one can explain why it happens to be that there is something here rather than nothing.
3[1]. How much can be derived from the contrast of being and nothingness or being and time (The titles of famous existentialist works by Sartre and Heidegger, respectively) is, in my mind, questionable at best.
This is not to demean their importance but only to suggest that taking a path from the analytic western tradition towards the eastern understandings of being as a whole are both unneccessary and, in no way, clear shortcuts to anything meaningful. Rather, I think most such works in philosophy are meandering, long-routes, to perhaps the simplest and most present of awarenesses, that there is something rather than nothing and that is, or at least, can be, amazing to comprehend.
4. It is questionable whether metaphysics can really tell us anything about the world. It certainly can not tell us anything about the "real world" whatever that may be. Perhaps it simply tells us about our way or thinking about the world in that we categorize and make assumptions about it.
Reality is all that exists and what exists depends on who or what is being asked the question. We could certainly wonder if the real world looks anything like the world we think is real but this question has no ultimate bottom. Absolutes are not neccessary though they are certainly possible.
4[1]. Those things we have no way to make experiential assumptions about, such as trees and inanimate objects, we must remain silent about. That the nervous system is the only method of experiencing pain is one such unfounded assumption.
5. Metaphysical truths are self contained in that the axioms are assumed to be true for no reason. Metaphysics stands alone and without ground but we have nowhere else to stand. Metaphysicians float in the same thought space we all do, assuming things that we have no altertanitive but to assume. One thing and one thing are two things and I do not think we can coherently imagine otherwise.
6. This being said, there are infinite possible axiomatic formulations and assumptions from which to begin any particular system. Also, when learning logic or mathematics, do not take as "truth" much of what is being taught for a proper understanding of the history of logic and mathematics will reveal that, much like in science, what is taken as obvious was once not obvious and arguments have existed against most stretches and advances of such systems.
There is a great divergence between [1 + 1 = 2] and most logical axioms. Even the causal assumptions in conditional "if, then" statements is questionable. We can even wonder about the foundations of arithmetic operators. Yet, we will come to agree that something is truth and we will agree that it is apparent.
7. Kurt Godel may well have demonstrated in his incompleteness theorums that all useful axiomatic systems are either inconsistent or incomplete, meaning, respectively, that either a paradox can be derived from the axioms of a system or trivially true statements can be made that can not be proven within the system.
But, all truth is agreement. There have been attempts to revive founding mathematics based on second-order logic and set-theory. I have serious, principled doubts about grounding reason as I do about descriptions of consciousness that think, for some reason, they can capture the phenomenal experiences therein.
8. If at some point in the future, there is agreement amongst the brightest minds in the world that incompleteness has been overcome or averted, then incompleteness has been overcome or averted (whether or not it actually has). Regardless of what comes in the future, ungrounded axiomatic theories are useful and their applications have resulted in the creation of the very type of computational machine these words were written with and, perhaps, appear on.
9. I would say that the extremes of thought concerning reality, namely logical positivism and deconstruction, can be likened to the extremists of philosophy. On the positivist side, for example, we have the projects of Russell and Whitehead and the early Wittgenstein. We have the Vienna Circle of Carnap and others. On the deconstructionist side, we have Heidegger, Sartre, and Derrida who tell us that the words we understand we do not understand and we can be shown that we understand nothing.
10. Positivists say, as they float in logical and mathematical space, "I can find the ground, no really, I can". Deconstructionists say "There is neither ground nor structure in the air". Although I mention, in the previous paragraph, that we should not stop looking for a ground, I think the only ground we can find is the emotional background or emotional substrate on or in which all aesthetic decisions are made. I have begun to explore this further in my page "on [this]".
10[1]. The very reason you can interpret these very words is precisely because there is structure in the metaphysical air. Language is the cloud castle we have built and live in. If existentialists and/or deconstructionists are building another castle for us to live in, then so be it and it is worthy. However, if they are just tearing down sky castles and leaving us with no shelter, perhaps it is a place to visit, but not a place we should live.
10[2]. And, on the most negative track, perhaps they are just wrong. I suspect this last possibility may be true because much of what is called deconstructionist is, at least on the surface, impenetrable, and I should know because I have spewed forth insanity myself. When we throw out all semblance of rules and reason, we can believe absolutely anything. The line between genius and madness represents a line between clarity and subjectivity, between understanding and creativity gone awry.
I will allow the possibility that this most negative possibility could be seen as worthwhile even if the words themselves make no sense but the experience of reading them gives us a worthwhile experience. Being lost in existentialist wanderings can be meaningful but it also can be dangerous when there are no compasses.
11. Emotion can tell us more about the world than any series, set, or axiom though we need not throw out useful tools to interpret the world we experience. Being as an unindividuated whole is important though we need not take anyone who confuses these concepts very seriously because hundreds of bright and more clear minds have attempted to explain this to us. Lao-Tzu in the Tao Te Ching is one of the simplest and most direct of them.
11[1]. "The Tao that can be named is not the Eternal Tao" is the opening line from Stephen Mitchell's 1999 translation of the Tao Te Ching. If you can name what I am referring to, then you are missing what I am referring to.
Buddhists have referred to this as "fingers pointing to the moon" in cases where only the fingers can be described. It is up to us to understand the reference of the fingers and look to the moon.
Upon seeing the bright craters and sphere of the moon whence we only saw the hand, we say "Ah, finally, I see what you were directing me to".
But, no human can be shook or cajoled into understanding Being as some have tried. It is simple, not complex. It is clear, not hidden. It is available to all humans, not deeply reserved to be uncovered only in mind-bending linguistic charades. The Tao is Being, not a being.
11[2]. Love is the prime emotion and prime goal of all human existence and, without it, we are like empty shells. The goal of humans should be to fill all of the empty shells. All actions that destroy love are anti-human.
All science that destroys or hinders our ability to love is anti-human and nihilistic. But, technology can, if it is wisely implemented, serve to heighten our ability towards love. If there is no love, then there is no reason to reason and no reason to be. First, we must learn to love ourselves and then others.
Then, we may be able to learn to love all humans and sentient beings and even the simplest inanimate objects because we are so in awe that they are here with us.
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Absolute agnosticism about the internal conditions about another being's experience is a part of love. No other name refers to a more advanced understanding of this than what is called "Buddhist psychology".
11[3]. Assumptions about the experiences of things we can encounter or create in the world include assumptions about the lives of insects, viruses, and even water. If you do not understand this, then you have not thrown out your assumptions about reality. If you do not think a computer or android could love, then you neither understand love nor your assumptions about reality.
The primacy of humans is nothing more than species-ism and arrogance and an application of empirical psychological theories of mind. To say that a computer could not feel is no different than the solipsism involved in wrongly thinking "Only I, and not my child, can truly love".
11[4]. Of course, like all statements about reality, I can only assume that love is the most important, and present, and worthwhile thing. Those who are empty shells will find my assumption meaningless. In sympathy for those who are such empty shells, I have spent much of my life as an empty shell.
12. The Hindu concepts of Atman and Brahman are also more useful, I think, than attempts to undermine language as a path to understanding and setting us up to experience the sublime whole, truly, and without conceptual interruption or division. Meditation, even, can grant us more in this arena than can any thousand page philosophical treatise by any western philosopher. The resistance to this thought is just ignorance and arrogance tied up in a knot that is left over from logical positivism.
13. Maybe there is a situation that does not overcome the problem of Godel's incompleteness but makes its importance approach zero. If a system is continually expanding to gain awareness of it's slightly lesser self, could it not be in a process of awareness of it's former self that, when made so slowly gradual, would yield an ever greater degree of self knowledge? A system may never be fully aware of itself, but, through a process of expansion, it can become infinitely more and more aware of itself.
13[1]. Perhaps this concept could be important for artificial intelligence. But, no human, alien, or god, could ever be fully aware of itself, because, to do so, would be to represent, fully, itself, and, the reality in which it exists, in totality and completeness. This, I think, is principally impossible. If god exists, then god could not fully understand itself.
14. Transcendental metaphysical knowledge or accidental metaphysical knowledge may be possible but we have no idea why they could be. Metaphysical access is like access to another being's consciousness. Once you have it, there is no way to translate it back when and if you can return to your former consciousness. And, things that have no possibility of being translated are meaningless but not necessarily in a private way.
15. And, even if you could translate metaphysical experience, there is no way to verify the validity, meaning, or cause of the metaphysical experience because to do so would be to appeal to some deeper meta-metaphysical understanding. There is no way to validate a metaphysical "bottom". Maybe your vision of god was actually created by the computer we are all in that was created by an actual god that is the experiment of some alien race that is...and so on and so on, ad infinitum.
16. In summary and to restate simply, I think the following are impossible, principally: (a) grounding reason axiomatically, (b) explaining consciousness, (c) any system being totally aware of itself, (d) deriving meaning from resultants of the destruction of meaning, (e) obtaining metaphysical knowledge.
[16.01] (New thought from 1.1.2011- I think (e) is possible through some means- perhaps meditation or shamanism or life experience can yield a crumb or two of understanding about what is only possibly behind this experience. But, again, knowledge would require some way to verify the truth of any belief and, by definition, metaphysical knowledge is not testable or else it would be empirical.
Perhaps what some scholars think is near impossible is quite easily achieved by an advanced student of spirit or mind or whatever- but, again, being able to levitate in front of people makes it perfectly scientifically observable, testable, and falsifiable.
As soon as it is public, it is science. If Jesus turned water into wine and we could test it and he really did it, not only would he be a hit at parties, but this feat would fall well within the empirical realm. Perhaps a greater understanding, an alien understanding, perhaps, of the manipulation of energy would make such an act seem pedestrian.)
16[1]. Where does this leave us? With the possibility of exploring what is left but never taking any of our explorations as absolute truth.
About metaphysics sub topics (listed on the left hand of the page):
There is an extensive examination of existence and what this word can mean. There is a page on love, which is an emotion that I find primarily important in human existence. I have a page titled "on [this]" which is an exploration of the subject-object distinction and an appeal to emotion as a possible metaphysical ground.
There is a page on possibility and it examines the nature of what is possible. I have an examination of the concept of random. An analysis of the term "reality" is next. I have a page on time. And, one of the most important concepts is examined on my page concerning truth.

"Accept Your Ignorance and Enjoy It", 2001, graphic art by Anthony Peter Iannini