You have, I believe, asked the same tough question on both our blogs so I am answering on both blogs. The question is about "the not-two aspect of reality". You note that Parmenides said "of necessity one thing exists, viz., the existent and nothing else."
If this is true then several questions arise: how to conceive of or represent this fact. And whether this is merely a conception or whether one can be aware of this fact in direct fashion.
I choose, as a physicist, to identify the "one thing" as the primordial gravity field, and attempt to show how our current universe, including us, can and did evolve from this one thing. To do so I necessarily include in the nature of the field the aspect of awareness, based partly on the self-interaction of the field that is necessary for the one thing to evolve (since nothing else existed!) and partly on a conclusion that I have reached that awareness, as I experience it, cannot be created from material building blocks, but must be inherent in the Participatory Universe that Wheeler intuited.
But this then implies, as Amos has noted, that we can, being evolved parts of the one thing, be directly aware of the one thing. Yet if this is the case, why is not everyone aware of this, and further, what does it mean to be aware of it?
In my essay I discuss how the existence of a threshold allows the creation of "two-state" systems, idealized as logic gates and the interconnection of these gates can produce numbers and such numbers can be generated by energy input to the 'counter'. I then discuss how we can, algorithmically, treat these numbers to derive 'feature vectors' which are the essential ingredients of physics. This process can be internalized in our brains to represent the world as "things", or what Zen calls "the Ten Thousand things".
Now whether awareness arises from the biological fact of putting the right building blocks in the right order, or from its inherent existence as a primordial field, in either case human beings identify as 'separate individuals', generally denoted by the term 'ego'.
If all we are is 'meat machines' then that's probably as far as we can go. But if awareness is the core property of the universe, then one might expect that it's possible to have some direct indication of this. Unfortunately, the nature of ego is to divide the universe into 'me' and 'not-me', an inherent two-fold reality.
Is it possible to transcend this? Many reports claim that is.
Abraham Maslow's studies, related in "The Peak Experience" claim that many people naturally have episodes wherein they experience the 'one-ness' of the universe, also termed 'being one-with the universe'.
William James in "Varieties of Religious Experience" came to the same conclusion.
Jill Bolte Taylor's "My Stroke of Insight" describes the state as she (a neuro-anatomist) experienced it while having a stroke.
Innumerable reports of LSD and psilocybin experience indicate the same thing.
All cultures have a mystical tradition based on experiences of this sort.
In my opinion every one of us was born with this general awareness, before our brains learned to distinguish 'me' from 'it' based on sensory input.
I also believe it is essentially a 'topological' awareness, based on *connectivity*, wherein the metric overlay of 'distance' is (almost) completely suppressed.
What is absolutely certain is that it can neither be adequately described in words (or math) nor can it be reached by talking, reading, or "thinking" about it. It is apparently reached through a biological state, either naturally, as Maslow and James report, or chemically induced, or stroke induced. Those who have never experienced it (or have forgotten the experience) tend to believe it's hogwash (or possibly codswallop). However it would appear that millions have experienced it, and the general consensus is that it's 1.) real, 2.) extremely positive, and 3.) has 'religious' overtones.
According to Zen and the Tao, it cannot be reached with words, but for a taste of the experience, I find D T Suzuki's translation of "Inscribed on the Believing Mind" to be exquisite.
I hope this adequately addresses the 'not-two' question.
Edwin Eugene Klingman
PS. This may seem to border on 'mystical', but I personally find it far less mystical than the belief that "math lives in some Platonic realm". It is based on direct experience, not abstract concept.