How do we ground ontological identity?

What is identity? Why and how do we assign and believe objects to be identical and identifiable? And this question is very difficult for mathematical "structures" which are identified in a a creative way: let X be such-and-such a structure and let a be an element of X.  How do we even identify and constitute ontological continuity for a symbol or a name? Why is "a" at one point of the page the same "a" at a different point? And why is its semantic and intentional aspect exactly the same? And when we say: let X be such-and-such a structure, exactly how do we point out (or create or bring out through ekthesis) this structure for a given context? Numbers, algorithms, structures - what gives them their abstract identity and identifiability? How are they named and pointed out and discussed? How do they subsist in the human mind and memory? How are they recalled and are they the same each time they are recalled? Does the human mind ever really act like the common notion of a rule-following algorithmic process? How can this be if we cannot identify the whole or the parts and do not know what guarantees their consistency or continuity? What really was so-called psychologism and was it really "refuted"? Does the very notion of "refutation" in this context even make sense? Can we point to a "structure" on a computer? The bare physical configuration of the hardware. It is we who "interpret" it.  But it does not really exist in a certain sense. Ethical perception can never be directed at an abstract "structure" but only at life. The machine is the counterfeiting of life, turning life into "structure".  The soul instead of seeing herself as an anagogic image became a stranger to herself and instead worships structure and the machine. Language is an aspect of mind which is in constant danger of being misperceived as an allegedly stable abstract structure when it is not. When the word arises the thought and meaning process has already ceased. All structure is the product of individual and social consciousness and can only be understood in terms of it. The most fundamental flawed and erroneous tendency of a human being is to confuse symbol (or image) with reality. Only the pure foundation of morality touches reality directly.

Also we must not forget the selective multi-layered and semi-unconscious nature and yet inseparably holistic nature of consciousness. With partial knowledge and horizons identity can only be probabilistic and there is no objective abstraction of what it means to carry out a rule. Our own identity is a momentary interpretation and construction patched together from open incomplete and partially submerged horizons. This enlightened view of logic, language and consciousness may well be the key to finally understanding quantum theory. The wave function limits horizons (which are always open and uncertain) and the collapse of the wave-function is an interpretation resulting from conflict and context, it is the rudimentary manifestation of the mind of the universe becoming momentarily conscious.

Inspired by traditional schools of Buddhist philosophy, aspects of Schopenhauer and Hegel,  the original philosophy of Brouwer (see Mark Van Atten's excellent book On Brouwer) and Gadamer's Truth and Method we propose to develop an authentic "phenomenological" account of logic, language and mathematics, which might be characterized as a "transcendental psychologism" based on hermeneutics, horizons, uncertainty, revisability and dialogue. With regards to mathematics this will involve a very radical ultrafinistic and constructivist critique. We challenge the epistemic claims, standard foundational frameworks of mainstream mathematics and philosophy and above all its human and social value - while at the same time furnishing a new foundation for the subset of mathematical results essential for all applications in science and engineering inspired by Simpson's Reverse Mathematics.

Yet these deep structures of consciousness can be given formal (symbolic, anagogic) illustrations - free of any subtext of metaphysical dogmas. This we find in Hintikka (see our note on the Dialectica interpretation), Beth semantics (forcing and sheaf semantics are based on early, perhaps philosophically superior, work), dialogical logic, dynamic epistemic logic and fuzzy logic.  

This is the Humean fallacy: how can fuzzy, semantically and cognitively constructed and selective consciousness obtain certain "objective" knowledge of its own functioning and mechanism (assuming of course that these notions are even legitimate at all)?  Such a "knowledge" can only occur fully at super-conscious states. The muddled and semi-opaque self-reflection of ordinary consciousness can only be approximate (a toy model) and at best essentially symbolic, analogic and anagogic (like a ladder), that is, it is not only an approximation, trace or shadow but serves as a motor and an energy for consciousness to transcend itself and to attain clear knowledge.