Hi Vladimir,
I'd like to first thank you for your lovely comments on my essay. Aside from some very pedantic grammatical errors, I thought that you wrote a wonderful essay. I really enjoyed this essay. Going through your essay, my comments are the following:
Philosophical works are highly dense and your essay demonstrates the magnitude of rigor necessary when an individual deals with such matters. You began by discussing the foundation crises that we are facing. These matters are substantially profound. There isn't much that I disagree with in your essay. For, indeed as you posited, "...The roots of the crisis lie in the initial cognitive attitudes of the ―Second Archimedean revolutionâ€-. Today, Fundamental Science rested in understanding the nature of the 'laws of nature', fundamental constants, space, time, number, information, consciousness..."
I thought you described the epistemic dilemma of the foundations of mathematics and the sciences very well. You said "...Long-standing problems are also in the philosophical foundation of the "Queen of Sciences" - Mathematics, which has been undergoing a crisis of foundations for more than a hundred years..."
I do agree that this crisis that we have been facing for over a century is a very recondite dilemma. The formidability of more foundational issues is stark. I must note an amusing irony that many (pure) mathematicians are of the mind that matters that are deemed "philosophical" are not to be taken seriously. Many 'real' scientists dismiss the credentials and (oftentimes) the entirety of the discipline of Philosophy as a whole, while [in the past at least] philosophers have yearned to be taken seriously. Many professional philosophers today still want to collaborate with mathematicians and physicists. I think we are starting to see it more. I think (and hope) that professional scholars are starting to realize that interdisciplinary work is the only way that we are going to be able to solve the world's most challenging problems. You briefly articulated this "...which mathematicians, as the philosopher S. Cherepanov notes, tried to overcome by inadequate methods.[11] A century of fuss and zero results! [12]..."
On Gödel's famous work you wrote "...The theorems revealed the limitations of the approaches of the Hilbert program..." This is very true, unfortunately. You worded this wonderfully "...Closing the problem of the justification of mathematics on mathematics itself, formalism replaced the question of the truth of its statements with the requirement of consistency.[13]..."
"...Mathematicians, logicians, and philosophers give very different, sometimes even polar, estimates of the historical significance of Godel's theorem. Bertrand Russell assessed the results of Goedel's logical discoveries: "Contrary to popular misconception, Goedel's incompleteness theorems do not imply that certain truths will remain forever unknown. Furthermore, it does not follow from these theorems that human cognition is limited in any way. No, theorems only show the weaknesses and shortcomings of formal systems."[13] To me, this can be related back to the fact that mathematicians do not (usually) respect philosophers on foundational mathematics- to the behest of philosophers whom have yearned to work alongside rigorous academic scholars. For instance, one place where a specialized mathematician (logician) has noticed that philosophers have committed acts of heresy is to extend Gödel's famous incompleteness theorems to be applicable to minds- you wrote "...Gödel's theorems reflect the fundamental feature of knowledge - openness and incompleteness of the cognition process, and on the other hand, the ontognoseological inferiority of formal systems..." Penrose was infamously guilty of doing just this in your quote, it applies to the former).
"...Due to the unsolved problem of justification of Mathematics, paradigm problems in Computational mathematics have arisen. Mathematician, expert in the field of artificial intelligence Alexander Narin'yani in the article "Mathematics XXI - a radical paradigm shift. Model, not an Algorithm" notes: "Computational mathematics is in a deepening crisis, becoming increasingly inadequate in the context of growing demands for practice. At the moment, Computational mathematics has no conceptual ideas for breaking this impasse....»[15]..." I think I agree with Narin'yani; I might not fully agree. It appears that (for instance) many of the open problems (of the Clay Institute- the Millennium Problems) are likely undecideable. However, I do think that computational mathematics will not get anywhere until the philosophy side is resolved ("metamathematics"). I suppose I am in agreement with A.Narin'yani.
Perhaps I am misunderstanding- "the ternary system of calculus and 'qutrits' will be used in future quantum computers..." Quantum computers take advantage of their ability to do multiple computations at once and a classical Boolean (hence, qubit) operation will result in only one of two 'things' (e.g. answers).
However, I do agree that "...So, the mentioned conceptual problems in Computational mathematics are the result of the unsolved problem of the 'foundations of mathematics'..." (p.2). I too agree with Bukin. I also can't help but think about other archetypical conceptions which too are binary in nature such as the Ying and Yang in Daoism. I think you developed the notion of triads very well. In quantum computing, the superposition has two and only two possibilities. It's very interesting to introduce another 'option'.
"...These triads have the form: limit, boundless, and harmony; odd, even, and even-odd number..." You mentioned an "even-odd: number- is this some sort of superposition between even and odd? Cantor demonstrated the wonderful idea of hierarchies of infinities. A notion of 'quantum mathematics' seems unusual and unnecessary- albeit it is interesting.
First of all, you wrote very clearly and you also added historical basis for your essay too. I think the following quote succinctly summed your thesis on page 4 "...Without solving this fundamental problem, taking into account the current state of computational mathematics and the development of computer technology, it is not possible to find an exact answer to dialectic questions: «Decidability â†" Undecidability», «Computability â†" Uncomputability», "Predictability â†" Unpredictability"..."
I also agree with "...The solution to the problem of the ―foundations of Mathematicsâ€-, and therefore knowledge in general, is the solution to the problem of modeling (constructing) the ontological basis of knowledge..."
You quoted Nikolai Kuzansky ―A part is not known without knowing the whole, since a part is measured by the whole.â€-[38] I think that I share this belief too. On page 5-6 you had many equation-like fragmented sentences as such "...Triune (absolute, ontological) space is the limit value (existential-extremum) of the absolute forms of the existence of matter (absolute states = ontological framework): linear state (absolute continuum) + vortex state (absolute discretuum) + wave state (absolute dis-continuum) = triune ( ontological, absolute) field. Its eidos (ultimate geo-geometric images-ideas): ―cubeâ€- + ―sphereâ€- + ―cylinderâ€- represent the absolute (natural) coordinate system of the Universum being as an eternal holistic process of generation of meanings and structures..." Perhaps this was a stylistic manner of writing. In conclusion, I think that you promoted Philosophy very well at the end of the essay. Rigorous philosophy is pedantic and very broad. I too agree with you that it is absolutely vital for scholars.
All in all, I thought that you wrote an excellent essay. It was very thought provoking. I will give it high marks; I'd like to email you. I'm interested to know about your credentials. Do you have a PhD in Philosophy or Physics?
Well done,
Dale Gillman