Dear Jorge, dear Rodolfo,
The first point having impressed me in your essay is your refusal of nihilism, a rare thing in modern philosophy of science. Of course, at first glance, philosophy of science is not directly concerned by nihilism, nor by anti-nihilism, respectively the denial and the defense of values, and even of the pertinence characterizing the notion of value as such. Or, in other words, notion of value seems to concerning moral philosophy, perhaps social sciences and so on, but not philosophy of science.
In fact, it is not so simple.
Alexey and Lev Burov - they posted a positive comment on your forum - show in their essay that the current or at least mainstream conception of science reducing the latter to "formal systems" and subsequently denying the value which makes the difference between arbitrary formal things and approaches having a sense, systematically leads to self-refutation Moira and Eileithyia for Genesis by Alexey and Lev Burov聽. Personally, I am glad to see that the back ground of your essay is the same.
Anyway, to begin, I entirely agree that modern science during a long time-period had been imprisoned by (in?) the mechanism paradigm, and that modern knowledge inciting us to go beyond this paradigm opens up new horizons. Yes, but the mechanism paradigm is not only a path leading to a deadlock. This paradigm, by its nature, denotes something deeper allowing us to transcend a mechanist vision of the world while remaining in a scientific vision of the world. I will try to explain the foregoing. Physical knowledge is characterized by symmetry in prediction and retro-diction. This point makes all the difference between physics and non-physics. Physical time-symmetry without which physics would not be physics is based on group-theoretic foundations. You can find more details in my own essay Daring Group-theoretic Foundations of Biological Evolution despite Group Theory? by Peter Martin Punin. Now, from a physical standpoint - with regard to purely algebraic motivations, it is another story - the epistemic roots of group theory initially, please pay attention to "initially", are inscribed in mechanics stricto sensu. Consider the watch on your arm. This mechanism expresses a superposition of several expressions of the same rotation group, and this point allows at least ideally to predict as well to retro-dict the passed and future states of the watch from any state chosen as t0-reference. In the same sense, Galileo referred to group-theory avant la lettre, without being aware of it. Nevertheless, that what we call today the Galileo-group is fundamental in Newtonian mechanics. Always from our present-day perspective, we say that the Galileo-group is a subgroup of the Lorentz-Poincare-group in SR, whereas at the RG-level, Riemann manifolds are to be interpreted as a transformations group over a set of local approximations formalized the Lorentz-Poincare-group. So, in macroscopic physics, you find a lot of group-theoretic structures expressed by mechanical systems and vice versa.
Now, saying that quantum physics essentially transcends the mechanism paradigm, you are absolutely right. Of course, quantum entities cannot be reduced to sole "objects", and even the notion of entity is reducing, but for human minds it is hard to find better. Of course, simplifying bit, a Hilbert space comprises connected potentialities potentially connected beyond their actual (?) realization. And so on, your statement is complete through its compactness; so repeating it would be useless. Well, but how to explain that we are not merely lost at the quantum level?
Here, the answer is compact: it is in turn for group-theoretic reasons. Hilbert spaces, the unique possible "environment" of Schr枚dinger's wave is a group-theoretic "entity". Heisenberg's matrix approach equivalent of Schr枚dinger's approach is based on non-commutative matrix group theory. Top-down causation as you evoke it from a rigorous physical perspective, is formalizable in terms of semi-groups; knowing that the latter, mathematically speaking, require previously given full groups. This point seems paradoxical, but it is not the case: To describe top-down causation to be formalized in terms of semi-groups, you need a Hilbert space being a full group-theoretic entity.
So, on the one hand, I think that the nature of physical laws necessarily comprises group-theoretic foundations, and on the other hand,these group-theoretic foundations allow physics to transcend its initial mechanist limits.
Now, an important point, must be clarified. If group theory funds mechanist as well as non-mechanist physics, it must precede ontologically any form of physics, and "preceding something ontologically" implies "preceding something in an immaterial and eternal way." The foregoing obviously leads to Platonism which does not please everyone. Well, but since you mention Leibniz, recall that Leibniz precisely says that our physical universe, instead of existing, also could not exist. So its existence is a mystery. The eventual existence Platonist mathematical universe would be mysterious, but not "more mysterious" than the existence of our material/physical one. Subsequently, if the ultimate consistency of physics is better ensured by Platonist presuppositions than by the current refusal of Platonism, there is no reason to reject Platonism at the cost of non-consistency.
In my paper you find certain paths denoting that anti-Platonism encounters serious deadlocks, among others the fact that within an universe considered as a historical process, law appearing "with" the phenomena they govern imply circularity.
Evoking yourselves the turtle paradox, you probably agree with the foregoing. Analogously to the turtle problem, an issue like "Where comes the law XX from?" perhaps receive the answer "From UVU." Yes, but the answer will be followed by a new question: "Where comes UVU from?" And so on. Ultimately, we have the choice between and endless, infinite series of questions and answers and questions and answers, or self-reference (implying circularity) or the reference to something transcendent/eternal. Reading your non-nihilist essay, and provided I understand well, I think, you are not opposed to a transcendent/eternal solution. Your reference to Leibniz seems to go in this sense.
But now, a real problem arises. Eternity is eternity, whereas aims and intentions are inscribed in time. So, our contest subject has to be reformulated: How can mathematical laws operate such a passage from eternity to temporality?
Or, more basically: given that physical laws are group-theoretic laws, and that biological evolution as such is not formalizable in terms of group theory, how can mathematical laws operate such a passage from group theory to non-group theory?
Even if both questions are not positioned at the same ontological level, they converge, and this interesting. But the answer is hard to find.
Saying that our universe is life-friendly, or fine-tuned for life, of course you are absolutely right. Nevertheless, fine-tuning is a necessary precondition for the appearance of life, but not a sufficient one. Fine-tuning belongs to physics, so to the aspects of our universe being formalizable in terms of groups and/or semi-groups. The next stages, the passage, under life-friendly conditions, from inert matter to living matter, and then evolution, and then the appearance of thought (this is again another story; see the essay of A.& L. Burov and my comment on their forum Moira and Eileithyia for Genesis by Alexey and Lev Burov globally are not formalizable in terms of group theory.
How to resolve this problem?
Have you an answer beyond the fine-tuning precondition?
In my own paper Daring Group-theoretic Foundations of Biological Evolution despite Group Theory? by Peter Martin Punin , I try to find this kind of answer: Given the essentially group-theoretic foundations of physics, evolution theory should be group theory embedded in non-group-theory. Such a model is possible. For the moment, there is a lot of speculation, but perhaps my approach is less speculative than other approaches doing as if physics would not be physics, or as if universal physical laws exceptionally could be forgotten in evolution theory.
Agreeing the essential of your vision, I would be glad to open a discussion with you.
All the best
Peter
