Vladimir Rogozhin
I believe that this is the problem of ontological justification /substantiation of the entire system of knowledge and cognition, and above all, the solution of the "millennium problem" No. 1 (not only for mathematicians) - the ontological justification /substantiation of mathematics (ontological basification).
Philosophical precepts have given us great minds.
"Philosophy is written in a majestic book (I mean the Universe), which is constantly open to our gaze, but only those who first learn to comprehend its language and interpret the signs with which it is written can understand it. It is written in the language of mathematics, and signs her - triangles, circles and other geometric figures, without which a person could not understand a single word in it; without them, he would be doomed to wander in the dark through the labyrinth "(Galileo "Assay Master").

Why in the first place "triangle"? Let us also recall Plato's "celestial triangle" (equilateral).
Fields Prize winner Vladimir Voevodsky (1966-2017): "What we now call the crisis of Russian science is not only a crisis of Russian science. There is a crisis of world science. Real progress will consist in a very serious fight between science and religion, which will end in their association."
I think there will be a "serious fight" here: Meta Axiom "In the Beginning was the Logos…/ Ἐν ἀρχῇ ἦν ὁ λόγος" VS. Hypothesis "In the Beginning was a" Big Bang ..."
where LOGOS is a META-LAW that governs the Universe (in the spirit of Heraclitus)
[http://baaltii1.livejournal.com/198675.html]
How to draw the LAW in the language of mathematics?
This is an equilateral triangle (since ancient Egypt).
On the other hand, to interpret in the modern language of physics (taking into account all the problems in its metaphysical / ontological foundations) the dialectical-ontological triad "being - nothing/otherbing - becoming" and draw it. By the way, G. Hegel tried to do this...
What is your attitude to the covenant of the mathematician Vladimir Voevodsky, the developer of the univalent foundations of mathematics?

"Truth should be drawn..." (A. Zenkin "SCIENTIFIC COUNTER-REVOLUTION IN MATHEMATICS")
[http://www.ccas.ru/alexzen/papers/ng-02/contr_rev.htm]

Understanding is "events of grasping the structure" (G. Gutner "Ontology of Mathematical Discourse"). So we need to grasp the "structure" (super-structure, primordial generating structure).
And the central problem: What holds, develops and directs this generative "super-structure"?

Therefore, here it is also necessary to take a fresh look (understand) matter: matter is that from which all meanings, forms of structure are born.
Good tips from A. Einstein:
“I like to experience the universe as one harmonious whole. Every cell has life. Matter, too, has life; it is energy solidified" and his famous metaphysical maxim: "God does not play dice with the Universe." How can this maxim be understood, taking into account the fact that quantum theory and the theory of relativity are phenomenological (parametric, operationalistic, "effective") theories without ontological justification (ontological basification)?
Tradition gives us the answer: God created the Universe according to the Logos (Law) (In the Beginning was the Logos..."
By the way, according to the ancient Greek dictionary, the word "Logos" has 34 clusters of meanings, that is, more than a hundred meanings. The main one for the "sciences of nature", "sciences of the spirit" and theology "Logos" = Law.
Here are its invariants to consider and draw an ontological frame, frame, foundation of Knowledge. By the way, the concept of "meta-law" is used by Lee Smolin in "Time Reborn: From the Crisis in Physics to the Future of the Universe".

    Dear PinkJunglefowl,
    Certainly, your essay deserves the highest marks. I subscribe to almost every word you say.
    In my essay, I drew attention to the fact that the generally accepted concept does not see new phenomena in experiments, due to its abstract representations.
    If you do not pay attention to the obviously biased imposition of distorted experimental results (for example, in the experiments of Michelson and the Pioneers), then, in my opinion, words will remain words, while facts may remain recognized facts and the basis of new theories of reality. Hope to see your comments.
    I wish you success!

      Andrew Beckwith Thanks for your attention, MagnoliaCentipede. To answer your Question A, as I mention in the essay (page 4, 2nd paragraph), our "postulation" of the initial, simplest possible (practically structureless) world system configuration is not a problem as it includes physically and logically inevitable interaction components, while all the observed world structures and dynamical laws are progressively and rigorously derived due to the unreduced interaction analysis (hence its crucial advantages). As to the causal, physical origin of that starting configuration itself, we can only make some reasonable guesses that can hardly be clearly confirmed, since the real world we obtain and can measure doesn't depend on those processes before it was born (we would obviously obtain here an infinite and practically senseless sequence of queries about what was before each past stage).

      Question B, about "Gödel’s incompleteness" and other uncertainties: See page 3, 1st paragraph and page 4, 2nd paragraph in the essay. Gödel’s theorem is correct, but physically it is even "too correct", i.e. trivial, because in real-system translation it means that you must know the initial system components to make any definite judgement about further system development, or in other words, that every structure has irreducible links to other structures. This is true, but in reality nothing prevents us from knowing the initial system or interaction process components, especially with the modern technology power. It is not always a trivial, but always feasible task. By contrast, the key point is to correctly obtain interaction products, without any simplification of the process. It is at this point that the essential extension of conventional science results is obtained within the causally complete interaction analysis. Another source of uncertainty that you may have in mind is the dynamically random, unpredictable behavior of real systems, but here again, it's not a problem if we can consistently derive all possible interaction outcomes and their respective probabilities, which is precisely the general result of the same unreduced interaction analysis. We cannot (and I think should not) avoid the probabilistic character of existence, but we must and can know the exact details in all those uncertainties (all real-interaction results and their probabilities). There should be no ad hoc guesses of usual science, in the form of its horribly simplified and abstract "models", but only totally consistent, causally complete analysis of each next interaction level, starting from the provably simplest (and thus inevitable) "primordial" interaction configuration.

          Peter Jackson Thank you for your positive attitude, ThistleLion. There seems to be a real problem with those Gödel’s theorems. 🙂 Too many people are inclined to think that they encode some mysterious and omnipresent "gap" in the knowledge we can acquire or maybe even obscure influences from unknown realities. As I explain in the above answer to another comment, the physical meaning of those purely abstract theorems is quite trivial: every real system or interaction level has irreducible links to other structures (typically by its components), and if we don't know those real interaction contributions, we can hardly obtain the correct interaction results. However, we practically can know all the necessary links and interaction components, using the power of modern technologies etc., up to the natural limits of our empirical capacities (like the "observable universe"), which do not create the essential knowledge incompleteness but only mark its permanently moving and quite causal border. In other words, all the "quantum mysteries" and other "dark matters" are not due to some fundamental science limits (as if encoded in Gödel’s theorems), but result only from artificial restrictions of the standard science framework (involving huge interaction simplification and ad hoc abstract models).

            Andrei Kirilyuk
            What is your view of “the provably simplest (and thus inevitable) "primordial" interaction configuration”? Does it involve a relationship (represented by an equation)? If so, where did the operators and the operands come from (or more correctly, the things that are represented by the operators and the operands)?

                Vladimir Fedorov Thank you, CeruleanJackal. Yes, irrespective of details, it seems we see the real possibility of modern science problems solution in the same direction of a much more "honest", i.e. truly consistent, analysis of real interaction processes involved in the world structure formation and evolution.

                  Lorraine Ford You can find a reference to this simplest interaction configuration in the lowest paragraph of page 5 of the essay. It is both physically and logically inevitable and then does produce all the observed world structures and dynamical laws, if we use the unreduced interaction analysis. This interaction is certainly described by a quite general formalism (well-known in its usual, empirically guessed and restricted interpretation), which involves no limits or special assumptions and represents a truly universal law self-consistently derived within the same interaction analysis. All the participating quantities or "operators" acquire now their essentially extended and universal meaning, applicable to any kind of system at any complexity level. This self-confirming universality and totally consistent problem solutions it provides form the intrinsically complete science framework that can be further tested and constructively used for any system of interest.

                      Andrei Kirilyuk
                      Your essay does not explain where categories/ operands (like energy, mass, momentum, position) come from. These categories only exist as relationships (represented via the use of operators) between other such categories. Relationships (categories, operators, equals signs) are fundamental. Numbers, that apply to these categories, are also fundamental.

                      These are specific aspects of the world. Can I pin you down to specifics? Specifically, are you saying that relationships and numbers are not fundamental?

                          incompleteness of a result, in terms of say proving all of its particular moving parts goes with the territory as what Kurt Goedel was trying to present. I.e. in the case of boundary value problems of physics, there are certain situations in which one has multiple solutions to a set of equations
                          quote
                          I’ll assume for now that by “answer on a math problem” you mean the solution to an equation or a set of equations. In this case, there are certainly instances where there are multiple solutions to a set of equations. For example, if I asked you to find the solutions of x2−4=0
                          , you would say x
                          could be either 2
                          or −2
                          , and both of these would be correct if I put no more restrictions on the problem. However, even with this problem there is a single, well-defined solution set. In this case, the solution set of x2−4=0
                          is given by {2,−2}.

                          For a similar example from calculus, consider finding the anti-derivatives of f(x)=x.
                          The solution set is any function of the form x2+C
                          where C
                          can be an arbitrary constant. Although this shows that infinitely many solutions to the problem exist, there is still a single solution set.

                          There are also many examples of problems which have no solutions. For example, find all real numbers, x,
                          such that x+5=3
                          and x+5=4
                          . The first equation shows us that x=−2
                          , and the second that x=−1
                          . Since these are not equal to one another, this set of equations has no solution. However, it still has a single, well-defined solution set. In this case, the solution set is the empty set, which is typically denoted {}
                          or ∅
                          .

                          For a problem to be considered “well-posed” there should be a single solution set, but this set could contain many solutions, a single solution, or no solution at all. Many algebra or calculus textbook examples are constructed so that there is exactly one, or only a few answers. However, with more interesting problems it’s often a challenge just to prove that a single solution exists, even if you can’t explicitly write that solution down.

                          In fact, many of the biggest unsolved problems in mathematics come down to questions about the existence of solutions to certain equations. For example, the Riemann hypothesis can be phrased as a question about whether or not any zeros of a (complicated) function exist which are away from a certain line in the plane. The Navier-Stokes problem can be phrased as a question about the existence (or non-existence) of badly behaved solutions to a set of partial differential equations. P vs NP can similarly be written as an existence question. In all cases, a single solution set should (hopefully) exist, but whether it is empty, filled by one element, or filled by many elements is a very difficult question. It just so happens that solving any of these problems can win someone a one million dollar prize.
                          end of quote
                          In a word there are many times in which evolution equations have MULTIPLE solutions but due to physics rules, assumed, we only take a few of them . I .e. I saw a liquid Helium Laboratory where only a few of the known Navier Stokes equations were implimented
                          We simply do not have the time in a lot of cases to generate ALL known solutions, and if several equally plausible solutions exist, we will focus on ONE particular solution, and that is the best we can do

                            Lorraine Ford Yes, sure, since I insist that all interaction details are important, this means that all relations and quantities matter. Everything is expressed in suitable mathematical form with extended interpretation and unrestricted universality. However, as I mention in the essay, this particular paper is not about those mathematical details but rather about the related general approach, which is essentially different from that of conventional science. This appears to be important too, as previous experience shows. When I use seemingly convincing mathematical arguments, they often remain without any reaction. The probable reason for that is the absence of a (desirably "accepted") abstract "model" from the start in my approach, while I'm trying to derive all the relations from scratch and in close contact with their reality-based meaning. Figuratively speaking, a "state-function" is a real-system configuration for me, rather than an abstract element of a mathematical "space" of state-functions. That is why there is a sense to discuss this difference in general science approaches (methods): it leads to qualitatively different results and thus a different kind of science, which is the theme of this competition.

                                Andrew Beckwith You say: "We simply do not have the time in a lot of cases to generate ALL known solutions, and if several equally plausible solutions exist, we will focus on ONE particular solution, and that is the best we can do." Apparently, you're approaching the truth by denying it.🙂 Why not, it's a known and quite efficient way, seriously. The point is that we can generate all the solutions, and we must take them all into account, which gives new science and the new, totally realistic picture of the world. I can't go into details here, but even intuitively, our real world is rather like that, isn't it? Many permanently, randomly changing solutions, everything's evolving, getting older, and the time irreversibly flowing. Those obvious features which conventional science can never properly explain, with its "necessary" (artificially imposed) uniqueness of solution.

                                  Andrei Kirilyuk
                                  In your above replies, you wrote about “the provably simplest (and thus inevitable) "primordial" interaction configuration”. However, to derive a “state-function”, either you or a computer need to perform not only conventional mathematical operations (involving lawful relationships with their associated categories, and numbers), but more importantly, you or the computer need to perform algorithmic steps. Are you saying that the “primordial” world is performing these algorithmic steps? After all, state-functions don’t just appear from nowhere: there are procedures that need to be followed to derive them. A “state-function” is hardly a primitive, “primordial” thing.

                                      Lorraine Ford "Primordial" refers just to the initial, practically structureless (but already interacting and potentially rich) state of our world and its state-function describing that real-world configuration at that starting moment. Then the system interaction leads to progressive emergence of our world structures, transforming it from that "primitive" initial configuration to ever more complex structure always described by its respectively more complicated state-function (including ourselves analyzing that state-function configuration🙂). Yes, all those ultimately complex structures of modern world emerge progressively from the deceptively simple initial interaction configuration, if only we do not neglect any "algorithmic steps" in this interaction development. The high structure-creation potential of the starting system configuration (not only this one) is also due to its many-body content, with its own internal interaction between elements (not resolved here, just constituent elements of an originally smooth material). It's basically about high structure-creation power of almost any real, unreduced interaction, ruthlessly "killed" by the technically "convenient" reduction of conventional theory. (However, it was semi-empirically emphasized in "self-organization", or "synergetics", but again within the fatal limitation of the standard description, just the empirical intuition about how external "simplicity" can contain and give rise to eventual complexity.) So, I just follow that unreduced "algorithm" of real-system development described by its respectively evolving state-function. I use analytical methods, but some computer application could help with further details. (In general, however, any real system dynamics is "noncomputable", as opposed to usual computer operation: one more source of usual science problems.)

                                          Andrei Kirilyuk
                                          Re “practically structureless”: But what exactly do you mean by structure; what are the basic elements of structure? Don’t the elementary symbols you would use, to represent structure, correspond to the basic elements of structure?

                                          If a state-function that describes the “real-world configuration at that starting moment” exists, then this state-function would seem to represent the knowledge of something with a God’s eye view, or a human being’s view, of the “configuration at that starting moment”. Sorry, but I would doubt that there would have been anything with a God’s eye view (i.e. an objective view) in the early world. Surely there would have been only primitive subjective views and interactions? The same with “self-organisation”: isn’t self-organisation just a God's-eye, bigger-picture view, that doesn’t exist?

                                          Re “In general, however, any real system dynamics is "noncomputable", as opposed to usual computer operation: one more source of usual science problems.”: When you say this, are you in effect agreeing with my essay that physics has no explanation for why the system is moving/ what is the source of movement/ what is driving the system, apart from laws of nature which really only say: “IF something moves THEN something else moves”? However, what is driving the system is also a part of the whole system, and so in order to represent the whole system, what is driving the system must be represented somehow, if only algorithmically. (I would say that, instead of nothing driving the system, it is matter that is driving the system; this can only be represented algorithmically.)

                                              Lorraine Ford By "practically structureless" I mean the absence of any real structures observable in this world, including all elementary particles. And since "structure" also means "space", there is no common 3D space at the starting system configuration: Even the "empty" (not really) space emerges (together with time) from this interaction development, in a well-defined form and process.

                                              The interaction unfolding towards real structure is encoded in its "potential energy" (and participating interaction components, with the "simplest" possible but certain, nonzero content), while the state-function describes rather the (evolving) result of the interaction process (it is our "unknown solution to be found", explicitly determined at the start only for that, starting, practically structureless configuration). So, the "God's eye" in your words "sees" that starting interaction potential (and the system components), or the "hidden complexity" of the later emerging world.

                                              For the "source of movement", the standard science would evoke "forces" or "energy distribution" (let alone "curved time and space"🙂), which are superficial and non-universal answers. Moreover, I state even that the problem is not only why the system is moving, but even what motion is, fundamentally and universally, beyond empirical definitions. I provide such motion definition and without going in details here, I would insist that any causally complete (i.e. truly rigorous) science must consistently derive such fundamental motion definition in terms of emerging process. Indeed, how can one discuss any reliable "laws of motion" without knowing what motion actually is, beyond empirical word plays like "motion is change"?

                                                  Andrei Kirilyuk
                                                  Thanks for explaining. But in my view, the basic elements of structure are categories and relationships, as well as numbers. It doesn’t matter what the particular categories are, or what the particular relationships are: it could be mass or position, or momentum or energy. The basic necessary types of element (the types of element are categories, relationships, and numbers) were there from the start.

                                                  Also in my view, “motion” could be defined as number jumps in one or more of the categories (e.g. the position category), the consequence of which is further number jumps in other categories, due solely to the existence of lawful relationships or ratios between categories. Number jumps are motion, because there is no such thing as smooth, infinitesimal number change. Thanks for answering my questions! 🙂

                                                    6 days later

                                                    Eckard Blumschein In the essay, I specify what it means, how it can be achieved and why it is critically important right now. Some of us can still remember times when fundamental science was great (or at least had that real social status). But now we need even much more than that previous high level…

                                                      Like the "Global Externalities" essay, "Causually Complete Science" does rightfully point out the dominant economic power structures that saturate our cultures and thus our science organization, this essay saying that feature more related to our social system and planetary civilization and thus the "socially important reason why we should make science great again by passing to the superior level of causally complete, intrinsically realistic and therefore truly reliable knowledge." Current essay is rather dense. Not sure how you pass into this truly reliable knowledge. Probably requires closer reading on my part.

                                                        James Hoover Thank you for your attention. It seems we agree on the necessity of essential changes in science content, organization and practice, leading to a superior level of results. I just propose a rigorously confirmed way of such changes, although presented in the form of a general description in this essay. However, this general presentation has its own importance, since it emphasizes the urgent need for such a change, if we want that the entire science enterprise preserves its role as the irreplaceable progress driver and truly objective, consistent and reliable form of knowledge.