## Some thoughts concerning a general theory of business 29: a second round discussion of general theories as such, 4

This is my 29th installment to a series on general theories of business, and on what general theory means as a matter of underlying principle and in this specific context (see Reexamining the Fundamentals directory, Section VI for Parts 1-25 and its Page 2 continuation, Section IX for Parts 26-28.)

I began this series in its Parts 1-8 with an initial orienting discussion of general theories per se, with an initial analysis of compendium model theories and of axiomatically grounded general theories as a conceptual starting point for what would follow. And I then turned from that, in Parts 9-25 to at least begin to outline a lower-level, more reductionistic approach to businesses and to thinking about them, that is based on interpersonal interactions. Then I began a second round, next step discussion of general theories per se in Parts 26-28 of this, building upon my initial discussion of general theories per se, this time focusing on axiomatic systems and on axioms per se and the presumptions that they are built upon.

More specifically, I have used the last three postings to that progression to at least begin a more detailed analysis of axioms as assumed and assumable statements of underlying fact, and of general bodies of theory that are grounded in them, dividing those theories categorically into two basic types:

• Entirely abstract axiomatic bodies of theory that are grounded entirely upon sets of a priori presumed and selected axioms. These theories are entirely encompassed by sets of presumed fundamental truths: sets of axiomatic assumptions, as combined with complex assemblies of theorems and related consequential statements (lemmas, etc) that can be derived from them, as based upon their own collective internal logic. Think of these as axiomatically closed bodies of theory.

• And theory specifying systems that are axiomatically grounded as above, with at least some a priori assumptions built into them, but that are also at least as significantly grounded in outside-sourced information too, such as empirically measured findings as would be brought in as observational or experimental data. Think of these as axiomatically open bodies of theory.

I focused on issues of completeness and consistency in these types of theory grounding systems in Part 28 and briefly outlined there, how the first of those two categorical types of theory cannot be proven either fully complete or fully consistent, if they can be expressed in enumerable form of a type consistent with, and as such including the axiomatic underpinnings of arithmetic: the most basic of all areas of mathematics, as formally axiomatically laid out by Whitehead and Russell in their seminal work: Principia Mathematica.

I also raised and left open the possibility that the outside validation provided in axiomatically open bodies of theory, as identified above, might afford alternative mechanisms for de facto validation of completeness, or at least consistency in them, where Kurt Gödel’s findings as briefly discussed in Part 28, would preclude such determination of completeness and consistency for any arithmetically enumerable axiomatically closed bodies of theory.

That point of conjecture began a discussion of the first of a set of three basic, and I have to add essential topics points that would have to be addressed in establishing any attempted-comprehensive bodies of theory: the dual challenges of scope and applicability of completeness and consistency per se as organizing goals, and certainly as they might be considered in the contexts of more general theories. And that has left these two here-repeated follow-up points for consideration:

• How would new axioms be added into an already developing body of theory, and how and when would old ones be reframed, generalized, limited for their expected validity and made into special case rules as a result, or be entirely discarded as organizing principles there per se.

• Then after addressing that set of issues I said that I will turn to consider issues of scope expansion for the set of axioms assumed in a given theory-based system, and with a goal of more fully analytically discussing optimization for the set of axioms presumed, and what that even means.

My goal for this series installment is to at least begin to address the first of those two points and its issues, adding to my already ongoing discussion of completeness and consistency in complex axiomatic theories while doing so. And I begin by more directly and explicitly considering the nature of outside-sourced, a priori empirically or otherwise determined observations and the data that they would generate, that would be processed into knowledge through logic-based axiomatic reasoning.

Here, and to explicitly note what might be an obvious point of observation on the part of readers, I would as a matter of consistency represent the proven lemmas and theorems of a closed body of theory such as a body of mathematical theory, as proven and validated knowledge as based on that theory. And I correspondingly represent open question still-unproven or unrefuted theoretical conjectures as they arise and are proposed in those bodies of theory, as potentially validatable knowledge in those systems. And having noted that point of assumption (presumption?), I turn to consider open systems as for example would be found in theories of science or of business, in what follows.

• Assigned values and explicitly defined parameters, as arise in closed systems such as mathematical theories with their defined variables and other constructs, can be assumed to represent absolutely accurate input data. And that, at least as a matter of meta-analysis, even applies when such data is explicitly offered and processed through axiomatic mechanisms as being approximate in nature and variable in range; approximate and variable are themselves explicitly defined, or at least definable in such systems applications, formally and specifically providing precise constraints on the data that they would organize, even then.

• But it can be taken as an essentially immutable axiomatic principle: one that cannot be violated in practice, that outside sourced data that would feed into and support an axiomatically open body of theory, is always going to be approximate for how it is measured and recorded for inclusion and use there, and even when that data can be formally defined and measured without any possible subjective influence – when it can be identified and defined and measured in as completely objective a manner as possible and free of any bias that might arise depending on who observes and measures it.

Can an axiomatically open body of theory somehow be provably complete or even just consistent for that matter, due to the balancing and validating inclusion of outside frame of reference-creating data such as experientially derived empirical observations? That question can be seen as raising an interesting at least-potential conundrum and certainly if a basic axiom of the physical sciences that I cited and made note of in Part 28 is (axiomatically) assumed true:

• Empirically grounded reality is consistent across time and space.

That at least in principle, after all, raises what amounts to an immovable object versus an unyieldable force type of challenge. But as soon as the data that is actually measured, as based on this empirically grounded reality, takes on what amounts to a built in and unavoidable error factor, I would argue that any possible outside-validated completeness or consistency becomes moot at the very least and certainly for any axiomatically open system of theory that might be contemplated or pursued here.

• This means that when I write of selecting, framing and characterizing and using axioms and potential axioms in such a system, I write of bodies of theory that are of necessity always going to be works in progress: incomplete and potentially inconsistent and certainly as new types of incoming data are discovered and brought into them, and as better and more accurate ways to measure the data that is included are used.

Let me take that point of conjecture out of the abstract by citing a specific source of examples that are literally as solidly established as our more inclusive and more fully tested general physical theories of today. And I begin this with Newtonian physics as it was developed at a time when experimental observation was limited for the range of phenomena observed and in the levels of experimental accuracy attainable for what was observed and measured, so as to make it impossible to empirically record the types of deviation from expected sightings that would call for new and more inclusive theories, with new and altered underlying axiomatic assumptions, as subsequently called for in the special theory of relativity as found and developed by Einstein and others. Newtonian physics neither calls for nor accommodates anything like the axiomatic assumptions of the special theory of relativity, holding for example that the speed of light is constant in all frames of reference. More accurate measurements as taken over wider ranges of experimental examination of observable phenomena forced change to the basic underlying axiomatic assumptions of Newton (e.g. his laws of motion.) And further expansion of the range of phenomena studied and the level of accuracy in which data is collected from all of this, might very well lead to the validation and acceptance of still more widely inclusive basic physical theories, and with any changes in what they would axiomatically presume in their foundations included there. (Discussion of alternative string theory models of reality among other possibilities, come to mind here, where experimental observational limitations of the types that I write of here, are such as to preclude any real culling and validating there, to arrive at a best possible descriptive and predictive model theory.)

At this point I would note that I tossed a very important set of issues into the above text in passing, and without further comment, leaving it hanging over all that has followed it up to here: the issues of subjectivity.

Data that is developed and tested for how it might validate or disprove proposed physical theory might be presumed to be objective, as a matter of principle. Or alternatively and as a matter of practice, it might be presumed possible to obtain such data that is arbitrarily close to being fully free from systematic bias, as based on who is observing and what they think about the meaning of the data collected. And the requirement that experimental findings be independently replicated by different researchers in different labs and with different equipment, and certainly where findings are groundbreaking and unexpected, serves to support that axiomatic assumption as being basically reliable. But it is not as easy or as conclusively presumable to assume that type of objectivity for general theories that of necessity have to include within them, individual human understand and reasoning with all of the additional and largely unstated axiomatic presumptions that this brings with it, as exemplified by a general theory of business.

That simply adds whole new layers of reason to any argument against presumable completeness or consistency in such a theory and its axiomatic foundations. And once again, this leaves us with the issues of such theories always being works in progress, subject to expansion and to change in general.

And this brings me specifically and directly to the above-stated topics point that I would address here in this brief note of a posting: the determination of which possible axioms to include and build from in these systems. And that, finally, brings me to the issues and approaches that are raised in a reference work that I have been citing in anticipation of this discussion thread for a while now in this series, and an approach to the foundation of mathematics and its metamathematical theories that this and similar works seek to clarify if not codify:

• Stillwell, J. (2018) Reverse Mathematics: proofs from the inside out. Princeton University Press.)

I am going to more fully and specifically address that reference and its basic underlying conceptual principles in a next series installment. But in anticipation of doing so, I end this posting with a basic organizing point of reference that I will build from there:

• The more traditional approach to the development and elaboration of mathematical theory, and going back at least as far as the birth of Euclidean geometry, was one of developing a set of axioms that would be presumed as if absolute truths, and then developing emergent lemmas and theories from them.

• Reverse mathematics is so named because it literally reverses that, starting with theories to be proven and then asking what are the minimal sets of axioms that would be needed in order to prove them.

My goal for the next installment to this series is to at least begin to consider both axiomatically closed and axiomatically open theory systems in light of these two alternative metatheory approaches. And in anticipation of that narrative line to come, this will mean reconsidering compendium models and how they might arise as need for new axiomatic frameworks of understanding arise, and as established ones become challenged.

Meanwhile, you can find this and related material about what I am attempting to do here at About this Blog and at Blogs and Marketing. And I include this series in my Reexamining the Fundamentals directory and its Page 2 continuation, as topics Sections VI and IX there.

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