## Some thoughts concerning a general theory of business 28: a second round discussion of general theories as such, 3

This is my 28th 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 and 27.)

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 Part 26 and Part 27, to add to the foundation that I have been discussing theories of business in terms of, and as a continuation of the Parts 1-8 narrative that I began all of this with. More specifically, I used those two postings to begin a more detailed analysis of axioms per se, 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 comprised of their particular sets of those 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 enclosed 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.

Any general theory of business, like any organized body of scientific theory would fit the second of those basic patterns as discussed here and particularly in Part 27. My goal for this posting is to continue that line of discussion, and with an increasing focus on the also-empirically grounded theories of the second type as just noted, and with an ultimate goal of applying the principles that I discuss here to an explicit theory of business context. That noted, I concluded Part 27 stating that I would turn here to at least begin to examine:

• The issues of completeness and consistency, as those terms are defined and used in a purely mathematical logic context and as they would be used in any theory that is grounded in descriptive and predictive enumerable form. And I will used that more familiar starting point as a basis for more explicitly discussing these same issues as they arise in an empirically grounded body of theory too.

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

And I begin addressing the first of those points by citing two landmark works on the foundations of mathematics:

• Whitehead, A.N. and B. Russell. (1910) Principia Mathematica (in 3 volumes). Cambridge University Press.

• And Gödel’s Incompleteness Theorems.

Alfred North Whitehead and Bertrand Russell set out to develop and offer a complete axiomatically grounded foundation for all of arithmetic, as the most basic of all branches of mathematics in their above-cited magnum opus. And this was in fact viewed as a key step realized, in fulfilling the promise of David Hilbert: a renowned early 20th century mathematician who sought to develop a comprehensive and all-inclusive single theory of mathematics as what became known as Hilbert’s Program. All of this was predicated on the validity of an essentially unchallenged metamathematical axiomatic assumption, to the effect that it is in fact possible to encompass arbitrarily large areas of mathematics, and even all of validly provable mathematics as a whole, into a single finite scaled, completely consistent and completely decidable set of specific axiomatic assumptions. Then Kurt Gödel proved that even just the arithmetical system offered by Whitehead and Russell can never be complete in this sense, from how it would of necessity carry in it an ongoing requirement for adding in more new axioms to what is supportively presumed for it, and unending and unendingly so if any real comprehensive completeness was to be pursued. And on top if that, Gödel proved that it can never be possible to prove with comprehensive certainty that such an axiomatic system can be completely and fully consistent either! And this would apply to any abstractly, enclosed axiomatic system that can in any way be represented arithmetically: as being calculably enumerable. But setting aside the issues of a body of theory facing this type of limitation simply because it can be represented in correctly formulated mathematical form, for the findings developed out of its founding assumptions (where that might easily just mean larger and more inclusive axiomatically enclosed bodies of theory that do not depend on outside non-axiomatic assumptions for their completeness or validity – e.g. empirically grounded theories), what does this mean for explicitly empirically grounded bodies of theory, such as larger and more inclusive theories of science, or for purposes of this posting, of business?

I begin addressing that question, by explicitly noting what has to be considered the single most fundamental a priori axiom that underlies all scientific theory, and certainly for all bodies of theory such as physics and chemistry that seek to comprehensively descriptively and predictively describe what in total, would include the entire observable universe, and from its big bang origins to now and into the distant future as well:

• Empirically grounded reality is consistent. Systems under consideration, as based at least in principle on specific, direct observation might undergo phase shifts where system-dominating properties take on more secondary roles and new ones gain such prominence. But that only reflects a need for more explicitly comprehensive theory that would account for, explain and explicitly describe all of this predictively describable structure and activity. But underlying that and similar at-least seeming complexity, the same basic principles and the same conceptual rules that encode them for descriptive and predictive purposes, hold true everywhere and throughout time.

• To take that out of the abstract, the same basic types of patterns of empirically observable reality that could be representationally modeled by descriptive and predictive rules such as Charles’ law, or Boyle’s law, would be expected to arise wherever such thermodynamically definable systems do. And the equations they specify would hold true and with precisely the same levels and types of accuracy wherever so applied.

So if an axiomatically closed, in-principle complete in and of itself axiomatic system, and an enclosed body of theory that would be derived from it (e.g. Whitehead’s and Russell’s theory of arithmetic) cannot be made fully complete and consistent, as noted above:

• Could grounding a body of theory that could be represented in what amounts to its form and as if a case in point application of it, in what amounts to a reality check framework of empirical observation allow for or even actively support a second possible path to establishing full completeness and consistency there? Rephrasing that, could the addition of theory framing and shaping outside sourced information evidence, or formally developed experimental or observational data, allow for what amounts to an epistemologically meaningful grounding to a body of theory through inclusion of an outside-validated framework of presumable consistency?

• Let’s stretch the point made by Gödel, or at least risk doing so where I still at least tacitly assume bodies of theory that can in some meaningful sense be mapped to a Whitehead and Russell type of formulation of arithmetic, through theory-defined and included descriptive and predictive mathematical models and the equations they contain. Would the same limiting restrictions as found in axiomatically enclosed theory systems as discussed here, also arise in open theory systems so linked to them? And if so, where, how and with what consequence?

As something of an aside perhaps, this somewhat convoluted question does raise an interesting possibility as to the meaning and interpretation of quantum theory, and of quantum indeterminacy in particular, with resolution to a single “realized” solution only arrived at when observation causes a set of alternative possibilities to collapse down to one. But setting that aside, and the issue of how this would please anyone who still adheres to the precept of number: of mathematics representing the true prima materia of the universe (as did Pythagoras and his followers), what would this do to anything like an at least strongly empirically grounded, logically elaborated and developed theory such as a general theory of business?

I begin to address that challenge by offering a counterpart to the basic and even primal axiom that I just made note of above, and certainly for the physical sciences:

• Assume that a sufficiently large and complete body of theory can be arrived at,

• That would have a manageable finite set of underlying axiomatic assumptions that would be required by and sufficient to address any given empirically testable contexts that might arise in its practical application,

• And in a manner that at least for those test case purposes would amount to that theory functioning as if it were complete and consistent as an overall conceptual system.

• And assume that this reframing process could be repeated as necessary, when for example disruptively new and unexpected types of empirical observation arise.

And according to this, new underlying axioms would be added as needed, when specifically faced and once again particularly when an observer is faced with truly novel, disruptively unexpected findings or occurrences – of a type that I have at least categorically raised and addressed throughout this blog up to here, in business systems and related contexts. And with that, I have begun addressing the second of the three to-address topics points that I listed at the top of this posting:

• 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 or discarded as axioms per se?

I am going to continue this line of discussion in a next series installment, beginning with that topics point as here-reworded. And I will turn to and address the third and last point of that list after that, turning back to issues coming from the foundations of mathematics in doing so too. (And I will finally turn to and more explicitly discuss issues raised in a book that I have been citing here, but that I have not more formally gotten to in this discussion up to here, that has been weighing on my thinking of the issues that I address here:

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

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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