## Some thoughts concerning a general theory of business 35: a second round discussion of general theories as such, 10

This is my 35th 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 26-34.)

I have been successively discussing three closely interrelated topics points and their ramifications in this series since Part 31, which I repeat here for smoother continuity of narrative:

1. Provide a brief discussion of the generality of axioms, and of how a theory elaboration can lead to reconsideration of the axioms underlying it, where that can lead to accepted axioms already in place, now being reconsidered as special cases of more general principles,

2. Adding, subtracting and modifying such axioms in general, and

3. The whole issue of consistency, and even just within constrained sets of axioms. And I will frame that discussion at least in part, in terms of outside data, insofar as that makes these theories open – and with a reconsideration of what evolutionary and disruptive mean there.

And as part of that still ongoing line of discussion, I have been addressing its Point 3 since Part 34. And to bring this preliminary orienting note up to date as a start to this posting and its discussion, I concluded that installment by stating that I would address the following more derivative points here, doing so in light of what I have been offering in response to Points 1 and 2 and my start to addressing the above Point 3 as well:

• Reconsider the issues of consistency and completeness, in light of the impact created by disruptive and seemingly disruptive examples, as for example raised in Part 33 (with its Higgs field example) and Part 34 (with its three physics sourced examples.)

• This will include, as noted in Part 34, my addressing the challenge of direct and indirect testing of potentially validatable assertions that would fit into an open general theory, and what those terms actually mean in practice: direct, indirect and validatable.

• Yes, that will mean reconsidering my Higgs boson/Higgs field physics example from Part 33 again too. And I will also mean my further considering what evolutionary and disruptive mean: terms and concepts that I have made at least case-by-case level use of throughout this blog. And I will also, of necessity connect back in that to my discussion of compendium theories as discussed in this series in its Parts 2-8.

And with that offered as orientation for what is to follow, I begin with the first of those three to-address topics points and with a bipartite distinction that I made in Part 34 as to how issues such as theory consistency could be considered: as overall and (at least ideally) all-inclusive body of theory spanning determinations, or as case-by-case determinations.

• For a closed axiomatic body of theory such as Euclidean geometry, any such effort, as conceived on a more strictly case-by-case, derivative theorem basis would be grounded entirely upon deductive analytical reasoning, and with a goal of determining adherence to and consistency with the set of underlying axioms in place.

• For an open axiomatic body of theory that begins with such an axiomatic framework but that also allows in (and even requires) outside evidentiary support such as experimental or other empirically sourced data, inductive reasoning becomes essential too – and the starting axioms that might be in place as this evidence is brought in, can be as subject to challenge as any newly proposed derivative theorem or lemma would be.

For purposes of this phase of this overall narrative, I simply point to closed bodies of theory for purposes of what might be considered simplest case comparison, as overall theories of physics, or of business, or of any other such empirically grounded reality are by their very nature, open. And that is where change and disruptive change enters this, where they might arise as a matter of planned intention and as conceived innovation, or without such a priori intention (even if inevitably so and as a result of a “law of unintended consequences.”)

What do simple change and disruptive change mean in this context? I would argue that the most salient general point of distinction between them lies in where they shed light on and differ from the routinely expected and predicted.

• Simple change does not challenge underlying theory, even if it can, or at least should be able to provoke the bringing of theory implementation into focus and even into question. For a physics example of this, I point to the so called paradox of radiation of charged particles in a gravitational field, as discussed in Part 34 (which at the very least had simple change elements to it.) For a business example, I would cite essentially any more-minor business process adjustment, as for example might become necessary if a manufacturer has to change where or how they source some essential supply that would go onto their own products and their production. This obviously does not in any way challenge basic underlying business theory. It most probably would not challenge that enterprise’s basic business model in place or its strategic or operational planning either – unless that is, this now needed change and any realistic accommodations to it would of necessity have disruptive consequences.

• But genuinely disruptive change, and certainly as an extreme for that, can challenge everything. In a physics context consider the accumulation of repeatedly verified and validated experimental findings that were both incompatible with classical physics and even incomprehensible to it, that led to the development of quantum physics. For a business and I add an economics context, consider the advent of electronic computers, and certainly as they transitioned from being rare “not for here” curiosities into their becoming ubiquitous business necessities – and for how that qualitatively transformed the role of and even the nature of information in a business context. Read something of business theory from right before World War II, and from today and look at the difference and at how much of that difference is explicable in information collection, organization, and usage terms.

And this leads me to two observational points, the first of which might be more obvious than the second and particularly for anyone who has been following this blog:

• The truly disruptive can raise red flags in how it sheds questioning spotlights on what was established theory. But it can be better at that than it is at presenting alternatives to those now-old and outdated understandings. Once again, consider quantum physics, or at last its early states where it was agreed to that those new and disruptive experimental findings were valid, but where there was no consensus as to what they collectively meant – no consensus or anything like it as to how they should be interpreted or understood, or of how a new generally stated axiomatically grounded weltanschauung could best be developed around them.

• And this old foundation-breaking without a corresponding at-least quickly following new foundation-making, can only lead to the arising of a (hopefully productive) compendium theory next step that would offer localized explanatory and predictive theory for specific types of disruptive outside data findings, that would at least hopefully serve as puzzle pieces for developing a new more general axiomatically grounded foundation around.

Note that while I have been discussing the above Point 3 here, I have also been explicitly discussing Points 1 and 2 here as well; they cannot in practice be separated from each other in any meaningful discussion of any of them.

And with this, I return again to my Higgs field example of Part 33. What axiomatic assumptions does the presumption of this mechanism for conveying the Higgs force carry with it, aside from the perhaps more obvious ones of its being selected and framed so as to be as consistent as possible with already established particle physics and related theory, and (hopefully) with a mainstream understanding of general relativity too? I have already cited one other axiomatic assumption in passing, with the notion of this serving as a reprise of the old theory of an aether for conveying the electromagnetic force. (OK, photons as force conveying boson particles do not need or use that type of mechanism, by maybe Higgs bosons do …. See this piece on the Michelson–Morley Experiment.)

I am going to continue this discussion in a next series installment where I will continue addressing the issues raised in the first three bullet points of this posting, with a continued focus that is (as here) largely aimed at the above-repeated topics Point 3. And I will continue addressing that, with Point 1 and 2 diversions, in terms of the three derivative topics points that I offered here. And with business theory and business practice in mind as I prepare to do so, I finish this posting by posing a few basic questions, the first of which is a reprise from the first general theory discussion of this series of its Parts 2-8:

• If a general theory, or rather an attempt to develop and organize one remains in effect, in perpetual turmoil from the consistent ongoing arising of the disruptively new and unexpected and from all directions,

• Does that effectively preclude any real emergence from a compendium model overall understanding there with the development of an overarching axiomatically grounded general theory?

• And what does this mean as far as overall body of theory consistency and completeness are concerned?

Addressing those questions will bring me directly into a discussion of the second of those derivative bullet-pointed topics, and a more detailed consideration of:

• Direct and indirect testing of potentially validatable assertions that would fit into an open general theory, and what those terms actually mean in practice: direct, indirect and validatable.

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.

## Some thoughts concerning a general theory of business 34: a second round discussion of general theories as such, 9

This is my 34th 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 26-33.)

I have been discussing three closely interrelated topics points and their ramifications in this series since Part 31, which I repeat here for smoother continuity of narrative:

1. Provide a brief discussion of the generality of axioms, and of how a theory elaboration can lead to reconsideration of the axioms underlying it, where that can lead to accepted axioms already in place, now being reconsidered as special cases of more general principles,

2. Adding, subtracting and modifying such axioms in general, and

3. The whole issue of consistency, and even just within constrained sets of axioms. And I will frame that discussion at least in part, in terms of outside data, insofar as that makes these theories open – and with a reconsideration of what evolutionary and disruptive mean there.

I have focused on and primarily addressed the first two of those points since then, leading me to the third of them: my primary point of consideration for this posting. And looking beyond that, I am planning on addressing the challenge of direct and indirect testing of potentially validatable assertions that would fit into an open general theory, and what those terms actually mean in practice: direct, indirect and validatable.

I add in anticipation of that discussion, that I proposed a possible reconsideration of a currently untested assumption of modern physics in Part 33: the presumed existence of a Higgs field, by suggesting an at-least theoretically possible approach to testing for its actual presence – where no current or foreseeable technologies could make that possible – at least directly. This line of discussion explicitly connects back to the issues of that third topics point and in fact to the first two as well and I will have more to add in addressing all of them in this emerging context. Then after continuing that case in point example, I will discuss two overarching, closely interconnected (if at times opposing) perspectives that enter into essentially every area of consideration here: the challenges of reductionism and of emergent properties.

And I begin addressing all of this by considering what at least in principle, might appear at first to be a simplest case context for discussing the above Point 3: consistency in a closed axiomatic body of theory.

There are two approaches that I could take here in pursuing that “simplest case” line of discussion:

• The question of whether absolute consistency can even be provable in principle, for axiomatic theories of any real complexity, as addressed for example by Gödel’s incompleteness theorem for bodies of theory that would include within them a theory of arithmetic, and

• The impact of apparent emergent inconsistency at a specific instance level, in an already largely developed body of theory.

I at least briefly and selectively addressed the first of those points in Part 30 (and recommend that you review that in anticipation for what is to follow here.) And I made note in that posting of a closely related second issue too: completeness, or rather the possibility of establishing whether any such body of general theory can ever be completely, comprehensively developed in some absolute sense. Both sets of issues: both sets of fundamental constraints prove to be crucially important here and in the context of this posting too.

Let’s at least begin to think about and clarify, and expand upon that point as already discussed here, by considering the second of the above two consistency issue bullet points and in the specific context of a particular widely accepted body of general theory: physics. And to be more precise here, let’s consider the discovery of and theory-based elucidation of some specific empirically observable, experimentally measurable phenomena as case in point examples.

• To bring this out of the abstract, let’s specifically consider solitons as a first such example.

• The unification of electrical and magnetic phenomena as mathematically described by Maxwell’s equations as a second, and

• The so called paradox of radiation of charged particles in a gravitational field as an historically relevant third.

I begin with solitons because their mere empirically observable existence served as direct challenge to some of the basic assumed tenants of classical physics, as then held sway.

• Solitons are stable waveforms that do not disperse or decay and even over extended periods of time. More precisely, if still somewhat loosely stated, they are self-reinforcing wave packets that maintain their general shapes and amplitudes while propagating at constant velocities, with their speed dependent upon their primary frequencies. And they arise and are maintained by mechanisms that serve to cancel out expected nonlinear and dispersive effects, as they would normally arise in the mediums that they exist in, that would be expected to dampen them.

• Mathematically, these solitons present themselves as solutions of a class of weakly nonlinear dispersive partial differential equations so while they did not comfortably conform with the physics in place when they were first formally characterized, they can be and in fact were precisely mathematically described and early on.

And this story does go back a ways. A Scottish civil engineer named John Scott Russell is credited with first formally describing them in 1834 in the professional literature, from when he first observed this phenomenon in the Union Canal in Scotland and then replicated it in a controlled laboratory setting.

On the face of things, solitons appear to violate numerous basic scientific principles, as laid out in classical physics. To cite a specific example there, they specifically appear to violate the second law of thermodynamics, and certainly as that is classically perceived. And in fact the only way to reconcile their existence with the rest of physics as that would serve as an overarching general body of theory, was to completely reframe them and in quantum mechanical terms. (See R. Rajaraman’s 1982 paper on that, as a foundational document as to how that reframing was reached: Solitons and Instantons.)

Why do I include this example here? It arose as a matter of replicably observed reality and it continued to exist as such as an unexplained, or rather an unacceptably explained observed phenomenon that did not fit into a large and seemingly exhaustively tested and proven body of theory, creating an at least apparent paradox problem if nothing else. And it was the accumulation of more and more such “outsider” phenomena that led to the development of quantum mechanics and the more expanded and nuanced body of theory that was finally able to include in this phenomenon too.

Maxwell’s equations offer a fundamentally different, but still parallel lesson here. Electricity and magnetism were well known and studied phenomena in the mid-nineteenth century, when Maxwell and others began an effort to unify them in some way, that led to his conceptual and mathematically stated breakthrough for achieving that goal (see this History of Maxwell’s Equations for further details on that.) And when that unifying description of what would now be called electromagnetism and electromagnetic properties was first published, it was heralded as a fundamental, groundbreaking advancement to the then-prevalent understanding of the physical world that held sway at that time. And it did offer very real and significant descriptive and predictive value. But this body of theory, it turns out, was not as fully consistent with mainstream nineteenth century physics as was then presumed. And some key aspects of it turned out, upon further study to be effectively independent of that physical theory’s basic weltanschauung, and I view that prevailing theory in that way because it did represent the world view of physicists and of scientists in general, of the time.

In a fundamental sense, the electromagnetic theory that Maxwell encoded in his equations, was a forerunner to what in the early twentieth century became known as the special theory of relativity. And his basic theoretical model, as he developed it, translates quite nicely to a general theory of relativity context too, as briefly outlined in Maxwell’s Equations in Curved Spacetime.

Why do I cite this example here? Solitons represent an example of how a replicably observable phenomenon can demand new basic theory and new basic underlying axiomatic assumptions that would underlie that, if it is to be meaningfully included in anything like a coherent and at least overtly gap-free general theory. Maxwell’s equations somewhat fit into the basic, accepted general theory then in place, but they could never fully fit in there because they were ultimately to prove more compatible with a more expansively inclusive, still yet to be discerned alternative axiomatically grounded vision of reality. And with that, I turn to my last such example here, as promised above: the at one time, painfully apparent paradox of radiation of charged particles in a gravitational field, as cited above.

According to every prevailing understanding of how particles at rest should behave in a general theory of relativity context, they should radiate energy and on a largely continuous or at least highly temporally structured ongoing basis. But observationally, they do not. Did this reflect a fundamental error or gap in the general theory of relativity per se? Did this reflect more fundamental problems in the basic, axiomatic assumptions held in place there? Ultimately this paradox was resolved by reconsidering frames of reference, and how those axioms and a range of conclusions derived from them should best be understood and interpreted. So this example adds in the challenge of how best to understand and make use of the axioms that are in place too.

All of the basic issues that I have sought to raise and highlight with my physics examples here, have their counterparts in other large general bodies of theory, any realistic possible theory of business included. And that certainly holds true when such bodies of theory are thought of in terms of the more general principles that I just raised in the context of the above three examples.

I am going to continue this narrative in a next series installment, as briefly outlined above. This will mean reconsidering the issues of consistency and completeness, in light of the impact created by disruptive and seemingly disruptive examples, as for example raised here. And this will include, as noted above, my addressing the challenge of direct and indirect testing of potentially validatable assertions that would fit into an open general theory, and what those terms actually mean in practice: direct, indirect and validatable. Yes, that will mean reconsidering my Higgs boson/Higgs field physics example from Part 33 again too. And I will also mean my further considering what evolutionary and disruptive mean: terms and concepts that I have made at least case by case level use of throughout this blog. And I will also, of necessity connect back in that to my discussion of compendium theories as discussed in this series in its Parts 2-8.

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.

## Some thoughts concerning this blog in our still rapidly developing COVID-19 pandemic crisis, and concerning this crisis itself

I began writing to this blog from my official launch of it on September 14, 2009, with a vast and incredibly audacious goal in mind: to offer at least one person’s perspective on what would amount to an overarching framework for thinking about businesses and the technology that supports them, as an at least relatively unified whole, and not just for a given here-and-now, and not just from the perspective of any one narrower topical context. And my goal there was to develop, write up and offer that as it might apply for the 21st century to come and not just as an historical retrospective. I am now just short of 2800 mostly essay length postings into that effort and I have addressed a fairly wide range of issues that I would argue connect into that. But that noted, I also admit that the more that I write to this and the further that I explore as I think about that vast collections of issues that I could write about here, the further I realize that I have to go to actually address anything like the range of intended coverage that I would ideally be able to approach in this.

One of the core axiomatic assumptions that I have built into all of this from the beginning, and that I have repeatedly explicitly stressed throughout it, is that any such span of time is absolutely certain to face disruptive change and both on a more localized and context-limited scale, but on a larger and more inclusive scale too. We all have to expect to see a steady flow of more gradual evolutionary change that with time can collectively lead to profoundly fundamental change and certainly when comparing across longer timeframes. But it is essentially just as predictable, at least as a matter of abstract principle that we will all face disruptively novel change too. And that can come very quickly. And disruptive by its very nature means unexpected too and certainly for when it arrives and for its specific details – and even when it is recognized as an ongoing possibility.

Such change can be positive or negative, or it can carry elements of both. Most anyone reading this can think back to technological examples of this from their own lifetimes and certainly where a disruptive change bears both positives and negatives,

• I for one, have born witness to the broad sweep of development of computers, starting in my case with early mainframe computers that were built around early solid state and integrated circuit technologies. Wise and knowing people presumed that there would never be any real need for more than a few dozen such devices and even worldwide. Very few indeed envisioned the development of physically small, inexpensive and powerful computers that would become essential commonalities for essentially all businesses, worldwide and for all people in general too – not then. Think of this as an example of an evolutionary pattern that has been punctuated by a succession of massively significant disruptive events too.

• And for a second example, or rather set of them, consider the rise of the internet, with that starting out as a simple network that could only be entered into if you had specific highly technical skills and special access rights. Then it went public and widely so, and even explosively so for how this increased expansion of use and involvement brought in more and more people. The development of the World Wide Web and the earliest web browser that went public was the disruptive fuse that led to that explosion.

• The second half of this example that I would make note of here, can be found in the advent of the interactive, web 2.0 internet in general, and of online social media in particular, with all of this arriving as a consequence of a seeming flood of earlier disruptive innovation and change.

It is all but impossible to step back into the mentality of a pre-computer age and certainly where that would mean throwing aside all of our basic assumptions and understandings that we have developed and both individually and societally. We all take our all but ubiquitous electronic information creation and sharing resources for granted; they are now fundamental to our day-to-day lives. And it is at least as impossible to slip back to the mentality that served as a more common zeitgeist of a pre-social media world too.

• Disruptive change is transformational and the more widely impactful that change, the more profoundly transformational it becomes and both for individuals and their communities and for societies as whole. And such transformation is not reversible.

And this brings me to the world that we are now all immersed in, and that continues to change, and more and more, and more profoundly and more profoundly, and every single day: the world of living with and seeking to survive and thrive in the presence of COVID-19. Right now, most of us are still just really focusing on the “survive” of that.

• COVID-19 is a profoundly transformative event. Will we return to a normal after it has passed? Yes, but it will be a new normal and one that we cannot even really begin to see the outlines of now. And that is fundamental to its being a globally impactful, highly consequential disruptive change per se, and at least as much as it is due to its specific features.

I wrote my first blog postings here thinking of the possible, likely and even inevitable sweep and flow of change and disruptive change to come, in the then nine decades still to come in this century. And I did so with a full awareness that while some of what I offer here might hold more enduring value, much would come to be quaint at best as those years and decades proceed. And now we are all facing and bearing witness to and seeking to endure what is probably the most impactful disruption and disruptive change since World War II. Arguably, you might want to offer a case that I should cite the realization that competitively adversarial development of nuclear weapons had made the doctrine of mutually assured destruction inevitable, as my most recent prior matching or exceeding event. But either way, you have to go back generations for a meaningful point of comparison. And we still have only seen just the beginning of this event: this pandemic. (For some of my thoughts on the globally impactful scope of what is all too likely to come of this, see my postings on this pandemic as offered at Macroeconomics and Business 2, as postings 365 and following. Note: I focus for the most part of the human costs of this as a public health challenge.)

• Life will return to normal after this. Businesses and economies will recover. Globalization and interconnectedness will continue to advance. More will be lost before that can happen than we might consider endurable, but we will endure that anyway. It is just that we will not, and will not be able to return to anything quite like our old normal – just like we can never go back to the pre-computer, pre-internet, pre-social media world that prevailed just a few short decades ago and within the lives of many still with us.

How much of the perspective that I have been offering in Platt Perspective will still hold specific value in that hopefully soon to emerge new world? I would expect some portion of it, and even a relatively significant portion of it to still offer at least some real value there. But the two core ideas – the two core principles that I feel certain will continue to hold true out of all of this are that:

• Change and disruptive change will continue to happen,

• And we need to be aware of, and we need to understand our basic underlying and even automatically presumed axiomatic assumptions.

We are currently facing COVID-19. Other large scale and tremendously impactful disruptive change will happen and yes, just within this century and probably within the next immediately coming decades of it. And yes, that might very well include another pandemic too. We just cannot predict anything like that now, and certainly where that means going beyond abstract principles and expectations. Disruptive is like that.

I finish this brief note by acknowledging that I find it comforting to write postings that address basic issues of business and technology as that affords me a sense of continuity and in ways that I find intellectually engaging. And at least some of my readers seen to appreciate what I continue to offer here, from how they continue visiting this blog, and from what I presume to be at least sense a of comfort that it might offer them from its continuity too. I at least hope that my continuing to address basic business and technology issues here offers at least some comfort and value in these trying times. So I will continue for now with a pattern that I have slipped into of alternately writing on-schedule, more usual postings that go live once every three days, and off-day supplemental postings that will for the most part address this crisis: this disruptive change.

And I end this with a promise. I wrote a series of brief postings in 2010 that I collectively gave the basic title: Reexamining Business School Fundamentals, that sought to at least sketchily map out how some of the basic concepts of business and of economies had changed in real world practice up to then. And I wrote and offered as update series to match that in 2017 that I perhaps unimaginatively titled Reexamining Business School Fundamentals (reconsidered). You can find them both at my directory page: Reexamining the Fundamentals as its sections II and VII. I said at the end of that second series, that I would with time, come back to add a third such update too. I will begin writing and posting that as I come to see what at least to me might be glimpses of that new normal to come, and certainly as it would enter into the areas of discussion that I seek to address here.

Meanwhile, you can find this and related postings at About this Blog and at Blogs and Marketing.

## Some thoughts concerning a general theory of business 33: a second round discussion of general theories as such, 8

This is my 33rd 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 26-32.)

I have been discussing three topics points and their ramifications in this series since Part 31, which I repeat here for smoother continuity of narrative:

1. Provide a brief discussion of the generality of axioms, and how a theory elaboration can lead to reconsideration of the axioms underlying it, where that can lead to accepted axioms already in place, now being reconsidered as special cases of more general principles.

2. Adding, subtracting and modifying axioms in general, and

3. The whole issue of consistency, and even just within constrained sets of axioms. And I will frame that discussion at least in part, in terms of outside data insofar as that makes these theories open – and with a reconsideration of what evolutionary and disruptive mean there.

And to be more specific in this connecting text lead-in to this posting and its intended discussion, I have addressed that complex set of interconnected topics up to here with a specific focus on the first of them. My goal here is to turn to the second of those topic points and address the issues of generality and of special case rules. “Axiom” as such and what that word actually means, is often thought of in a manner that can only be considered to be axiomatic, with that very concept more usually taken as an unexamined given than anything else.

The concept of axiom only really comes to the forefront of thought when specific axioms and the conceptual systems that they serve as intended foundations for, come into question. And that can happen in closed axiomatic theory contexts, as in the case of alternative approaches to geometry as briefly discussed in Part 32, or in open axiomatic theory contexts as in the case of Newtonian and Relativistic general theories of physics as also touched upon in that posting. (See Part 32 for summary definitions of axiomatically closed and open bodies of theory, as I will continue making use of that point of distinction in what follows.)

I begin addressing the above-repeated second topics point of the list of them that I am working on here, by repeating a couple of what would probably be fairly obvious details that I have already made note of:

• When axioms come into doubt and are challenged in an axiomatically closed body of theory, as for example when parallelism and its axiomatic description has been challenged, they are most often replaced in total with new alternatives, or eliminated as to type entirely. They are not reduced to special case or similar status and retained as such; they disappear – and with their replacement forming part of the axiomatic underpinnings of a new, alternative general body of theory.

• And when axioms come into doubt and are challenged in an axiomatically open body of theory, as happened in a physical sciences context when observations began to be made that did not fit into a Newtonian axiomatic explanatory model, they tend to be reduced to special case rules and rule of thumb approximations, even if very useful ones.

Newtonian physics based calculations of all sorts are still routinely used and essentially wherever their theoretically more precise relativistic calculation alternatives would offer corrections that are so minor in scale as to be dwarfed by any possible measurement errors that might obtain in a given empirically based setting. So classical Newtonian physics can be and is used as a font of special case wisdom and utility. But that noted, theory opening outside data (such as experimental data) can also lead to the complete repudiation of and elimination of what was once considered an inviolable axiomatic law of nature principle too. Just consider the theory of the ether, and particularly in the context of electromagnetic energy and its propagation. It was axiomatically assumed that there had to be an all-pervasive space-filling medium that electromagnetic energy would travel through as an essential requirement for its propagation from any point A to any point B in space. Then the Michelson–Morley experiment: initially intended to detect and characterized this assumed medium, demonstrated that it did not in fact exist.

• That led to the repudiation and elimination of an axiom-level assumption, of the possible and even likely existence of necessary all-pervasive space-filling medium in space and time, and apart from space and time per se that would play an essential role in energy transfer and in fact in causal connectivity insofar as that would depend on energy transfers.

• And then Peter Higgs developed his theory of mass as a consequence of the action of a special categorical type of boson (subatomic particles with whole integer spin), now known as the Higgs boson, that serves as a conveyer of and a conferrer of mass and for all subatomic particles that can interact with it, and for all matter that would be comprised of those particles. And that particle was detected and in replicated experiments, proving its actual, empirical existence. And that particle and its action, at least according to current theory, requires that another form of all-pervasive space-filling medium exist instead: the so called Higgs field (see above-linked piece on the Higgs boson for a quick introduction to this too.)

• So the electromagnetic ether disappeared from physical theory, but an essentially-ether based theory came back to replace it – at least for now, absent any realistically possible way to validate or disprove its existence as a next step counterpart to the Michelson–Morley experiment.

Electromagnetic energy and photons (as boson particles that contain and convey it) experientially exist as validated empirical realities. The concept of the ether in their context was taken as a (not yet) testable axiomatic truth, and a de facto part of the non-empirically based side to physical theory as held, until that experiment moved its possible existence into the open side of that body of theory. Mass, and more specifically the Higgs bosons that manifest it at its most basic subatomic level have now been experientially proven to exist too. I would argue that its ether: the Higgs field might or might not exist, and certainly as it is currently understood. But either way it is part of the as of now at least, closed side to current physical theory as far as understanding mass at that fundamental level is concerned.

• So a fundamental axiom can go away in an axiomatically open body of theory – and then come back again as has happened here with presumptions of all-pervasive space-filling mediums that might exist as separate from space and time itself.

Nota bene: As an idly considered alternative, or rather as a source of such an alternative to the Higgs field as an all-pervasive medium for transmitting the effects of Higgs bosons, it crosses my mind that a reconsideration of the nature of space-time itself might be in order. And I see possible value arising there, in addressing this problem by carrying out this reconsideration with a particular focus on a scale approaching that of the Plank distance as a spatial unit, and for time, the interval that it would take a photon to traverse one Plank distance unit, or at least some small number of them. I cite this as a possible source of insight while thinking through the story of electromagnetic energy and its photonic bosons, and in the light if you will of the Michelson–Morley experiment and its confirmation, and with a tight linkage established between minimum paths between points in a general relativistic space-time, and the paths that photons in fact follow there. Light, to focus on one band of the electromagnetic spectrum, travels through space itself and in a way that both shapes and in many respects defines space-time and certainly as a source of observable phenomena. Does the same basic principle that I loosely articulate there, apply to the phenomena that we associate with mass too?

That, of course, leads me to the question of experimental validation and the issues of moving what amounts to axiomatically closed conjecture into the realm of empirically testable axiomatically open territory. And I would suggest that given the tools available and in use in modern physics, that testing anything like my conjecture here would call for call for a particle accelerator that would at minimum be orders of magnitude larger than any that have ever been built or even seriously considered. Developing a mathematical theory that would descriptively and predictively model a direct, aspect of space alternative to the Higgs field (and that would prove or disprove the existence of that too), would be the easy part and certainly when compared to actually carrying out this type of direct test.

And that leads me to the issues that I would at least begin to address here as I conclude this posting: the issues of what constitutes direct tests, or indirect ones in general in an axiomatically open body of theory context. I will simply note for now that any determination of direct, as that term would be used here and in similar contexts, would depend on what is axiomatically assumed and on what is routinely done, and on what types of tools are available and in routine use.

Setting that side note aside, at least for now, I am going to continue this discussion in a next series installment where I will turn to consider the third and final topics point of my above-repeated list:

3. The whole issue of consistency, and even just within constrained sets of axioms. And I will frame that discussion at least in part, in terms of outside data insofar as that makes these theories open – and with a reconsideration of what evolutionary and disruptive mean there.

Then I will consider two areas of consideration: the challenge of reductionism and of emergent properties, and the challenge of direct and indirect testing and what they actually are.

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.

## Some thoughts concerning a general theory of business 32: a second round discussion of general theories as such, 7

This is my 32nd 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 26-31.)

I have been discussing two categorically distinctive, basic forms of axiomatic theory since Part 28 that I will continue to delve into here too:

• Axiomatically closed bodies of theory: entirely abstract, self-contained 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. And the axioms included in such overall theories might be selected because they are at least initially deemed to be self-evident in some manner. Or they might be selected in a more arbitrary manner as far as that is concerned, and with a primary selection-determining criteria being that they at least appear not to be in conflict with each other.

• And axiomatically open bodies of theory: 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. Their axiomatic underpinnings are more likely to be selected on the basis of their appearing to be self-evident truths.

And with those basic definitions repeated for smoother continuity of narrative, I begin the main line of discussion of this posting by repeating the to-address list of next topic points as offered at the end of Part 31, as an anticipated précis of what I would discuss here:

• Provide a brief discussion of the generality of axioms, and how a theory elaboration can lead to reconsideration of the axioms underlying it, where that can lead to accepted axioms already in place, now being reconsidered as special cases of more general principles.

• Adding, subtracting and modifying axioms in general, and

• The whole issue of consistency, and even just within constrained sets of axioms. And I will frame that discussion at least in part, in terms of outside data insofar as that makes these theories open – and with a reconsideration of what evolutionary and disruptive mean there.

As a final point of orienting detail that I would repeat from Part 31, I appended the following point of observation to the three basic topics points of that to-address list:

• Axiomatic assumptions should always be considered for what they limit to and exclude, as well as for what they more actively include and in any axiomatic body of theory.

And with that offered, I turn to the first of three topics points under consideration here, and issues of generality and specificity in the underlying assumptions that are selected for an organized body of theory, and that are presumed to hold so much system-wide significance there as to qualify for axiom status. I begin doing so by proposing a metatheory axiom (an axiom that would inform a general theory of general theories) that I would argue offers a reasonable starting point for essentially any axiomatically grounded body of theory:

• An assumed axiomatic truth need not always be true and under absolutely any and all possible circumstances; it need not be presumed to be universally valid and without exceptions and in all contexts. It only has to present itself as being widely enough valid, for it to be presumed to hold universally true within the confines of a specific body of theory that it is assumed to be axiomatic to.

• Turning that point of detail around in the context of this series, it means that the range of applicability of such a body of theory as a whole, should be wide enough so that it would make sense to think of it as a general theory per se. This means that it should be wide enough reaching for its descriptive and predictive applicability and impact so that its axioms would not just present themselves as special case rules of thumb for understanding some specific type or category of phenomena.

• Think of that as an admittedly loose definition that represents a line of demarcation distinguishing between compendium theories and their assemblages of narrowly applicable special case components, and more widely stated general theories per se (see this series for its Parts 2-8 for a more detailed discussion of compendium theories as a categorical type.)

To take those points of detail out of the abstract, consider the concept of parallelism as it is addressed in elliptical and hyperbolic geometries. From a closed body of theory perspective, it is quite possible to develop an entire, extensive body of geometric theory around either of those approaches to that concept, so their range of valid applicability is sufficiently large so that they can offer significant value – even as they flatly contradict each other, proving that neither approach to parallelism can be held to be absolutely universally true in any sense. They can, however, still serve as reasonable bases for developing their own respective bodies of theory. And given the history of parallelism and how it has been variously considered, that type of axiom-level postulate has in fact served as the driving impetus for essentially all non-Euclidean geometries at least as their origins are traced back to their Euclidean “ur-geometry” origin.

And from an open body of theory perspective, I would cite the general theory of relativity and its use of non-Euclidean geometries as well as empirically grounded axioms, with experimental and observational findings serving to open that body of theory up beyond the constraints of its more strictly axiomatic underpinnings. That body of theory of necessity contains within it the axiomatic underpinnings of the mathematics that it is developed in terms of: that body of mathematics’ interpretation of parallelism included.

The first topics point, as I am addressing it, includes within it “… and how a theory elaboration can lead to reconsideration of the axioms underlying it.” And to continue with my use of the physical sciences as I take this line of discussion out of the abstract, I coordinately cite classical Newtonian physics and Einstein’s theories of relativity, and his general theory of relativity in particular. Newtonian physics is predicated upon an assumption of Euclidean geometry and its axiomatic foundations with that mathematical theory as a key aspect of its overall axiomatic framework. And that means it presumes as a given fundamental truth, Euclid’s fifth postulate: his fifth axiom as it defines and specifies parallelism. That assumption does not and cannot realistically apply to the wider range of physical phenomena that Einstein sought to address in his theories, but Newtonian calculations still hold true as meaningfully valid, empirically useful approximations for objects that are large relative to the size of individual atoms and that are observed to be moving slowly relative to the speed of light. So as a matter of practicality, Newtonian assumptions can be considered as if special cases in an Einsteinian theoretical framework – where they appear as absolute truth, axiomatic assumptions in a strictly Newtonian setting.

I am going to continue this line of discussion in a next installment to this series, where I will at least begin to address the second of three topic points as offered towards the start of this posting:

• Adding, subtracting and modifying axioms in general.

Then after addressing that I will turn to and consider the third of those points:

• Consistency, and even just within constrained sets of axioms. And I will frame that discussion at least in part, in terms of outside data insofar as that makes these theories open – and with a reconsideration of what evolutionary and disruptive mean there.

## Some thoughts concerning a general theory of business 31: a second round discussion of general theories as such, 6

This is my 31st 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 26-30.)

I have been discussing general theories per se in this series, as well as the more specifically focused general theories of business that I hold out as the primary topic here. And as a part of that I have been discussing axiomatic theories per se in this series, since Part 26. And more specifically, I have been discussing two categorically distinctive, basic forms of axiomatic theory since Part 28 that I will continue to delve into here too:

• **Axiomatically closed bodies of theory:** entirely abstract, self-contained 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. And the axioms included in such overall theories might be selected because they are at least initially deemed to be self-evident in some manner. Or they might be selected in a more arbitrary manner as far as that is concerned, and with a primary selection-determining criteria being that they at least appear not to be in conflict with each other.

• And **axiomatically open bodies of theory:** 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. Their axiomatic underpinnings are more likely to be selected on the basis of their appearing to be self-evident truths.

I addressed the issues of both of these approaches to analytically ordered and organized knowledge and understanding in Part 30. And I then stated at the end of that posting that I would turn here to focus more fully on axiomatically open theories. And more specifically, I said that I would discuss:

• The emergence of both disruptively new types of data and of empirical observations that could generate it, as that would impact on axiomatically open theories,

• And shifts in the accuracy and resolution, or the range of observations that more accepted and known types of empirical observations might suddenly come to offer in that context.

And in the course of delving into that complex of issues, I will also continue an already ongoing discussion here 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 that would be presumed in a given general theory, and what such optimization even means.

And I begin this posting and its line of discussion by posing an axiomatic assumption in what I would presume as a valid metatheory of general theories as a whole, that I would argue to be a reasonable starting point for thinking about open and closed bodies of theory, and axiomatically open theories in particular:

• In the real world, people make observations and seek to organize them in their minds and in their understanding, for how they connect into and create larger consistent patterns.

• But in genuinely closed axiomatic theories, the a priori accepted and presumed axioms in place in them take precedence, and apparently conflicting empirical findings would at most call for the creation and elaboration of competing theoretical models and systems.

• In open axiomatic theories, empirically derived or otherwise-accepted data can and does compel evolutionary, or for more disruptively novel new data, revolutionary change in existing underlying theory – and change that can reach down to the level of the axioms in place themselves there.

• There, and as a defining point of difference, separating axiomatically closed systems of theory from open ones, outside-sourced data and insight does take precedence and even over the root axiomatic assumptions in place.

And with this noted, I turn to and begin discussing the topics list for this posting, starting with a more detailed consideration of the outside-sourced information that would be collected, interpreted and used here: the outside-sourced data that would explicitly enter into an open axiomatic system, making it open as such. And I begin addressing that by making note of a fundamental point of distinction. Data that does not simply support an already tested and validated, organized body of theory can arise from two different types of direction:

• It can arise as new types of, and even disruptively new types of outside-sourced information as new and novel types of experimental, quasi-experimental or observational tests are carried out, that could have an effect of validation testing that theory, or

• It can arise as output of already established tests and of any of those three categorical types, as they are carried out with greater levels of precision or under new and expanded ranges of application that might not have previously been possible to achieve.

• Put perhaps simplistically, think of this as making a distinction between disruptively new types of data that would be brought into an axiomatically open theory and both to expand its effective applicability and to expand its range of validation, and

• More evolutionarily developed data and its sources. (Nota bene: there is an assumption implicit in this, that I will challenge as I proceed in this line of discussion.)

To take that out of the abstract, let’s consider a source of examples from physics. Newtonian physics was first proposed, experimentally tested, and further refined and expanded as a body of general theory, and extensively so, on the basis of the ongoing study of objects that are very large when compared to individual atoms, and that travel very slowly in comparison to the speed of light. That is all that was possible at the time.

Then when it became possible to observe and measure phenomena that fell outside of those constraining limitations, and gather and analyze data that could be used to validate or disprove Newtonian assumptions and predictions in these now-wider ranges of empirical experience, systematic deviations were found between what classical Newtonian physics would predict and what was actually observed. And entirely new types of phenomena were observed that Newtonian physics on its own, did not seem capable of addressing at all. And while Newtonian physics and its descriptive and predictive modeling are still used and with great precision and accuracy where they do apply and where their calculations work, they have been supplanted by newer bodies of theory with different underlying axiomatic assumptions built into them: the special and general theories of relativity, and quantum theory, where they observationally and predictively break down.

• Newtonian physics with its mathematical and computational complexities, is quite sufficient for calculating rocket engine thrust and burn times and all of the other factors and parameters that would be needed to launch a satellite into a precisely intended orbit, or even to send a probe to a succession of other planets as for example was called for with the Voyager probes.

• The special and general theories of relativity and quantum theory come into their own in an ever-increasing range of applications that violate the basic and now known-limiting assumptions that Newton made and that classical physics came to elaborate upon.

And with this noted, I return to the parenthetical comment I just made concerning assumptions. And the example from physics that I just raised here, highlights why that assumption can be problematical. As validating examples in support of that contention, consider the Michelson–Morley experiment, as an example of what a classical physicist would consider a disruptively novel experiment and the findings that it generated, challenging Newtonian physics and providing explicit support for Einstein’s theory of special relativity. And at the same time, consider Brownian motion: first formally described in the physics literature in 1828 and finally explained as to mechanism by Albert Einstein as a body of observable findings that classical physics could not explain. The first of these two examples was disruptively novel and both for the data derived through it and for how that data was arrived at; the second was observational in nature and did not by all appearance involve or include anything new or novel at all. But predictively describing the phenomena involved in it called for new axiomatic level understandings and new fundamental theory.

New evolutionary-level change and even new revolutionary change in how data is arrived at and in what it consists of, does not always compel fundamental change in a theory, or a more general body of theory in place. Both can in fact serve to bolster and support it. But both can come to fundamentally challenge it too. But even so, it can seem self-evident that the disruptively new and novel might be more likely to serve as a source of challenge than of support and certainly for a body of general theory that is still, in effect coming together as a newly organized whole.

I begin addressing that conjecture by posing a pair of questions:

• Is it always possible to draw a sufficiently clear cut distinction between disruptively new and revolutionary, and step-by-step evolving and evolutionary in the type of context raised in my above-offered bipartite distinction of data, that would enter into axiomatically open theories?

• And if the goal of making that type of distinction here, is to categorically identify meaningfully distinctive drivers of change in axiomatically open theories, is this the right point of distinction for pursuing that?

Is it always possible to draw such degree-of-novelty distinctions in ways that would always, unequivocally hold as valid? It would make things easier to be able to identify and label as such, bodies of supporting or refuting evidence, and methodologies for arriving at that data that are disruptively new, and ones that are simply next-step evolutionary elaborations of already established data patterns and data generating methods, and simply look for the disruptively new forms of experiment and the data they lead to, in order to identify fundamental theory-level changers. But ultimately, simple step-by-step data collection and analysis cannot offer a conclusive, or even necessarily a meaningful distinction there. (As a working example there, consider Brownian motion again and the gap of understanding that prevailed for it from 1828 as noted above, and Einstein’s insight most of 80 years later in 1905.) So I in fact do more than just question my implicit assumption as noted above; I fundamentally challenge it with a goal of rethinking what disruptively new and emergent even means in this type of context.

• So how would one categorically think through axiomatically open theories here, for how outside-sourced information would impact upon them, and ideally in ways that would allow for meaningful, independently replicable determinations of novelty and newness as well as of impact? And that is where I come to the issues of scale, as touched upon by name towards the top of this posting.

I am going to at least begin to delve into that complex of issues, and also the issue of optimization, in the next installment to this series. And in anticipation of that, this will mean my considering:

• The generality of axioms, and how a theory elaboration can lead to reconsideration of the axioms underlying it, where that can lead to accepted axioms already in place, now being reconsidered as special cases of more general principles.

• Adding, subtracting and modifying axioms in general, and

• The whole issue of consistency, and even just within constrained sets of axioms.

And I will frame that discussion at least in part, in terms of the outside data that makes these theories open – and with a reconsideration of what evolutionary and disruptive mean here. And with that noted, I conclude this posting by offering a basic point of observation, and of conclusion derived from it:

• Axiomatic assumptions should always be considered for what they limit to and exclude, as well as for what they more actively include and in any axiomatic body of theory.

## Platt Perspective at ten years

This posting, as its title indicates, is a marker that makes note of what has now become ten years of my writing to this blog. This is also posting number 2689 to go live in this still ongoing effort. And I offer it with no specific ending to this in sight, or under consideration. And to complete my putting this into some sort of numerical perspective, I add that if you count postings that I have finished writing and editing for publication, as of this morning, that have not gone live yet but that I have uploaded and that are on the server queue and waiting to go live, I have accumulated 2705 postings in total here so far. And the vast majority of those postings: those essays and notes are organized into longer and even book length series.

Why do I start this update to this blog, and this note on what I am doing here with that opening? I am trying to put what I offer here into an at least briefly stated overall perspective for what for me, has become an ongoing imperative in all of this. I write; I have been doing that and both in general, more “executive summary” terms and in more analytical detail, for a long time and certainly as part of my ongoing work and professional life. And I have variously addressed separate parts in all of that, of what I have come to see as a single larger interconnected puzzle. My writing effort here has taken on extra meaning for me from my effort at developing and assembling all of this as an organized, overall effort, and with a goal of outlining and discussing more of that comprehensively organized puzzle as a whole. And my developing and offering this blog and its ongoing flow of postings has become a part of me and of who I am. And right now I am making note of that and more openly acknowledging it because of the perhaps accident, or at least coincidence that a tenth anniversary happens to present itself as a type of round number that would make it notable. I come from a culture that uses a decimal based counting system so ten and tenth stand out.

I have offered what would probably present themselves as similar types of update assessments in earlier anniversary and related updates, as I have looked back at what I have offered here and as I look forward to what I intend to offer. And as a part of that collective if only occasionally augmented narrative, I have generally at least attempted to figure out how much of my “core” material that I would share here, I had already completed and posted. I have decided that is a less than productive approach to take as I do not have any meaningfully valid answer to that type of question – as my earlier predictions and then retractions of what I have offered on that would probably indicate. So I simply note here, that there is more that I would write about that I find to be important enough to organize and include in this effort, and that hopefully at least some others might come to value too. And yes, that includes my still actively unfolding series on general theories of business, as well as at least some of my still actively developing series that I do in fact see as being more core than peripheral, or supplementary to my thinking and to my hands-on practice from my own direct experience. (You can find my series on general theories of business at Reexamining the Fundamentals, Section VI and its Page 2 continuation, Section IX.)

So what should I focus on here, as I look back to what I have offered in this blog, and as I look forward with thoughts of what is to come in it? I already knew when I wrote my ninth anniversary posting that I would start writing more material here that is more explicitly grounded in history and historical narrative. I knew that I would do that as I seek to put more of what I write about of the here-and-now and more of what I write about that I see coming, into a longer-term and contextually richer perspective. And I have been doing that; I have offered historical narratives that connect into specific posting and series contexts on an ongoing basis and for a much longer period of time than that one year observation might suggest, but I have more consistently followed that approach in the last year. And I anticipate continuing that trend in my ongoing writing to this blog too. And at the rate that I am going, I do expect to reach the triple Scheherazade number of 3003 postings and in just a few more years. What can I say? Scheherazade was a lot more succinct in her story telling than I am. And she almost certainly edited better than I do too. (See Platt Perspective at 1001 Postings, and the Scheherazade Number and Some Thoughts Concerning a General Theory of Business 1: a series offered to mark Platt Perspective reaching 2002 postings.)

I still see this endeavor as being open ended, and with no specific goal or end point in mind that once reached would mark and end to it. I still enjoy writing it and at least some people seem to enjoy reading it too, which I find very gratifying. And with that, I continue on here into what will now be year 11.

Timothy Platt, Ph.D.

## Some thoughts concerning a general theory of business 30: a second round discussion of general theories as such, 5

This is my 30th 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 26-29.)

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 26-29 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. As a key part of that continued narrative, I offered a point of theory defining distinction in Part 28, that I began using there in this discussion, and that I continued using in Part 29 as well, and that I will continue using and developing here too, drawing a distinction between:

• 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.

And I have, and will continue to refer to them as axiomatically closed and open bodies of theory, as convenient terms for denoting them. And that brings me up to the point in this developing narrative that I would begin this installment to it at, with two topics points that I would discuss in terms of how they arise in closed and open bodies of theory respectively:

• 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.

I began discussing the first of these topics points in Part 29 and will continue doing so here. And after completing that discussion thread, at least for purposes of this digression into the epistemology of general theories per se, I will turn to and discuss the second of those points too. And I begin addressing all of this at the very beginning, with what was arguably the first, at least still-existing effort to create a fully complete and consistent axiomatically closed body of theory that would address what was expected at least, to encompass and resolve all possible problems and circumstances where it might conceivably be applied: Euclid’s geometry as developed from the set of axiomatically presumed truths that he built his system upon.

More specifically, I begin this narrative thread with Euclid’s famous, or if you prefer infamous Fifth postulate: his fifth axiom, and how that defines and constrains the concept of parallelism. And I begin here by noting that mathematicians and geometers began struggling with it more than two thousand years ago, and quite possibly from when Euclid himself was still alive.

Unlike the other axioms that Euclid offered, this one did not appear to be self-evident. So a seemingly endless progression of scholars sought to find a way to prove it from the first four of Euclid’s axioms. And baring that possibility, scholars sought to develop alternative bodies of geometric theory that either offered alternative axioms to replace Euclid’s fifth, or that did without parallelism as an axiomatic principle at all, or that explicitly focused on it and even if that meant dispensing with the metric concepts of angle and distance (where parallelism can be defined independently of them), with affine geometries.

In an axiomatically closed body of theory context, this can all be thought of as offering what amounts to alternative realities, and certainly insofar as geometry is applied for its provable findings, to the empirically observable real world. The existence of a formally, axiomatically specified non-Euclidean geometry such as a an elliptic or hyperbolic geometry that explicitly diverge from the Euclidean on the issue of parallelism, does not disprove Euclidean geometry, or even necessarily refute it except insofar as their existence shows that other equally solidly grounded, axiomatically-based geometries are possible too. So as long as a set of axioms that underlie a body of theory such as one of these geometries can be assumed to be internally consistent, the issues of reframing, generalizing, limiting or otherwise changing axioms in place, within a closed body of theory is effectively moot.

As soon as outside-sourced empirical or other information is brought in that arises separately from and independently from the set, a priori axioms in place in a body of theory, all of that changes. And that certainly holds if such information (e.g. replicably observed empirical observations and the data derived from them) is held to be more reliably grounded and “truer” than data arrived at entirely as a consequence of logical evaluation of the consequences of the a priori axioms in place. (Nota bene: Keep in mind that I am still referring here to initial presumed axioms that are not in and of themselves directly empirically validated, and that might never have even been in any way tested against outside-sourced observations and certainly for the range of observation types that that perhaps new forms of empirical data and its observed patterns might offer. Such new data might in effect force change in previously assumed axiomatically framed truth.)

All I have done in the above paragraph is to somewhat awkwardly outline the experimental method, where theory-based hypotheses are tested against carefully developed and analyzed empirical data to see if it supports or refutes them. And in that, I focus in the above paragraph, on experimental testing that would support or refute what have come to be seen as really fundamental, underlying principles and not just detail elaborations as to how the basic assumed principles in place would address very specific, special circumstances.

But this line of discussion overlooks, or at least glosses over a very large gap in the more complete narrative that I would need to address here. And for purposes of filling that gap, I return to reconsider Kurt Gödel and his proofs of the incompleteness of any axiomatic theory of arithmetic, and of the impossibility of proving absolute consistency for such a body of theory too, as touched upon here in Part 28. As a crude representation of a more complex overall concept, mathematical proofs can be roughly divided into two basic types:

• Existence proofs, that simply demonstrate that at least one mathematical construct exists within the framework of a set of axioms under consideration that would explicitly sustain or refute that theory, but without in any way indicating its form or details, and

• Constructive proofs, that both prove the existence of a theorem-supporting or refuting mathematical construct, and also specifically identify and specify it for at least one realistic example, or at least one realistic category of such examples.

Gödel’s inconsistency theorem is an existence proof insofar as it does not constructively indicate any specific mathematical contexts where inconsistency explicitly arises. And even if it did, that arguably would only indicate where specific changes might be needed in order to seamlessly connect two bodies of mathematical theory: A and B, within a to-them, sufficiently complete and consistent single axiomatic framework so as to be able to treat them as a single combined area of mathematics (e.g. combining algebra and geometry to arrive as a larger and more inclusive body of theory such as algebraic geometry.) And this brings me very specifically and directly to the issues of reverse mathematics, as briefly but very effectively raised in:

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

And I at least begin to bring that approach into this discussion by posing a brief set of very basic questions, that arise of necessity from Gödel’s discoveries and the proof that he offered to validate them:

• What would be the minimum set of axioms, demonstrably consistent within that set, that would be needed in order to prove as valid, some specific mathematical theorem A?

• What would be the minimum set of axioms needed to so prove theorem A and also theorem B (or some other explicitly stated and specified finitely enumerable set of such theorems A, B, C etc.)?

Anything in the way of demonstrable incompleteness of a type required here, for bringing A and B (and C and …, if needed) into a single overarching theory would call for a specific, constructively demonstrable expansion of the set of axioms assumed in order to accomplish the goals implicit in those two bullet pointed questions. And any demonstrable inconsistency that were to emerge when seeking to arrive at such a minimal necessary axiomatic foundation for a combined theory, would of necessity call for a reframing or a replacement at a basic axiomatic level and even in what are overtly closed axiomatic bodies of theory. So Euclidean versus non-Euclidean geometries notwithstanding, even a seemingly completely closed such body of theory might need to be reconsidered and axiomatically re-grounded, or discarded entirely.

I am going to continue this line of discussion in a next series installment, where I will turn to more explicitly consider axiomatically open bodies of theory in this context. And in anticipation of that narrative to come, I will consider:

• The emergence of both disruptively new types of data and of empirical observations that could generate it,

• And shifts in the accuracy resolution, or the range of observations that more accepted and known types of empirical observations might suddenly be offering.

I add here that I have, of necessity, already begun discussing the second to-address topic point that I made note of towards the start of this posting:

• 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.

I will continue on in this overall discussion to more fully consider that set of issues, and certainly where optimization is concerned in this type of context.

## 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.

## 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.)

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