## Zeno’s paradox, Moravec’s paradox and rethinking how we project forward in our planning 3

This is my third posting to a series on paradoxes, and both as philosophical constructs and as the concept of paradox is applied to business and technology contexts (see Part 1 and Part 2.)

I began this series by relatively systematically examining two specific paradoxes:

• One historically ancient example: Zeno of Elea’s dichotomy paradox (see Part 1),

• And one late 20th century example: Moravec’s paradox (see Part 2.)

And in the course of that discussion I proposed and developed three fundamental reasons as to how and why an assertion might be deemed to be paradoxical, which I repeat here as foundation material for this installment’s discussion too:

1. A seeming paradox can arise when a logically consistent and formally valid line of reasoning is offered in description of an empirically observable circumstance, but where its underlying axiomatic assumptions are not applicable to the circumstances or conditions for which it is being applied.

2. Alternatively, a seeming paradox can arise and even when the application of its reasoning to a specific empirically observable circumstance would seem valid, when there are perhaps subtle and unobvious but still significant flaws or gaps in its underlying logic.

3. A seeming paradox can also arise when axiomatic assumptions are made that simply reflect the limitations of some current state of the art, and for the technology available, for current practices in using that technology, or both. And this can, among other things arise because of implicit assumptions that the particular path that technology is developed in historically, could be the only one possible.

At the end of the second installment of this series I noted that Point 3, above might be construed to represent a special-case example of Point 1. But I note here that the axiomatic assumptions held onto in Point 3 might have taken on that status because they were valid and applicable and even compellingly so when originally arrived at. And then circumstances changed and these assumptions lost some or all of their validity and perhaps quite gradually. But because they had taken on automatic default-assumption status: axiomatic status, this fact was not noticed or acted upon even as they lost their validity. So they keep being assumed true and used as a foundation point for further assumptions and for lines of argument and decisions made. In this sense, Point 3 might be seen as being all about timing problems, at least for a wide swath of its range of applicability.

Then I ended Part 2 of this series by noting that:

• The complexities and I add “alternative simplicities” of the real world can and do intrude in any analysis as wide-ranging as the ones that I have been attempting here.

I was initially going to add to that: “…and particularly for my Moravec’s paradox example” but the point raised here applies equally to my first posting’s more classical example too and certainly when it is viewed from the perspective of my Part 1 discussion of it. And I said that I would at least begin to peel back the lid from over the issues that the above bullet pointed assertion adds to this discussion in a next series installment. And I begin doing that here by fundamentally challenging all three of the reason-for-paradox points that I have been building this series around, and that I have just repeated above. And I begin that by citing the fundamental crisis in what constitutes proof and provability that led to the rise of what came to be known as modern mathematics:

• The formal proof-based end to any realistic attempt to prove or to assume that axiomatic systems of any real complexity (e.g. at least as complex as arithmetic) could be either complete as a logical system, or demonstrably fully consistent and even just within the underlying logical framework that defines mathematical proof per se.

In anticipation of discussion to come, this posting is all about completeness and consistency, and about absoluteness or rather its alternative as these concepts would be applied to the logical frameworks that a seeming paradox would be offered and understood within.

I started this series discussion with an example that I presented and argued from a mathematical perspective, with Zeno’s paradox. And I then turned to a more modern example, provided by Hans Moravec and others in the 1980’s that can also be framed in at least semi-quantitative terms in its analysis. But I am not writing about mathematical paradoxes in specific, or assertions in general here that would obligatorily be reducible to strictly mathematical description. The issue here is that these assertions be subject to logical statement and analysis with a clear elucidation of the underlying assumptions that underlie them and of the reasoning that would show those assumptions to be relevant. I cite mathematics and mathematical systems of proof and their underlying logic as the analysis of those systems have revealed underlying truths about logical systems per se that have much wider ranges of applicability – including here.

So I begin my seeming mathematical digression in this narrative from that point, and from a point that historically arose immediately before the break from the past that gave us modern mathematics, that I briefly noted above (and that I will briefly explain, at least for details relevant to this discussion.)

Mathematics has been viewed as fully logically grounded at least since Euclid, and quite arguably since the Pre-Socratic Greek philosophers of the Pythagorean school who saw number and numerically explicable pattern as the prima materia – the fundamental underlying first element of the universe, out of which all is constructed.

Turning explicitly to the examples set by geometry as a rich source of insight for purposes of this discussion, Euclid developed a rigorously consistent axiom, theorem and proof -based system that would allow for validation of geometric statements and assertions that are valid within his logical framework, with disproval of any that would be inconsistent with it. And he built this system around a set of axiomatically assumed fundamental statements that he viewed to represent self-evident truths. But one of these assumptions: his so called fifth axiom, or parallel postulate came into question for this, and from early on. Was Euclid’s assumption as to the nature of parallel lines and geometric parallelism always valid as an absolute self-evident truth?

And alternative system of geometry was developed around Euclid’s approach, that contained essentially all of the elements of his system but that one axiom, that came to be known as affine geometry (with that name affixed to this approach in 1748 by Leonhard Euler.) It is important to note that affine geometry very definitely addresses the issues of parallelism and of parallel lines, planes et cetera – but without making specific axiomatic assumptions about this geometric property of the type that Euclid built his system of geometry around. The goal here was to develop a clear and consistent geometry that did not hold as self-evident and absolute, any particular axiomatic assumptions as to the nature of parallel as a geometric property.

And alternative non-Euclidean geometry systems were also developed that axiomatically defined parallel, but in ways that were quite divergent from the way that Euclid did. Hyperbolic geometry and elliptic geometry come immediately to mind here. And one consequence of this progression of geometric systems proliferation, and parallel mathematical developments as well, was the ending of any presumption that axioms per se are self-evident truths. They are starting point assumptions, and what you can validly prove in an axiomatic system depends entirely on what you choose to select as your axiomatic starting point: a statement or theorem can be valid in one system but invalid in another. But the basic and yes, “self-evident truth” axiomatic assumption underlying all of this was that once you arrived at a non-contradictory, consistent set of axioms you could build a complete, logically consistent and logically fully provable system of resulting theorem statements around them.

Mathematics as a whole was in a fundamental sense rigorously developed around that underlying assumption, and both as a systematic study of numbers, shapes and more exotic abstract entities, and as an underlying system of logic: mathematical logic. And this approach in effect reached its apotheosis with Alfred North Whitehead’s and Bertrand Russell’s four volume work: the Principia Mathematica: an attempt to develop one of the foundation elements of mathematics as a whole: arithmetic as a fully described complete and consistent abstract axiomatic system. But cracks were already beginning to appear in this vision of mathematics as a whole even before they completed their fourth volume to this work in 1913, and certainly by the publication of their 1927 second edition.

David Hilbert famously sought to build mathematics around a stable, consistent, fully provable and complete core in what came to be known as Hilbert’s Program. And then in 1931, Kurt Gödel proved his two incompleteness theorems, proving once and for all that Hilbert’s Program and any effort of its type could never succeed, except when limited to very simple and even exception-based contrived axiomatic systems. Axiomatic-based logical systems of any complexity cannot be presumed to be complete, or completable with any finite number of axioms added to expand their reach.

And this brings me back to the issues of paradoxes, as they seem to plausibly violate the axiomatic systems they are couched as being paradoxical within. And this also brings me to a fourth reason-for-seeming-paradoxical point:

4. A statement or assertion might appear to be paradoxical if it can only be understood and explained by either altering or generalizing an axiom already in place in new and unexpected directions, or by adding in one or more new supporting and validating axioms to the logically framed system of understanding that they are evaluated within.

I would argue that this point diverges from the first three as numerically listed above, for a variety of significant reasons. But among the most important of them is that the first three points are all predicated in actual usage on two assumptions.

• One is that in keeping with the classical sense of what an axiom is as posited by Euclid, the axioms underlying a system that would be used to evaluate a possible paradox are all generally deemed to be self-evident truths.

• The other is that the logical systems that statements are evaluated under when determining that they are paradoxical, are generally if not essentially always viewed as logically complete and fully consistent, and capable of serving as a basis for establishing any valid statement: any valid theorem in their purview as such. And similarly they are deemed capable of identifying any false statement within their system as such too.

This fourth point does not require or rest upon either of those metalogic assumptions. And by its very nature, this fourth point cannot simply be assumed to be an absolute in the pre-Gödel sense either.

I am going to follow this posting with a next series installment where I will more fully discuss the now four rules that I have offered for evaluating putative paradoxes, not for their individual meaning or application, but in more general terms as to what statements like them are. Then after that and on the basis of this developing narrative, I will discuss evolutionary change, disruptive change, and descriptive and predictive understandability from the perspective of the conceptual framework that I have been developing in this series up to here. Meanwhile, you can find this and related postings at Ubiquitous Computing and Communications – everywhere all the time 2 and in my first Ubiquitous Computing and Communications directory page. I also include this in my Reexamining the Fundamentals directory as an entry to a new Section V: Rethinking Underlying Assumptions and Their Logic.

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