## Zeno’s paradox, Moravec’s paradox and rethinking how we project forward in our planning 1

Zeno of Elea was a fifth century BC Greek philosopher who is now known to a significant degree from paradoxes that he proposed – statements that appear to be both self-contradictory and at the same time true. One of them that I would cite and discuss in this posting is his so called dichotomy paradox: one of a series of lines of argument that he presented with a goal of arguing that motion is illusory. Moravec’s paradox by contrast is a modern invention, first formulated by the artificial intelligence researcher Hans Moravec and cited and articulated by others in the 1980’s AD and beyond. I offer this posting as a first step to a more general discussion on rethinking what paradoxes are, and as a thought piece on projecting forward in our planning and expectations, and certainly when predicting and planning for technology advancement and its possible emergent consequences. And I begin at the beginning with Zeno and his thought piece paradox, which I quote here (as cited in the above Wikipedia reference to Zeno’s paradoxes):

• “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.”– as recounted by Aristotle, Physics VI:9, 239b10.

A man seeks to walk across a room. Before he can complete this he first has to walk half way, dividing his overall journey into two distinct actions. Before he can do that he has to walk half of that first half and so on leading to a requirement of completing an open-ended and even infinite number of consecutive actions. And it is essentially argued that since it is impossible to actually perform an infinite series of consecutive actions, actually completing this journey across that room becomes impossible.

Zeno’s logic divides the span of this room mathematically into an infinite series of progressively smaller summed fractions of distance to be traversed with the smaller bunched into the beginning. Mathematically, and I add logically as well, this is equivalent to starting with one half-way-long segment then dividing the remaining portion in half and so on, loading progressively smaller fractions onto the end of the series. And I note that here as this detail will come up again later in my discussion.

If a man walks across a room at a constant velocity (a fixed unchanging speed and in a straight line) they move forward in relatively fixed increment steps – and even if this process is mathematically described via calculus in terms of infinitesimally small subunits in describing all of the details of this action. The logic behind the dichotomy paradox is sound. It is just that it is being applied here in a context for which it does not apply. So I would at least begin my line of reasoning and discussion here by proposing two fundamental reasons why a paradox might appear reasonable and valid but why it might in fact not be:

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.

I would identify scenario 1, above, as representing an **external logical failure** and scenario 2 an **internal logical failure**. And I would argue that Zeno’s paradox fits the pattern of point 1.

Not to belabor this, let’s consider an empirically real situation where the logic that Zeno offered is both valid and also readily assumed – so no one would find any realistic or reasonable grounds for seeing paradox in them. And my working example here involves repeatedly playing a simple game of solitaire – a one person card game.

A player: call him Bob, learns of a solitaire game that catches his interest so he gets a deck of playing cards and tries it and he loses. So he has played one time and has a 0% success rate. He plays again and this time he wins so his success rate jumps up to 50%. And he keeps playing and he keeps winning. And every next victory increases his overall percentage success rate by a significantly smaller incremental fraction. Game two increased it by 50%. Games three through ten collectively only bring it up to 90% for a total increase of just an additional 40% for those eight additional games. Games 11 through 100 collectively can only add another 9% to that, to bring his win rate up to 99%. And the logic continues from there with 999 wins out of 1000 games played only adding another 0.9% to that score and so on. In this case empirical reality really is fractionally divisible exactly as Zeno proposed and his logic is pretty much automatically assumed and expected so no paradox potential is visible or reasonably perceived.

To perhaps at least begin to belabor this point now, walking across a room according to the rules of Newtonian physics makes application of Zeno’s reasoning seem paradoxical, but motion according to Einstein’s theory of relativity can make it seem very reasonable and empirically valid. Consider a fast walker, and particularly one who seeks to accelerate his pace until he reaches the speed of light. A fixed, unmoving observer tracks and measures his effort. At first and at lower speeds and certainly when compared to the speed of light, he simply accelerates smoothly. But as his speed enters a range in which relativistic corrections are needed for the math involved in describing his behavior and properties, things begin to happen. The observer begins to see the walker’s apparent mass increase, and the overall mass/energy in him as a system. This observer sees him seemingly foreshortening along the axis on which he is moving and if the observer can see his wrist watch he sees time for the walker seemingly slowing down too. And in order to actually reach the speed of light, time for the walker would have to stop, at least from the outside observer’s perspective, his observed mass would expand out to become infinite and therefor larger than the mass of the entire observable universe and a range of other limiting events would also occur. So even when you look at observable motion, Zeno’s underlying reasoning as to how action is partitioned might not lead to seeming paradox but rather to simple explanatory confirmation.

There are a number of ways mathematically, for distinguishing between problems where Zeno’s logic and his underlying assumptions do and do not apply. One that I would note here is to cite the larger set of at least most possible applications for Zeno’s reasoning as representing mathematical boundary value problems. And the relevant distinction in determining applicability of this reasoning to the specific circumstance is one of whether the boundaries that would be approached can be best represented as being open or closed (see open and closed sets as discussed for example in this piece on point set topology.) In one case the outer boundary approached is within the reachable range of action and the set that describes the boundary limits there is closed. In the other that boundary goal is not reachable and the set that describes it is open. And this brings me to a third fundamental point that I would add to my numbered list from above:

3. When you face what seems to be a paradox, reframe your reasoning and your understanding of its applicability into a wider context and reconsider it from a more open perspective to see if you are in fact dealing with an internal logical failure, an external logical failure, or perhaps seemingly both. And resolution of the seeming paradox will fall out as you reframe the basic problem that you are addressing in more appropriate terms and with a clearer understanding of what axiomatic assumptions you are making, and what of set of them would make the most sense.

And with that 1,300 plus word preamble, I move forward from Zeno to Moravec and to his seeming paradox. And as I did for Zeno, I will quote Moravec’s paradox from its Wikipedia entry and from his own words for its succinctness and clarity as cited there:

• “It is comparatively easy to make computers exhibit adult level performance on intelligence tests or playing checkers, and difficult or impossible to give them the skills of a one-year-old when it comes to perception and mobility.”

Higher level cognitive reasoning is computationally simple while seemingly much more basic and lower level skills of perception and sensory data analysis, balance, motion and timing are computationally very challenging. And this seems to be so counterintuitive as to be starkly paradoxical. I am going to follow this posting with a second same-topic installment where I will more fully look into this proposed paradox, and in a manner that at least parallels my discussion of Zeno’s as outlined 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.

leave a comment