### The Greek Bay Packers

The New York Giants were inside the Green Bay five-yard line, maybe at around the two. The Packers jumped offside. Normally, that would be a five-yard penaty, but inside the defensive team's five, it's half the distance to the goal line. The teams line up again. Green Bay jumps offside again. Half the distance.

"Hey," I said to my friend Sandy, with whom I was watching the game, "the Packers are implementing the Zeno strategy!" I figured, they knew the Giants were going to score and take the lead unless they did something desperate, so they intended to continue jumping offside forever, allowing the Giants ever closer to the goal line but never able to cross it.

That whimsy brought up an interesting point: What could be done about a team really trying to implement such a strategy. Say, there's twenty seconds left in a game, and the team trailing by four has a first-and-inches to the goal line. The defensive team figures there's no way to stop each of four successive QB sneaks from getting the ball in the endzone, in which case they lose. And so they jump offsides the first moment the offensive team lines up, intending to do so forever -- meaning, until everyone gives up.

More realistically, what if a team just has a tired offensive unit it wishes to rest for a while, and a desire to flummox their opponent. On x-and-inches to the goal, why not jump offsides 10 or 20 times in a row? Certainly, the odds of giving up a touchdown are only changed minutely by a series of sub-centimeter advances towards the goal line.

By the way, the Wikipedia entry is excellent in explaining why the notion of a limit from calculus does not solve Zeno's paradox:

"A suggested problem with using calculus and mathematical series to try to solve Zeno's paradoxes is that these solutions miss the point. To be precise, while these kinds of solutions specify the limit point of infinite series, they do not explain how such a series can actually ever be completed and the limit point be reached. Thus, calculus and mathematical series can be used to predict where and when Achilles will overtake the tortoise, assuming that the infinite sequence of events as laid out in the argument ever comes to an end. However, the problem lies exactly with that assumption, as Zeno's paradox points out that in order for Achilles to catch up with the Tortoise, an infinite number of physical events need to take place, which seems to be impossible in and of itself, independent of how much time such an act would require if it could actually be done.

"Indeed, the problem with the calculus and other series-based solutions is that these kinds of solutions beg the question. They assume that one can finish a limiting process, but this is exactly what Zeno questioned. To be precise, Zeno started with the assumption that a finite interval can be split into infinitely many parts, and then argued that it is impossible to move through such a landscape. For calculus and other series-based solutions to make the point that the sum of infinitely many terms can add up to a finite amount therefore merely confirms Zeno's assumption about the landscape (geometry) of space, but does nothing to answer Zeno's question of how we can actually (dynamically) move through such a space."

1. This is why the infinitesimal approach to calculus is more beautiful than the somewhat deceptive limit approach. The notion of an infinitely small quantity is needed to make some sense of Zeno's paradox, something like the notion of quana, except infinitely more plastic.

In regard to the football rules, I've thought about that. I think the refs do have the power to charge unsportsmanlike conduct for something like that, and/or the league may levy fines on a discretionary basis. There is probably some general rule by which the ref may be allowed to make executive decisions in times where people are trying to exploit loopholes at the expense of the spirit of the game.

2. That's 'quanta', not 'quana'.

3. Quana are rigid nondeformable coextensive immensities, each occupying the entire universe. Their reciprocals are quanta.

There is a "nonstandard" version of the calculus in which infinitesimals and their infinitely large reciprocals are as real as integers. It's better than the version they taught you.

4. Isn't it up to the referees as to when they blow a play over? I believe if a defensive player steps offside, the refs have the option of allowing the play to continue. This would then turn into an advantage for the offense, since if the play has a desirable outcome for the offense the penalty can be declined. If the play is unfavorable to the offense, the penalty can be accepted and the down replayed.

My guess is that the writers of NFL rules anticipated the possibility of a Zeno strategy, without ever having heard of Zeno of Elea.

5. I think if they make contact before the snap the whistle is always blown, but I'm not sure.

6. I don't think infinitesimals, even if properly formalized, really handle the problem. It is one thing to devise a coherent, formal handling of some mathematical entity, but another to say it applies to real space/time. In terms of real space/time, either an infinitesimal occupies some amount of space, or it doesn't. If it does, it can be sub-divided per Zeno. If it doesn't, then no number of those fellows can add up to any finite amount of space.

7. Point taken, Gene. Cantor's classification of different types of infinity may be of some help, though I am still trying to tease out how.

8. Thank goodness Aristotle solved this problem.

9. I don't see Aristotle's solution as anything more than a version of the limit solution, which I've given a reason to believe doesn't go through. Am I wrong in my understanding of his solution?

10. I always thought the solution was that, as you divide into smaller units of length, you increase the number of units traversed in the same amount of time. 10 seconds is 100 tenths, 1000 hundredths, 10,000 thousandths, etc. So, even as the units continue to shrink, the units covered in the same length increase.

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