Bet on Achilles to Beat the Tortoise

Many are the times I’ve thought about Achilles and the tortoise:

Zeno concluded from this and other paradoxes that motion is an illusion, so it’s important to show why Achilles will beat the tortoise. Otherwise, we’ll all have to sit very still.

Zeno’s argument goes something like this:

1) When Achilles starts running at point A, the tortoise is already at B.
2) By the time Achilles reaches point B, the tortoise is at point C.
3) By the time Achilles reaches point C, the tortoise is at point D.
4) This series can be extended forever.
5) In order to catch up, Achilles will have to perform an infinite series of tasks in a finite amount of time.
6) It’s impossible to perform an infinite series of tasks in a finite amount of time.
7) Therefore, Achilles can never catch up to the tortoise.

But we know that mighty Achilles is going to catch up to the tortoise and win the race. Fast runners beat slow walkers. Hence, the paradox.

In his book A Brief History of the Philosophy of Time, Adrian Bardon suggests that Aristotle had a good response to Zeno. Aristotle apparently argued that Zeno’s paradox rests on confusing “an abstract value (i.e. time), which is mathematically divisible into instants, with actual change, which is not literally composed of infinitesimal units of change”. In other words, Aristotle distinguished between “the rules for time (as a mere abstraction) and those for change (as a real phenomenon)”. Maybe Aristotle was right, but I’m not sure Zeno would have been convinced.

Bardon also discusses the mathematical concept of a “limit”, which “allows for an infinite number of finite quantities to add up to a finite sum”. That’s the idea mentioned at the end of the video. Many have concluded that Zeno can be answered using this concept, although Bardon asks: “Can a limit be a real endpoint to a real process, or is it just a new mathematical convention that disregards the metaphysical question about time and change with which Aristotle and Zeno are struggling? Does it really help matters to say that [motion or change] represents convergence on a limit? That wouldn’t have sounded like real motion to either Zeno or Aristotle”.

Maybe Zeno’s argument does require a subtle and sophisticated response. But what struck me as weak about his argument was one of the premises listed above:

6) It’s impossible to perform an infinite series of tasks in a finite amount of time.

Really? Who says? Isn’t that what we do every time we move from point A to point B? Having arrived at point B, haven’t we also traveled 1/2 of the distance, 1/3 of the distance, 1/4 of the distance, 1/5 of the distance and so on? Isn’t this a clear example of performing an infinite series of tasks in a finite amount of time? Granted that the tasks overlap, but it seems fair (albeit boring) to describe what we’ve done this way, without having to explain what the mathematical concept of a “limit” is or draw a distinction between the rules for speaking about time and the rules for speaking about change.

Perhaps this is mere sophistry and Zeno would have considered it such, since he and the Sophists were contemporaries. I think it’s a simple truth that moving around involves doing many little things by doing one big thing. Take that, Zeno!

On a similar note, the English philosopher G. E. Moore wrote a famous article called “Proof of an External World”. In that article, he said he could prove the existence of the world outside our minds by drawing our attention to his two hands: “by doing this, ipso facto, I have proved the existence of external things”. Whether that’s a great argument or not is an open question. Moore defended his argument, however, by pointing out that skeptical philosophical arguments (such as proving motion to be an illusion) often rely on philosophical intuitions or generalities (such as premise 6 above) that we have much less reason to accept than the common sense beliefs they supposedly refute.

All right, it’s safe to start moving again.

Philosophers and the One Right Answer

A very bright guy named Richard Marshall interviews academic philosophers at a site called 3 A.M. Magazine (as of his latest interview, Mr. Marshall is still “biding his time”).

This is from an interview with Thom Brooks of Durham University:

“Hegel’s Science of Logic reveals a fascinating insight into the philosophy of punishment. He writes that punishment should not be considered as either retribution, deterrence or rehabilitation. Instead, punishment is grounded in retribution – those punished must deserve it and cannot be innocent – but retribution is only one part of a larger view. Punishment is not retributivist, preventative or rehabilitative, but rather all three in one. Three in one. Why would we expect to find anything different in Hegel than this anyway?”

With all due respect to Hegel (one of the most influential yet incomprehensible philosophers ever), is it really a fascinating insight to note that punishment is justifiable for different reasons? Philosophers are too often prone to seek a single justification or analysis for some phenomenon. Philosophical arguments can sometimes remind you of that beer commercial (“Tastes great! No, less filling!”). 

The entry on “Punishment” in the Stanford Encyclopedia of Philosophy provides an example of this tendency:

“The practice of punishment must be justified by reference either to forward-looking or to backward-looking considerations [or both?].

If the former prevail, then the theory is consequentialist and probably some version of utilitarianism, according to which the point of the practice of punishment is to increase overall net social welfare by reducing (ideally, preventing) crime.

If the latter prevail, the theory is deontological; on this approach, punishment is seen either as a good in itself or as a practice required by justice, thus making a direct claim on our allegiance. A deontological justification of punishment is likely to be a retributive justification.

Or, as a third alternative, the justification of the practice may be found in some hybrid combination of these two independent alternatives. Recent attempts to avoid this duality in favor of a completely different approach [such as saying that neither theory captures the whole truth?] have yet to meet with much success…”

Practically speaking, meeting with “success” in this case means being found convincing by other academic philosophers. It might be that recent attempts to avoid this duality haven’t been successful because most philosophers interested in punishment (and ethics in general) are committed to an either/or solution, finding the single right answer. There is relatively little to be gained, professionally speaking, by pointing out that both sides of a traditional argument reflect part of the truth.

What sometimes happens in philosophical argument (aka “combat”) is that philosopher A offers his theory and philosopher B suggests a counterexample. Philosopher B then offers her theory and philosopher A suggests a counterexample. Philosopher C then concludes that neither theory is successful. An alternative approach would be to agree that the counterexamples show that neither theory captures the whole truth, although they each capture some of the truth. Then everyone could pack up and go have a drink (or a slice of pie).