Shared Article from Slate Magazine

Zeno's Paradox Is a Trick—But a Very Interesting T…

The Greek philosopher Zeno wrote a book of paradoxes nearly 2,500 years ago. "Achilles and the Tortoise" is the easiest to understand, but…

David Plotz @ slate.com

O.K., so, briefly: If you think that the point of Zeno’s Paradoxes of motion is to prove that the arrow never will reach its target, or that Achilles never does pass the tortoise, &c. — then I think that you are mistaken about the point of raising the paradox in the first place. Of course, it’s hard to be confident about the motives of dead philosophers who have no surviving books. But what we do know is that Zeno was a student of Parmenides; and Plato tells us that his books were written to defend Parmenides’s doctrines, by negative means,^[1] showing that the views of his opponents led to contradictions.

So the most charitable understanding of Zeno’s aims is not that he’s trying to show you that Achilles can never catch the tortoise. Of course he does; just watch them race and you’ll see it happen. His point is to ask, given that Achilles passes the tortoise, well, how is that possible? And, for good or for ill, to argue from the paradox that you can only make sense of Achilles passing the tortoise if you reject presentism, and accept eternalist and Parmenidean conclusions about the nature of time and being.

Maybe he’s right about that, and maybe he’s wrong. (I’m inclined to think he’s wrong.) But note that if your solution is to try and settle the issue by introducing a lot of mathematical notation and conceptual apparatus from modern calculus — for example infinitesimal limit processes, convergent and divergent series, etc. — as is done in the Slate article here, and as is probably the overwhelmingly most common first response to Zeno’s paradoxes by mathematically-trained writers — then probably you are doing a better job than any pre-classical Greek philosopher could do in elaborating the precise nature of the problem.^[2] But you’re not obviously refuting Zeno’s claims in any way, at least not yet. At the most you’re kicking the can down the road, and really you’re sort of strengthening Zeno’s own position. After all, naive formulations of mathematical notation are more or less always going to involve you in all kinds of specifically eternalist language, for example about moments in past and future time actually existing, instantiating the value of functions, etc. You cannot normally take the limit of ?@ce;201d;S(t) over values of t that don’t exist (no longer exist, do not yet exist).^[3]

Or perhaps you can. But if you can, then doing so, and explaining what you’re doing when you do it, will take some very non-naive reinterpretation of ordinary mathematical language — and some nice metaphysics, too, to justify your reinterpretation. In any case the solution is going to have to be deeply philosophical, not just a matter of applying a technical innovation in maths.

1. ^[1]In the Parmenides: I see, Parmenides, said Socrates, that Zeno would like to be not only one with you in friendship but your second self in his writings too; he puts what you say in another way . . . You affirm unity, he denies plurality. . . . Yes, Socrates, said Zeno. . . . The truth is, that these writings of mine were meant to protect the arguments of Parmenides against those who make fun of him and seek to show the many ridiculous and contradictory results which they suppose to follow from the affirmation of the one. My answer is addressed to the partisans of the many, whose attack I return with interest by retorting upon them that their hypothesis of the being of many, if carried out, appears to be still more ridiculous than the hypothesis of the being of one.↩
2. ^[2]Since the 19th century, we’ve done a lot to really nicely rigorize the mathematics of infinites and infinitesimals, in ways that sometimes anticipated by but never fully available to ancient mathematicians.↩
3. ^[3]If anything, this is even more true of late-modern mathematics than it was of classical mathematics. Contemporary mathematics constantly helps itself to a lot of the language of existence, actuality, etc., for mathematical objects, in areas where Euclid and other classical mathematicians were typically much more circumspect about making existence claims for mathematical objects that hadn’t yet been constructed.↩

Happy Tyrannicide Day (observed)! To-day, March 15^th, commemorates the assassination of two notorious tyrants. On the Ides of March in 2014 CE, we mark the 2,057 anniversary — give or take the relevant calendar adjustments — of the death of Gaius Julius Caesar, ruthless usurper, war-monger, slaver and military dictator, who rose to power in the midst of Rome’s most violent civil wars, who boasted of butchering and enslaving two million Gauls, who set fire to Alexandria, who battered and broke through every remaining restraint that Roman politics and civil society had against unilateral military and executive power. Driving his enemies before him in triumphs, having himself proclaimed Father of His Country, dictator perpetuo, censor, supreme pontiff, imperator, the King of Rome in all but name, taking unilateral command of all political power in Rome and having his images placed among the statues of the kings of old and even the gods themselves, he met his fate at the hands of a group of republican conspirators. Led by Marcus Junius Brutus and Gaius Cassius Longinus, calling themselves the Liberators, on March 15, 44 BCE they surrounded Caesar and ended his reign of terror by stabbing him to death on the floor of the Senate.

Die Ermordung C?@c3;a4;sars, Karl von Piloty (1865)

By a coincidence of fate, March 13^th, only two days before, also marks the anniversary (the 133^rd this year) of the assassination of Alexander II Nikolaevitch Romanov, the self-styled Imperator, Caesar and Autocrat of All the Russias. A group of Narodnik conspirators, acting in self-defense against ongoing repression and violence that they faced at the hands of the autocratic state, put an end to the Czar’s reign by throwing grenades underneath his carriage on March 13^th, 1881 CE, in an act of propaganda by the deed.

Das Attentat auf Zar Alexander II. am 13. M?@c3;a4;rz 1881 in St. Petersburg. Anonymous.

In honor of the coinciding events, the Ministry of Culture in this secessionist republic of one, together with fellow republics and federations of the free world, is happy to proclaim the 15^th of March Tyrannicide Day (observed), a commemoration of the death of two tyrants at the hands of their enraged equals, people rising up to defend themselves even against the violence and oppression exercised by men wrapped in the bloody cloak of the State, with the sword of the Law and in the name of their fraudulent claims to higher authority. It’s a two-for-one historical holiday, kind of like President’s Day, except cooler: instead of another dull theo-nationalist hymn on the miraculous birth of two of the canonized saints of the United States federal government, we have instead one day on which we can honor the memory, and note the cultural celebrations, of men and women who defied tyrants’ arbitrary claims to an unchecked power that they had neither the wisdom, the virtue, nor the right to wield against their fellow creatures.

My favorite collectible coin. This silver denarius was actually minted and circulated in Macedonia by M. Junius Brutus after he and his fellow conspirators stabbed Caesar to death. The obverse features Brutus’s head in profile. The thing in the middle, above EID MAR (Ides of March) and flanked by the two daggers, is a Liberty Cap, traditionally given to emancipated slaves on the day of their freedom.

It is worth remembering in these days that the State has always tried to pass off attacks against its own commanding and military forces (Czars, Kings, soldiers in the field, etc.) as acts of terrorism. That is, in fact, what almost every so-called act of terrorism attributed to 19^th century anarchists happened to be: direct attacks on the commanders of the State's repressive forces. The linguistic bait-and-switch is a way of trying to get moral sympathy on the cheap, in which the combat deaths of trained fighters and commanders are fraudulently passed off, by a professionalized armed faction sanctimoniously playing the victim, as if they were just so many innocent bystanders killed out of the blue. Tyrannicide Day is a day to expose this for the cynical lie that it is.

There are in fact lots of good reasons to set aside tyrannicide as a political tactic — after all, these two famous cases each ended a tyrant but not the tyrannical regime; Alexander II was replaced by the even more brutal Alexander III, and Julius Caesar was replaced by his former running-dogs, one of whom would emerge from the carnage that followed as Imperator Gaius Julius Son-of-God Caesar Octavianus Augustus, beginning the long Imperial nightmare in earnest. But it’s important to recognize that these are strategic failures, not moral ones; what should be celebrated on the Ides of March is not the tyrannicide as a strategy, but rather tyrannicide as a moral fact. Putting a diadem on your head and wrapping yourself in the blood-dyed robes of the State confers neither the virtue, the knowledge, nor the right to rule over anyone, anywhere, for even one second, any more than you had naked and alone. Tyranny is nothing more and nothing less than organized crime executed with a pompous sense of entitlement and a specious justification; the right to self-defense applies every bit as much against the person of some self-proclaimed sovereign as it does against any other two-bit punk who might attack you on the street.

Every victory for human liberation in history — whether against the crowned heads of Europe, the cannibal-empires of modern Fascism and Bolshevism, or the age-old self-perpetuating oligarchies of race and sex — has had these moral insights at its core: the moral right to deal with the princes and potentates of the world as nothing more and nothing less than fellow human beings, to address them as such, to challenge them as such, and — if necessary — to resist them as such.

How did you celebrate Tyrannicide Day? (Personally, I toasted the event at home, watched the Season 1 finale of Rome, posted some special-occasion cultural artifacts to Facebook, and re-read Plutarch’s Life of Brutus from a nice little Loeb edition that I picked up from Jackson Street Books in Athens, Georgia.) And you? Done anything online or off for this festive season? Give a shout-out in the comments.

Toasting the Ides at home. . .

Thus always to tyrants. And many happy returns!

Beware the State. Celebrate the Ides of March!