For want of a nail...

... the shoe was lost; for want of a shoe the horse was lost; and for want of a horse the rider was lost’. Or in the words of Benjamin Franklin’s own synopsis (in his Maxims Prefaced to Poor Richard’s Almanac, 1758), ‘A little neglect breeds mischief’. It is a simple truth, concisely and graphically put; however, one risks plunging wildly overboard as soon as one extends the chain of reasoning to more than a handful of terms. How far can one take it?- to a message, a battle, a kingdom, planet Earth, Milky Way South East, universal heat death?

Isaac Asimov once suggested (I forget where) an interesting analogy for how history really works, as a kind of 'bow wave' in time. At first there is immense tumult as the immediate effects of a cause (an event, a great personality, and so on) spread and proliferate; but then a phase of sublimation supervenes, assimilating those effects to the broader, deeper currents of Universal history. The same applies for Great Men. What if Hitler had won World War II? It would certainly still have mattered in 2000 and even 2100, but what difference would it have made by 20,000 AD? Or 200,000 AD or 2,000,000? Indeed, looking at the history of Europe since the end of the Cold War, it is striking that, at least in terms of international politics, things look remarkably as they might have in 1925, had the First World War not intervened.

The whole logic of 'For want of a nail' rests on two false assumptions: that the world is basically atomistic, and that what changes is more vital than whatever remains the same. They are equally curious opinions, although entirely compatible with a world of fragmentation and distraction we inhabit now. But the universe is not just a concatenation of atoms but is on the contrary deeply and universally structured, local quantitative variations may be rapidly countered and any potential qualitative effects generally suppressed. And although we may not be interested in the same old same old, the universe and its human expression, history, takes it all into account.

Popular (mis)interpretations of chaos theory make a similar error. Contrary to the way in which it is sometimes alluded to, chaos theory does not say that chaos is fundamental. On the contrary, it says that apparent chaos is often actually the expression of profound (if also somewhat obscure) order and simplicity. That is why the old saw about a butterfly in Rio de Janeiro causing a thunderstorm in Peking is so misleading: not because the effects of its wing beats could not trigger a storm on another continent, but because they are only one of a massive number of other more or less profound effective causes. Especially on the global scale, weather consists, by and large, of immense and highly stable systems controlled by forces and laws operating on an incomparably vaster scale than all the life forms on the planet put together. Of course, a single butterfly may tip a system that is either exceptionally sensitive or already in extremis into a new mode or phase, but that is hardly the same thing. Having the trigger does not mean you don’t need the rest of the howitzer.

So what is this really about? My own view is almost equally simple: it is about a popular ideology designed to accept the unintelligibility and uncontrollability of the world, be it through popular wisdom or pseudo-scientific explanation or (in a previous age) the Will of God, or indeed by any other means. The reasons why the world is fragmented and full of distraction are beyond both the understanding and the control of individuals, and there are vast forces ranged against anyone trying to create either a popular understanding of exactly what those forces are or a political organisation capable of combatting them.

Zeno’s paradoxes

Zeno’s paradoxes are amongst the most venerable and to many impenetrable of all philosophical puzzles. There have been many attempts to explain Achilles and the tortoise, the arrow paradox and the so-called dichotomy paradox, often illustrated by 'proving' the impossibility of a frog ever managing to hop its way out of a pond. Apparently there were about 40 in all, and were originally created by Zeno of Elea to illustrate Parmenides’ conception of the world as logically changeless by showing that much of what we believed we witnessed in the world simply could not be happening.

This view has undergone a lot of humorous parody and satire, from Zeno’s contemporaries to Terry Pratchett’s occasional sketches of ‘Ephebian’ (i.e., Greek) philosophers deducing all sorts of wonderful things. All in all it tends to remind one of the wonderful line in Monty Python and the Holy Grail, ‘and that, my lord, is how we know that the earth is banana-shaped’.

There have been plenty of attempts to refute Zeno too, involving ideas like infinitesimals (roughly speaking Aristotle’s answer), which works quite well. For we have known since Archimedes that the sum of an infinite number of progressively smaller amounts will add up to a finite total. Or rather, it will tend towards a finite result, which is a problem, since it is not obvious that this is not actually simply a mathematical way of simply restating the original paradox, because this reasoning does not, in itself, conclude that the frog ever actually reaches the side of the pond.

Another use of the infinitesimals inherent in Zeno’s original paradoxes is to reject them. If it simply isn’t true that there is always a point lying between two other points, no matter how close together, to which the frog/ arrow/ Achilles/ whatever can move next, then it must make definite progress and eventually Achilles will be the winner. You can also reject the idea that things are anywhere definite or finite at any given point in time, but I think it’s a bit much to have to rely on relativity or the uncertainty principle to beat up classical philosophers!

I have never quite understood why these paradoxes seem to attract solutions like this. Perhaps I am missing something, but hasn’t Zeno actually defined the problem in such as way that it cannot be solved? He describes all these events in terms of them never being concluded. For example, if Achilles’ movements are defined in terms of reaching where the tortoise was last – a place from which it must have moved on – then plainly he can never overtake it in this framework. But that is only to day you can define a problem in terms of failure, which is what Zeno has done. If at each step Achilles' motion is defined as reaching only to where the tortoise last was, he cannot have reached where it is now. So the problem is actually stated in terms that permit only failure. In other words, failure is not a paradoxical result of this overall situation but a simple deduction from its definition.

There is no need to appeal to any fancy mathematics. All you have to do is to realise that this is not a paradox at all, but rather the description of an activity in negative terms that cannot be shaken off without seeming to disregard the original problem. Zeno’s paradoxes strike me as valid but uninteresting – except for what they tell us about how philosophers and mathematicians reason.