Sorites and the mean - not for the fainthearted

It is said that Chryssipus the Stoic held that there were, for all problems, true solutions. But he also held that at times, we can’t see them – and those times called for a morally disciplined silence. It is in this spirit he approached the paradox of the heap – the sorites. The paradox is as follows: if we construct a heap from seeds, say, we can, by adding seeds successively, reach a point where we might say that we have a heap, and identify that with the number of seeds we have used – say, 200. And yet, when we subtract one seed, we are disinclined to say that we no longer have a heap. Given that fact, we might play the game by claiming that we haven’t reached a heap no matter how many seeds we use in order to avoid identifying the heap with a certain number of seeds – but then, paradoxically, we will never achieve a heap. In fact, we don’t really seem to be able to quantify a thing like a heap; neither do we want to say that the heap is a quality when clearly it can be analyzed into its separate parts. To borrow a term from contemporary logic, there is no “heapmaker” – so how can there be a heap? Chryssipus, according to Sextus Empiricus, recommended that “when the Sorites is being propounded one should, while the argument is proceeding, stop and suspend judgement to avoid falling into absurdity.” Analytic philosophers, such as Mario Magnucci, who wrote a seminal paper on the stoic response to the sorites, have attempted to incorporate Chryssipus’s response into standard Western logic. To me, the stoic response is closer to the notion of Mu in Rinzai Zen. The famous Mu Koan goes like this: a disciple of Zhaozhou, a Chinese zen master, asked him if a dog has the Buddha nature. Zhaouzhou answered Wu – Mu in Japanese – which means no, empty, vacant, and – it is said – applies in different ways to the question: that there is no dog, that there is no Buddha nature, that the dog does not have Buddha nature, and so on. In other words, the answer is meant to break the mental habit of thinking that the way of assembly – where distinct parts are put together – and the way of disassembly, where distinct parts are separated, are grounded in the real. Indetermination is neither a fact of the real nor not a fact of the real.

My friend Owen Goldin has pointed out that the Aristotelian response to events like the heap - events that seem to defy any natural division into parts – is to postulate threshholds. In Eubulides, Aristotle and the Sorites, Jon Moline claims that Eubulides first developed the sorites paradox. Eubulides was taught by a pupil of Socrates and disputed (successfully) with Aristotle. According to Moline, the ancient tradition put the sorites in opposition to the Aristotelian concept of the mean. Moline quotes a second century figure, Aspasius:

“For it is no different with any other sensible thing from which they arrive at Sorites. For at what point is a man wealthy? When he has deposited ten talents, they inquire? And if one should take away a drachma, is he no longer wealthy? And if two? For there is no one of such things, since it is a sensible thing, which it is possible to define accurately, whether a poor man or a rich. And concerning the bald man they ask whether by one hair one can become bald, or by two, or three? Whence the arguments said that bald men were also Sorites. For concerning a heap (soros) they ask the same thing, whether by one grain of wheat the heap is made smaller, and then whether by two, and so on according to the pattern; and it is not possible to say where first there is no longer a heap because no sensible thing can be understood accurately, but only broadly and in outline. Thus this holds even of actions and of feelings. For it is not possible to say by how much anger one attains a mean in anger or surpasses it or fails to come up to it because of deviation towards too much or too little. Wherefore there is need of practical wisdom for discovering the mean in feelings and in actions.”

Moline constructs a plausible version of a Eubulidian argument regarding, say, generosity, which uses the heap method to show that what is generous can be reduced to what is extremely mean if we do not grant an autonomy to the quality in regards to the quality.

“For suppose the man of practical wisdom in my circumstances would give n drachmas. Giving just so much as the man of practical wisdom would give and would declare to be the mean is, you would say, generous. But it is not. For suppose I give just one obol less. Is my gift not generous ? Surely you must concede that giving just one obol less than the generous amount is generous, for an obol is a trifle. Yet if we apply the principle you concede a sufficient number of times, it follows that it is generous to give nothing. But this, clearly, is not generosity, but the extreme of meanness. And suppose that I give one obol more than the mean as specified by the man of practical wisdom. I am giving one obol more than the generous amount, but surely my gift is still generous, for again, an obol is a trifle. But if we apply a sufficient number of times the principle that giving one obol more than what is generous is generous, it follows that it is generous to give not n obols, but one's entire fortune. And this is not generosity, but the extreme of prodigality.”

It in this way that the Stoics attempted to drive a wedge between what Vico would call geometric knowledge and discursive knowledge.

Myself, I think this has particular pertinence to a pattern in historiography: the rise and fall arc. Too often, disputes among historians about the rise or decline of some historical property fail to acknowledge that rise and fall are sorites. Hence, arguments become very vicious about what the risemakers or the fall-makers are, and when they occur. Accepting that historical narratives have a sorites paradox at their center helps us clarify the half-fictitious nature of the business. Even if one doesn’t stop and fall silent, like Chryssipus, one has to accept the possibility that finally, withdrawal is the correct response.