Consider an alternative version of Chrysippus’ story. Imagine Dion as before, but rather than imagining Theon to be that part of Dion which omits only Dion's foot, now assume Theon to be only Dion's foot. Again, Theon is a part of Dion. (True, Theon cannot – in this example – be easily thought of a potential persisting individual in his own right, but this is itself an interesting point to bear in mind for later.) Now, rather than considering what will happen if the foot is removed, we might ask what would remain if everything other than the foot is removed. That is to say, let us remove all that constitutes Dion but which is not also part of Theon. Is what is left Dion or Theon? (It cannot be both.) Now, if we have any intuitive response to this admittedly peculiar position, I think that the more likely answer is that the single disembodied foot before us is more likely to be considered to be Theon than Dion. Now this thought experiment was just like that provided by Chrysippus, but it produces quite the opposite result. In both cases the part discarded is the 'overlap', the portions of Dion which are not shared by Theon, and in both cases what is left was at one time both part of Dion and part of Theon. Indeed, in both cases what is left was once part of Dion and is the whole of Theon.
So if these two examples are relevantly similar, how can we explain the different reactions to them? Chrysippus' original example offers us a picture of two conceivable and viable individuals and focuses on one small part which one has and the other has not: a part which is, we would agree, inessential to the larger individual. (But: Polly Low once pointed out to me that for some people it may be the case that their feet are so essential to their persona (if not their identity as a persisting individual) that for them the removal of a foot may be a more telling loss. What if David Beckham’s right foot were removed?)
But the Stoics assert that the part is neither other than (ἕτερον) the whole nor the same; for the hand is neither the same as the man (for it is not a man) nor other than the man for it is included in the conception of the man as man (σὺν αὐτῇ γὰρ ὁ ἄνθρωπος νοεῖται ἄνθρωπος).
SE M 9.336, trans. R.G. Bury
The first part of this is straightforward. A hand is not the same as a man (presumably the man whose hand it is) since one is a hand and the other is a man. But there is of course a link between hands and men, and this is what the Stoics try to characterise in the second half of this text. A hand is not 'different from' a man, since when you think of a man you think of a man with hands. Hands are not, in other words, merely optional accessories for humans.
But the Stoics do not make so clear exactly what this last claim amounts to. Does it, for example, make 'handed-ness' an essential property of a human, so that anything which does not have hands cannot be a human? I assume that the Stoics would have known of cases of people losing their hands in accidents or in battle, and if so then they would have to give an account which allows these too to count as humans. Perhaps 'having at some point had hands' is an essential characteristic of a human.
In any case, what does this mean for Dion and his foot? It might explain why it is a foot which is removed rather than a hand, since the footless Dion is not on anyone's account in danger of failing to be a human. And a foot on its own is on no-one’s account likely to be thought of as an individual. Having feet, after all, is not a peculiar characteristic of humans as having hands might perhaps be thought to be; lots of creatures have feet, but not many have hands.
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First, a thank you for getting me to look up "sorites" in "sorites argument", which led me to the wonderful entry on teuэ- at Bartleby:
http://www.bartleby.com/61/roots/IE531.html
relating thigh, thousand, thumb, butter, tumescent, quark, and creosote.
So, when is pile not a pile, a beard not a beard, a man not a man?
It's interesting to me that the practical engineer and mathematician Archimedes had a description of this kind of movement - growth and negative growth - in his pre-calculus. "The greatest mathematician of antiquity, Archimedes of Syracuse, displayed two natures, for he tempered the strong transcendental imagination of Plato with the meticulously correct procedure of Euclid." (C.Boyer, History of the Calculus, Dover 1959:48) Further: "In the seventeenth century, however, the infinitesimal and kinetic methods of Archimedes were made the basis of the differential and fluxionary forms of the calculus." (ibid:59) To calculate changing surfaces and volumes he imagined them as made up of "mathematical atoms" or thin sheets and pursued their interaction to exhaustion, so to say. BUT Boyer explains, in relation to A's work on the parabolic segment that: "In order to define 4/3 A as the sum of the infinite series, it would have been necessary to develop the general concept of real number. Greek mathematicians did not possess this, so that for them there was always a gap between the real (finite) and the ideal (infinite).
"It is not strictly correct therefore to speak of Archimedes' geometrical procedure as a passage to the limit, for the essential part of the definition of the limit is the infinite sequence." (ibid:53)
So not even A was able to give a clear and explicit answer to the problems raised by Zeno's paradoxes.
"The notion of the limit of an infinite series is essential for the clarification of the paradoxes; but Greek mathematicians (including Archimedes) excluded the infinite from their reasoning. The reasons for this ban are obvious: intuition could at the time afford no clear picture of it, and it had as yet no logical basis. The latter difficulty having been removed in the nineteenth century and the former now being considered irrelevant, the concept of infinity has been admitted freely into mathematics." (ibid)
Boyer emphasizes that in his "Method" A "realized that it is advantageous to have a preliminary notion of the result before carrying through a deductive geometrical demonstration..." (ibid:49)
And this is where Hegel's disembowelling of Kant's antinomical dragons starts. A notion of the Whole, that works (eppur si muove), of movement, change, growth, in a word of the interaction of Being and Nothing generating Becoming, the full understanding of which is an understanding of the Whole and all its parts.
If we take a subsidiary Whole (one considered as such just for the illustration) and instead of calling it Dion or Swiss Roll or Capital we call it A, then we should perhaps not exclude from our intuitive consideration of it the existence of other similar As, as this helps us get a "preliminary notion of the result" before our analysis.
Then we add or subtract a bit to A, leading to positive or negative growth of some kind. So we can call the result A', with the addition of an absolute bit (plus or minus) to the original entity.
Boyer mentions Democritus and the Platonic school "groping" towards "infinitesimal considerations", and this is what happens here - a series of A', A'', A'' etc arises, and regardless of the size of the [increment] sooner or later the quantitative change turns qualitative, and A becomes different from the other As not merely in degree but in kind.
And here the collective judgement of those considering the matter comes in. Consensus in human communities rules, and is codified sooner or later into habits, concepts, laws, and given linguistic expression, and proceeds dialectically in conflict or harmony with the consensus results of other communities to generalize itself or shrivel.
The Greeks reached no consensus about the growth problem, and the gap between the real and the ideal remained unbridgeable. The potential solution of fusing "Archimedes" with "Democritus" never happened. Even Descartes insisted on having an unbridgeable!! Even Kant!
And even today most maths teachers go by rote and flog the algorithm rather than visualize the result and give the maths its head.
Perhaps our philosophical unbridgeable today has moved on to the gap between our behaviour as humanity and our behaviour as human individuals or limited collectives (states, say).
The foot of a man is a more drastic example than a slice of Swiss Roll. Perhaps the more drastic the better and we should present Dion with his balls chopped off. Popular culture grapples with these problems asking when is a man not a man in terms of man-like creatures - replicants, zombies, vampires, aliens... Political culture grapples with them by adding or subtracting various defining bits (veils, hair, sex organs, skin colour, ethnic or geographical origin, missing or damaged chromosomes or genes) and sanctioning perceived non-As. History grapples with them by making certain As work side-by-side with "their" non-As for common goals and discover their original definitions were wrong.
For me now, the question of what bit of which (foot of Dion, slice of Swiss Roll, surplus value of Capital) takes on a new relevance. Because what's at issue is understanding ourselves and what we become and how.
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