• J.E. Annas, “Stoic Epistemology,” and G. Striker, “The Problem of the Criterion,” in S. Everson, Epistemology (Cambridge: 1990).
• S. Bobzien, “Chrysippus and the Epistemic Theory of Vagueness,” Proceedings of the Aristotelian Society 102 (2002), 217-238.
• M. Frede, “The Stoic Notion of a Lekton,” in S. Everson, Language (Cambridge: 1994), 109-28.
• M. Frede, “Stoics and Skeptics on Clear and Distinct Impressions,” in Frede, Essays in Ancient Philosophy (Oxford: 1987), 151-78.
• M. Mignucci, “The Liar Paradox and the Stoics,” in K. Ierodiakonou (ed.), Topics in Stoic Philosophy (Oxford: 1999), 54-70.
• F.H. Sandbach, “Phantasia Kataleptike,” in A.A. Long, Problems in Stoicism (London: 1971).
The Sorites Paradox and a 5 year old
I was listening to your podcast on the Stoics and Epistimology on the way home from work one day, and when I got home, I told my 9 year old daughter and 5 year old son the Sorities paradox. As I went through it, my daughter said that adding the 18th grain would make a heap, so I asked, of course, how can adding just one grain make a heap, and my son Ollie said, "no dad, you're not adding the one grain to the pile, you're adding the whole pile to the one grain; that's why it's now a heap". I may be as impartial as I am classically educated, but that's the best solution I've heard. If nothing else, it shows an amazing grasp of the commutative property of addition, and is a lot more ballsy than Chrysippus's solution!
You may not feel the same, but I can't wait for your summer break to end!
Thanks again for your podcasts.
Out of the mouths of babes
Thanks, that's a great story! Maybe you have a future philosopher in the making there. I agree, it's admirable that unlike certain Stoics we could name, Ollie was at least ready to take a stand.
The next episode goes up Sunday!
every hair is numbered ...
I think the answer to this 'adding a grain of sand to create a heap problem' is simply to say ...
"you asked me to decide when I would call this a heap if you added sand grain by grain so I did. in this situation x grains isn't a heap but x+1 is ... i'm not saying I would apply the same principle in a different situation.
for instance, i'm not going to ask someone ... 'who left that heap of sand on my desk? well at least there's a lot of grains there. i'm not sure because obviously I haven't counted the grains ...' "
That's a nice idea: it sounds a bit like just biting the bullet and accepting one of the supposedly absurd claims, namely that n grains is not a heap but n+1 is. But you might be right that the context in which one would admit this could be crucial. Actually I recently heard an episode of the Elucidations podcast that argued for a version of this solution, arguing that one could perhaps solve the paradox by invoking context as you are doing. Worth a listen. Here is the link.
If adding or subtracting a grain does not change heap status, then it is possible to disperse any heap without changing the number of grains in the heap by repeatedly adding a grain at the outer rim and removing one from the center.
Does this not show that the sorites paradox is wrong in assuming that the number of grains solely defines what constitutes a heap?
Oh, that's clever! I like it. The only thing is that I guess it is less a solution than a way of reframing the paradox: just as we can ask when the heap is no longer a heap because the grains are too few, we can ask when the heap has lost its "heap" status by being dispersed. Clearly not from moving just one grain to the rim, or two, or three... but at some point.
Thus I think that your point shows not a flaw in the sorites reasoning, really, but that "heaps" are defined by two features: quantity (the number of grains) and arrangement (as a pile, not scattered all over the room or whatever). And the argument can be applied to either of these two features.
Apples and Oranges
My example shows conclusively that any number of grains capable of forming a heap are also able to form a non-heap at the same time.
This does not mean that the concept of heap is paradoxical in nature.
It means that the paradox makes an error by trying to replace an indefinite quantifier like heap with a concrete number because it fails to consider the impact of other characteristics of the heap like distribution and outside factors such as the observer.
Those variables are not negligible.
Another error occurs when the paradox wants to attribute all change to the last step performed.
If you add one grain to a pair, you cannot claim that it is only the last grain added that makes it a trio and ignore the other two.
All three grains contribute equally to form a trio, but the paradox wants you to forget that you were already two thirds of the way.
As my example has shown that part is unquantifiable in nature for heaps since the contribution of a grain hinges on others factors than pure numbers.
When the paradox claims over and over again that adding yet another grain is not enough, it is essentially moving the goalposts.
I think you may be missing the point of the paradox. The whole idea is indeed that there is no concrete number that makes a heap, and that adding just one grain cannot make a non-heap into a heap - yet, at some point between (say) 1 grain and 1 million grains, we somehow have made a transition from non-heap to heap. In other words you are treating one half of the paradox (heaps cannot be sharply defined) as if it were the solution to the paradox.
adding a grain of sand to a pile when does it make it a heap. I can see where a impressionist who is looking at it ,not knowing the cognitive process would say at a certain ( unknown advanced stage ) would say "thats a heap of sand "(again that would be as recognizing the form(Platonic ) as being large (Aristotle ) in comparable to form of small and middle ( in this case Aristotle would not get to the heap as he would keep a middle way . The impressionist at a earlier point would say "thats a pile of sand " or a hand full of sand A scientist would use a scale as would a seller at the marketplace to determine a agreed upon definition .
another paradox is "thats the straw that broke the camels back " as not the first or any other one but the last one "
( meaning to me as losing knowledge as going over the edge causing a collapse . This than indeed have a tipping pt ,as to metaphysical cause and effect of matter to collapse .
Anyways , so far i have listened to all 61 podcasts and enjoyed them all . I feel " without gaps " is the important part .as is the golden means ones knowledge advances in equal proportions with each podcast . Thanks for your tremendous efforts ,
Thanks, glad you like the series! Actually I think the straw that breaks the camel's back is crucially different from the heap case. In the straw case, the camel's back does break at some point (or if this example is too hard to really imagine, then adding one gram weights to a table until it falls). So there is a sharp cut off point there, where we go from not breaking to breaking. By contrast, vagueness cases like heaps, baldness, etc are, well, vague: there is no one step that makes the difference.
I think that vague terms like "heap" are statistical terms. They do not refer to a specific amount at all but to an average amount with a standard deviation around that mean that gradually dissipates, a normal distribution. The world is an inherently vague place, that is, it is an inherently statistical place. In statistics, for example, we would say a man is "tall" if his height is more than one standard deviation above the average height of all men of his age. Perhaps this helps?
Well, I'm no expert on statistics but it sounds a little like your solution is simply to declare an arbitrary line ("more than one standard deviation") which just bites the bullet on the paradox. So if we have a man who is half a millimeter below that height which the statistician takes as the beginning of "tall," he is not tall, but if he grows that half millimeter, he is tall. This seems counterintuitive as a description of any real property out in the world. Of course you might be willing to do this for practical purposes but it seems arbitrary. And you might not want the concepts you use to describe the world to be arbitrary; as you say, pretty well all our empirical concepts are subject to vagueness in this way, and so it could suggest that all our descriptions of the world are in some sense unmoored from any external reality. That is the worry, at least.
Vagueness is now a whole sub-branch of philosophy so it gets to be a very complicated topic, which I am not really competent to explain but you can check out this overview: https://plato.stanford.edu/entries/vagueness/. The Stoics are important, I think, not because they had a good solution already but because they are the first European philosophers we know about to have grappled with the problem. (I say "European" because I think vagueness was thematized in classical Indian thought from early on too, especially by the Buddhists.)
The "more than one standard deviation" is not at all arbitrary. It arises "naturally" out of the mathematical investigations of randomness. I suppose by natural I mean that mathematically it is inevitable, automatic, obvious. And the proof is in the pudding. Adopting the standard deviation as the right measure is what makes statistics work, and it is spectacularly successful, it underpins all our technological achievements. It is true that it will still seem counter-intuitive, but that is frequently the case with mathematical conclusions. A mathematician is comfortable with the belief that intuition is not reliable. Intuition is based on sensory impressions and as the Stoics realized this means that intuition can't be infallibly reliable. Since our language is created based on sense impressions, it is inevitable that language can't have precise meaning either. Our empirical concepts are subject to vagueness and are unmoored from any external reality, unless something like statistics can be used to moor them to external reality. Then we are left with the rather unsatisfying conclusion: we know it is true because it works.
But this brings up the question: what did the Stoics think of the reality of such mathematical ideas as circles, etc? I assume they would deny that mathematical concepts are real. Are they just sayables? Is circle in the same category as majestic? It seems to me that mathematical ideas are in between in some sense the real and the sayables like majestic, etc. There would seem to be a whole range of classification necessary to delineate the various kinds of reality. Then having sorted out this spectrum of realities, would there be also vagueness in specifying what kind of reality we meant? Perhaps we might need to statistically specify the range of various types of reality? This could easily get out of hand, but it is amusing to speculate.
I have had cognitive impressions in the sense that I have had impressions that I find it impossible to doubt. I am not sure that knowledge is the right term to use for those impressions, but I don't have an alternative suggestion..
Thanks for the link about vagueness.I am very interested.
Stoics on math
That's an interesting question about the Stoics on mathematical objects; they are materialists of course so they would have to reduce things like triangles and numbers to being either features of bodies or concepts. Here is a paper on the issue:
Add new comment