No, as a heap should contain at least 3 objects, two to go below and one above.
A "heap" has only an existential definition; therefore, I find the assertion, "adding a single grain of wheat surely doesn't chnage the 'heap-ness' of the wheat", a fallacy!
In fact... If one excepts the axioms
"Some M grain(s) of wheat is not a heap",
"Some N > M grain(s) of wheat is a heap",
"adding or removing a single grain of wheat surely doesn't change the 'heap-ness' of the wheat",
then you can prove:
for any k grains such that M ≤ k ≤ N, the wheat is both a heap and not a heap; therefore I Win!
Hi, Mathhead200! Thanks for playing.
Yes, you win. A prize to be had here is consistency, and you can have it by rejecting one of the 3 'axioms'.
You're right, of course, that the puzzle is designed so as to make all of them attractive. But you have found that by considering the logical consequences of accepting them all you can defuse your attraction to one of them.
You pick on the third, although you don't say why. Perhaps it was, for you, the less attractive of the 3.
By calling it a 'fallacy' you acknowledge its superficial attractiveness, but then (by means of the proof) show it in what you hope is a less flattering light.
Like Questioner, you have found you can reject the implication (which was coming in my second question) that we should never feel comfortable going from non-heap to heap with the addition of a single grain.
Unlike them, you don't suggest a threshold. Maybe you aren't obliged to. There are such things as fallacies: sets of reasonable-looking assumptions that we have to remind ourselves (by flagging up unwanted implications or consequences) to reject.
But what Questioner suggests is more constructive: a way of refining our use of the word 'heap' so that, next time around, the puzzle's second question might lose its force.
Boo to that! I think the puzzle is on to something, wants to challenge us to somehow resolve the absolutism of the first question to the relativism (or maybe 'gradualism') of the second.
So, for me, the later in the game before either player confesses total lack of sympathy with either of the first two questions, the better the game.
So far, then, two games. Both broke up with the second player apparently immune to the relativist/gradualist charm of the second question. But supporting the absolutist charm of the first.
Thank you, both.
Tell me, do you think that a single grain of wheat is a heap?
Heap and non-heap probably the wrong way round, but I think you're just confirming that a threshold is nowhere feasible. Agreeing, then, to my second question ('do you also agree that adding a single grain could never turn a non-heap into a heap?'). Re-iterating, that is, your third 'axiom'.However, I do believe that adding to a heap (or removing from a non-heap) doesn't change this attribute of the wheat.
This is like answering 'yes' to my first question. Your first axiom.Furthermore, I assert (without formal proof) that there is some minimum value M that all parties agree is not a heap
Your second axiom.... and some maximum value N that surly must always be defined as a heap
... because, presumably, the two just-mentioned (first and second) axioms have such a ring of truth...Therefor
i.e. the one to fall was the relativist/gradualist worry about specifying a threshold - allegiance to which you recently renewed, but which had again proved most vulnerable (of the 3) to the doubts thrown up by consideration of unwanted logical consequences.... the assumption ("axiom") that I found most likely to be fallacious was the third.
So the game ends just as before, with you rejecting (after earlier accepting) the puzzle's second question.
What's this though...
Ah, so maybe you can refine the relativist/gradualist assertion so that consistency with absolutism (existence of indubitable heaps and non-heaps both) is maintained?(Although, maybe only a refined version of it.)
I do hope that's what you mean! If so...
Ok, my second question... do you agree that (or agree to some approximation of the notion that) adding a single grain could never turn a non-heap into a heap?
No, I meant was what I said. My original statement was "adding or removing (1 grain) from either state didn't change it", and I called that false. I was just pointing out that part of that assumption is in fact true. You can add to a heap, and remove from a non-heap "safely". Just pointing out my focus was only on adding to a non-heap, or removing from a heap.Heap and non-heap probably the wrong way round, but I think you're just confirming that a threshold is nowhere feasible. Agreeing, then, to my second question ('do you also agree that adding a single grain could never turn a non-heap into a heap?'). Re-iterating, that is, your third 'axiom'.However, I do believe that adding to a heap (or removing from a non-heap) doesn't change this attribute of the wheat.
Ah, so maybe you can refine the relativist/gradualist assertion so that consistency with absolutism (existence of indubitable heaps and non-heaps both) is maintained?Well, let's assume I agree with the assumption. We see that the three axioms together are inconsistent. But, perhaps I was making an assumption within my argument I shouldn't have. I assumed that wheat being both a heap and not a heap was a contradiction. But maybe the two state are merely contrary. Perhaps there is a (at least one) state that wheat can be in that is both not a heap and not a non-heap. What shall was call this (one of these) state? A half-heap, a part-heap, a maybe-heap, a vague-heap, ...? If this were true then we could indeed make all the original assertions whilst maintaining consistency.Ok, my second question... do you agree that (or agree to some approximation of the notion that) adding a single grain could never turn a non-heap into a heap?
I should mention though that this would perhaps abolish the current usefulness of the (relative) term heap; where a considerable part (if not most) of are every day usage of the word falls in this new category, this new state allows for only trivial non-heaps and heaps to be defined as such.
Hi there, Mathhead200, thanks for playing some more!
Splendid!I assumed that wheat being both a heap and not a heap was a contradiction. But maybe the two state are merely contrary. Perhaps there is a (at least one) state that wheat can be in that is both not a heap and not a non-heap. What shall w call this (one of these) state? A half-heap, a part-heap, a maybe-heap, a vague-heap, ...?
(I'm assuming that after choosing appropriate labels you'll clarify 'non-heap' as 'non-full-heap' or whatever, so that heap and non-heap-proper (i.e. not a heap) are still kept apart. 'Contraryness', as you say, but still avoiding contradiction.)
Well... the consistency, yes. But shouldn't we then risk betraying absolutism? I like to think the puzzle has succeeded in stirring in you a fairly robust absolutist conscience...If this were true then we could indeed make all the original assertions whilst maintaining consistency.
Well said! And does it even allow for a single grain to be a non-heap?I should mention though that this would perhaps abolish the current usefulness of the (relative) term heap; where a considerable part (if not most) of are every day usage of the word falls in this new category, this new state allows for only trivial non-heaps and heaps to be defined as such.
Perhaps that privilege (trivial, as you say) is to be reserved for zero grains? Then you will have to abandon your axiom no. 1, perhaps concluding, as you did before for axiom 3, that it must be a 'fallacy'. And I shall record that game number 3 ended with player 2 insufficiently fired with absolutism, though a relativist/gradualist. (I will record your impressions, too, if you like!)
Or perhaps you will extend the privilege of heap-hood to one grain but not two. Again, I wouldn't be able to fault your relativist/gradualist sensibility. Embarassment about locating a threshold comes if we delay it any more... but not here, at the step from no wheat to wheat, or from singularity to plurality, where a reason for the location is clear enough.
But still, I would have to judge this choice of threshold insufficiently absolutist. And so will you, if you have set N in your axiom no. 1 to a value greater than 1. Good for you if you have. You 'get' the absolutist message of the puzzle. Not that ideological solidarity is admirable in itself. But we need it to carry on the game.
Someone untouched by the charms of the puzzle might protest that a decisive threshold at any number is perfectly absolute - distinguishing as it does a (minimal) heap from a non-heap. But then, I can accept that the puzzle doesn't always charm. I might, however, invite that person to play the iterative version of the game, where I delay the generalised second question till after 3 or 4 versions of the first (for 1, 2, 3 and perhaps 4 grains). Or I might start, as with my second start with Questioner, at 3 or 4.
We aren't necessarily faced with that problem, though. You may be able to refine your present position once more, so that absolutism isn't compromised. Let's see...
So, my third question.
I like your idea of recognising 'shades of grey' between non-heap and heap, but I worry that in answering to the challenge of the puzzle's second question (the relativist/gradualist one) you have compromised your commitment to the first (the absolutist one). So, tell me, do you think that a single grain of wheat is even minimally a heap?
The problem is in the wording (or perhaps the question it self.) I will say that a heap of wheat implies at least one grain of wheat, or perhaps more generally at least some N grains of wheat, (i.e. to answer your question I assert that N > 0.) However, does 1 grain of wheat imply a heap of wheat? Well that is a different question all together.do you think that a single grain of wheat is even minimally a heap?
Before I go too far, letís write some assertions down:
Define to mean wheat is a heap.
Define to mean wheat is a non-heap.
Then I assert (contraries)
I also assert there exists an and with such that
If we were to let (contradictions) then we could prove that , i.e. there must be a threshold from wheat to non-wheat. If we don't then given some , at most we can prove it either is not a non-heap or it is not a heap (unless I missed something.) Perhaps "heap" is just a term we draw too much inference from? If we avoid giving "heap" a strict limit it "makes sense" deductively, but it loses it normal usefulness. But, can we really come to a numeric threshold such that every amount above it is a heap?
Hi again Mathhead200, thanks for wrestling some more with this puzzle.
Possibly my latest question was worded wrong, and that might be because I read just slightly too much into this:
My third question should be...
Tell me, do you think that a single grain of wheat is a...
... and I want to refer to the first, most un-heap-ish category. Or else to its complement, which is the sum total of heap and the middle shade.
I'm happy to use your terminology, except that, if it's ok with you I'd like to replace NH with ENH (for 'extreme-non-heap' or 'definite-non-heap'). Then we can keep 'not a heap' and 'a non-heap' interchangeable. We'll also be less likely to feel that consistency obliges us to say anything like...
... which would, as you point out, cause the middle ground to disappear.
Alternatively, we could refer to an extreme-non-heap as a 'pittance' of grains. ( .)
Anyway, my third question...
Tell me, do you think that ?
Do you mean heap to non-heap? Or please enlarge.then we could prove that , i.e. there must be a threshold from wheat to non-wheat.
it seems to me as if you are asking: how many is "many"?
that is, at what point do we pass from a singular, individualized description of quantity, to a collective notion of quantity?
in general, this seems to depend on context...how fine a mesh we can discern, before saying "continuous". for example, a speed of 16 frames per second (or thereabouts) is enough to fool our visual centers into to thinking we are seeing "actual events" instead of a series of individual photographs in sequence. in the context of the bible, it appears that this number (the threshold between "manyness" and "amountness" (if those are even words)) was once "40". in richard adam's wonderful novel "watership down", in the culture of the rabbits, it was the number "5" (a character (a rabbit) in the novel is named "Fiver", carrying with it the connotation that "he sees beyond", five for all intents and purposes being infinity to the rabbits).
there is a certain linguistic tension between our "everyday gradation" of: 1...2...3........k...k+1......oh, i don't know, a whole bunch! and our mathematical modes of enumeration. for example, the number of hairs on my head is finite. for logical purposes, this may as well be N. but in perception, i perceive it as a kind of density: somewhat sparser at the crown, and at the places my hairline used to be. at the apex of what used to be the front of my hairline, there is now a single stubborn hair that refuses to acknowledge it has lost the dignity of being a "lock", while some wispy clusters only scant inches away can barely keep up the "manifold" illusion, the clearly visible pink of scalp below giving the lie to their "heapness", suggesting a patient counting is possible.
the hair sprouting above my ears, on the other hand, steadfastly refuses to be trivialized as "hairs", and is still "heapy".