Re: Sorites only joking...

Quote:

tom: Tell me, do you think that a single grain of wheat is a heap?

me: I think that no (zero, the absence of) grains of wheat is always a pittance.

Just to make sure we get the lower threshold right. (Although, I believe that one grain could hardly be called a heap in our planet sized context; in another context, like say I'm deathly allergic to wheat, one grain might just be a whole heap of wheat.)

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tom: On the other hand, do we need to conduct an opinion poll (or whatever kind of survey you had in mind) to establish whether a single grain could ever be called a heap?

me: If we replace single grain in the question to any particular $\displaystyle k$ grains; and, we want to know is $\displaystyle k$ grains a pittance, heap, or neither, then a survey, although a seemingly reasonable approach, doesn't give us a practical answer to the question. (Remember, I called these variables "volatile". They are extremely dependent on context, and __time__ is a factor in that context. By the time the survey is over, the values may have changed, even if nothing else (other then the passing of time) has.)

Although values exist for $\displaystyle M, N$, I doubt we can ever know them (at least if we gain the information through traditional means like a survey.)

Re: Sorites only joking...

__Game 6.__

Quote:

Tom: Tell me, do you think that a single grain of wheat is a heap?

Mathhead: Certainly not.

Tom: And do you agree that adding a single grain could never turn a non-heap into a heap?

Mathhead: Kind of. I think there is a middle ground between 'heap' and an emphatic grade of 'non-heap', which we shall call a 'pittance' of grains, at the other extreme. And I do declare that adding a single grain will never turn a pittance into a heap.

Tom: Ok. Now, is a single grain anything other than a pittance?

Mathhead: I think that no (zero, the absence of) grains of wheat is always a pittance.

Tom: Well, good. But I didn't mention zero. What about the single grain? Is that not equally a pittance?

Mathhead: [Please confirm, yes, a single grain is a pittance!! Or I'm very confused.]

Tom: And are you comfortable with the addition of a single grain turning a pittance into a non-pittance? If so, are you willing to locate the lower threshold M (going from pittance to non-pittance)?

Mathhead: I'm comfortable with the idea of a threshold in principle, but it's naive and, perhaps, missing the point to think that we could or should try to locate it. It's too volatile for that.

Tom: On the other hand, is it so volatile that we can't assume it would never reach down to a single grain?

Mathhead: [hoping for an absolutist answer here, but I shan't presume it this time...]

Tom: [... and you can guess where I want to go from there...]

Re: Sorites only joking...

Quote:

Tom:... What about the single grain? Is that not... a pittance?

It depends on the context; more precisely, the specific values of $\displaystyle M, N$ determine if that is true or not.

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Tom: On the other hand, is it so volatile that we can't assume it would never reach down to a single grain?

Yes. (I know that might not be the answer you want.) We can **not** assume that *in an arbitrary context* one grain of wheat is not a heap. That is to say, there could exist a context in which $\displaystyle M = 0$. (Again I point to the "deathly allergic to wheat" counter-example.)

In the context of our planet (specifically), I feel we can assume $\displaystyle M > 1$.

Re: Sorites only joking...

To develop the possible reading of 'context' that I mentioned in post 38...

Quote:

- Tell me, do you think that a single grain is a heap?

- Could you possibly tell me more about the context of this question, so that I know if I can answer it in an absolute way?

- Sure. I hoped it was tacitly implied, but I don't mind spelling out, that the context is to be the very most general and obvious one.

That is, 'heap' as it functions in the English language to classify and compare quantities of grains of wheat, but focussing, as you'll have become aware, on comparisons that specify whole numbers of grains.

This 'discrete' focus is useful not only in excluding (harmlessly I hope) the huge class of continuous examples, such as those involving quantities of wheat mulch, but also in creating an appropriate analogy with other discrete examples such as the smallest number of hairs on a hairy head, or the smallest number which is a large number, or (if you really want to go there) the smallest number of harvests making a 'heap' of harvests for the planet. Also we would definitely want to factor in the use of 'heap' counterparts within other languages.

Such exclusionary and inclusionary references, however implicit, usually help to set the 'context', and make us feel we have the semantic linguistic competence to answer the first question.

If I asked, instead, 'tell me, do you think that a single grain of wheat is a whole heap of wheat for someone afflicted with a serious wheat allergy?', you would of course judge the question in this explicit context, and expect to locate completely different thresholds. You would be classifying the same numbers of grains within a different category system. One related to but distinct from the original, literal or un-hyperbolic one. (See here, pp81 - 85.)

I can think of other contexts that would modify the opening question in other directions, indicating different transfers of the original categories. For example, 'tell me, how far into the heap spectrum is a single grain?'

But, without such explicit indications, you are invited to show your competence in using just the literal, un-'transfered' categories.

As, after all, even a wheat-allergic person is perfectly free to do. He knows perfectly well that a single grain is (considered literally) a very obvious example of a non-heap.

I suspect, though, that your 'contexts' are not just a few competing category systems like this, but a rather dense array of perspectives that will form the kind of 'spectral' distribution seen in graph 3. (Wondering)

Re: Sorites only joking...

A single grain cannot be a heap and removing a single grain from any heap does not make it a non heap. You have to remove a smaller heap from the total heap to make it a non heap. Eh?

Re: Sorites only joking...

Quote:

Originally Posted by

**Nervous** removing a single grain from any heap does not make it a non heap. You have to remove a smaller heap from the total heap to make it a non heap.

OK... but if you *do* remove a single grain from a heap, what are you left with? A heap?

If so, let's do it again...

(Thinking)

Re: Sorites only joking...

What's wrong with there being a heap left?

Maybe heaps are like infinities. After all, infinity minus one is still infinity. Infinity plus one is still infinity.

And, since the human mind cannot imagine numbers greater than three without grouping, wouldn't it be likely that our brains interpret a heap as an infinite amount of numbers?

So "heap" may just be our brain's saying "Screw it, let's call it infinity because it's so big, besides it would operate like infinity if you where to add or subtract a finite amount to it, unless that "finite" amount where something else that was so large, I -the brain- would consider as a heap or infinity as well."

Still, I'm afraid I don't get your original point, why is there a problem with removing a single grain from a heap and it still being a heap? Or am I so confused that this was not even your point?

Re: Sorites only joking...

Quote:

Originally Posted by

**Nervous** Maybe heaps are like infinities.

That would solve a few problems in the world...

But, of course, after enough iterations of the apparently harmless subtraction, you are going to contradict your earlier premise that 'a single grain cannot be a heap'.

Maybe you read 'Let's do it again...' as meaning 'you think again!'. No, it meant 'let's subtract again'.

Don't get me wrong. I agree that for a good while we can go on subtracting without causing the remaining collection of grains to lose heap-status. But not forever. I know, I probably sound like your bank manager.

So the puzzle is how to avoid drawing a line or else how to decide where to draw it.

Not everyone is impressed by the puzzle, however. Some people are happy to specify a particular cut-off number. How about you?

Re: Sorites only joking...

So I guess I'm stuck here. Even with the "standard dictionary definition", *"heap"* is (at least somewhat) context dependent. So I find it hard to come up with a "better" answer then what I've already said here: http://mathhelpforum.com/math-philos...tml#post729208

To reiterate, whether some $\displaystyle k$ grains of wheat is a heap of wheat depends on context. And in a given context there is a absolute answer; either $\displaystyle k \le M$ or $\displaystyle k \in (M,N)$ or $\displaystyle k \ge N$. But, it's impractical to know the answer, because it can change so "easily". When the word is used in every day life a context is created, one that both the "sending" and "receiving" party understand (sometime ambiguously). Now, as soon as you pose the question "well, let's remove a grain and ask is it still a heap" (resp. "let's *add* a grain and see if it's still a *pittance*") the problem become that we have memory, and that memory become part of the new context. We know that it was a heap a second ago, and so now why shouldn't it be? But should a new person come along, or should those people go to sleep and come back in the morning, all of the sudden the slightly smaller pile may not look like a heap anymore. Perspective is part of the context... And should you stand there long enough, removing grain after grain, at some point your brain will say: "Hey wait a minute... This isn't a heap anymore! What are you trying to pull?"

Re: Sorites only joking...

Quote:

Originally Posted by

**Mathhead200** at some point your brain will say: "Hey wait a minute... This isn't a heap anymore! What are you trying to pull?"

But why should this psychological double-take be worthy of mention? Why shouldn't you expect your brain to simply acknowledge a slightly lower probability or proportion of heap-recognising judgements with every loss of a grain?

OK, so your own threshold when forced repeatedly to choose one may spring back rather dramatically at some moment, but that doesn't explain the double-take. "What are you trying to pull?" doesn't square with your current insistence on the answer to the puzzle's first question being a matter of degree, i.e. yes for some tiny proportion of possible contexts or perspectives or (psychological) subjects. Contrary to the double-take, you before the spring-back (in this latest thousand-edging-down-to-zero scenario) could quite well count as one of that tiny proportion. It could be a datum in the tip of the tail of the distribution in graph 3. Your brain should merely say: "Funnily enough, I feel I want to withdraw my recent votes for certain numbers of grains being a heap, or to put it another way, me-now doesn't contribute to the tiny degree of heap-ness bestowed on these various small numbers, even though me-a-moment-ago does."

Deveno is quite content to admit that a single grain is, to some tiny degree, a heap. As are most people who claim that 'heap' is 'context-dependent'. And I keep thinking I'm going to have to give you up to that position. But you won't let me! Your "what are you trying to pull?" outburst betrays (yet again) a very healthy absolutist conscience. Whereby you can imagine realising that some of your recent heap-affirming judgements weren't only relatively extremist but patently absurd, inconceivable, linguistically incompetent, mistaken, obviously false etc.

Everything you and Deveno say about the fluidity and vagueness and context-dependence of judgements at the border I agree with. But the magic of the puzzle (when it does charm) is in reminding us that the border is nowhere near zero. So where?

Quote:

Originally Posted by

**Mathhead200** First I'll assume "pittance" to mean an amount of wheat which could __never__ be called a heap of wheat,

(Bow) (Bow) (Bow) (Bow) (Bow)

Re: Sorites only joking...

one can perform this experiment, with some unpopped pop-corn, and a measuring device of some sort (like a cup).

measure out a cup. without counting the kernels, one has no idea of "how many" kernels the cup contains (you can, if you like, try to form an estimate, it should not be an extremely large number). we can, provisionally say it is "a small heap".

now, instead of subtracting kernels one at a time. instead, form a pile one at a time. every time you add one, see if you can "count the kernels on sight". perhaps (in order to avoid the "but i know how many are there" dilemma), you might do this several times, with a helper, who randomly adds between 1-4 kernels each time.

perhaps you see where i am going with this, perhaps you do not. if not, i will be explicit: we seem to have a built-in "logarithmic scale" of size: the difference between 1 and 2 is HUGE, the difference between 98 and 100, not so huge.

at what point does the difference become "imperceptible"? it depends on the person, in general. but we are not dealing with x-y, which has a minimum of 1, but log(x/y), which has no minimum for x > y. which explains why we are having such trouble, when does a positive number ε > 0 become "small"?

for example, with our eyes/brain, more than 16 fps (approximately) is "wasted information"...there is only so small of a time interval we can directly perceive. we can say for this instance ε = 1/16.

log(2/1) ~ 0.69, very few people would consider that "negligible".

log(100/98) ~ 0.02, which is much smaller.

log(1,000,002/1,000,000) ~ 0.000002, well below the threshold of perception of most people.

i think the "confusion" this puzzle engenders is a kind of "bait and switch". surely "one grain" is "one grain" no matter if we add it to another single grain, or a million. but actually, it's NOT. our perceptions of size are logarithmically scaled, not linearly scaled. what is imperceptible in large numbers is not imperceptible in small numbers. our mathematics (and logic) has progressed to the stage that we can appreciate this, our language has not.

so what is this ε i am talking about? well, it does depend on context, but it is not "absolutely variable". for example, i can only see differences in thickness down to 1/64 of an inch or so (perhaps 1/128). what does this mean?

log(x/y) < 1/128

log(x) - log(y) < 1/128

log(x) < log(y) + 1/128

let's say i have widgets which are 1/128 of an inch big. when do they become a "clump" of widgets? easy, when:

log(x+1) < log(x) + 1/128, or equivalently:

(x+1)/x < 1.00784 (approx.)

solving for x, we get when x > 1/(0.00784) ~ 127.5, 128 widgets become a clump.

of course, there *are* problems with this model. perhaps i have the logarithmic base incorrect, or the limits of my perception wrong. i merely want to point out that: "can a non-heap ever be made into a heap by adding one grain" is entirely the wrong question. it assumes an "absolute linear scale" which is at odds with our actual perceptions of size.

Re: Sorites only joking...

Hi, Deveno! Thanks for having another splash.

Quote:

- Tell me, do you think that a single grain of wheat is a heap?

- And do you agree that adding a single grain could never turn a non-heap into a heap?

- I think that's entirely the wrong question. It assumes an "absolute linear scale" which is at odds with our actual perceptions of size.

- Oh, ok. Do you agree that doubling the cardinal size of a non-heap could never turn it into a heap?

- Then would you care to identify which is the largest whole power of two that is a non-heap?

- Certainly. I would say, the fifth. That is, 32.

- That's a good clear answer! Thank you. Of course, someone might ask you why 16 isn't also (in some contexts, or in some people's primitive number sense) a heap...

- Well I don't see any harm in admitting that could be the case.

- ... And in a tiny minority of contexts or perceptions, a single grain is a heap? Then mustn't you qualify your answer to the very first question?

I admit that, proceeding by a set ratio (or, for that matter, a set number of grains) rather than one grain at a time, you might find it easier to choose a threshold that you can insist is firm, and doesn't lead back to a single grain heap. I'd like to look at that. But, as I say, it would be useful to know if you have yet aquired an intuitive aversion to the single grain heap.

I thought you were completely untouched by that intuition. But maybe here you're willing to affirm, yes, one widget and a hundred widgets are both certainly not a clump. That would be great...

Quote:

- Tell me, do you think that a single widget is a clump?

- And what is the highest whole number (or interval of whole numbers) of widgets which you can deny is a clump?

And you have an actual formula, potentially, for the smallest heap... something like, the reciprocal of the just-noticeable difference, if it could be measured, of the approximate number sense that we've been hearing about. (On this forum, I vaguely recall.)

Very interesting! (Clapping)

Please tell us more! And forgive if I have to insist on putting it all through the sorites test (which you'll have gathered I'm quite keen on).

Re: Sorites only joking...

I'm happy to utilize your language, except that, if it's ok with you I might such as to change NH with ENH (for 'extreme-non-heap' or 'definite-non-heap'). Then we can remain 'not a heap' and' a non-heap' interchangeable. We'll additionally be less most likely to feel that uniformity requireds us to point out everything like ...

Re: Sorites only joking...

Okay, here's my attempt.

(Thinking)

Grateful for objections or other input.

(Happy)

Re: Sorites only joking...

Just bumping this over the spam.