I'm sorry, but your suggestion doesn't make any sense to me. Saying "I have zero coconuts" would mean that you have a coconut? Would you have to say you had "negative 1" coconuts if you didn't have any? At a sandwich shop you would have to order a "no feet long" sub? And this is your plan to prevent confusion?
All that would really do swap the meaning of a bunch of words around so that "zero" means "one", "one" means "two", etc. Then we'd have to call it a "Two three four" right triangle but remember to make it with "three" (four), "four" (five), and "five" (six) stones and we're right back where we started.
The problem here is confusion about what quantity is actually being counted: the "fence posts" or the "fence lengths"? That's always going to depend on context - the speaker and listener have to be on the same page. There's no way to fix that by changing the number we count from.
Where do I say anything about zero becoming 1? What I am talking about is that counting arbitrary periods in a given quantity always starts with zero whether you count it or not. If you ground all of those coconuts up and decided to count how many cups of coconut you have in that mass, well you would be starting with an empty cup measure, which is zero. It’s not just fence posts
Sorry about the above comment. I'm being way too snarky in this thread. I hope the following makes up for the snark in my other comments.
What I meant above is that, in set theory, there are ordinal numbers and cardinal numbers, both of which are natural numbers. The easiest way to understand them is in terms of arrays: the number of elements in an array is a cardinal whereas an index into the array is an ordinal.
Ordinals start at 0, because that's the first natural number, and that's prooobably? why array indexing traditionally begins at 0 (I'm totally guessing). Cardinals also start at 0. So, 0 is the "cardinality" of the empty set, {}, which has no elements and so no ordinal. In the set {+}, 0 is the ordinal that corresponds to the position of the single element of the set, in the set {+,+} 1 is the ordinal for the last element in the set etc.
How this is relevant to the article above is that the artist very clearly intended the number of nuts in the "squares" at the sides of the triangle to be understood as cardinals: the squares represent sets and the cardinality of each set corresponds to the surface of that square. Sets are not ordered so the artist is free to place their elements in any arrangement, including a square, and placing a set in the form of a square on the side of an edge of the triangle clearly signals that its cardinality corresponds to the square of that edge. So arranged, the three squares necessarily overlap, so the nut at each of the three points in the triangle must be counted twice, once for each square it participates in. Seen that way, the image is a visual representation of Pythagora's theorem for a triangle with sides 3, 4 and 5, with squares 9, 16 and 25, where 9 + 16 = 25 (so the bottom edge is the hypotenuse).
On the other hand, the person who commented on the blog interpreted the number of nuts as ordinals, denoting the position of vertices in three lattices, represented by the "squares". The same person therefore interpreted e.g. the three nuts on the left edge of the triangle as standing for vertices indexed by ordinals 0, 1 and 2, and so representing a square lattice of side "2"; and so on for the other "squares". Seen that way, the image is a visual representation of a triangle with sides 2, 3 and 4, where the squares of the sides are 4, 9 and 16, where 4 + 9 ≠ 16 and so the triangle is obtuse rather than right (so the lower side is no longer the hypotenuse, since its square is no longer the sum of the squares of the other two sides) and the image is not a correct visual representation of Pythagoras' theorem.
I want to say that even having written down the latter interpretation it still sounds deliberately obtuse to me, but the real lesson I think is that there are always multiple interpretations of the same statement, or formula, etc, and it's useful to be able to see as many of them as possible. At the same time, there is usually one intended interpretation and that's the one that should be preferred. All that is formalised in First Order Logic, in the concept of an interpretation, that is an assignment of truth values (true or false) to all the atoms of a predicate. A FOL interpretation is uniquely identified by the set of atoms to which it assigns the value true, therefore the number of possible interpretations is equal to the cardinality of the powerset of the set of atoms of a predicate. That's a lot of a interpretations! That's why you need an intended interpretation (also a concept in FOL).
On the other hand, I might be wrong. Maybe it is deliberately obtuse to count the surface of a square by ordinals, rather than cardinals. Well I don't know.
Bottom line is that in art, like in maths, one must always look for more than one way to see things and not assume that they know all the answers before they have asked all the questions.
More than just tradition, it’s because (at least in C) if a is an array, it’s effectively just a pointer and so a[i] = *(a + i) (which means the i’th element of a is just the contents of memory address a + i). In particular, we have a[0] = *a.
The first element of the array lives at zero offset from the address pointed to by a, so is considered the ‘zeroth’ element.
Yeah. It has nothing with intuition or mathematical sense, quite the contrary. It’s just a quirk of a language that got transformed as “the natural way” after decades of reinforcement. There were other languages before C where array indices started at 1.
That's right. Before, and after too. R for example has indices starting in 1. But I was thinking about the specific 0-based convention. I guess I was wrong about it though.
It's not a problem with confusing ordinals and cardinals; the artist's problem is, as others have said, a fencepost error.
I suspect that what has annoyed people is that if all the hazelnuts are evenly-spaced, then the triangle the artist created is not a 3-4-5 triangle, it's a 2-3-4 triangle, which isn't a right triangle. To make it look like a right triangle, he's had to arrange the nuts with unequal spacing. He must have noticed that, and it should have annoyed him too.
I suppose there's some geometry in which a 2-3-4 triangle is a right triangle; but I doubt the artist was exploring non-euclidian geometries.
FWIW, I didn't notice the error immediately - but I did notice the uneven spacings.
>> It's not a problem with confusing ordinals and cardinals; the artist's problem is, as others have said, a fencepost error.
Yeah, my argument is that it's a fencepost error only if the numbers of nuts are interpreted as ordinals indexing the vertices of a lattice on the Cartesian plane, rather than cardinals counting the elements of a set, while the artist intended them to stand for cardinals. That is what the anonymous contributor to the 360 blog, mentioned in the article above, seems to have seen:
"To my eye," the commenter continued, "the hazelnut grids look exactly like pins on a Geoboard, or lattice points in the plane. And given that perspective on this image, we see a 2-3-4 triangle, an obtuse triangle, and squares of area 4, 9, and 16."
But this is clearly only one way to see things and there's nothing to make it more valid than the other, except of course that this one leads to an error which strongly implies it's not the right view.
>> I suspect that what has annoyed people is that if all the hazelnuts are evenly-spaced, then the triangle the artist created is not a 3-4-5 triangle, it's a 2-3-4 triangle, which isn't a right triangle. To make it look like a right triangle, he's had to arrange the nuts with unequal spacing. He must have noticed that, and it should have annoyed him too.
Some other comments say something similar, but I don't understand it. What is the issue with spacing? As you point out it would be difficult to get a perfectly mathematically correct spacing with irregularly shaped solids, like hazelnuts. Which suggests that spacing was not part of the intended interpretation. Can you explain?
> while the artist intended them to stand for cardinals.
Oh stop it with that BS already. He fucking obviously intended them to stand for line lenghts, or he wouldn't have arranged them in a right (but wrong) triangle.
No, but if you wanted to count how many coconuts you had, you would start with 0 (pointing at nothing) and then you would continue by 1,2 etc pointing to each other coconut. This would allow you to use the same counting process to count something, even if there was nothing to count.
All that would really do swap the meaning of a bunch of words around so that "zero" means "one", "one" means "two", etc. Then we'd have to call it a "Two three four" right triangle but remember to make it with "three" (four), "four" (five), and "five" (six) stones and we're right back where we started.
The problem here is confusion about what quantity is actually being counted: the "fence posts" or the "fence lengths"? That's always going to depend on context - the speaker and listener have to be on the same page. There's no way to fix that by changing the number we count from.