$1$ heap of sand $+\ 1$ heap of sand $= 1$ heap of sand?

Question

My uncle, who barely passed elementary school math (which leads me to believe he read this in some kind of joke magazine), once told me this when I was very young.

$$1 \text{ heap of sand } + 1\text{ heap of sand } = 1\text{ heap of sand} .$$

It does sound like a joke, but as I learned more and more math (still at basic college math), I still couldn't (and can't) disprove/prove it.

So we all know this is true "linguistically," since if you add 1 heap of sand to another one, it will still be a heap of sand, but mathematically, 1 + 1 cannot equal 1.

I expect there to be, and wouldn't be surprised if there weren't, similar questions to this one but I couldn't find it using the search feature.

I guess I'm going out on a limb here because of the stupidity (and perhaps the obvious answer) of this question, but I seriously cannot disprove this based on what I know.

Answer 1

The issue is a misunderstanding of the words involved. One number plus one number is still one number; the numbers involved are not the same, but they're all numbers. Similarly, one heap of sand plus one heap of sand is another, completely different heap of sand. Being "a heap of sand" is a property, not an identifier. If you said "heap of sand A plus heap of sand A is heap of sand A", I would say no, that's false - adding "heap of sand A" to itself should duplicate each grain of sand, resulting in heap of sand B which is twice as large.

Answer 2

The resolution here is that $+$ does not refer to addition, but to composition, or more mathematically union of sets. The "1" here does not refer to the coefficient alongside the unit "heap of sand", neither is "heap of sand" actually a unit. The "1" is actually just a standin for an indefinite article.

So a mathematical model of your uncle's statement goes something like this:

The union of a set of sand and a set of sand is a set of sand.

More formally:

For any $A, B \subset S$, $A \cup B \subset S$.

Clearly this is a completely different statement than $1 + 1 = 1$.

Answer 3

One thing you must notice here is that:

$$1\, \text{heap of sand} + 1\, \text{heap of sand} = 1\, \text{heap of sand} \not \large\Rightarrow 1+1=1$$

The reason is that $1$ heap is not well-defined or quantitative or mathematical enough for being used to perform mathematical operations like addition,subtraction and make equations. If you had mentioned $1$ Kg. or perhaps $1$ quintal or even, in some special case, $1$ bag of sand, then also you can perform addition,subtraction etc. and equate both sides of the "$=$" sign. Otherwise this equation makes no sense.

It is similarly wrong to say that $$1\, \text{straight line} + 1\, \text{straight line} = 1\, \text{straight line} \implies 1+1=1$$

Instead as pointed out by McFry, you should try to use set theory to clarify your problem since it is a clear case of composition.

Answer 4

Mathematically, there is nothing wrong with the statement, $$ 1\text{ heap of sand} + 1\text{ heap of sand} = 1\text{ heap of sand}. $$ For example, in the extended real numbers, $\infty + \infty = \infty$. This does not imply $1 + 1 = 2$ on its own.

However, I think you have understood your uncle's statement wrong in writing it down that way. A better way to write it down mathematically -- one that seems more true to the meaning -- is to say, $$ \text{If } x \text{ and } y \text{ are heaps of sand, then } x + y \text{ is a heap of sand}. $$

Now if we define a "heap of sand" to be a natural number which has size at least $1000$ (representing the number of grains of sand), then we are simply saying that two natural numbers which have size at least $1000$ add to a natural number of size at least $1000$. Which is perfectly true and correct.

Finally, I must comment on:

I still couldn't (and can't) disprove/prove it.

You couldn't prove or disprove it because it's not a mathematical statement until you define what a "heap of sand" is, and what it means to add two of them. That is the crux of the problem here. Start out by defining what a "heap of sand" is, and then we may be able to see if the statement is correct or not.

Answer 5

You can avoid the paradox by stating "Addition does not preserve the number of heaps".

Answer 6

Linguistically speaking, sand is a mass noun like milk and money.

a noun with the syntactic property that any quantity of it is treated as an undifferentiated unit ... In English, mass nouns are characterized by the fact that they cannot be directly modified by a numeral without specifying a unit of measurement, and that they cannot combine with an indefinite article (a or an). ...

Cumulativity and mass nouns

An expression P has cumulative reference if and only if for any X and Y:

If X can be described as P and Y can be described as P, as well, then the sum of X and Y can also be described as P.