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Basic arithmetic teaches us the value of $1$ by counting (i.e. apples or oranges). A more advanced teaching of the count reveals that the count is a concept where each count of $1$ is exactly identical. This is different from the real world. Reapplying this means each apple we count is identical to all the other apples.

As I understand it the count would break down if we were counting varying things (i.e. $1$ apple, $2$ oranges, $3$ planets) yet we might say we have $1,2,3$ things. To explain that better I am saying I have $1$ orange $1$ apple and $1$ planet and the count is $1,2,3$ things.

Then when we advance a little further we begin to assign variables $x=1$ for example.

$x=1$ is interesting to me because if $x=1$ then does it not follow that $1=x$

If $1=x$ are we not at that point assigning a value of $x$ to $1$

Is the value of $1$ relative?

01nexium
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    $x=1$ is not (necessarily) an assignment of the variable $x$, but an equality, and as such, it is equivalent to $1=x$. The value of $1$ is not changed. In geometry, however we have to determine a unit (e.g. 1m or 1cm, etc.) for measures. – Berci Nov 24 '18 at 02:11
  • Okay but how do we or how did we historically determine what 1cm is? Of course you might reference smaller units of measurements and that’s fine. Yet using other measurements as a point of reference to determine your current unit creates the same situation infinitely. I could simply continue to ask the question down to the smallest known unit of measurement at which point you have to reference higher units. This is the equivalent of circular reasoning. – 01nexium Nov 24 '18 at 02:23
  • Note that most programming languages take great care to make a distinction between == and = for equality and assignment, respectively, because those are two distinct concepts. Compilers tend to raise warnings if you use = in a context where it seems you wanted ==. When you read x = 1 in a programming language, I advise you to read it as "x receives 1" and not as "x equals 1", because they are not the same thing. Using the same word for two distinct concepts will just foster confusion. – Stef Dec 08 '23 at 21:12

2 Answers2

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The value of 1 as an integer is fixed, but its value multiplied by units of measurement depends on whether the measurement is absolute or relative, and what its measurement is.

For example, what is 1 foot in the old imperial system of measurement? Someone was telling me that it used to depend on the size of the foot of the British monarch, and that here in America we're stuck on King Edward's foot. I'm not sure if that's true or just some plausible misinformation from Wikipedia.

If you're a Web designer, whether by profession or as a side job, you probably know about the em in CSS. Given a particular font, 1 em is the width of a lowercase M in that font. But if you change the font, the em changes accordingly. Ignoring tracking and kerning, two lowercase Ms will take up 2 ems, three lowercase Ms will take up 3 ems, etc.

But the thing is that ultimately 1 is an abstraction, which you can derive from 0 or you can derive by set-theoretic means thus: $$\varnothing = 0, \{\varnothing\} = 1, \{\varnothing, \{\varnothing\}\} = 2, \ldots$$

Robert Soupe
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This is an interesting and confusing question, likely to be closed because it's not precise enough for an answer. That said, I will try to clarify some things.

When you use the numbers $1, 2, 3, \ldots$ to count, you can count whatever you like - all apples, or apples and oranges. Then you are doing "applied mathematics" and it's your responsibility to make clear what you are counting. But it's not the number $1$ that changes its meaning.

When you use the number $1$ in an equation like $x=1$ you are probably planning to do algebra. Then the $1$ is just one of the possible values for $x$, and might not be counting anything at all. But whatever it means there, $x=1$ and $1=x$ say the same thing, because the equal sign means precisely that the things on either side of it are the same object, perhaps named differently.

If you were writing a computer program rather than doing mathematics the equal sign might say "assign a value to a variable", and $1=x$ would be nonsense.

In arithmetic both the equations $$ 2 + 3 = 5 \text{ and } 5 = 2 + 3 $$ say the same thing. (Sometimes kids in the early grades don't think so, because the second one is not what they are usually asked about, and you can't use a calculator that way.)

Edit in response to comment.

You ask

What is the value of 1 alienated from the count of 1?

As a mathematician, I look at the numbers as abstract things we're calling "$1$", "$2$", and so on. The collection of numbers has many beautiful properties. You can do arithmetic with numbers. Of course mathematicians created these mathematical abstractions to mirror the everyday behavior of numbers defined only informally. (Plato would argue that the numbers exist before mathematicians created them, but that's another discussion.)

So the number $1$ does not need any "reference point". It just is.

When you want to use numbers to model something in the everyday world - perhaps to count - then what you decide to "count as the reference point" depends on how you frame your problem. Sometimes you will want to count only apples. Sometimes you count fruit. In either case you can add or subtract quantities of whatever you are counting.

The numbers are just waiting there, unchanging, free of context, for you to use whenever and however it's convenient.

Ethan Bolker
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  • This answer clarifies that you can count apples and oranges when you strictly count. It denies the ability to do so when adding, subtracting, etc. The problem I see is that we use the concept of counting an apple as a reference point for the value of 1. What is the value of 1 alienated from the count of 1? – 01nexium Nov 24 '18 at 02:36
  • @01nexium See my edit. – Ethan Bolker Nov 24 '18 at 13:44