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Consider the classic statement 2 + 2 = 4. Do the symbols 2 and 2 here refer to the same object or not? I have always believed that they are designations for the same object. If we look at the definition of numbers in ZFC, we see that numbers are defined as concrete sets. This understanding did not cause me any problems before, however, I do not quite understand this image in the article. enter image description here Here we see that the operation has two different inputs. How will this work in the case when we perform + on 2 and 2?
Of course, if + is an operation in some computer, then there is no problem. We are adding two different memory locations. But there are no memory locations in arithmetic. This gave me the idea that number is a class of objects. So, when we write 2 we actually mean some number 2. However, it looks strange and it's hard for me to think that way. I don't know how else we can give some operation the same numbers.This problem is easily solved if the input of any operation is a vector, but in this case each operation has one input, not several.So, do numbers exist in a single instance or are there many of them?

Александр Чайков
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  • "Here we see that the operation has two different inputs." Don't be silly. The physical limitation of not being able to supply an actual physical machine the same object to two different entrances at the same time does not apply to abstractions. The picture is a metaphor, and metaphors tend to come with limitations. – anon Jun 24 '22 at 07:43
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    It has two different slots for inputs. But there's nothing that says the inputs themselves have to be different. (Just assume numbers have the super-power that they can appear anywhere in the universe and be in two places at once because..... well, if you think about that is exactly what numbers can do. If I can have 2 popsicles in my apartment at the exact same time George around the world is petting his 2 cats at 2 oclock then surely 2 sliding down two chutes at the same time is no problem.) – fleablood Jun 24 '22 at 07:47
  • There is only one "$2$" – Peter Jun 24 '22 at 07:48

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Consider the following even more duplicated, tautological, piece of natural language:

"This table, and this same table, are the same table."

It appeared 3 times in the sentence. Did 3 tables appear magically in the world to say this? So did we need 3 tables to make this statement?

Instead of thinking of "$2 + 2$" as a "machine", it is best to think of it as a language element. It names another number, and names that number just like $2$ itself names a number. We can use a process to find out which number, just as we can use a process to find out what the label "the letter in the middle of this sentence" refers to, but still, it itself is just a label.

Of course, then still, how do you get the "2"s to carry out the identification process if you want to do so? Are they the same, or different? Your question here is very similar to the following question on Philosophy:

https://philosophy.stackexchange.com/questions/91406/why-cant-numbers-be-used-up

One way to look at it is that 2s are mental objects for which we can just create as many copies of them as we want whenever we need from an internal prototype stored in our brain. Another way to look at it is we have the same 2, but we pass two references to it to the process. Given that the "2" itself is unaltered by the process, but rather that process is only one of identification, these two interpretations yield no disagreement, but the distinction can be relevant in other areas of application - particularly computer coding. There, you may encounter having to actually make this distinction explicit - it's called "pass by value" versus "pass by reference". When you pass an argument by value, the computer actually creates a copy. When you pass by reference, it does not.

Or here may be a simpler way to say it. You see "2" appear in many contexts. In this post, in the news, on your phone, and so forth. Why should that same "2" also not be referred to twice while being separated only by a single symbol, to create a phrase which refers to another number that is not necessarily 2?

The_Sympathizer
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