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Elementary arithmetic tells us these two expressions are equal. Here what puzzles me about this. $1 + 1$ is symmetric in the sense we have two identical copies of the same entity (unity) and $0 + 2$ is obviously not. So, in certain sense these two expressions should be regarded as qualitatively different (we can prescribe physical meaning to the numbers, say, talk about some elementary particles, that makes the question a bit less abstract).

What kind of symmetry is this? What exactly is lost when we go from $1 + 1$ to $0 + 2$ and gained when we travel back?

Shaun
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S. N.
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    One could definitely argue that $0+2$ is also 'symmetric' as it equals $2$, and $2$ can be split into these identical copies you were speaking of. Mathematically, I wouldn't say there's any difference. $2$ apples looks no different to $1+1$ apples or $0+2$ apples. – Joe Jun 17 '20 at 19:09
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    We have $2=1+1$, so I don't see why you want to consider $2+0$. So $1+1$ is symmetric, and we change it to $2$ - so what? That's also symmetric. – Dietrich Burde Jun 17 '20 at 19:09
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    I think you're more-or-less interested in the mechanics of partitions, so Young's lattice may give you some insight: https://en.wikipedia.org/wiki/Young%27s_lattice – Alex R. Jun 17 '20 at 19:10
  • @Joe The reason I used 2 + 0 is because it is the same amount of entities as 1 + 1. Okay, let's stick with apples, I have one apple and so have you. I give you mine. Now the situation is clearly asymmetric. – S. N. Jun 17 '20 at 19:55
  • @DietrichBurde Please see my comment above. – S. N. Jun 17 '20 at 19:55
  • Depending on how pedantic you want to be, it could take a few hundred pages to nail down. – Shaun Jun 17 '20 at 19:56
  • @AlexR. Thank you. I should check it. – S. N. Jun 17 '20 at 19:56
  • @Shaun I trust this is too much for me, my mind is certainly no match for that of Russell. – S. N. Jun 17 '20 at 19:59
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    What's the difference between "Abraham Lincoln" and "the signer of the Emancipation Declaration". Or between the "capitol of France" and "the city where the Louvre is"? – fleablood Jun 17 '20 at 20:10
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    In my opinion, the expressions themselves may be distinct, yet the value of the expressions are equal. This is much like how if we were to say "Donald J Trump" this is a different phrase with different connotations than if we were to say "The president of the united states from 2016 to 2020." They both describe the same thing but in different ways. So, here we get to the heart of the matter... how we define equality and whether we consider things equal if they describe the same thing or if they are equal only if their expressions are the same. – JMoravitz Jun 17 '20 at 20:12
  • Now... certainly... if you are interested in treating $1+1$ as different than $0+2$ and are interested in questions like "in how many ways can you add positive integers up to get a total value of $4$ where order matters" (counting $4,1+3,3+1,1+1+2,1+2+1,2+1+1,1+1+1+1$ as all distinct) that is certainly something that is done, but it is worth noting that the objects we are counting here are such that despite their values may be considered the same the expressions are different. As Alex pointed out, Young's Lattices are a good starting point. "Partitions" and "Compositions" are another – JMoravitz Jun 17 '20 at 20:14
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    @fleablood Were there no difference, there would be no point of having two sentences. We could live with only one. – S. N. Jun 17 '20 at 20:15
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    There is absolutely a point of having separate sentences. They both bring attention to certain aspects of the thing we are describing despite both describing the same thing. – JMoravitz Jun 17 '20 at 20:17
  • An object (such as the number $2$) can be represented in an infinite number of ways. Why don't you just go look at Monet's Haystacks? What you consider special (such as symmetry) in any particular representation is in your mind, not inherent in the object. – Keith Backman Jun 17 '20 at 20:22
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    It doesn't work so well in English, but consider the following sentences. "Fleablood (as opposed to someone else like Alex) threw a ball to me" versus "Fleablood threw a ball (as opposed to something else like a book) to me" versus "Fleablood threw a ball to me (as opposed to someone else like Shaun)." Now, all three of these sentences can be written the same way: "Fleablood threw a ball to me" however in some languages we can have this sort of italicization effect, where we call attention to which aspect of the sentence requires extra attention, and this is a useful thing to do. – JMoravitz Jun 17 '20 at 20:23
  • @JMoravitz Thank you! I have to take some time have to ponder over what you have written. Surely, it is the same from the viewpoint of arithmetic. You are right it has everything to do with how we understand or interpret the notion of equality. – S. N. Jun 17 '20 at 20:24
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    @JMoravitz I think you have inadvertantly answered OP's original question. The expressions draw attention to different things. Just like $0+2$ might draw your eye towards the additive identity, or $1+1$ might draw your eye to (arguably) the most basic question in arithmetic. – Joe Jun 17 '20 at 20:25
  • In the end though, they all describe the same actions as having occurred. Despite having italicized a specific part of the sentence it remains true that all individual parts of the sentence were true... that it was fleablood, that the action was throwing, that the object being thrown was a book, etc... We have that despite the differences in connotations and where to draw our attention, the final action described by all three sentences was in fact the same action. In the same way, $0+2$ and $1+1$ both describe (in different ways) the number "Two" despite describing in different ways – JMoravitz Jun 17 '20 at 20:29
  • we can claim equality only for the value of expressions or normalized form. the words are different and express different things. arithmetics kind of factorizes. if we apply the operation of taking the first letter than they wouldn`t be the same. this might be too syntaxis based example, but in terms of logic, the result of the expression is just a list of reductions. and you can give any meaning to syntaxis. – grigorii stepanov Jun 17 '20 at 20:35
  • I used to the thought, why to use numbers. there is nothing identical in our world, why should we count? but actually we count not objects, but their projections on for example on being human. for example when we claim "there is 4 people in the room" we claim that there are 4 different objects, but with a set of properties, which can define them as human beings. not sure that it is the answer. but might be relevant for your reflection. – grigorii stepanov Jun 17 '20 at 20:38
  • "we can claim equality only for the value of expressions or normalized form. the words are different and express different things. arithmetics kind of factorizes..." Yeah, but you can't deny $\sqrt{4} = 1+1$. – fleablood Jun 17 '20 at 22:36

1 Answers1

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Your question is largely metaphysical and relates to the map-territory relation.

JMoravitz has given an interesting perspective in the comments that the difference between the expressions is how they semantically emphasise different properties. I would further demonstrate this with the fact that $4=3+1=2\times 2$. Each expression refers arithmetically to the same number but the second emphasizes the position of $4$ amongst the integers and the third emphasizes its factors.

But I also think this shows another interpretation of the distinction. Your two expressions are effectively describing different algorithms that can be used to compute $2$. In the same way that a thing can be referred to by different names (e.g., "my grandma" is "my grandpa's wife" is "my mother's mother" is...) or two roads can lead to the same destination, a mathematical object can be defined via different procedures. For instance, $\frac12$ could be constructed as $\frac24$ or $\frac36$ or via infinitely many other ways. (MSE Question 1819718) asks this from the perspective of how different sets of numbers are constructed and whether they really refer to the same underlying object. In particular, look at hmakholm left over Monica's answer.

The most common interpretation is that the object we understand as "$2$" and that can be defined as either $2+0$ or $1+1$ exists Platonically. In this sense, we don't care how it is expressed since all expressions refer to the same abstract object - they are models of the same thing. Another view you could have is that even if the expressions refer to distinct objects, the distinction is moot since all properties (apart from those relating to the expression itself) hold for one iff they hold for the other, since we can relate the expression $2+0$ to the one $1+1$.

In sum, the math that we communicate is generally on a level above the expressions themselves. So both expressions are at heart $2$ and the distinction is unimportant.

Jam
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  • I must admit you (together with @JMoravitz with his valuable comments and Keith Backman whose comment I like) have answered my question. I shall be glad to accept it. However, I realised I put the question in too abstract manner (sorry for unintentional confusion). – S. N. Jun 19 '20 at 18:58
  • Let's consider some objects like apples in stead of numbers. You have one apple and so do I. I give mine to you (now it is $2 + 0$ situation, let's agree to believe it to be the same as $0 + 2$). It is no more symmetrical. What describes this loss of symmetry? Should we talk about distribution of apples between us (that probably leads us to the notion of entropy and the realm of statistical physics)? – S. N. Jun 19 '20 at 19:00
  • I don't see what you mean by "It is no more symmetrical". Why do you consider is less symmetrical? – Jam Jun 19 '20 at 20:15
  • Perhaps, I misunderstand yuor question. Well, the situations where each of us has one apple and where you have two and I don't have any are certainly different, are they? I would call the first one symmetrical and the second lacking symmetry. – S. N. Jun 20 '20 at 07:43
  • @S.N. You seem to be confusing the mathematical expression for the thing it can represent. $1+1$ doesn't necessarily have a physical interpretation like "one object and another object". It just as well means "the next thing in a list after the first" or even as abstract as "the number after the number $1$". – Jam Jun 20 '20 at 14:55