# Is '=' a relationship between the objects or their expressions?

The Wikipedia definiton of equality gives it as a 'relationship between two expressions'

This confuses me as when we define mathematical expressions like 2+2=4 it makes no sense to say that '=' or 'equals to' relates the two expressions as it would mean that '2+2' and '4' are representing themselves as expressions as opposed to denoting or naming the objects.

I think in mathematical contexts when we use expressions it is always to unambiguously name an object, for example: '2 is an element of N' is meaningless if '2' represents itself as an expression.

I understand there is a relation between the expressions whose value are the same but is defining this relationship as 'equality' and saying it is denoted by '=' correct? It seems that '=' should denote a relationship an object has with itself.

Take x where x is any real number,

2x=x+1 is true when x=1 but not for any other values, this is a constraint how are the two strings of symbols related by equality? Whether we can conclude that the two have the same value depending on the interpretation.

If we define '=' as a relation between objects we have no issue with this.

• In the typical manner: under the given variable assignment, 2x= x+1 is denoted as 2 = 2, which holds in every model, hence is true. It seems to me that many of your questions could be answered by picking up a text on mathematical logic. Oct 20, 2022 at 17:25
• "2x=x+1" is a relationship between the expressions "2x" and "x+1". As you noted, this relationship is a constraint, but it is conditional on a context, not universal - typically a context where x represents an unknown quantity. If the context is "in this algebra problem" there's a goal of finding a value (or values) that "satisfy" the constraint, meaning make the constraint true.
– user62966
Oct 20, 2022 at 23:52
• Again a misunderstanding on the way language "works"... If I say "Plato is a philosopher" what I'm saying? That there is a man named Plato and that he is a philosopher. I'm not speaking about words but with words. When in mat we say 1+1=2 we are using numerals, i.e. name for numbers, to say something about numbers: specifically, that the result of the operation of adding one with one one produces as result two. Oct 21, 2022 at 6:07
• We can't know how to interpret the description of equality without context. It might be a description of the syntactic class of the "=" symbol, or it might be an attempt to solve the problem of equality that Frege discussed, but by using syntactic categories rather than concepts. Oct 21, 2022 at 9:04
• "2x=x+1 is true when x=1 but not for any other value" That doesn't matter. It is still a relationship. The value of the equality statement, whether it turns out to be true for all, only some or no possible values of the strings, plays no role for the question in your title. It's a relationship between two strings, and it may or may not hold for a given interpretation. Oct 21, 2022 at 12:16

One way to understand equality in mathematics is via Type Theory: see e.g. Ansten Klev, The Justification of Identity Elimination in Martin‑Löf’s Type Theory (Topoi, 2017).

The basic notion is that of evaluation:

In the explanation of the forms of categorical judgement the notion of evaluation plays an important role. [...] The following example suggests what one should understand by evaluation: (3 + 2)! x 4 evaluates to 480. Thus, ordinary arithmetical computation as well as computation in an extended language of arithmetic are instances of evaluation.

Ordinary mathematical equations can thus be interpreted as computations, using the formal definitions of the arithmetical operations of sum and product.

If we adopt the definitions of numerals (names for numbers): 1 = s(0) (the successor of zero) and 2 = s(1), we have that equality 1+1=2 holds because when we evaluate the expression 1+1 according to the rules of arithmetic what we get is s(1).

In conclusion, when in mathematics we state an expression like1+1=2 we are using numerals to say something about numbers (whatb else?); specifically, we are stating that the result of the operation of adding one with one produces as result two.

• So when we write the computation '1+1' that string '1+1' becomes a name for the object that it evaluates to? Hence there are 'two names' and we state something about them? Nov 30, 2022 at 10:38
• @Confused - We write expressions (what else?). Expressions has reference (if they are terms, i.e. "names") or meaning (if they are sentences)). In arithmetic a "complex" expression like 1+1 must be interpreted as a "recipe" for a computation: to evaluate the expression amounts to perform it. The result of computation in arithmetic ia a number (in e.g propositional logic is a truth value). Nov 30, 2022 at 10:59
• A sentence uses names (for object and relations) to state "facts" about objects. Thus, an arithmetical sentences uses names for numbers to express facts about them: that the result of the corresponding computation will be... Nov 30, 2022 at 11:01
• Is it acceptable to describe an expression like 1+1 describes the value of an operation by essentially describing the function and inputs it is a value of? Nov 30, 2022 at 16:25
• @Confused - yes, + in arithmetic is a binari operation and in terms of logic is formalized with a function symbol. Nov 30, 2022 at 17:16

It is patent that in the expression "2x = x + 1", equality is not equality between the strings of characters since the two strings are clearly not identical.

Equality is identity applied to values, which are elementary properties.

To assert that 2x equals x + 1 is to assert that they have the same value. Given the mathematical language used, we normally understand that the value in question is numerical.

However, there is clearly no reason that 2x and x + 1 should have the same value, so equality cannot possibly be a relation between the possible values of the two expressions 2x and x + 1.

Instead, the expression 2x = x + 1 implies that x has the value 1. Thus, asserting 2x = x + 1 is a way of constraining x to have the value 1. The expression 2x = x + 1 can only be true if x = 1 is true. But x may not be equal to 1. If x is not equal to one, we can deduce by transposition that 2x = x + 1 is false.

However, if x = 1, then 2x = x + 1 is true, so x = 1 is a solution to The expression 2x = x + 1.

So asserting the expression 2x = x + 1 is one way to imply that x has some particular values, in this case just the one value 1. So the assertion works as a constraint on the expression 2x = x + 1 and on the value of x. The assertion requires the expression 2x = x + 1 to be true (whether or not it is true), and so requires x to have a certain possible values, namely 1 in this case.

• Is it the case that we can say the two 'strings' are related such that 'under a given interpretation these strings have the same value in this problem' Oct 25, 2022 at 18:58

Take x where x is any real number, 2x=x+1 is true when x=1 but not for any other values, this is a constraint how are the two strings of symbols related by equality?

The simplest form of 2x=x+1, algebraically, is x=1 and that means these two strings are equivalent "2x=x+1" = "x=1". So the simplest string of symbols related by equality are 'x' and 1. The constraint appears to exist because the equation 2x=x+1 is not in its simplest form.

• I get where you're going with that but I'd still argue that you should use an AND statement rather than an equal sign to chain them. I mean yes if you evaluate the inside of the parenthesis first you'd only yield TRUE = TRUE for x =1, but outside of that you'd yield TRUE = FALSE and if you are unaware of parenthesis first you'd get x+1 = x = 1 which is downright impossible. So the idea is good but maybe use a more clear formalism. Oct 21, 2022 at 14:11

I don't know if you can get all the following using the Windows character map program (maybe under the Unicode font?); I got them using a Chrome "Insert Symbol" function. I've included descriptions of some for "flavor" but the others you could probably find the definitions for by copying them into the Google search bar (or if you have Chrome and go under "Math" via "Insert Symbol," then check the description attached to each).

1. I think this one is sometimes used as the most generic for "an equivalence relation of some sort or other". At least, I've seen it used that way in some very deep-level set theory texts.

2. This one is often used in lieu of (21) for "is defined as."

3. "Identical to."

4. "Strictly identical to," whatever that means... (maybe something to do with this?)

There are also special arrows for mappings overall, or individual types of mappings, e.g. injective, bijective, surjective, etc. but these have an "equivalence-like" flavor to them, arguably.

• For some of these I think it is difficult to describe them as a relation between objects, for example the idea of 'defined to be' doesn't make sense between objects but they are essentially defining the equality between objects with a little extra context, like 'these two objects are always equal for any value'. Would you agree with this? I'm not sure if 'identity' or 'defined to be' are technically equivalence relations. Dec 1, 2022 at 19:27
• @Confused see the distinction between so-called "real vs. nominal definitions". Also, that whole article might be worth a look! As for identity, see "Identity". Dec 1, 2022 at 23:07