What is the exact definition of equality?

["Don't panic!"]

I've been grappling with the idea in my head as to how I would explain to someone exactly what equality between two mathematical objects actually means. This maybe a very stupid question, so apologies in advance, but if I'm honest I struggle to come up with an answer that doesn't involve using the notion of sameness which seems very unsatisfactory as sameness is not a well-defined concept.

So far what I've come up with is that if we have two mathematical quantities $A$ and $B$ that each possesses particular mathematical properties, then the expression $A=B$ (assuming that $A$ and $B$ are constant mathematical quantities, i.e. they don't change their form) is the statement that the two quantities do not describe distinct mathematical objects are in fact different representations of a single mathematical object - they are identical in all respects (the fact that $A=B$ is essentially saying that $A$ and $B$ share an identical set of mathematical properties and neither has additional properties not shared by the other (note that I'm trying to avoid using the word same)).

If $A$ and $B$ are not constant quantities however, then the expression $A=B$ is actually a proposition that is only true for specific values of $A$ and $B$. In this case, we may say that if $A=B$ then $A$ and $B$ simultaneously denote a single mathematical property.

Finally, we can set $A$ and $B$ to a particular numerical values, such that, if $A=B$ then they are just different labels for a unique numerical value.

I hope this is somewhat clear. Any insight into the situation would be much appreciated.

 

[HallsofIvy]

To say that "A= B" simply means that "A" and "B" are names (possibly different) for the same thing.

When you say "So far what I've come up with is that if we have two mathematical quantities A A and B B that each possesses particular mathematical properties, then the expression A=B A=B (assuming that A and B B are constant mathematical quantities, i.e. they don't change their form)" you are thinking of A and B as "quantities" or "objects"- they are not, they are simply labels or names for some mathematical object. And saying that "A= B" means that they are names for the same thing. When I write "x= 3" I simply mean that "x" and "3" are naming the same thing.

(Of course, we also have the more general concept of "equivalence relation". A and B are "equivalent" if they are alike in some manner- the manner being specified by the given equivalence relation.)

 

[lavinia]

Generally two mathematical objects are considered to be the same if there is a mapping between them that preserves all of their properties. For instance two vector spaces are considered the same if there is a linear map between them that is bijective.

This idea of sameness is relative to which properties are under consideration. Two topological spaces are them same if their is a homeomorphism between them. But if they are also Riemannian manifolds then as Riemannian manifolds they may not be the same because there may not be a homeomorphism that is also an isometry. For instance an egg and a golf ball are the same as topological spaces but not as Riemannian manifolds.

Interestingly every object is the same as itself, since the identity mapping always preserves structure. But there may be other structure preserving mappings as well. All of them verify sameness. For example a rotation of a sphere is an isometry and thus preserves its structure as a Riemannian manifold.

 

[micromass]

I've been grappling with the idea in my head as to how I would explain to someone exactly what equality between two mathematical objects actually means. This maybe a very stupid question, so apologies in advance, but if I'm honest I struggle to come up with an answer that doesn't involve using the notion of sameness which seems very unsatisfactory as sameness is not a well-defined concept.

So far what I've come up with is that if we have two mathematical quantities $A$ and $B$ that each possesses particular mathematical properties, then the expression $A=B$ (assuming that $A$ and $B$ are constant mathematical quantities, i.e. they don't change their form) is the statement that the two quantities do not describe distinct mathematical objects are in fact different representations of a single mathematical object - they are identical in all respects (the fact that $A=B$ is essentially saying that $A$ and $B$ share an identical set of mathematical properties and neither has additional properties not shared by the other (note that I'm trying to avoid using the word same)).

If $A$ and $B$ are not constant quantities however, then the expression $A=B$ is actually a proposition that is only true for specific values of $A$ and $B$. In this case, we may say that if $A=B$ then $A$ and $B$ simultaneously denote a single mathematical property.

Finally, we can set $A$ and $B$ to a particular numerical values, such that, if $A=B$ then they are just different labels for a unique numerical value.

I hope this is somewhat clear. Any insight into the situation would be much appreciated.

You should be looking at foundations of math for this. There are two routes:
1) The first route is that equality is an undefined notion that satisfies some axioms. https://en.wikipedia.org/wiki/First-order_logic#Equality_and_its_axioms Whatever equality means is then just metamathematical.

2) The second route is to go and check set theory. Everything in mathematics is defined as a set, whether it be groups, topologies, etc. So if we define equality for sets, then we have defined equality in entire math. Now the extentionality principle defines equality by saying that two sets $A$ and $B$ satisfy $A=B$ if and only if for each $x\in A$, we have $x\in B$ and for each $x\in B$ we have $x\in A$.

The thing lavinia is talking isn't really about the strict equality, but about when we consider objects the same. Yes, we might consider (philosophically) two objects to be the same when they are not strictly equal.

 

["Don't panic!"]

So when we write $x=y$ for example, what is this actually "saying"? That $x$ and $y$ represent the same mathematical object (if this equality holds for all $x$ and $y$), or that $x$ and $y$ have the same value for a particular mathematical property? For example $$(x+y)^{2}=x^{2}+y^{2}+2xy$$ are two different expressions for the same mathematical quantity (as this is true $\forall x,y$). Whereas, $$x^2-a^{2}=0$$ is a conditional statement, as equality holds for only particular values of $x$ (such that $x^{2}-a^{2}$ and $0$ represent the same numerical value, when $x=a$ or $x=-a$). Also, we could choose to set $x=1$, for example, in which case we are saying that $x$ and $1$ are different expressions for the same number.

I feel that I've gotten myself into a complete mess as to what the notion of equality actually means?!

 

["Don't panic!"]

2) The second route is to go and check set theory. Everything in mathematics is defined as a set, whether it be groups, topologies, etc. So if we define equality for sets, then we have defined equality in entire math. Now the extentionality principle defines equality by saying that two sets AA and BB satisfy A=BA=B if and only if for each x∈Ax\in A, we have x∈Bx\in B and for each x∈Bx\in B we have x∈Ax\in A.

So is this saying that two sets are said to be equal if and only if they contain the same elements? How does this relate to a simple mathematical case between two numbers $x$ and $y$ say. In this sense, what does it mean to write $x=y$?

 

[micromass]

For example $$(x+y)^{2}=x^{2}+y^{2}+2xy$$ are two different expressions for the same mathematical quantity (as this is true $\forall x,y$).

Yes, if you write that, then it is understaad as saying

$$\forall x,y\in \mathbb{R}:~(x+y)^2 = x^2 + 2xy + y^2$$

Or it could also be interpreted as an equality of polynomials.

Whereas, $$x^2-a^{2}=0$$ is a conditional statement, as equality holds for only particular values of $x$ (such that $x^{2}-a^{2}$ and $0$ represent the same numerical value, when $x=a$ or $x=-a$).

In this case, we indeed do not interpret it as $\forall x,a\in \mathbb{R}: x^2 - a^2 = 0$. How we choose to interpret it depends on context. This is not a problem with the definition of equality, but with the way statements are written. Indeed, it is just that mathematicians choose not to write the entire statement because it is too tedious.

 

[micromass]

So is this saying that two sets are said to be equal if and only if they contain the same elements?

Yes.

How does this relate to a simple mathematical case between two numbers $x$ and $y$ say. In this sense, what does it mean to write $x=y$?

Then you will need to study set theory to see how numbers are defined as sets. For example, we have the following definitions:
$$0 = \emptyset,~1 = \{\emptyset\},~2 = \{\emptyset,\{\emptyset\}\},~...,~n = \{0,...,n-1\},...$$
Now we have reduced equality of (natural) numbers to an equality of sets. For example $1 = 0$ is not true because $\emptyset = \{\emptyset\}$ is not true. Indeed, $\emptyset \in \{\emptyset\}$, but $\emptyset\notin\emptyset$.

I agree that this is a weird definition, but its only purpose is to reduce all of arithmetic to sets. That way we don't need to give separate definitions of equality for separate classes of numbers or objects, but everything follows from the set theoretical equality.

 

[Stephen Tashi]

So when we write $x=y$ for example, what is this actually "saying"?

Unless you have defined what kind of mathematical object $x$ and $y$ are, it isn't saying anything mathematically precise. For example, the definition of two matrices being equal is different than the definition of two complex numbers being equal.

It may be philsophically helpful to discuss some universal notion for the use of "=", but such a concept is nothing you can use in a proof. There are different uses of "=" because there are different equality relations in mathematics.

Part of defining a mathematical object, such as a matrix, is giving a definition for what it means for two objects of that kind to be "equal". In other words, you should not think about the organization of mathematics as the procedure: 1) Define a mathematical object and then 2) Apply a universal definition of "=" to that object. The task of defining what "=" means is part of defining the mathematical object.

 

["Don't panic!"]

In this case, we indeed do not interpret it as ∀x,a∈R:x2−a2=0\forall x,a\in \mathbb{R}: x^2 - a^2 = 0. How we choose to interpret it depends on context. This is not a problem with the definition of equality, but with the way statements are written. Indeed, it is just that mathematicians choose not to write the entire statement because it is too tedious.

So is equality technically the case in which two mathematical quantities "contain the same information" in that they share all and only the same properties?

If one writes $f=g$ (where $f$ and $g$ are two functions) is this the statement that $f$ and $g$ define the same function, whereas $f(x)=g(x)$ is the statement that the two functions have the same numerical value for that particular value of $x$, but that $f$ and $g$ don't necessarily define the same function?

 

[Stephen Tashi]

So is equality technically the case in which two mathematical quantities "contain the same information" in that they share all and only the same properties?

If it were technically that, you'd run into the technicality that the "two" mathematical quantities are not "two" mathematical quantities, they are "one" mathematical quantity.

 

["Don't panic!"]

The task of defining what "=" means is part of defining the mathematical object.

So equality is a relative concept then, depending on the particular set of mathematical objects one is studying?
When we see in texts expressions like "$\text{something} = \text{something else}$" is it just implied from the context what equality between two objects means?

Heuristically, would it be correct to say that given any mathematical property $P$, if two mathematical objects $A$ and $B$ satisfy the condition that whenever $A$ has property $P$, $B$ has also has property $P$, then the two objects are equal and are in fact not two distinct objects but one mathematical object described in two different ways?!

 

[Stephen Tashi]

So equality is a relative concept then, depending on the particular set of mathematical objects one is studying?
When we see in texts expressions like $\text{something} = \text{something else}$ is it just implied from the context what equality between two objects means?

Yes and yes.

Whether to use the symbol "=" for an equality relation or to use some other symbol is a matter of tradition and culture. For example, in logic it is traditional to speak of two propositions being "equivalent" instead of saying they are "equal" and to use a symbol such as $\iff$ for the equality relation of being "equivalent".

A mathematically sophisticated person can usually infer from context what "=" means. There are contexts where the meaning of "equal" is not clear. For example, suppose we read "Let X and Y be random variables and assume X = Y". One might say that two random variables are "equal" if they have the same distribution. On the other hand, "X=Y" might mean that each realization of the random variable X is in 1-to-1 correspondence with a realization of the random variable Y and that the same value is realized for both.

 

[micromass]

So is equality technically the case in which two mathematical quantities "contain the same information" in that they share all and only the same properties?

Yes. I don't agree with Stephen Tashi at all since there is a universal definition for equality that works well, namely the equality of sets. All other equalities can be derived from that. There is no need for a relative notion of equality.

 

[Stephen Tashi]

Yes. I don't agree with Stephen Tashi at all since there is a universal definition for equality that works well, namely the equality of sets. All other equalities can be derived from that. There is no need for a relative notion of equality.

I'll agree with your disagreement in the sense that mathematics could, in principle, be rewritten that way. My point of view is that, in practice, it is not presented with such logical perfection. Tradition and culture of a many mathematical subjects result in "=" being defined explicitly, not inferred from the basic definition of equality for sets.

 

["Don't panic!"]

Whether to use the symbol "=" for an equality relation or to use some other symbol is a matter of tradition and culture. For example, in logic it is traditional to speak of two propositions being "equivalent" instead of saying they are "equal" and to use a symbol such as ⟺ \iff for the equality relation of being "equivalent".

So for example, if we write $\frac{1}{2}=\frac{2}{4}$ are we saying that the two expressions are equal with respect two their numerical value (i.e. they have the same numerical value), but they are distinct objects as fractions.

HallsofIvy's comment about $A$ and $B$ being different labels for the same thing if $A=B$ makes a lot of sense to me (at least in how I've heuristically thought of equality between mathematical expressions). In this context, if I've understood it correctly, when considering a set of numbers, the expression $x=y$ is the statement that $x$ and $y$ are different labels for the same number. For example, putting $x=1$ is the statement that $x$ and $1$ are labelling the same number. I assume this translates to other contexts, as in $f=g$ means that $f$ and $g$ are two different representations for the same function, and $M=N$ means that $M$ and $N$ are two different representations for the same matrix, etc?!

 

[Stephen Tashi]

So for example, if we write $\frac{1}{2}=\frac{2}{4}$ are we saying that the two expressions are equal with respect two their numerical value (i.e. they have the same numerical value), but they are distinct objects as fractions.

That depends on how you define the mathematical object "fraction". Do we want to define "fraction" so $\frac{1}{2}$ and $\frac{2}{4}$ are different fractions?

I have no axe to grind on particular ways of defining "fraction". The important point is that the question exists. You can't answer whether two "fractions" are "distinct" until you have completely defined what "fraction" means - i.e. specified what it means for two "fractions" to satisfy the equality relation of being "the same".

HallsofIvy's comment about $A$ and $B$ being different labels for the same thing if $A=B$ makes a lot of sense to me (at least in how I've heuristically thought of equality between mathematical expressions).

It's a useful way to think about "=", but you may still have to resort to reading the fine print because the meaning of "same thing" depends on the meaning of "same", which depends on context. For example, in computer science, some mathematical objects are themselves labels for other things - e.g. a string variable in a computer program can contain a value that is a label for a number and in the program there are labels such as "X" for string variables.

 

["Don't panic!"]

How does one define equality in terms of an equivalence relation (for example between numbers)? In doing this, is it possible to prove the transitive property of equality (or is this taken as an axiom)?

Would it be correct to express equality as $$\lbrace (x,x)\vert x\in S\rbrace$$ where $S$ is some set. Then from this we have that $$(x,x)=(y,y)\iff x=y$$ which follows from the equality of ordered pairs (as $(x,x)=\lbrace\lbrace x\rbrace ,\lbrace x,x\rbrace\rbrace$ and so $\lbrace\lbrace x\rbrace ,\lbrace x,x\rbrace\rbrace =\lbrace\lbrace y\rbrace ,\lbrace y,y\rbrace\rbrace\iff x=y$)?

 

[Stephen Tashi]

How does one define equality in terms of an equivalence relation (for example between numbers)? In doing this, is it possible to prove the transitive property of equality (or is this taken as an axiom)?

Often a definition of the symbol "=" is given for a type of mathematical object (such as a matrix) and then a proof is given (or left to the reader) that this particular type of "=" is an equivalence relation. Proving a relation is an equivalence relation involves proving the relation is transitive.

Another approach would be to incorporate an statement in the definition of the mathematical object that says "We shall assume there exists an equivalence relation denoted by "=" on this set of mathematical objects". That is an abstract approach that doesn't specify any procedure for determining whether two such objects are "=". If somone wishes to apply the theory of such abstractly defined objects to a practical problem then they have to define an an equivalence relation for the objects that applies to that particular situation.

How things are done "for numbers" is not a simple question. If you want to do things with logical perfection, "numbers" can be defined (as micromass suggests) from more elementary notions involving sets. That is a graduate level approach. If you are reading an undergraduate text then you are liable to see "numbers" defined by the abstract approach (mentioned above) of simply stating the assumption "There exists an equality relation denoted by "=" that is defined on the set of numbers".

 

["Don't panic!"]

How things are done "for numbers" is not a simple question. If you want to do things with logical perfection, "numbers" can be defined (as micromass suggests) from more elementary notions involving sets.

So can equality be defined as an equivalence relation as in my previous post (using the notion of set equality)?

 

[Stephen Tashi]

So can equality be defined as an equivalence relation as in my previous post (using the notion of set equality)?

I think a skilled logician could define a particular type equality that way. His problem would be to define the set $S$ and this involes defining the meaning of the relation $\in$ for that set. In practice, when you read mathematics instead of trying to force the meaning of "=" into that viewpoint, it's best to pay attention to how particular articles define "=".

 

["Don't panic!"]

Ah ok, I guess I'm probably trying to overcomplicate things and this is just hindering my understanding.

 

[WWGD]

So when we write $x=y$ for example, what is this actually "saying"? That $x$ and $y$ represent the same mathematical object (if this equality holds for all $x$ and $y$), or that $x$ and $y$ have the same value for a particular mathematical property? For example $$(x+y)^{2}=x^{2}+y^{2}+2xy$$ are two different expressions for the same mathematical quantity ! <Snip>

Careful here: $(x+y)^2 = x.x+ xy+yx+y.y=x^2+xy+yx+x^2 = x^2+2xy+y^2$ assumes commutativity, i.e., that $xy=yx$ ,which is not always present. Try square matrices $A,B$ ( non- multiples of the ID) to find a counterexample.

And I would say that equality between Mathematical objects depends on the way the objects are defined. If a matrix is defined as an array $(a_{ij})$ of numbers, then the defining property is the ordered $i \times j$-ple of quantities {$a_{ij}$} and if these are equal for all $i,j$ then two matrices are equal.