# Why do we equate a mathematical object with what denotes it?

Consider a matrix,

We denote matrices by an uppercase letter of the English alphabet like A, B, C, etc. Let this matrix above be denoted by A.

I can write,

I have often seen an equal to sign introduced between A and the matrix,

An equal to sign in my understanding means is the same as, I'm confused about how a mathematical object like a matrix (on the right) is the same as the symbol that denotes it? How 'denotes' is interchanged with '='?

• It's because of the axiom scheme of replacement, on the logical (not just mathematical) level. It's a way to keep track of replacement instances of substituted identity over the glyphs. But we do have equivalency glyphs like := or the equals sign with "def" over it, to track such distinctions more sharply. Nov 13, 2022 at 16:14
• You are equivocating on the interpretation. If you read "A" as a mere symbol without meaning, then you should read the representation of the matrix and the "=" the same way. The entire equation is just a string of meaningless symbols. But if you are going to read the meanings of "=" and the matrix notation, then you have to read the meaning of "A" too. What is the meaning of "A"? Well, it's what the equation says, "A" means the same thing as the matrix notation. Nov 13, 2022 at 16:51
• '=' in mathematics asserts that two 'mathematical objects' (numbers, sets, matrices etc) being discussed are the same object, having two symbols or expressions on each side simply means that the mathematical objects they represent are the same (2+2,3)=(4,3) is an equality because the pairs have the same elements in them, just written differently. Symbols themselves are not generally talked about in mathematical language, but the objects are, and we describe them using symbols, in the same way we would define Socrates using the word 'Socrates' in sentences. Nov 13, 2022 at 22:22
• tl,dr: Because they both denote the same thing. What you are calling "the actual matrix" is technically not the actual matrix any more than the A is, it is just a more concrete and constructive way of denoting the actual matrix, which is a conceptual mathematical object, not a series of written technical notations. Nov 14, 2022 at 15:16
• You are not okay with the name A. Are you okay with the name 9 (as in the top left of your matrix)? If so, can you say carefully what you feel the difference between them is? Nov 14, 2022 at 16:23

Why do we equate a mathematical object with what denotes it?

In your example with the matrix, A is what mathematicians think of as a variable, while what you call the matrix, namely

that you present as what is denoted by A is what mathematicians think of as a formal expression denoting the value of the variable A.

The label "matrix" has the same mathematical role as, for example, the label "natural number" or the label "real number".

Thus, your example can be interpreted in the same way as we have to interpret the expression x = 2, and we would not say that x denote 2. The idea is that x = 2 specifies a restriction to the possible values of x. If x has been introduced as a for example a natural number, it is given has having a priori a potential infinity of possible values taken from the set of natural numbers. The expression x = 2 is then used to restrict the possible values of x to one value, namely 2. We could restrict the possible values of x in many different ways, for example, x ∈ {2, 3} or x ∈ [0, 9]. We could also do it through equations: for example the expression x² – 1 = 0 restrict x to the values which are solutions to the equation x² – 1 = 0, namely 1 and -1, so the equation is equivalent to, for example, x ∈ {-1, 1}.

In x = 2, x does not denote the value 2, it denotes the concept of numerical variable, which is itself a complex notion, somewhat underappreciated. What denotes here the value 2 is the figure '2'.

Similarly, in the example used in the question, A does not denote a matrix, it denotes a matrix variable, one which is then restricted to one particular matrix, itself denoted by the following expression:

We would all, mathematicians included, typically talk of this expression as a matrix or the matrix, but this is similar to confusing the figure '2' with the number 2.

This is not the question but it is true that mathematical expressions can be considered alternatively as denoting and as not denoting. This is true but this is not specific to mathematics. We routinely do it with natural languages as well: "Snow is white" is true if snow is white.

EDIT

We also know that x = 2 is not the same as using x to denote 2 because x = 2 is either true or false, while the use of for example the expression "Donald Trump" to denote some person is neither true nor false. It is just a fact.

• You say that x does not denote '2', this is one thing that confuses me slightly, x denotes the value of a variable which we can see as being the object '2' in the assignment givent, it depends on context but if we are showing a restriction on x limiting it to '2' surely the symbol 'x' denotes that object, I might be a mistunderstanding though. Nov 13, 2022 at 22:14
• When I say 'value of a variable' I kind of mean an undefined possible object. Nov 14, 2022 at 16:49
• @HarshitRajput "how A can be a variable" Presumably because like 'x', 'A' does not mean any particular thing in the mathematical language. We couldn't use '2' as the name of a variable because '2' means one particular thing, namely, the number 2. "the matrix (...) is symbolically (...) denoted by A" The actual matrix is denoted by what you call "the matrix", like mathematicians do. A denotes a matrix variable. Your equality says that the value of this variable A is the value denoted by what you call the matrix, which is in fact not a matrix but a few squiggles on the page. Confusing? Nov 15, 2022 at 17:09
• @HarshitRajput No, what you call the matrix is an array of numerical figures, not of numbers. We talk of numbers but this is confusing the symbol with what the symbol is used to denote. A matrix is an abstract relation between variables or constants thought of as taking numerical values, i.e., numbers. So the actual numbers are far removed from the array written on the page, which is just squiggles on paper. This is like words, which are squiggles on paper denoting something else. The confusion is natural because we all talk as if the array was the matrix. Nov 22, 2022 at 17:06
• @HarshitRajput You can try "use-mention distinction". When we use symbols, we don't put quote marks: 1, 2 and 3 are numbers. When we mention symbols, we put quote marks: "1", "2" and "3" are numerical figures, that is, figures representing numbers. Nov 23, 2022 at 11:44

Generally speaking, we don't equate a mathematical object with what denotes it. The symbol A is one thing, and any object that may be denoted by A is a different thing.

Let's consider an analogous situation in plain English. Suppose that I say the sentence, "John is my brother." I am not claiming that the word "John" is my brother, or that John is the phrase "my brother," or that the word "John" is the phrase "my brother." I am claiming that the person referred to by the word "John" is the same person as the person referred to by the phrase "my brother"—in other words, I am claiming that John is my brother.

Exactly the same is true when it comes to mathematical equations. Suppose that I write the equation "A = [9 13]." I am not claiming that the symbol A is the matrix denoted by the expression [9 13], or that the object denoted by the symbol A is the expression [9 13], or that the symbol A is the expression [9 13]. Instead, I am claiming that the object denoted by the symbol A is the same object as the matrix denoted by the expression [9 13].

(Or, put much more tersely, I am claiming that A is [9 13].)

Denotes suggests an interpretation function, that is, some function between the syntax and semantics of your theory. Underneath this function, call it I, the symbol A is mapped to the matrix you have given.

There are many notions of equality in logic, ie, the symbol is overloaded. Here, the equality is likely definitional equality.

The two aren't exactly interchangeable, but given that two symbols are definitionally equal, their denotations should be as well. And given that a symbol denotes a matrix, and supposing that matrix has a syntactic representation (which maps semantically to "itself", we should be alright with introducing the two piece of syntax as definitionally equal

• Please excuse my blunders, I'm a rookie. Are you suggesting that, since we have assigned a symbol to the matrix (the symbol A) and since we can in writing use A instead of the entire bulky matrix, they can be treated equal? Nov 14, 2022 at 13:42
• As I said above, there are many notions of equality. Under one such notion- defintional- they are equal. Nov 14, 2022 at 15:14
• The key thing here is that your interpretation function I is also required to interpret the symbols on the right hand side as meaning a matrix value. The symbol `=` typically means that the value on the left is the same as the value on the right; so one needs to evaluate (i.e. interpret and produce a value from) the expressions on the left and right of the symbolic equation in order to know the truth value of the equation as a logical proposition. Nov 16, 2022 at 0:15

This is an amazing question and the answer is hinted at in the Logic book by Stephen Cole Kleene (*). Here is the exact extract from page-4:

To put in short: The "equality" is something in the Mathematical language, and, the symbol "A" and "the matrix with entries" are just short hand for the logical formulas which exist in the foundation language with which we do our Mathematics in a simpler observer language.

I do not remember seeing A=3 used to convey "A denotes 3" in any graduate-level or higher mathematics books or articles. If this is the definition of A, mathematicians communicating with other mathematicians would use := instead of =, which means "is defined as". In school and some undergraduate textbooks, = is used as a shorthand for :=.

The statement A=3 might appear in research mathematics if A is defined in some other way (e.g. "Let A be the positive solution to A * A = 9"). In that case, "A=3" should be understood as pointing back to the definition of A. You could expand the statement as "The positive solution to A * A = 9 is 3".

(The equals sign might be used for definitions in schools because students do not properly distinguish a definition of A from a sentence stating the value of A. They might interpret "Let f(x)=x+3" as meaning "Let f be the function such that f(x)=x+3 for all x".)

• I know many mathematicians who never use `:=`. Besides, the fact that `A = 3` does not necessarily mean that `A` was defined as `3`. Maybe A has another definition, but then it was found out that this definition was equivalent to `A = 3`, just like your `A * A = 9` example. In that case, `A` does denote `3`, even though the definition of `A` wasn't `A = 3`.
– Stef
Nov 15, 2022 at 15:52
• I also do not understand what you're trying to imply with your last paragraph, the one in parentheses. What do you mean "They might interpret "Let f(x)=x+3" as meaning "Let f be the function such that f(x)=x+3 for all x"." ? Mathematicians also interpret it that way. Not just undergraduate students.
– Stef
Nov 15, 2022 at 15:55
• Saying two definitions are equivalent gets into some philosophical questions that I had hoped to avoid. I think that `A` is defined a priori as the solution to `A * A = 9`, and it's true a posteriori that this definition is equivalent to `A = 3`. Can you send me a link to some of those mathematicians' publications which use `=` in that way? Nov 15, 2022 at 22:48

I think maybe what the OP is asking comes down to the difference between assignment vs equality test (as a boolean predicate). This distinction of course exists in many (most?) programming languages. In languages descended from ALGOL (like PL/pgSQL), `:=` is assignment and `=` is an equality test. This will be rudimentary to those with any programming background, but to summarize: An expression like `x := 2` means "make the name `x` 'refer to' the integer `2`". On the other hand, `x = 2` is a boolean predicate (equality test) that evaluates to `TRUE` or `FALSE` depending on whether `x` is in fact equal to `2` (thus it can be used as part of control-flow logic expressions like `IF x = 2 THEN RETURN TRUE`).

So I'm inferring that the original question is: why doesn't this distinction exist in mathematical notation? I do see it occasionally: for example, for assignment you sometimes see an equals sign with a "df" subscript (which I can't type here due to lack of support for math notation on the Philosophy site)

But this still leaves the question of how it makes sense to say `let x = 2` instead of something equivalent to `x := 2`. I guess the way you might explain it is that `let x = 2` is in effect shorthand for "let's require that `x = 2` be a true statement", and the only way for it to be a true statement is if the symbol `x` does in fact "denote" or "refer to" the number `2`.

As for why the comparison `x = 2` makes sense, in math notation it is simply understood by convention that the left hand side of the equality means "the mathematical object that the name `x` refers to". Just like when I say "the fine was \$100", it doesn't mean that the sequence of letters 't','h','e','f','i','n','e' is the same thing as a quantity of money: it means that the thing referred to by the phrase "the fine" is equal to 100 US dollars.

What you have hold of there is representation theory.

It's a lot easier to do this with something that has a physical interpretation simpler than a matrix. So I will do that first. Then I will do one example of a matrix.

Consider a 2-Dimensional system describing location. It is based on x and y coordinates. So you have the origin (0,0) and you have the set of points (x,y) representing the 2-D plane.

A location in this plane, say (3,4), is thus separated from the origin by a vector. We could call this vector P1 for position 1.

Now suppose we rotated our coordinate system by 90 degrees. The coordinate system, note. Not the points in the plane. The coordinate system is a mapping we put on top of this set of points. We now take every (x,y) point and in the new system, we have the new X axis pointing "up" along the old y axis. And the new Y axis pointing in the negative of the old X axis. So the old y value becomes the new x value. And negative of the old x value becomes the new Y value.

Apologies for the crudity of the diagram. It is intended to show that there are two coordinate systems overlaid on the same set of points. That is, we have the same point in two different (one rotated from the other) coordinate systems. It's the same point but its is represented in two different coordinate systems.

So, in the old coordinate system we would say P1 = (3,4). And in the new one, P1=(4,-3). That is, there is an actual point in the (x,y) plane, and there are different representations of it. We need to specify which representation we are in for the equality sign to hold.

Another way to say this is, the point P1 is an arrow that goes from the origin to the point. And it is represented by an ordered pair of coordinate values. The arrow does not change when you rotate the coordinates, but the coordinate values do.

So what about your matrix? There is a corresponding coordinate system, depending on what system your matrix fits into. Consider rotations. You can write a rotation around the z-axis (say it's the vertical axis where you are) like so. So if you stood up and rotated around the vertical axis by an angle theta, that is how you could write it.

But it is not the only way you could write it. You could lie flat on your back, then while laying on the ground you could rotate the same theta. Then you could stand up by "rotating" youself 90 degrees. You would come back to your feet rotated by the same angle. So this weird little dance of lying down first, rotating, then getting back up, let's you represent the rotation in the x-axis. There it looks like so. It's the same rotation, just represented in a different coordinate system.

Generally this is what is happening with your matrix. In a particular representation there are particular numerical values. There is a mathematical object, the matrix, and in a particular representation it is equal to those numerical values in that arrangement.

Mathematicians have a way of presenting this where such things are mappings. So the point P1 is a mapping from the origin to a point in the plane. It is the arrow. Physicists usually want the coordinates, and so will talk about P1 as having x and y components. These two views are equivalent. That equivalence is why the equality sign is acceptable.

• This has nothing to do with the question. If the OP had asked about A = 2 it would have been more clear. Representation theory? Nov 13, 2022 at 18:53
• @user4894 Wow, I'm sorry you can't see the principle. What you have hold of is fixation on concretes. Nov 13, 2022 at 20:11
• @user4894 is correct. Your answer has nothing to do with the question. Your condescending reply was not appropriate. Nov 13, 2022 at 22:47
• I’m not as convinced as my colleagues that this is “nothing to do with” - I can see some of what you’re getting at in terms of a pan-mathematical identification of mathematical things - but I do think Speakpigeon’s answer is probably closer, in that if all we were interested in was linear algebra, we could potentially get by with just logical quantification and variables. Nov 14, 2022 at 7:29