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Recently my brother in high school and I were talking about some math when he said

If we know two things are the same i.e. equal why do we need to state that they're the same? What he was trying to say was, why do we need to say a thing is itself?

This is what I answered

When we start, we really don't know whether those things are equal i.e. that they're the same. We start by seeing them as different, not knowing whether they're indeed the same. This is evident in the fact that we also give them different names at first. It is only after some observations we find that these two things which we named different and were perceived as different were indeed the same thing. Stating two things are equal is important because we started off by assuming they're different.

I'm not sure this is what the correct answer is. I believe mathematics teachers here would've encountered such a situation with their students at some point.

I'm not a mathematics teacher but I think this question is a fit for the community.

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    $\begingroup$ I think one or two SPECIFIC EXAMPLES of this in which your brother finds puzzling would help a lot. As written, this could be anything from applying (or simply stating) the reflexive property of equality to a theorem statement in which two seemingly different numbers/expressions (or more generally, mathematical objects) are asserted to be equal. $\endgroup$
    – Dave L Renfro
    Feb 6 at 15:27
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    $\begingroup$ +1 for specific example. Also this kind of "Reasoning" will lead to wrong results if you start with a false assumption. You should rather start at something that's "obviously true" and from there derive new, "interesting" facts. $\endgroup$
    – Jasper
    Feb 6 at 19:29
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    $\begingroup$ Why do we bother explicitly stating any piece of knowledge? $\endgroup$
    – Daniel R. Collins
    Feb 7 at 16:36
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    $\begingroup$ @XanderHenderson I am not sure that every question a student asks is related to a specific example. What I imagine happened here, is the student hearing the word "equals" and sees the equals sign a lot during a class. So it is natural for the question to enter his mind: "After all equal means same, why do we have to work so hard on distinguishing things which are known to be the same?". I think that's an important question in and of itself, regardless of specific examples $\endgroup$
    – Amit
    Feb 8 at 13:01
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    $\begingroup$ Without specific examples I don't even know what the question means. Echoing what Daniel R. Collins wrote, why tell someone you're going away for the weekend when you already know that and they'll learn after? Why put price tags on things if their price is what it is anyway? I'm not being facetious, I really don't get the question. $\endgroup$
    – Thierry
    Feb 8 at 14:35

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Your answer is good for showing that indeed we sometimes arrive at an equality via an experimental approach: We try to relate different quantities we observe in day to day life to each other and express their relationship, very often via an equality. The simplest example is probably $x=v \cdot t$, which can be stated as "The distance an object traverses equals to its speed times duration of travel". Of course one can go deeper into this, and talk about cases where for example the speed varies with time, and then that particular form of the equality does not hold.

Then, you can explain equality in a more abstract sense, as a relation. So in math we can express all kinds of relations, some of them are rather trivial: $5=5$, others slightly more interesting: $$1+2+3+\dots+n = \frac{n(n+1)}{2}$$

and so on. This can be a particularly nice example for the usefulness of an equality relation: it is a lot easier and quicker to compute the right hand side than the left hand side.

It can also be explained that while our rule for distances travelled $x=v\cdot t$ is an empirical (stemming from observation) one, the rule for summing the integers written above can be derived without having to rely on any physical observations.

This demonstrates the difference between what we usually call physical laws, which are based on our experiments and observations, and what are called mathematical theorems. Both may involve equality relations, but the "justification" and meaning of the equations is very different in both worlds (albeit deeply interrelated!).

Also it is interesting to distinguish a symmetric relation like $=$ from an asymmetric one. For a symmetric relation like $=$ we have $a=b$ iff $b=a$ but for $c<d$ we have $d \nless c $ (hence $<$ is asymmetric). It is also fun to observe that if we have $A\leq B$ and $A \geq B$ it implies $A=B$.

Upshot: there are many ways to address such a question, both from an experimental science perspective and also from a mathematical perspective. I would try to go along with what the student finds most interesting and explain it from that angle :)

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    $\begingroup$ I like this answer, but part of it irks me in a certain way which I must voice. It very well may be true that $x = vt$ is an empirical law, but it is also very possible that it is the definition of constant velocity. For it to be an empirical law you have to tell me how to measure $x$, $v$, and $t$ independently. Then see, experimentally, that $x \approx vt$ is true in most experiments. However, if you can only show me how to measure $x$ and $t$, with $v$ being defined as $\frac{x}{t}$, then this is not an empirical law but just a trivial rearrangement of the definition. $\endgroup$
    – Steven Gubkin
    Feb 8 at 23:14
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    $\begingroup$ I had a similar experience in undergrad physics. The first lecture was Coulomb's law. The professor stated that it had been tested experimentally that the force between two charged objects was directly proportional to each charge and inversely proportional to the distance between them. I challenged them on their definition of each term, but they could not define charge. $\endgroup$
    – Steven Gubkin
    Feb 8 at 23:18
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    $\begingroup$ I said that it seemed to me that the inverse square law for distance was something observable (distance, time, hence force being observable), and then we would define the charge of a given object by taking some charged object, defining its charge at 1 meter to be "1 charge unit" and defining the charge of all other objects according to the force induced by this unit charge at a distance of 1 meter. If so, the inverse square law was an empirically tested law, while Coulomb's law would follow deductively from the inverse square law and the definition of charge. $\endgroup$
    – Steven Gubkin
    Feb 8 at 23:20
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    $\begingroup$ @StevenGubkin You are absolutely right. It's an interesting fact that kinematics employs such simple mathematics, that it's almost tautological to write $x=v \cdot t$ and claim that it is an empirical fact. But actually that happens in all mathematical formulations of physics: they have their own mathematical self consistency, where the laws can be derived from certain simple axioms just as much as they can be confirmed or derived by experiment. But I absolutely agree that this example is particularly simple, hence even more important to explicitly separate the empirical from the mathematical $\endgroup$
    – Amit
    Feb 8 at 23:22
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    $\begingroup$ BTW I guess the easiest way to show that the "simple" math of $x=vt$ isn't that simple (in reality), is to mention special relativity. The driver and the pedestrian won't really agree on neither $x$ nor $t$ :) $\endgroup$
    – Amit
    Feb 8 at 23:32
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If I write $$ 202384569+4765923845-141243678 = $$ then go off and think a bit, do a hand calculation, or consult a calculating device, and come back and complete it to $$ 202384569+4765923845-141243678 = 4827064736 $$ then likely my main reason for using the equals sign is to record the knowledge just obtained for future use. This allows me to avoid having to repeat the work I just did.

If I'm in elementary school and the teacher asks me to add $2$ and $2$, and I then write $$ 2+2=4 $$ then I'm using the equals sign to communicate to the teacher that I know the value of the sum.

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  • $\begingroup$ +1, but If we decide to let $f(x)$ be $x^2$ for purposes of an exercise, it's not like a really smart person would know in advance that $f(x)$ and $x^2$ are eternally equal to each other. We are not recording the labor of a calculation. We are making a temporary definition by writing $f(x)=x^2$. This use of equality seems fundamentally different from the communication of a calculation that you describe. $\endgroup$
    – user52817
    Feb 8 at 20:11
  • $\begingroup$ @user52817 The equals sign is part of mathematical language. As with natural language, two examples cannot hope to capture all of the different ways that language is used, and I did not intend to be exhaustive. I did, however, choose the examples that I did because they seemed closest to the usage the OP's brother asked about: "If we know two things are the same i.e. equal why do we need to state that they're the same?" I'm pretty certain the OP's brother doesn't believe that "knowing they're the same" applies to the equation $f(x)=x^2$. $\endgroup$
    – Will Orrick
    Feb 8 at 20:17
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    $\begingroup$ Will Orrick: the OP states the student is in high school, so I expect the student is at least at the level of pre-algebra, certainly beyond arithmetic. $\endgroup$
    – user52817
    Feb 8 at 20:36
  • $\begingroup$ @user52817 I agree with you that it is likely the student has at least seen equations like $y=2x+3$. My assumption, based on the way it was phrased, was that the question was concerned with more basic notions. It is common for students when they reach algebra to have to come to grips with the multiple related but distinct uses of the equals sign. For many, however, this may be the first time they have thought carefully at all about equality, which they may never have properly understood. If this question is related to more advanced issues, I hope the OP will clarify that. $\endgroup$
    – Will Orrick
    Feb 9 at 3:46
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A usage of "=" that can be confusing for learners is when we use the equality symbol to assign something to a variable. Perhaps we should use ":=" in these cases but this is not standard. For example, we might write,"If $x=5$ then $x^2=25$." In this case, the variable $x$ and the integer 5 are not literally equal. We are just saying if $x=5$ then...

Another related example is what we see in typical proofs that $\sqrt2$ is irrational: Suppose $\sqrt2$ is rational. Then $\sqrt2=\frac{p}{q}$ for some integers $p$ an $q$.

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  • $\begingroup$ Not sure I understand. $\quad$ Saying that $x=5$ and $\sqrt2=\frac pq,$ where = is the usual equality relation on the set of numbers, doesn't preclude $x$ from being a number representation and is in fact saying that they are literally equal. $\quad$ That conditional statement doesn't claim that $x=5.$ $\quad$ "$x=5$" is not necessarily assigning a value (it might be asserting a constraint, a result, or a falsity); "if $x=5$" is provisionally assigning the value $5$ to $x;$ "x:=5" doesn't merely assign a value, because the assignment is immutable within its scope. $\endgroup$
    – ryang
    Feb 7 at 19:29
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    $\begingroup$ -1 I think this answer actually runs into the "trouble perceiving equality as a relational statement" problem. $\endgroup$
    – Daniel R. Collins
    Feb 7 at 20:04
  • $\begingroup$ @ryang do you think "Let $x=5$. Then $x^2=25$" and "If $x=5$ then $x^2=25$" are different usages of the equal sign? $\endgroup$
    – user52817
    Feb 7 at 21:22
  • $\begingroup$ Maybe a better example of the issue might be this: Let $f(x)=x^2$. OK...for how long does this equality last? What is the scope of the equality? When I have first introduced functions to students, some have puzzled over how long the definition lasts. Until the next problem, when all of the sudden $f(x)=2x+1$? $\endgroup$
    – user52817
    Feb 7 at 21:30
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    $\begingroup$ @StevenGubkin As a professional mathematician I do understand this. But the OP refers to a high school student. At this level, some have yet to encapsulate the concept of variable into their mental schema. Using equality between variables and numbers in this new domain of algebra results in cognitive disequilibrium. $\endgroup$
    – user52817
    Feb 8 at 23:57
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Stating two things are equal is important because we started off by assuming they're different.

You'd only assume that they are different if you are attempting a proof by contradiction.

why do we need to say a thing is itself?

Writing $$V_S=\frac43\pi r^3$$ and $$x^2-y^2=(x+y)(x-y)\tag#$$ doesn't merely equate objects/expressions that are "obviously" the same, but conveys equations that may not be self-evident (yet, anyway). So much so that $(\#)$ benefits from being rewritten as $x^2-y^2\equiv(x+y)(x-y),$ which specifically asserts that $(\#)$ holds for every combination of $x$ and $y$ such that the equality is defined. (In contrast, notice that not every point of the Cartesian plane satisfies $(V_S,r).$) We are not stating the obvious with a tautology like $$x=x,$$ but communicating and using formulae and identities that we have discovered.

On the other hand, = is useful also for conveying information that cannot be derived, for example to impose constraints, specify definitions and issue labels: $$x+4=10\\\sqrt{x^2}:=|x|\\f(x)=2x+4\\y=2x+C.$$

Another way to classify equations.

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I think there might be some kind of language barrier because of the similar but fundamentally different words in OP's question:

If we know two things are the same i.e. equal why do we need to state that they're the same? What he was trying to say was, why do we need to say a thing is itself?

The symbol $=$ is used in mathematics to express identity (two things actually being just one thing) as well as equality (two things really are distinct but can be considered equal in at least some aspect) as the following two examples might show:

  1. Pythagorean Theorem: $c^2 = a^2 + b^2$ states that the square over the hypothenuse is equal (in size) to the sum of squares over the other sides. But of course those squares are not identical ("the same thing"). But their equality is still an important fact to state.
  2. $\{ x \in \mathbb N ~|~ x \text{ ends in 0 or 5} \} = \{x \in \mathbb N ~|~ x \text{ is divisible by 5} \}$ states that both sets are actually the same thing, the set of the numbers 0, 5, 10, ... Again, this identity is state-worthy.

For pure numbers, the distinction between equality and identity breaks down.

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  • $\begingroup$ I’m not sure what your final sentence is explaining. $\quad$ Both your examples are equalities, and calling $(1)$ an identity additionally informs that it is not merely conditionally true (or a constraint), but universally true on $\mathbb R^3;$ on the other hand, $(2)$ has no free variable, so calling it an identity is correct though a teeny tad hot-aired. $\endgroup$
    – ryang
    Feb 9 at 10:31

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