It's easy to forget math is a language for communicating ideas. As words, "two and three is equal to five" is cumbersome. Replacing numbers and operations with symbols helps: "2 + 3 is equal to 5".

@Gabr: You bring up a great point. The internal mechanics of how “x = x + 1” works is actually pretty complicated. Inside a computer, x is really a memory location (or register), so we’re saying “Read the value stored in location x, add 1 to it, and store that back into location x”.

“x=x+1” if you are familiar with programming - certainly all of us are - then you came across this statement in your life. Well I remember the first time I saw this statement I got the meaning very easily and I was astonished since it is mathematically wrong ! but why ? it sees x as a box/container and as a value as well, it says take the box that it is named x and add to what is inside it a “1” ! … the first x in the statement means the box named x, while the second after the equal sign means what inside the box, or in another way … x (now) is the x (of the past) + 1.

I’d disagree with the equating of math as a foreign language. I think mathematical notation is a foreign language, but math itself is some entity separate from its notation.

@anonymous@kalid Oh, I though you were referring to the different equals signs in most computer languages (JS has all three)—didn’t know it originated from a math norm. BTW, is this a weird way to refer to the conversation between you and @anonymous? I thought it looked like nesting.

For some reason, the insights pane was not working for me. It just freezes after clicking the “Post Insight” button. So here it is, copied out:

Aha! The insight that helped was:
the equal sign in Euler’s formula

Details:
Till now I was viewing the Euler’s formula as more of a transformation. But the insight that -1 was just getting there and $ e^{iπ} $ was going around the circle. For your article on co-ordinates and some recently acquired trig, I also got the $ cos x $ and $ i sin x $ part.

And thought about adding Live Previews for the comments yet? If that’s too hard, then let us edit comments or at the very least, delete them. But then, that would require some sort of an accounts system which would be hard too. You’d have to extend the Aha! account to BE.

The curveball I remember was in my first computer class on the IBM 1130 studying Fortran IV. The stanza x=10 was explained to mean “replace variable x with value 10”, or something like that. Keep up the good work. Very interesting!

I would guess that one of the things we can take from this is that the “equals sign” gets its meaning from content. After fighting the == === = for the last few days this post couldn’t have come at a better time for me. Keep it up!

@Anonymous: Math has had to come up with different types of equals (like triple equals, etc.) but they’re not well known.

The general-purpose X “is the same thing as” Y might be true, but isn’t helpful in getting to the subtleties (2 + 2 = 4 has a different meaning from the Pythagorean theorem, even though both use the equal sign, a^2 + b^2 isn’t really “the same thing as” c^2, it’s more “they are entirely different things which have the same magnitude”).

i always thought of it as ‘x = y’ means ‘is is the same thing as y’. for the ‘set x to be y’ type of this, i usually add ‘let x = y’ or ‘if x = y’ when i’m doing maths. and isn’t the fundamental thingy meant to use the triple parralel lines ‘identity’ sign?

@mark: Heh, nice coincidence! Yep, the contents being compared impact the equals sign for sure. Even in math, you’re not supposed to compare vectors and scalars (1 vs [1]).

@Harold: Glad it clicked. I find “is” to be a little too generic for my tastes, so maybe “is another representation of” or similar. But good point about the importance of this, it’s the same central concept referred to by different names.

This is a tremendous post. The connection to two fingers pointing at the same moon has a real chance to enlighten.

What’s missing, to me, is the reading of = that I use most often, which is “is”. As in, 2+3=5 read as “two plus three is five.” The idea that the two distinct seeming things on opposite sides of the equal sign are actually the same thing is a very powerful one. Many students I encounter don’t recognize the gravitas.

I suspect that all of these forms of equality you have listed fall under the umbrella of “they are the same, modulo one’s interpretation”. For instance, $$3+4=7$$ is a useful operation if we say that the original group of seven is being partitioned into a group of three and a group of four, i.e. on the left there are tight rules for admission into each of the two groups, on the right the rules are relaxed to allow all seven members into a single group. The important part, however, is that such partitions are figments of our imagination. When we look around, we partition the space around us into familiar objects; not at all unlike what is happening in a simplification.

However, equality can sometimes be too strong for our purposes. For instance, say we want to reason about all chairs at once. The usual way is to work modulo an equivalence relation, which behaves a like equality but allows more things to be equal. Keeping with the example of chairs, you look up the definition of a chair (check that it agrees with your intuition) then say two objects are equivalent ($$\sim$$) if and only if they share these defining properties. Modulo $$\sim$$ we can pick an arbitrary chair and describe the whole class. Where does this form of reasoning actually come up? Certainly in abstract algebra, where two structures may have the exact same internal structure, but have different names for their constituents (like calling a “window” various other names or using various languages to name the structures comprising a house). This kind of equivalence is called an isomorphism ($$\cong$$) and allows us to pick any “representative” by naming its constituents however we like. A final example is in (elementary) geometry, where it is generally accepted that one is to reason about a whole class of figures by first choosing a representative.

Finally, in the formalities of set-theory, equality has a very strict form: for a given set $$X$$, the relation $$R={(x,x):x\in X}$$ is the equality relation on the elements of $$X$$ (the careful reader will notice that I have used $$=$$ to define $$R$$; this is allowed because we are really using the axiom of extensionality–the elements of $$R$$ are defined to be the same as those in set on the right). Of course the elements of $$X$$ may be interpreted in various ways, which is why one might say $$(1,2-1)\in R$$ if $$X$$ is the set of integers, rational numbers, real numbers, complex numbers, etc. So the concept of equality being the same “up to interpretation” takes a very strict meaning in foundational mathematics.

Your teaching on understanding “=” and the Harold post on 20120924 referring to which is “is” finally helped me to understand why someone would say, “It depends on what the meaning of the words ‘is’ is.”