5

What is the name of the symbol $=$ in English? Wikipedia says people use "equals sign" than "equality". Is it right? Is it to avoid ambiguity between the equality formula $(x = y)$ and the equality symbol $=$?

The main my question is the following: can I call the symbol $=$ equality when it is obvious from the context?

I'm writing a text like:

The language of set theory uses the following symbols:

  • truth: $\top$
  • falsity: $\bot$
  • negation: $\neg$
  • conjunction: $\land$
  • disjunction: $\lor$
  • implication: $\Rightarrow$
  • biconditional: $\Leftrightarrow$
  • universal quantifier: $\forall$
  • existential quantifier: $\exists$
  • equality: $=$
  • membership: $\in$

Does this sound strange? I'm not sure because I have seen some mathematicians use the word "equality" for the symbol but perhaps they are not native English speakers.

Reply to some comments:

To @xander-henderson: For example, there are many famous propositions like Cauchy-Schwarz inequality. It is a proposition or a well-formed formula, not a symbol itself.

To @jmoravitz: In Japanese, the proposition $x = y$ is only called "等式" or "方程式", and the symbol $=$ is only called "等号". The proposition $x \le y$ is called "不等式" and the symbol $\le$ is called "不等号". So I don't have any idea whether it is okay or not. Calling both "inequality" is like calling both "不等". So I'm afraid "equals sign" feels like too colloquial, but I'm not a native English speaker, it's unreliable. I have seen at least Japanese and Ukrainian mathematician uses "equality" as a symbol in some obvious contexts. But Wikipedia says "equals sign" is popular. So I asked.

dxiv
  • 76,497
Paalon
  • 89
  • 2
    I don't understand the distinction you are making here. Can you explain the differing contexts in which you believe that there is ambiguity? – Xander Henderson Jun 08 '22 at 16:50
  • 2
    "When it is obvious from the context" it would be obvious from context, so go ahead. Language is fluid and there are no hard rules. That said, I personally would feel weird referring to the literal character = in the context of its existence as a character as "equality" rather than "the equals symbol" or "the equals sign" but it really shouldn't matter in the long run. – JMoravitz Jun 08 '22 at 16:51
  • 2
    "There are many famous propositions like Cauchy-Schwarz inequality" That is referring to the entirety of all of what follows in red: $\color{red}{|\langle u,v\rangle |^2\leq \langle u,u\rangle \cdot \langle v,v\rangle}$. That is not referring just to what follows in blue: $|\langle u,v\rangle |^2\color{blue}{\leq} \langle u,u\rangle \cdot \langle v,v\rangle$. If talking about the symbol as it appears on a keyboard and not talking about the contextual statements in which it appears, it is better understood to still call it a symbol. – JMoravitz Jun 08 '22 at 17:21
  • 1
    I can't grasp why my post is closed. Some people use the word equality as a symbol, but I don't know it is a common practice or will be accepted in native English speakers. Some people use "equality" as a proposition itself. There is a room to be ambiguous. My question is about it. – Paalon Jun 16 '22 at 14:26
  • 3
    Your context disambiguates the two meanings, viz. you state that you are listing set-theory symbols, so you can simply write "equality" instead of "equality symbol". Ditto for: negation, conjunction, disjunction, implication, biconditional. – Bill Dubuque Jun 18 '22 at 10:18
  • FWIW, the Unicode name for U+003D = is "equals sign". But they also call the backslash "reverse solidus", which nobody in real life does. – Dan Jul 18 '22 at 16:35

3 Answers3

3

As a practicing mathematician, I'd call it "the equality symbol". "The equals sign" sounds clumsier. As some people have commented, "equality" would suggest to me "an equality", which would be a phrase like $x=y$, not just the symbol $=$.

paul garrett
  • 52,465
3

In Logic, almost all logicians do use "equality" to refer to "=" as in "FOL with[out] equality", as short for "equality symbol". Mathematicians in general prefer to refer to "=" as "equals sign" or "equality sign" or "equality symbol" (with the explicit "symbol"). Yes, "equality" can also refer to a formula of the form "$s = t$". Somewhat amusingly, a formula of the form "$s ≠ t$" is instead called an "inequation" and not an "inequality".

To answer your question, no, it is not a grammatical mistake. This is why we say "FOL with[out] equality". If "equality" can only refer to equations rather than the symbol "=", then this phrase would be ungrammatical. But it is perfectly fine. For an example right here on Math SE, see this post where Noah Schweber repeatedly unambiguously uses "equality" to refer to "=".

And I would prefer not to think of this as 'just a matter of context', because one cannot just drop arbitrary words in general; ultimately it still comes down to whether it is acceptable in current mathematical vernacular. In English, it is as I said above. In other languages, it may be a totally different story.

user21820
  • 57,693
  • 9
  • 98
  • 256
  • 2
    Another example: in ring theory "identity" can mean the multiplicative neutral element (usually denoted $1)$ or a (ring language) equality that holds true for all free variables, e.g. $,x + y = y + x.,$ We study rings with both types of identities, but the latter is usually disambiguated by the name "rings with polynomial identity", e.g. here, and Jacobson's famous proof that rings with identity $,x^n - x,$ are commutative. – Bill Dubuque Jun 18 '22 at 19:06
  • @BillDubuque: Yes, and in logic some people even refer to "=" as "identity" as in "FOL with[out] identity", though it is less common now. Funnily, google search results yield ( "first-order logic with equality" : "first-order logic with identity" ) = 16.6 million : 17.3 thousand but ( "first-order logic without equality" : "first-order logic without identity" ) = 4780 : 5290. As if "identity" was more common for "=" in the past along with FOL without equality, whereas now FOL with equality has almost entirely taken over together with the modern use of "equality" for "=". – user21820 Jun 19 '22 at 03:15
2
  • truth: $\top$
  • falsity: $\bot$
  • negation: $\neg$
  • conjunction: $\land$
  • disjunction: $\lor$
  • implication: $\Rightarrow$
  • biconditional: $\Leftrightarrow$
  • universal quantifier: $\forall$
  • existential quantifier: $\exists$
  • equality: $=$
  • membership: $\in$

To me, this is not merely a list of symbol names, but a list of syntactic elements. So, $=$ is the equality relation, $∨$ is the disjunction connective, $∃$ is the existential quantifier, etc. This answers the question of whether to call item 10 ‘equals sign’ or ‘equality’.

And could we please call item 6 ‘conditional’ rather than ‘implication’, particularly since you are calling item 7 ‘biconditional’? (Also, many modern texts use the symbols $→$ and $↔,$ rather than $⇒$ and $⇔,$ for the material conditional and the material biconditional.)

ryang
  • 38,879
  • 14
  • 81
  • 179