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I am interested in what the definition of a mathematical expression is. For example $x_1 + x_2$ is a different expression than $x_2 + x_1$, even though they both evaluate to the same function on the reals. Has anyone rigorously defined what a mathematical or algebraic expression is, so as to distinguish expressions like $x_1 + x_2$ from $x_2 + x_1$?

user107952
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  • Note: $x^p-x$ is $0$ for all $x\in\mathbb Z/p\mathbb Z$ – J. W. Tanner Aug 31 '20 at 19:35
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    Look for an introductory text in mathematical logic. Things like terms, expressions and sentences are defined rigorously. – John Douma Aug 31 '20 at 19:36
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    In my mind, as someone who doesn't do algebra or logic, the term "expression" is not given a precise definition. I only use it informally, and if I want to be precise then I avoid the term altogether. – littleO Aug 31 '20 at 19:37
  • You may be interested in the way that expressions form the foundation of the Wolfram Language implemented in Mathematica. There is a lot of information about expressions at the Wolfram website. – Somos Aug 31 '20 at 23:44
  • @DietrichBurde I don't think it is too generous to interpret the question as "how can I precisely define the expression $x_1+x_2$?". That is, I think part of the question is "how can I make my question more precise"... – user1729 Sep 01 '20 at 13:13

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Expression is a grammatical term in the mathematical language. It can be formally defined when it's needed for formal treatment, but normally one keeps it open so that new constructions can be added. One then first defines variables and numeric constants, and then recursively build up expressions: an expression can be a variable, a numeric constant, an expression between parentheses, an expression followed by an operation and then another expression, and so on.

You can see examples in this Wikipedia article.

md2perpe
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  • Variables are expressions
  • If $f$ is an $n$-ary function and $a_1,\ldots, a_n$ are expressions, then $f(a_1,\ldots, a_n)$ is an expression

Notes:

  • The second point includes constants if we consider $0$-ary functions.
  • For some (in particular, 2-ary) functions, we have special notations, such as $x_1+x_2$ for the sum-of-two-numbers function applied to $x_1$ and $x_2$
  • If we introduce indexed variables (as we informally did above), that can be considered as a special notation of a function of the index as well
J. W. Tanner
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Hagen von Eitzen
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  • Hi, I have a quick follow up question to your answer. I noticed that Wikipedia defines an equation (https://en.wikipedia.org/wiki/Equation) as follows: "...an equation is a formula that expresses the equality of two expressions...", does this mean that an expression can be one term? The reason for me asking is that we define an expression to be a finite combination of symbols (plural) but surely we have equations with one term on the left and or right hand side of the equals sign (e.g., $3x+2=4$)? – Taylor Rendon Jul 28 '22 at 16:08