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According to wikipedia:

"In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, punctuation, grouping, and other aspects of logical syntax."

https://en.wikipedia.org/wiki/Expression_(mathematics)

To me, this definition seems kind of vague, I wonder if there exists a precise definition of "mathematical expression".

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    The obvious answer seems to be "no" for all practical purposes. You can get a precise definition of certain kinds of mathematical expressions in certain contexts, but if you tried to precisely enumerate all kinds of expressions used in all disciplines, not only would your definition be completely incomprehensible and unusable, it would quickly be made obsolete when someone needs to invent a new notation for their original work. – David K Jan 29 '16 at 22:40
  • It's an informal term, and the definition given isn't trying to meet mathematical standards for introducing a new term which can in principle be eliminated by substituting the definition. But "we know one when we see it". Can you cite any example of a phrase which you're not sure is a "mathematical expression"? :) – BrianO Jan 29 '16 at 23:10
  • The only place where the term is well-defined is in formalisms, where those "rules that depend on the context" make it precise. – Paul Sinclair Jan 29 '16 at 23:23
  • Despite reputation to the contrary, rigor solely for rigor's sake doesn't persist in mathematics. Mathematics is largely a social exercise, and without something for a given notational rigor to do, people will simply stop using it. ¶ Imagine for a moment that such a rigorous notion of "mathematical expression" existed, so that one might say, of a given string, "Oh yes, that's an expression," or "No, that's not." To what use could such a distinction be put? We sometimes say (as David K implies) that such-and-such is a particular kind of expression, and we can therefore do something to it. – Brian Tung Jan 29 '16 at 23:26
  • (cont'd) But even in such a situation, there's a clear delineation that this kind of expression is distinguished by the things we can do to it. Other expressions might not be so distinguished, but are no less valid for all that. – Brian Tung Jan 29 '16 at 23:27
  • To be honest, I thought that definition was rather precise and explicit, so far as definitions for something that needs to be so broad and universal. – fleablood Jan 29 '16 at 23:39
  • @fleablood: It's as precise as it needs to be, and no more. :-) – Brian Tung Jan 29 '16 at 23:41
  • If I were asked to provide an definition of a "mathematical expression" I would probably so say something like "a statement that expresses a concept that is mathematical in nature" so be thankful that I'm not in charge of the dictionaries. – fleablood Jan 29 '16 at 23:44
  • Does this answer your question? Is there a rigorous definition of mathematical expressions that can distinguish them from their values? – user182601 Feb 04 '24 at 03:46

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