Talk:Expression (mathematics)

Algebraic expression

Should this be moved to algebraic expression? Septentrionalis 15:42, 20 September 2005 (UTC)[reply]

I think not. There are many expressions such as

\int _{0}^{t}f(x)\,dx

which are usually not called "algebraic". --Aleph4 16:03, 20 September 2005 (UTC)[reply]

Yes, but they're not really discussed in this article. Septentrionalis 16:58, 24 September 2005 (UTC)[reply]

i think we are all missing the point. by our clumsy definitions of the basic algebraical terms and making them too technical the essence of mathematics is being taken away from the masses. it is high time we make proper changes. i request all to refer to hall and knight's elementary algebra for schools and then build up from there.

Should the link to axiomatic theory of expressions be removed? It's a poorly organized page that describes some guy's new "theory" about the foundations of mathematics. He's still developing his theory. I don't have the knowledge to adequately evaluate his claims, but it looked pretty sketchy to the untrained eye. I expect the link was added by the fellow himself.

I added the article algebraic expression. Any comments are appreciated. Isheden (talk) 11:16, 28 October 2011 (UTC)[reply]

I'm not sure about your article algebraic expression. According to your usage, expressions involving pi and e are not algebraic expressions because pi and e are not algebraic numbers. That's not a usage I've ever heard. Rick Norwood (talk) 17:41, 28 October 2011 (UTC)[reply]

It's a very good point you make. The sources I found indicate that expressions involving transcendental numbers are not considered as algebraic expressions. It's at least consistent since the algebraic numbers form a field, the algebraic closure of the rational numbers. It also fits to the informal definition in the lead of the article algebraic function. Actually, the reason I've been considering this is that I wanted to define what an algebraic fraction is—another article I've created. The background was that I wanted to find out whether all examples in the article Fraction (mathematics) should really be called fractions. The results would indicate that e.g. √2/2 and

{\frac {2+i}{2-2\cdot i}}

are rational algebraic fractions since they involve only algebraic numbers, whereas π/4 is not, since π is transcendental. Do you know of sources claiming otherwise? Isheden (talk) 21:59, 28 October 2011 (UTC)[reply]

Numbers or numerals?

Oleg just reverted this, but I think the proper term is "numerals". An expression is a combination of symbols. A number is not a symbol, but a numeral is. Therefore, expressions contain numerals; they do not contain numbers. Other thoughts? I'll re-revert if there are no objections in the next day or so. capitalist 02:53, 14 June 2006 (UTC)[reply]

Why do you say that a mathematical expression is a combination of symbols? Do you have a reference for this? I think it is a combination of numbers, not numerals, together with functions and variables (see my comment on Talk:Complex number). -- Jitse Niesen (talk) 05:17, 14 June 2006 (UTC)[reply]

Yes, the reference is the Merriam Webster's Online Dictionary which gives this definition:

1 a : an act, process, or instance of representing in a medium (as words) : UTTERANCE <freedom of expression> b (1) : something that manifests, embodies, or symbolizes something else <this gift is an expression of my admiration for you> (2) : a significant word or phrase (3) : a mathematical or logical symbol or a meaningful combination of symbols (4) : the detectable effect of a gene; also : EXPRESSIVITY 1

So given that definition, we can conclude that a mathematical expression is a group of symbols. Given that conclusion and the fact that numerals are symbols while numbers are not, we can conclude that mathematical expressions contain numerals, not numbers. QED! :0) capitalist 03:28, 15 June 2006 (UTC)[reply]

The Oxford English Dictionary has a similar definition—unfortunately there is no Mathworld article on it. However, with the example given at Talk:Complex number of Polish notation of "+ 1 1" being the same expression, with just a different notation, as the common infix "1 + 1", I do not know that this is sufficient. This has the same numbers operated on by the same operator; and both mean what is possibly another notation "one plus one". Is this different because the symbols are different, where the Polish notation has the same symbols in a different order? Still, certainly, a expression is still a representation, and "3 - 1" is not the same expression as "1 + 1". —Centrx→talk 07:14, 15 June 2006 (UTC)[reply]

It is possible to be right and wrong at the same time. Yes, there is a distinction to be made between numbers and numerals and, yes, an expression contains the latter, not the former. However, it is equally true that 2 plus 2 is not 4. Rather, we should say that the number represented by the numeral 2 added to the number represented by the numeral 2 yields the number represented by the numeral four. That sort of excessive precission is called pedantry, and is to be avoided. Make the technical distinctions only in cases where they matter. Rick Norwood 15:34, 15 June 2006 (UTC)[reply]

No, when a word is used in language, the meaning to which it refers is always implied, that is the prime and default purpose of language. When a term is used as term, it is the exception and has special formatting to indicate it, in such examples as — Depend derives from the same word as pendulum. — and — The word "cat" designates a feline. — but that does not mean that it is false to say that "A cat is a feline." It is the reason why on Wikipedia each article begins with something like "The United States is a country" rather than "The United States is the name of a country". The fact remains that we are defining what an expression is. If an expression is, in fact, a combination of numerals,etc. rather than numbers, then defining it as number would be equivalent to saying "The United States is a continent" or the "United States is a population", not the pedantry of defining it as as term. —Centrx→talk 19:20, 15 June 2006 (UTC)[reply]

You are splitting hairs. If I did the same, I could object that "The United States" is not the name of a country. The name of the country is "The United States of America". Rick Norwood 21:09, 15 June 2006 (UTC)[reply]

You are incorrect. If an expression does not consist of numbers, then defining it as such would be false. The analogue would be stating that the name of the United States is "Northern Hemisphere". If indeed, this were simply splitting hairs, then it should be perfectly acceptable to you either way. —Centrx→talk 21:35, 15 June 2006 (UTC)[reply]

So is the number 1 or a lone variable, say x, a mathematical expression? The MW dictionary quote seems to imply yes. It says symbol, singular, not symbols, plural. The article is unclear. — Preceding unsigned comment added by 71.220.62.231 (talk) 04:37, 2 August 2011 (UTC)[reply]

Yes. The number (numeral) 1 and the monomial x are mathematical expressions. Rick Norwood (talk) 13:21, 2 August 2011 (UTC)[reply]

Well that is a lot there Redneck121 (talk) 21:16, 10 February 2016 (UTC)[reply]

explicit vs implicit expression

In Golden_ratio, there is a redlink to "explicit expression". Is this a well defined math concept? If so, it might be useful to define it in this article?  Xiutwel ♫☺♥♪ (talk) 21:08, 22 December 2007 (UTC)[reply]

See also: Closed-form expression 21:10, 22 December 2007 (UTC) —Preceding unsigned comment added by Xiutwel (talk • contribs)

Division by zero not an expression?

"... is not [an expression], because the parentheses are not balanced and division by zero is undefined." Does that make sense? That x / 0 is not mathematical expression just because it's undefined? Saeed Jahed (talk) 10:19, 15 March 2009 (UTC)[reply]

A nonsense string of symbols is not considered a (well formed) mathematical expression. As another example "3 @ 4" would not be considered a mathematical expression because the operation "@" is undefined. On the other hand, if a definition of "@" were given in the preceding text, then "3 @ 4" would become a mathematical expression. There is a large body of literature on what constitutes a "well-formed formula". Rick Norwood (talk) 14:45, 15 March 2009 (UTC)[reply]

Not sure if what Rick said is relevant. Certainly the unbalanced parens discount it from being an expression. However, division by zero (x / 0) is certainly a perfectly valid expression -- its value is undefined, but it is an expression. Therefore, I have changed the counter-example to ")x)/y", and removed the statement about division by zero. (There is even a later example of an expression being undefined due to division by zero). EatMyShortz (talk) 07:22, 31 August 2009 (UTC)[reply]

Support: Division by 0 is, correctly, undefined. However, "(x ÷ 0)" is correct when defined as a mathematical expression (which has been defined in Expression_(math).) ACredibleLie (talk) 15:15, 18 February 2010 (UTC)[reply]

Relations?

Can expressions include relations? Is x < y an expression? It seems the distinction that expressions cannot include the equals sign (often considered a relation) should at least be included in the second sentence. I usually think of expressions as more or less equivalent to "terms" in first order logic, and terms cannot include relations... Is there a source that includes relations as part of expressions? Dmcginn (talk) 16:12, 2 July 2010 (UTC)[reply]

Here is an example of an inequality being called an expression:

"The expression

a\geq b

is synonymous with

b\leq a

." (p. 66)

Real Analysis: A Historical Approach By Saul Stahl

--50.53.50.57 (talk) 06:47, 2 October 2014 (UTC)[reply]

In this example, a function definition is called an expression:

"... the set

B

in the expression

f:A\to B

is sometimes called the codomain of

f

." (p. 128)

Introduction to Real Analysis By Michael J. Schramm

--50.53.50.57 (talk) 07:25, 2 October 2014 (UTC)[reply]

Isheden's complaint

I agree that this article (and all other articles on elementry mathematics) must be as easy for a non-mathematician to read as possible. I've shorted and simplified the first paragraph.

As best can see, the second paragraph, beginning "In algebra..." does not say anything that needs to be said here. Do I hear any objection to removing it?

On the other hand, we might want to add a little bit about the rules for well-formed formulas.

Rick Norwood (talk) 17:59, 28 October 2011 (UTC)[reply]

I think the paragraph you mention can be removed. Possibly algebraic expression could be merged into this article instead. Isheden (talk) 22:02, 28 October 2011 (UTC)[reply]

"Meaningless jumble"

This pearl of prose is hardly encyclopedic language. It should be rephrased. FilipeS (talk) 18:55, 5 April 2012 (UTC)[reply]

Agree. Possibly algebraic expression could be merged into this article if it becomes more encyclopedic. Isheden (talk) 08:12, 6 April 2012 (UTC)[reply]

Contradictory as to whether an equation is an expression

The section Semantics: meaningful expressions says

...for instance, an expression might designate a condition, or an equation that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules.

But this is contradicted in paragraph 2 of the section Variables where it says

Thus an expression represents a function whose inputs are the value assigned the free variables and whose output is the resulting value of the expression.

By saying it's a function it precludes it from being an equation. Also, the last paragraph of the lead in the article Formula says

Expressions are distinct from formulas in that they cannot contain an equals sign (=).[6] Whereas formulas are comparable to sentences, expressions are more like phrases.

again contradicting the first quote above. I'll leave it to others to decide whether/how to fix this. Loraof (talk) 20:15, 29 September 2014 (UTC)[reply]

The two first quotes are not contradictory, as "=" may be considered as an operator which takes its values in {true, false}. This the case in general purpose computer algebra systems. However the second quote is wrong for at least two reasons. Firstly, an expression may represent a function only when the set of possible values of the variables (range of the function) is defined, or in other words, if the semantic of the expression is defined. Secondly, an expression may not contain any variable and its evaluation may not result in a number. For example, a matrix of integers is an expression which does not represent any function and cannot represent any other mathematical object than itself.

I agree that the quote of Formula deserve to be edited: this is the opinion of one author, not a common convention. D.Lazard (talk) 21:48, 29 September 2014 (UTC)[reply]

Inconsistency in table

In the section Different forms of mathematical expressions, the table says that arithmetic expressions can have factorials but not integer exponents. This seems contradictory: each is simply a sequence of multiplications, in one case like 4×3×2×1, and in the other case like 4×4×4×4. So they both ought to be allowed or not allowed.

Also, since the table says that polynomials can contain an "integer exponent", I think that row heading should be renamed "Positive integer exponent" or "Non-negative integer exponent".Loraof (talk) 20:25, 29 September 2014 (UTC)[reply]

Good points. Another problem with the table is that it says polynomials can have elementary arithmetic operations, but polynomials do not have division. BTW, the table is a template: Template:Mathematical expressions. --50.53.46.203 (talk) 14:29, 1 October 2014 (UTC)[reply]

Dictionary definitions of "expression"

expression: "A very general term used to designate any symbolic mathematical form, such, for instance, as a polynomial."

The Mathematics Dictionary edited by R.C. James

expression 1b(3): "a sign or character or a finite sequence of signs or characters (as logical or mathematical symbols) representing a quantity or operation"

Webster's Third New International Dictionary of the English Language, Unabridged edited by Philip Babcock Gove

expression 1b(3): "a mathematical or logical symbol or a meaningful combination of symbols"

Merriam-Webster's Collegiate® Dictionary, Eleventh Edition

--50.53.50.57 (talk) 08:14, 2 October 2014 (UTC)[reply]

"Definition 1: Let S be a set of symbols. An expression in S (or word in S) is a finite sequence of symbols of S. For example, if the set of symbols is {a, b, c}, then aabc and cba are both expressions in S." (p. 7)

An Introduction to Mathematical Logic By Richard E. Hodel

--50.53.46.137 (talk) 06:03, 3 October 2014 (UTC)[reply]

"We call a finite sequence of (occurrences of) formal symbols a formal expression." (§ 38. Formal number theory.)

Mathematical Logic By Stephen Cole Kleene

--50.53.60.76 (talk) 14:42, 3 October 2014 (UTC)[reply]

Margaris gives a definition of "string" that appears to mean the same thing as what Hodel and Kleene mean by "expression":

"The construction of a formal axiomatic theory begins with the specification of a finite set of formal symbols, and a string is defined to be a finite sequence of formal symbols." (p. 14)

Later, in the context of Gödel's incompleteness theorem, Margaris says of N, a formal number theory:

"An expression of N is a string or a finite sequence of strings of N." (p. 185)

First Order Mathematical Logic By Angelo Margaris

--50.53.60.76 (talk) 19:27, 3 October 2014 (UTC)[reply]

The definitions from logic, where an expression is any string, differ sharply from the more general definitions, where an expression is a meaningful string. In general mathematics x + 2 is an expression, while +=@@#%$ is a string but not a mathematical expression. Rick Norwood (talk) 11:12, 4 October 2014 (UTC)[reply]

Thanks for pointing that out. Logicians appear to define the term "expression" differently from mathematicians, and the article should say so. Do you have any sources other than the dictionary definitions above that define "expression" in the sense that mathematicians use it? --50.53.61.13 (talk) 14:09, 4 October 2014 (UTC)[reply]

Complete rewriting needed

I agree with preceding posts that the article has many issues. Here are several ones that have not been quoted in the preceding quotes, or have only been partially quoted. Here are some of these issue

The article does not contains anything beyond the informal dictionary definition
The article is unreferenced
The classification of the types of expression seems original research
This classification is incomplete, as it does not mention logical expressions and expressions that cannot been evaluated to numbers, as matrices.
It uses without definition "arithmetic expression" (the wikilink provided is a self redirect)
It does not mention computer algebra at all, while this field is the basis of the modern understanding of the concept (computer algebra systems use to not manipulate mathematical objects, but only their representation as expressions, forgetting most of the associated semantic.

This list is incomplete, and shows that the article deserves to be completely rewritten. D.Lazard (talk) 11:00, 2 October 2014 (UTC)[reply]

"The classification of the types of expression seems original research"

Are you referring to the table in the section Different forms of mathematical expressions? --50.53.47.11 (talk) 14:12, 2 October 2014 (UTC)[reply]

I refer to the template and to the remainder of the article. The term "arithmetic expression" seems to not have a standard meaning. The distinction between analytic and closed form expression is also dubious. As far as I know, "analytic expression" is an old term of what is now called "closed form expression", which is no more in common use because the possibility of confusion with the "analytic" of the analytic function. D.Lazard (talk) 17:17, 2 October 2014 (UTC)[reply]

No History

There is no section describing how and when symbolic expressions were developed and replaced text. FreeFlow99 (talk) 09:49, 6 November 2015 (UTC)[reply]

The problem for that is the lack of WP:Reliable sources. In fact, as far as I know, there are two periods. The first one is the introduction of mathematical notation, which is described in History of mathematical notation. However, the concept of "expression" appeared much more recently. I suspect that it has been rarely used before the second half of the 20th century, when it has been popularized by its use in computer algebra and other aspects of the computerization of mathematics. In fact, that is the computerization that requires a clear distinction between a mathematical object and its various representations as expressions. Moreover, in relation with computer proofs, a formal definition of "expression" is often needed, and there is not yet a general agreement of what should be such a formal definition. This explains the lack of reliable sources for the history of the notion. D.Lazard (talk) 15:18, 6 November 2015 (UTC)[reply]

Missing from table

grouping (parentheses)

Equations (=), inequalities, and inequations (not =) as expressions (e.g. x = x, treated as an expression, reduces to true)

power towers, finite and infinite

nested radicals, finite and infinite

hyperoperations

multivariable and vector calculus

integral transforms and other transforms

operators (differential and integral, vector)

probability and statistics operators

sets and set operations (union, intersection, ...)

logic (and, or, not, implication, truth values, quantifiers, higher order logic...)

vectors and matrices and their operations

roots of polynomials that cannot be expressed as radicals (e.g. roots of x^5 + x + 1)

complex numbers

combinatorics (combinations, permutations, ...)

functions (e.g. f(x + 1) as an expression)

and many more...

A more general notion of mathematical expression

Objects from literally all areas of mathematics, physics, and computer science can be treated as mathematical expressions, because they can be manipulated as such. Indeed, mathematical software such as Mathematica and Maple do exactly this. Groups, graphs, geometric shapes, and computer programs are just as much mathematical expressions as 1 + 1.

75.46.182.198 (talk) 02:07, 6 June 2018 (UTC)[reply]

I totally agree with the first sentence of the last paragraph. However, the last sentence is wrong, as a mathematical object is not an expression by itself. If you omit the section "Forms", this is exactly what the article says.

I agree also that the section "Forms" is problematic. This is a classification that is far to be complete, as you have pointed. It is also wrong. For example

x+y

is not an arithmetic expression if

x

and

y

do not denote numbers. Also, it is misleading, as suggesting that a specific named function (such as "gamma function") is an expression. Also, the distinction between closed form expressions and analytic expressions seems WP:OR. Also, the table seems WP:SYNTHESIS. My opinion is that section must be removed. However, this needs a consensus. So I will first edit the section and moving it toward the end of the article. D.Lazard (talk) 08:58, 6 June 2018 (UTC)[reply]

Expression, formula, equation, function

The second paragraph of the current article lead says that a formula can be evaluated to true or false. This does not aid understanding of the use of formulas in elementary maths or physics. Consider the well-known quadratic equation ( $ax^{2}+bx+c=0$ ) and quadratic formula ( $x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$ ). It is possible to substitute a, b, c and x into the formula and get a true result if x is one of the values of the expression. But to test a value of x, it is simpler to use the equation. The true use of a formula is to provide instructions for calculating a value. Formulas could use words or an assignment operator, but they are usually written starting with a variable name and equals sign, so they are often confused with equations when the term formula is being defined. For this reason, it is hard to find useful reliable sources. I also think the link to formula (see following note) is unhelpful for most readers of the article.

An expression is an anonymous formula (by analogy with an anonymous function) in the sense that it also provides instructions for calculating a value, but without giving a name for the instructions or result. It would also be useful to contrast expressions and functions (like the distinction between polynomial expression and polynomial). JonH (talk) 23:55, 15 April 2022 (UTC)[reply]

Note that when I wrote this comment, formula redirected to well-formed formula which was unhelpful, but now it redirects to formula#In mathematics which does help. JonH (talk) 19:37, 24 September 2023 (UTC)[reply]

The paragraph says that a formula can be evaluated to true or false, depending on the values that are given to the variables. So the text does require an assignment of a value to each variable before evaluation. You have a point in that the phrasing could be more clear.

Moreover, you are right that one of the meanings for the word "formula" is "receipe to calculate a value (given values for its 'input' variables)". This meaning is quite different from what is meant in the article, and we should clarify this, too. The latter meaning is the one used widely in mathematical logic, and the inequation example illustrates it perfectly: no receipe whatsoever can be obtained from it, but it evaluates to true or false, depending on the value of x.

A reliable source for this meaning of "formula" (and "expression") is likely to be found in an arbitrary logic textbook. Unfortunately, I only have German textbooks at hand. I found the source Bergmann.Noll.1977,^[1] p.28, before Def.6.4 and 6.5, but this book has no English translation. On the other hand, in Hermes.1972,^[2] translated as Hermes.1973,^[3] I didn't find a concise remark that could be used as source. - Jochen Burghardt (talk) 17:07, 16 April 2022 (UTC)[reply]

Also, good sources for "expression" could be found in textbooks on computer algebra. In fact, this is computer algebra that has popularized the term of "expression", because of the need of distinguishing mathematical objects from their representations. Indeed, a large part of computer algebra consists of transforming expressions without changing the represented object (for example, for simplification). D.Lazard (talk) 09:01, 17 April 2022 (UTC)[reply]

That's a good point. If you'd like, please start a History section talking about this. Farkle Griffen (talk) 21:59, 9 August 2024 (UTC)[reply]

References

^ Eberhard Bergmann and Helga Noll (1977). Mathematische Logik mit Informatik-Anwendungen. Heidelberger Taschenbücher, Sammlung Informatik. Vol. 187. Heidelberg: Springer.
^ Hans Hermes (1972). Einführung in die mathematische Logik. Mathematische Leitfäden (4th ed.). Stuttgart: B.G. Teubner. ISBN 3-519-12201-4.
^ Hans Hermes (1973). Introduction to Mathematical Logic. Hochschultext (Springer-Verlag). London: Springer. ISBN 3540058192. ISSN 1431-4657.

Removal of citation alerts

The article still needs general work, but there seems to be enough general citations and inline citations to justify removing the current alerts. Farkle Griffen (talk) 21:52, 9 August 2024 (UTC)[reply]

Expression definition in logic textbooks

@Farkle Griffen: I'm repeating my previous edit summary here: A formula contains expressions as constituents, e.g." $2+2$ " is an expression, it is part of the formula " $\exists x:2+2\leq x$ ". Therefore, a formal definition of "formula" always includes a formal definition of "expression". Hence every logic textbook contains a definition of "expression", as part of a definition of "formula". See e.g. Hans Hermes, Introduction to Mathematical Logic, 1973, ISBN 3540058192, sect. II.1. - Jochen Burghardt (talk) 07:10, 10 August 2024 (UTC)[reply]

History section removal on Oct. 10, 2024

The history section was removed citing WP:Content Forks. I'm not sure I understand this. By the "page in a nutshell" at the top, "Most types of content forks are acceptable. The two types that are not are POV forks, in which articles are split up so that one can advocate a different stance on the subject, and pages of the same type on the same subject". Since this certainly isn't a POV fork, and it’s just a section rather than a whole new page, I thought it would be appropriate, especially given the different scopes of the articles.

While, yes, it contained much of the same information as History of mathematical notation, it would be impossible to create a history section that doesn't overlap given all mathematical expressions use mathematical notation. Though, as a defense for how these are different, the other article contains the history of notations for Equality/Equations in the development of Algebra, inequality, other relations, and general mathematical formulas, whereas (in compliance with, and as supported by the lead) this only includes notations that describe mathematical objects. And further (although it currently doesn't) this section could reasonably include the etymologies of terms rather than just their symbols, which that article doesn't cover. [Edit: For example, the popularization of the term "Expression" itself due to its use in computer algebra as mentioned in a previous discussion.]

Given this, it seems reasonable that the history section should stay. Is there something that I'm missing? Farkle Griffen (talk) 22:20, 10 October 2024 (UTC)[reply]

An alternative might be to focus on portions pertinent to expressions specifically (I'm not sure what those might be), and then include a hatnote such as {{broader|History of mathematical notation}}. –jacobolus (t) 03:18, 11 October 2024 (UTC)[reply]

That's what I did, or at least, what I was trying to do. Most of the content in that section was about the topics supported by the definition in the lead. It wasn't a total copy of the other article, only those that refer to specific objects or functions, up to common mathematical notation (so nothing recent like General relativity/Quantum mechanics notation mentioned). I'm not sure what I included in that section that could be considered considered not an expression.

Though, there were some redundant paragraphs not directly related to the topic (like the paragraph mentioning analytic geometry). Would it be worth it to re-add it and prune some of the "dubious" paragraphs? Farkle Griffen (talk) 03:44, 11 October 2024 (UTC)[reply]

For example, everything in "Early written mathematics" seems somewhat off topic here, and could probably be summarized in at most a short paragraph. –jacobolus (t) 20:31, 11 October 2024 (UTC)[reply]

Yeah, I can do that Farkle Griffen (talk) 21:34, 11 October 2024 (UTC)[reply]

Hatnote to "formula"

Hi @Farkle Griffen, I removed the hatnote you just added:

"Mathematical expression" redirects here. For the grammatical meaning of "expression", see Formula.

I don't understand what this is trying to say, or why you think Formula is a likely target readers are looking for and won't be able to find. There is already a prominent link to mathematical formula, italicized, at the start of the second paragraph here. –jacobolus (t) 17:57, 25 July 2025 (UTC)[reply]

The term "expression" has two common meanings. Informally (though not uncommonly) I've seen the phrase "mathematical expression" meant in the grammatical sense, roughly meaning a "Mathematical phrase" such as "

y=ax+b

". I've also seen a few articles link here implying the wrong sense. Figured that merited a disambiguation hatnote. It could be changed to:

to be a bit clearer, but given the amount of times I've seen that mistake, I think there needs to be something at the top. Is there a way to make it clearer? – Farkle Griffen (talk) 18:10, 25 July 2025 (UTC)[reply]

There's already a prominent link to mathematical formula in the second paragraph here. I don't think an out-of-context hatnote pointing there at the very top of the page is going to be meaningfully helpful to readers. I also still don't understand what distinction "the grammatical sense" is supposed to make; this phrasing seems confusing to me, and will plausibly be even more confusing to someone with no experience in this topic. Adding a wikilink to phrase doesn't help me to understand better. None of these usages seems to be all that closely related to the lay meaning of "expression". Aside: the use of the term "mathematical expression" to mean "any chunk of mathematical notation" (including equations, etc.) is also a valid and common one, and should probably be discussed here more clearly: the current language "Expressions are commonly distinguished from formulas" is true, but also potentially misleading, since in other contexts they are not really distinguished. What bit of notation counts or doesn't count as an "expression" per se depends on context. Sometimes people want to operate on whole equations or systems of equations etc. treated as "expressions". –jacobolus (t) 20:54, 25 July 2025 (UTC)[reply]

The issue is this article is about the subject "a symbolic representation of a mathematical object", not the word "Expression" per WP:NOTDICT. The more general sense has a distinct meaning, and is not the subject of this article, hence the hatnote.

The lead is already out of line with MOS:LEAD since it doesn't summarize the body and includes information not mentioned in the body, so trying to shoehorn a detail like that seems like a bad idea. I suppose if we did want to, we could have a section on "Expression vs Formula vs Notation" just going over semantics.

Personally, I think a hatnote is the best option, but if there's no way to concisely distinguish them, a new section about the word might be the only option. – Farkle Griffen (talk) 22:29, 25 July 2025 (UTC)[reply]

I don't understand which readers you are targeting with a hatnote like that. (Basic rule: "Mention other topics and articles only if there is a reasonable possibility of a reader arriving at the article either by mistake or with another topic in mind.") Someone who doesn't know that there is supposed to be a difference between the terms "expression" vs. "equation" vs. "formula" (etc.) in one or another context is not going to know they are possibly looking for a different page, or is going to almost immediately find a link to mathematical formula when it is mentioned in the second paragraph. Someone who is closely familiar with these terms is not going to have trouble looking up their various pages or reading down in the article body. The hatnote was also not, in my opinion, disambiguating between two different subjects with the same name, but rather talking about a related concept (or in some contexts perhaps a sub-topic). The various definitions of these terms can be discussed much more clearly in the article's prose than in a few words in a hatnote.

"The more general sense has a distinct meaning" – I'm still not precisely clear on which sense/meaning you are talking about. Can you unpack it more explicitly? –jacobolus (t) 02:15, 26 July 2025 (UTC)[reply]

"I'm still not precisely clear on which sense/meaning you are talking about. Can you unpack it more explicitly?" – I believe the same one when you said: "the use of the term "mathematical expression" to mean "any chunk of mathematical notation" (including equations, etc.) is also a valid and common one"

"Mention other topics and articles only if there is a reasonable possibility of a reader arriving at the article either by mistake or with another topic in mind." – Yes, there is the subject of this article, a symbolic representation of a mathematical object, and there is the more general "any chunk of mathematical notation (including equations, etc.)", which is also (sometimes) called "Mathematical expression".

I'm attempting to add a hatnote because the first is the subject of the article, and the second is not. The point of the hatnote is to say "The phrase mathematical expression is sometimes also used to mean formula, but they are not the subject of this article" which is not made explicit in the lead (nor should it be). – Farkle Griffen (talk) 02:52, 26 July 2025 (UTC)[reply]

But: (1) both meanings of the term, which are both technical jargon, substantially overlapping but slightly different with intended meaning dependent on context, should be discussed on this page; and (2) the article mathematical formula does not describe either of these two meanings. –jacobolus (t) 03:06, 26 July 2025 (UTC)[reply]

There are generally there are three kinds of "chunks of mathematical notation". Ill-defined expressions,

3,\{+7

, which are meaningles, Well-defined expressions,

3x+7

, which describe an object, and well-formed formulas,

3x+7\geq -2

which describe properties of / relationships between objects. There are no other kinds.

The first two are talked about in this article § Well-defined expressions, and the last one already has two articles about it (formula, well-formed formula). – Farkle Griffen (talk) 03:19, 26 July 2025 (UTC)[reply]

As far as I can tell nothing about the definition given at Well-defined expression excludes "formulas" from being expressions. –jacobolus (t) 03:37, 26 July 2025 (UTC)[reply]

Read § Formal definition here – Farkle Griffen (talk) 03:40, 26 July 2025 (UTC)[reply]

Can you be more specific? I don't immediately see anything there which would exclude a "formula" from being a type of "expression". –jacobolus (t) 03:43, 26 July 2025 (UTC)[reply]

The definition of expression is built up from constant symbols, variables, and function/operation symbols over the domain. Well-formed formulas need to contain a relation symbol (predicate) to combine expressions to express a truth value like = or < (called atomic formulas), and they can possibly contain quantifiers, logical connectives, etc. – Farkle Griffen (talk) 03:50, 26 July 2025 (UTC)[reply]

Are you sure that binary relations don't count as "operations" under the definition at Expression (mathematics) § Formal definition. To be honest the description at these pages is technical enough that it's hard for me to know at a glance exactly what does or doesn't quality there.

I'll be happy to leave it to experts what these terms should mean in very specific technical contexts.

In ordinary mathematical writing, the term "expression" commonly also covers equations, inequalities, etc., and I'm not convinced this is a fundamentally different concept/topic than what you are talking about or that authors using it that way are just being sloppy. –jacobolus (t) 05:15, 26 July 2025 (UTC)[reply]

"Are you sure that binary relations don't count as "operations" under the definition" yes, otherwise, for example,

(3\geq 5)+2

would be well-formed in standard arithmetic.

Replying to "As another example to consider, how would you treat the Iverson bracket? It explicitly turns "formulas" into "expressions".

There are some "weird cases" where, where you have to do some "gymnastics" to allow for certain notations. Formally, (at least in first order logic, where ZFC is defined), you can never "turn formulas into expressions". But there are ways to get around it. For example, to define the notation

\{x\in A|\phi (x)\}

for some arbitrary formula

\phi (x)

you have to define a new function symbol, esssentially

F(A)=\{x\in A|\phi (x)\}

for each formula

\phi (x)

in ZFC, so that technically the expression doesn't take in a formula as a parameter, and the formula is just there as part of the notation.

I haven't seen how the Iverson bracket itself is defined formally, but my guess is it's defined similarly as a class of functions

[\phi (x)]

for each formula

\phi (x)

. – Farkle Griffen (talk) 06:07, 26 July 2025 (UTC)[reply]

I am also not sure I fully agree with your recent change of "expressions are a kind of mathematical object" to "expressions denote mathematical objects". Both of these statements are true. Our article here could perhaps do a better job of explaining that (disclaimer: I haven't read it closely from top to bottom). –jacobolus (t) 02:20, 26 July 2025 (UTC)[reply]

Can they be? Sure, in a meta-linguistic sense, but when giving the definition is not the time for belaboring about metamathematics. – Farkle Griffen (talk) 02:57, 26 July 2025 (UTC)[reply]

Your change is probably an improvement, but we should discuss the topic in more detail and more explicitly in this article (currently it's at best hinted at), describing how mathematical chunks of mathematical notation can themselves be treated as objects and operated on, separately from evaluating them. It is the basic idea underlying e.g. computer algebra systems and symbolic integration, but also seems relevant much more broadly to some extent. For example a formal power series is, in a certain sense, a way of treating the formal expression of a polynomial function to be an object per se. –jacobolus (t) 03:09, 26 July 2025 (UTC)[reply]

These seem to be mentioned in ~~§ Variables and evaluation~~ § Computer science and § Formal expression, but Expression (computer science) has its own article. In any case, I don't understand what the point of this reply is. I agree there are some cases where expressions are used like objects, but the definition isn't the time to talk about it. – Farkle Griffen (talk) 03:35, 26 July 2025 (UTC)[reply]

@Jacobolus, I've lumped together a bunch of the "computation" sections that were randomly spread throughout the article into a "Computer science" section. I think this gives it better structure. § Formal expression probably still needs to mention polynomial rings and formal power series explicitly. Does this seem better to you? – Farkle Griffen (talk) 05:02, 26 July 2025 (UTC)[reply]

I'll try to take a closer look tomorrow if I get a chance.

As another example to consider, how would you treat the Iverson bracket? It explicitly turns "formulas" into "expressions". –jacobolus (t) 05:19, 26 July 2025 (UTC)[reply]

Here's what James Tanton (2005) Encyclopedia of Mathematics says (p. 182):

expression Any meaningful combination of symbols that represent numbers, operations on numbers, or other mathematical entities is called an expression. For example, ⁠ $2+x$ ⁠ and ⁠ $\textstyle a^{b+c}$ ⁠ are expressions. One could argue that ⁠ $x+y=2$ ⁠ is an expression, although mathematicians may prefer to call it an equation. Similartly ⁠ $\textstyle {\sqrt[{3}]{8}}$ ⁠ could be called an expression even though it is equivalent to a single number. In formal logic, compound statements are sometimes called expressions. For example, ⁠ $\lnot (p\land (q\lor r))$ ⁠ is an expression. [...]

This seems like a pretty reasonable initial description, and I think our article here should say something similar. It can then go on to say that in some contexts the word expression more specifically describes chunks of mathematical notation that don't describe an equation or other relation. @Farkle Griffen, I don't think your use of the word "formula" is necessarily reflective of how that word is used broadly in mathematical writing. Here's Tanton (pp. 199–200):

formula Any identity, general rule, or general expression in mathematics that can be applied to different values of one or more quantities is called a formula. For example, the formula for the area ⁠ $A$ ⁠ of a circle is: $A=\pi r^{2}$ where ⁠ $r$ ⁠ represents the radius of the circle. The quadratic formula for the roots of a quadratic equation of the form ⁠ $\textstyle ax^{2}+bx+c=0$ ⁠ is: $x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

In my opinion more specific technical definitions in particular contexts should be subordinated to these broader, less restrictive, definitions. –jacobolus (t) 20:04, 26 July 2025 (UTC)[reply]

I don't see how either of those disagree with what I've said. To quote myself:

"The definition of expression is built up from constant symbols, variables, and function/operation symbols over the domain. Well-formed formulas need to contain a relation symbol (predicate) to combine expressions to express a truth value like = or < (called atomic formulas), and they can possibly contain quantifiers, logical connectives, etc."

Both of those examples for formula have the relation symbol "=". – Farkle Griffen (talk) 20:30, 26 July 2025 (UTC)[reply]

To respond to the first half alone:

I don't have an issue with clarifying that the word "expression" sometimes also applies to formulas. That was the whole point in adding a hatnote. My issue is this article is not about the word "Expression" it is about a symbolic representation of a mathematical object. If the article needs a more specific name, then the name can be changed. If this article were about "any chunk of mathematical notation (including equations, etc.)" then we should just redirect the article to Mathematical notation.

The lead is supposed to summarize the body. Disambiguation is usually done in the hatnotes. – Farkle Griffen (talk) 20:46, 26 July 2025 (UTC)[reply]

I don't agree with you about the proper scope of this article. I think it should cover the topic as broadly considered, something like Tanton's "Any meaningful combination of symbols that represent numbers, operations on numbers, or other mathematical entities". We can also more specifically consider a narrower concept of "expression" that excludes what might be called "statements", "propositions", "relations", or the like. It's even fine if most of the article is about that more specific concept. But these are not so far apart that they need to be split into separate articles. A hatnote is not a clear way to establish any particular distinction. –jacobolus (t) 01:01, 27 July 2025 (UTC)[reply]

Theres already an article about that: Mathematical notation – Farkle Griffen (talk) 01:02, 27 July 2025 (UTC)[reply]

We could certainly plausibly merge those articles and redirect to there. I wouldn't necessarily recommend that though. –jacobolus (t) 01:10, 27 July 2025 (UTC)[reply]

Why wouldn't you reccomend that? I wouldn't want that because these are fundamentally different. If they are the same, then they should be merged per WP:NOTDICT.

In any case, Tanton's definition is exactly the one I'm suggesting: "Any meaningful combination of symbols that represent numbers, operations on numbers, or other mathematical entities"

This does not mention relations. – Farkle Griffen (talk) 01:17, 27 July 2025 (UTC)[reply]

Tanton explicitly mentions that an equation could be considered a type of expression under his expansive definition. I looked through the Princeton Companion, and there are several practical examples in there of the word "expression" being used to refer to an equation. –jacobolus (t) 01:21, 27 July 2025 (UTC)[reply]

That doesn't address the first half of my reply at all.

This could be a completely different conversation if we could instead talk about how to deal with the extraneous usage, but instead we seem to be discussing whether or not to delete this article entirely.

Please just accept that the use that includes formulas is extraneous so this can be a productive conversation. – Farkle Griffen (talk) 01:27, 27 July 2025 (UTC)[reply]

Seperate to my above comment, in the sentence "One could argue that ⁠x+y=2 is an expression, although mathematicians may prefer to call it an equation." It seems more like Tanton is appending "equations" to the definition, not asserting it follows from the definition, and still he is very hesitant to do so.

"x+y=2" doesn't represent an object. It's a statement. "x+y" does. – Farkle Griffen (talk) 01:48, 27 July 2025 (UTC)[reply]

I don't agree that such usage is "extraneous"; it seems quite ordinary and common. Rather, I think you want to narrow the scope of the article by importing a more restrictive definition from certain technical niches, and I don't think that's necessarily a good idea. Substituting that for a broader definition has a potential to cause unnecessary confusion and mislead people. Maybe it would be helpful to try to write a short summary of the question and ask someplace like WT:WPM. –jacobolus (t) 02:09, 27 July 2025 (UTC)[reply]

"Extraneous" does not mean uncommon. I'm saying is is not the subject of the article. Both of these are valid and common uses of the word. But their definitions are distinct. They are seperate things.

If you're asserting that that "a symbolic representation of a mathematical object" is not a valid definition, and the correct definition of the word is synonymous with Mathematical notation then this article should be deleted and redirect there per WP:NOTDICT. – Farkle Griffen (talk) 02:19, 27 July 2025 (UTC)[reply]

Wikipedia is treated as an authoritative source. If you make an article that says something like "In mathematics, an A is an XYZ", many readers are going to take this as the definition of the word. If the same term is commonly used with multiple related but slightly different meanings, then that implication is going to be quite misleading, and the distinctions should instead be clearly and explicitly described. In many cases, covering both such topics/concepts in the same article works fine, especially if there isn't an overwhelming amount to say about them. In other cases, it makes sense to split them. In yet other cases, both turn out to be subtopics of some overarching third topic and there is little enough to say about all three that they can be effectively combined. Sometimes it makes the most sense to make a disambiguation page. Sometimes it even makes sense to create an article somewhere in between a disambiguation page and a content page, as in the example of Range of a function. –jacobolus (t) 03:09, 27 July 2025 (UTC)[reply]

That doesn't change the point. The way I see it, there are two options:

(a) Assert the only correct definition is the broader sense, delete the article and redirect to Mathematical notation.

(b) Use the "restricted definition".

Though, to be honest, I would assert that as "the definition in mathematics" because I don't think the broader sense is meant as a mathematical term, (see my reply at the bottom of the thread), whereas the "restricted" one is. – Farkle Griffen (talk) 03:21, 27 July 2025 (UTC)[reply]

In reply to "Sometimes it even makes sense to create an article somewhere in between a disambiguation page and a content page, as in the example of Range of a function."

I'm not sure that is a correct way to go. Is there a Good- or Featured-rated article like that? In general, it's probably not a good idea to use ~~random~~ articles not formally reviewed as examples. – Farkle Griffen (talk) 03:38, 27 July 2025 (UTC)[reply]

In case (b) above, it could make sense to just update Expression with a section on mathematics:

Mathematical expression: A symbolic representation of a mathematical object.

Mathematical expression: Any sequence of mathematical notation.

Or something like that. – Farkle Griffen (talk) 03:43, 27 July 2025 (UTC)[reply]

Are there multi-topic articles which are good or featured pages? Probably. Are there semi-disambiguation overview pages which are good or featured? No idea but probably not many.

But not every pages should aspire to be a good or featured article. Some pages are destined to be permanent semi-stubs, and that's fine. The goal should be to accurately help readers answer their questions / learn about topics they are interested in, not to win internet badges.

However, I don't agree with you that either sense of "expression" here is synonymous with "mathematical notation".

Anyway, you might find Wikipedia:Disambiguation § Broad-concept articles useful. –jacobolus (t) 04:18, 27 July 2025 (UTC)[reply]

If it's not synonymous with "Mathematical notation" then you may have a different definition still from the broader definition, possibly violating WP:Verifiability. – Farkle Griffen (talk) 04:38, 27 July 2025 (UTC)[reply]

Again seperate from my reply above:

IMO the point of official GA/FA articles is to say "This article has been thoroughly vetted by the community and should serve as a model for related articles" not to "win internet badges".

I agree some pages will happen to be stubs forever, but I don't agree that they are stubs by necessity, and any article on an encyclopedic topic is capable of being a GA given a dedicated editor. – Farkle Griffen (talk) 04:41, 27 July 2025 (UTC)[reply]

WP:Broad-concept article seems to be about about concepts that are.. well... broad. Like History of France or Philosophy. Not articles about "words with multiple meanings" – Farkle Griffen (talk) 04:50, 27 July 2025 (UTC)[reply]

No, it's specifically about this situation, of a word that has closely related meanings / slight variants of meaning that should be discussed together in common, and aren't really effectively disambiguated with a bare list. –jacobolus (t) 04:54, 27 July 2025 (UTC)[reply]

I also don't agree that our current description at formula, "an equation or inequality relating one mathematical expression to another", is reflective of the way people understand that word, except perhaps in narrow specialized contexts. I vote with the sources using a term such as "statement" or "proposition" to go with that meaning. Calling any equation or inequality whatsoever a "formula" does not at all match my experience of how that word is used in practice. (Frankly, I don't think "formula", in common usage, is a very well defined concept.) –jacobolus (t) 01:19, 27 July 2025 (UTC)[reply]

@Jacobolus, for what it's worth, though I can't prove it, I believe the use of "expression" in the more general sense is meant grammatically/linguistically, synonymous with "phrase" and not as a mathematical term.

For example: "The mathematical expression/phrase "x+y=2" ... " and analogously "The English expression/phrase "The sum of x and y is two." ..." are the same use of "expression", neither of which is necessarily mathematical.

I think this is where the confusion comes from. – Farkle Griffen (talk) 02:57, 27 July 2025 (UTC)[reply]

I don't think this seems right. The usage I am talking about is definitely referring to chunks of mathematical notation per se. There are even about 2000 results in Google scholar for the phrase "equation is an expression" and another 400 for the phrase "equation is a mathematical expression", with examples like "An equation is a mathematical expression that contains an equals symbol" or "A differential equation is a mathematical expression that contains the derivatives of an unknown function" (some of the search results are in the sense you have in mind, of "the X equation is a mathematical expression of Y phenomenon" or whatever). –jacobolus (t) 03:21, 27 July 2025 (UTC)[reply]

That's still consistent with the linguistic sense, it's just that instead of chunks of latin letters in English grammar to get an "English expression", you are using chunks of mathematical notation in formal grammar to get a "mathematical expression". As in "a thing expressing information". – Farkle Griffen (talk) 03:28, 27 July 2025 (UTC)[reply]

Every sense of this word is more or less consistent with the linguistic sense. The sense you want to focus on is roughly a synonym for "phrase"; some sources even make this explicit, as in a book I found aimed at a (high school?) student audience which said something like "an expression is a mathematical phrase; an equation is a mathematical sentence." –jacobolus (t) 04:51, 27 July 2025 (UTC)[reply]

Agreed? I don't understand what the point of this reply is. – Farkle Griffen (talk) 04:58, 27 July 2025 (UTC)[reply]

Well, the one I want to focus on is analogous to noun phrase, to be specific, since, otherwise, it should just redirect to Mathematical notation. – Farkle Griffen (talk) 04:58, 27 July 2025 (UTC)[reply]

Logical expressions

This article seems to me to be rather complicated. I am not sure about all the recent changes, but the removal of the section on logical expressions has made it a bit simpler and that is good. There are other articles about them:- Boolean expression, Well-formed formula and the disambiguation page Logical expression. There is also Category:Logical expressions which contains several articles (including this one) which do not mention the phrase, but not Boolean expression which does. This adds to the confusion. Also the disambiguation page Expression lists this article twice, under "Linguistics" and under "Symbolic expression". JonH (talk) 16:23, 27 July 2025 (UTC)[reply]

[1] Eberhard Bergmann and Helga Noll (1977). Mathematische Logik mit Informatik-Anwendungen. Heidelberger Taschenbücher, Sammlung Informatik. Vol. 187. Heidelberg: Springer.

[2] Hans Hermes (1972). Einführung in die mathematische Logik. Mathematische Leitfäden (4th ed.). Stuttgart: B.G. Teubner. ISBN 3-519-12201-4.

[3] Hans Hermes (1973). Introduction to Mathematical Logic. Hochschultext (Springer-Verlag). London: Springer. ISBN 3540058192. ISSN 1431-4657.

[1]

[2]

[3]