Definition for Mathematical Formula

Question

Imagine that you were writing an elementary book, for example for high school learners, and at the beginning you had a glossary where you wanted to write the definitions for common mathematical words (e.g. numerical expression, algebraic expression, variable, equation, etc.) to help the reader make sense of those words when they appeared throughout the text. In this setting, what would you write to explain/define what a mathematical formula is?

Imagine that you wanted to put an entry for mathematical formula in a table like in these screenshots, taken from the book Algebra and Trigonometry: Structure and Method, Book 2 which you can read here, if you have create an account.

I would like to avoid circular definitions.

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Some context

The term formula seems to raise confusion with some frequency and this can be seen by the number of Math Stackexchange questions that appeared over the years:

1. what-is-the-difference-between-equation-and-formula
2. what-is-the-difference-between-a-function-and-a-formula
3. what-is-the-correct-meaning-of-term-formula
4. in-the-context-of-mathematics-what-is-the-difference-between-equation-and-formula
5. what-is-the-difference-between-formula-and-function
6. difference-between-formula-and-algorithm

After some investigation it seems that people use the term mathematical formula to mean (mainly) two things:

1. Any mathematical sentence that uses symbols (e.g. equations, inequations/inequalities, identities). Which makes equations a kind of formula.
2. A rule for expressing a subject in terms of some other variables (as stated in this comment). Or a procedure to compute some value of interest. Usually written as two expressions connected by an equals sign. Which makes formulas a kind of equation.

Mathworld seems to define a formula as the first. Many books use the term formula for the rules used to compute areas and volumes, and there is also the quadratic formula which provides the solution(s) to a quadratic equation. These are instances of the second definition. Less frequently, I've also seen people call formula to any string of mathematical symbols, making an expression a kind of formula. Another example is the book Introduction to Algebra that defines formula as a synonym for expression in page 23 (according to this comment).

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EDIT After reading some of the comments I feel the need reword the first part of my original question to:

Imagine that you were writing a book that revisits elementary mathematics subjects/areas but from an advanced standpoint, for example for advanced high school learners or people preparing to an undergraduate mathematics heavy course. One example of such book might be Serge Lang's Basic Mathematics.

Now consider that in the beginning of this book you had a glossary where you wanted to write the definitions for common mathematical words (e.g. numerical expression, algebraic expression, variable, equation, etc.) to help the reader make sense of those words when they appeared throughout the text.

In this setting, what would you write to explain/define what a mathematical formula is?

I personally think that the answers given up to the point in time of this edit fit both the original question and this reword, but would prefer if new answers take this revision into consideration.

EDIT 2 I accepted @ryang's answer because it has all the information requested by the Question, but also found Charles Wells' work, which helped me interiorize stuff:

• Website
• A report on mathematical discourse, which has an entry for 'formula'.

Answer 1

In formal logic, ‘formula’ has a standard and precise definition. Roughly speaking, take a constant and/or variable, apply a non-logical operation (e.g., addition) if desired, repeat if desired, this gives a ‘term’ (notice the deviation from its common meaning in mathematics); relate two terms using the equality relation to obtain an ‘atomic formula’; apply a logical operation or quantifier to atomic formulae finitely many (including zero) times to obtain formulae; a formula is then called a sentence/proposition precisely when it has no free variable.

But formal logic is tangential to your question.

1. Any mathematical sentence that uses symbols (e.g., equations, inequations/inequalities, identities).

   Which makes equations a kind of formula.

What's missing from this characterisation is some mention that a formula expresses a rule or required relationship or somesuch.

After all, the equation $3x^2+7x=8$ is typically decidedly not a formula.

2. as stated in this comment:

   Perhaps: A formula is a rule for expressing a subject in terms of some other variable(s); for example, $V=πr^2h$ expresses a cylinder's volume in terms of its radius and height.

   Which makes formulas a kind of equation.

The keyword here is rule, which is rather more specific than “equation”, and switching it out alters the meaning.

And certainly, $V=πr^2h$ is not an identity, as triples like $(1,1,1)$ do not satisfy it. The example says that for each cylinder with volume, radius and height $V,r,h,$ respectively, the attributes obey the relationship $V=πr^2h.$

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Addendum

Would you say that there exist formulas that contain an inequality symbol?

I'd typically call such an object an ‘inequality constraint’ rather than a ‘formula’.

In the language of first-order arithmetic on the other hand, $x<y$ is an ‘open formula’ [side note: it can be rewritten as $\exists r\: (x'+r=y)].$

Maybe don't try to conflate the formal-logic and informal usages of ‘formula’.

For example, in Spivak's Calculus, 4th Edition, page 33 he writes: "When does equality hold in the formula $G_n \le A_n$?"

It is apparent from the context that Spivak is using the term 'formula’ in the formal-logic sense. (But since this is an Analysis, not Logic, text, I’d probably rewrite the question as “When does equality hold in the sentence/statement/assertion $G_n \le A_n$?”)

There's similarly no ambiguity in my various usages (in the third line and the previous sentence) of ‘term’: (1) in formal logic, $7y+8$ is a term (2) in mathematics, the expression $7y+8$ is not a term (3) the word ‘term' is a term.

In mathematics, the word ‘formula’ is not part of the theory, so does not have a rigorous (or at least widespread rigorously consistent/matching) definition. Possibly, Spivak's text contains a definition of ‘formula’ in the preface or glossary, and the quoted usage above goes with it?

At the end of the day, don't forget about the context; I mean, have you ever found the word ‘formula’ ambiguous in spite of its surrounding context?