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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.


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).


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:

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    $\begingroup$ To me, the 2nd definition you give feels like it's headed in more of the right direction. I'll be interested to see what others think. $\endgroup$
    – Sue VanHattum
    Commented Jun 28, 2022 at 17:39
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    $\begingroup$ (1 of 2) I doubt the importance of a precise definition for this particular term. I'm thinking about my own high school math experience and I don't remember the different usages of that term (as you've shown) ever troubling me. Didn't help me crunch the quadratic equation. Or help me figure out what subtracting a negative number means. $\endgroup$
    – guest
    Commented Jun 28, 2022 at 22:15
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    $\begingroup$ (2 of 2) By the way, "high school math" is awfully imprecise. What subject, grade, track does your book cover? P.s. See Feynman's criticism of new math definition obsessions: fs.blog/richard-feynman-teaching-math-kids In all seriousness, I would write the book, then the glossary, then check your own usage. Not play Bourbaki with definitions-first approach. (But I'm not too worried about the impact on kids. Very few book projects like this get finished, and if finished sold/implemented.) $\endgroup$
    – guest
    Commented Jun 28, 2022 at 22:17
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    $\begingroup$ I agree with “guest”’s comments. This question is not something that troubles students in a practical sense. An example of something that is important that troubles students is correctly rearranging equations, for example in order to make a variable the subject of a formula/equation. If one of my students where to ask me what a formula is, then $(2)$ would be my default answer: it’s an equation with $y=f(x)$ or $u=f(t),$ etc. $\endgroup$ Commented Jun 28, 2022 at 22:49
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    $\begingroup$ With all due respect, it does bother me, or I wouldn't be asking it. And it seems to bother other people, as many have asked before. I understand your point and it is true that it can be considered a pedantic question. However, your comments are not really adding anything to the table, as I am a student with a doubt and after reading your comments the doubt persists. $\endgroup$
    – fire-bee
    Commented Jun 28, 2022 at 23:34

1 Answer 1

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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.

  1. 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 $V=πr^2h.$


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?

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    $\begingroup$ As my answers to your previous two comments are all in the above addendum, I don't wish to have a protracted chat reiterating the same. $\quad$ But to elaborate: (1) when writing mathematics and one wishes to use formal-logic terminology calling $z \leq x + y$ a 'formula', then to be consistent, shouldn't one also call $7y+8$ a 'term'? (2) I would call the triangle inequality a 'formulation' rather than a 'formula', but— insert here the final paragraph of the addendum —I wouldn't find it jarring if an author calls it a 'formula'. $\endgroup$
    – ryang
    Commented Jun 29, 2022 at 13:23
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    $\begingroup$ Please understand that because the terminology under discussion is not within some theory (since we aren't discussing formal logic), your question belongs just as much to English-language SE as it does to mathematics SE. While the ontology is important, so is context! $\endgroup$
    – ryang
    Commented Jun 29, 2022 at 13:33
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    $\begingroup$ Yep. Remember that when you initially misquoted me when creating this post, I'd privately pointed out that the word "equation" does not appear in my proposed explanation / informal definition? My elaboration (in the form of the above Answer) still doesn't make such assertions, and I'm careful to use words like "typically". Have you seen anyone call the triangle inequality a formula? (No.) Is it invalid to call the triangle inequality a formula? (No, because no axiom, law, widespread standard, or definition, for that matter, is really being violated either.) I made several references $\endgroup$
    – ryang
    Commented Jun 29, 2022 at 14:24
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    $\begingroup$ I guess that you are right, people seem to use both formal logic definition and the "rule/law" definition, but don't feel the need to differentiate between the two. $\endgroup$
    – fire-bee
    Commented Jun 29, 2022 at 14:56
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    $\begingroup$ Your confusion stems from falsely attributing rigour to an informal explanation (MathWorld's), and then falsely attributing formality to Spivak & Tao's informal usages of the word (which has no problem because there is no ambiguity, even as there may be slight inconsistency if they selectively call $x−y=x+(−y)$ a formula without calling $x-y$ a term). I did point out from the outset that the formal-logic perspective is tangential to your Question, and subsequently reminded not to conflate the two perspectives. $\endgroup$
    – ryang
    Commented Jun 30, 2022 at 6:18

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