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I'm teaching trigonometric identities next unit, and realized that I don't understand the relationship between a handful of mathematical terms.

  • Properties. For example, the commutative property that $a + b = b + a$.
  • Theorems. For example, the Pythagorean theorem that $a^2 + b^2 = c^2$.
  • Laws. For example, the Law of Cosines that $c^2 = a^2 + b^2 - 2ab \cosθ$
  • Identities. For example, the Pythagorean identity that $\sin^2x + \cos^2x = 1$.

Oh, and then there's the "Multiplicative Identity Property" that $1*a=1$

My sense is that identities are mathematical equations that are always true. Which, I think means the property, law, and theorem listed here are all also identities? I believe properties are statements about mathematics that are foundational, while theorems are proved from properties, but am not sure how laws and identities fit in with either of them.

So, in general, how do mathematicians decide whether something is a property, law, theorem, or identity? Or, are they somewhat arbitrarily decided? For example, could we have called in the Cosine Theorem or Cosine Identity instead of the Law of Cosines?

Xander Henderson
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Becky Poore
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    Theorems are more general; the Pythagorean theorem asserts an identity, other theorems can assert many other things. The distinction between "law" and "identity" is informal and, I would guess, mostly historical. Properties are basic facts you use to prove other stuff but this is also somewhat informal and historical. – Qiaochu Yuan Feb 02 '25 at 04:56
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    Identity is a formula with the equal sign where the two expressions to the right and to the left always have the same value when evaluated. – Mauro ALLEGRANZA Feb 02 '25 at 09:59
  • Being pedantic, the identity $a^2 + b^2 = c^2$ is not a theorem. It is an identity which holds in certain cases. The Pythagorean theorem is a theorem about the relationship between the sides of a right triangle. Specifically, if a right triangle has legs of length $a$ and $b$ and hypotenuse of length $c$, then $a^2 + b^2 = c^2$. – Xander Henderson Feb 02 '25 at 16:03
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    I notice all of your examples are equations. You might want to consider mathematical statements that have a more complex structure than just a single equation, involving logical connectives (if-then; and; or; not) and quantifiers (for all; there exists). – Lee Mosher Feb 02 '25 at 16:03
  • Note, also, that you might be parsing the "multiplicative identity property" incorrectly. Parse the phrase as "(multiplicative identity) property". The "multiplicative identity" (this is one noun phrase) is an object $e$ in a set with some kind of multiplication $\star$ such that $e\star a = a = a\star e$ for any other $a$ in that set. That is, the multiplicative identity is an object which has this property. In the real numbers, $1$ is the multiplicative identity (and $0$ is the additive identity). – Xander Henderson Feb 02 '25 at 16:08
  • Related: https://math.stackexchange.com/q/2925898/ , and https://math.stackexchange.com/q/640843/ . – Xander Henderson Feb 02 '25 at 16:10

2 Answers2

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  • Theorems. For example, the Pythagorean theorem that $a^2 + b^2 = c^2$.

A theorem is strictly speaking just a standalone true statement (even a triviality), but conventionally refers to useful/noteworthy results (as opposed to definitions— from which they are derived). As such, notice that the Pythagorean theorem is $\color\red{\text{not}}$

  • $a^2 + b^2 = c^2,\tag1$

but something like

  • Given a right triangle with hypotenuse of length $c$ and remaining sides of lengths $a$ and $b,$ we have that $a^2 + b^2 = c^2.$

If we comb finer, we can distinguish between lemmas, propositions, theorems and corollaries, with the classification and choice of labels depending on the context and taste.

Law and property are really just synonyms for theorem, except that they are frequently presented in a list with overarching conditions/restrictions, which are frequently left tacit when the "law" is subsequently quoted or referenced. For instance, the full statement of the "law" $$(ab)^x=a^xb^x$$ contains attendant restrictions (e.g., "for positive bases and a real exponent" or "for nonzero bases and an integral exponent"), and disregarding these conditions is a common source of mistakes, akin to disregarding equation $(1)$'s restrictions on $a,b$ and $c.$

  • Identities. For example, the Pythagorean identity that $\sin^2x + \cos^2x = 1$.

Generally, an identity is an equation that holds for every possible set of values in its domain. For example, $$(a+b)^2=a^2+b^2+2ab$$ is an identity—we might emphasise its universality by writing $(a+b)^2\equiv a^2+b^2+2ab,$ which means "for every complex number $a$ and $b,$ it holds that $(a+b)^2=a^2+b^2+2ab$"— as is $$\log(xy)= \log(x)+\log(y),$$ which is true for all values that don't "break" the equation (i.e., such that the entire equation is actually defined).

As such, the phrase "Pythagorean identity" brings to my mind $$\sin^2(θ)+\cos^2(θ)=1$$ $\color\red{\text{rather than}}$ the conditional equation $$a^2 + b^2 = c^2,\tag1$$ which pertains specifically to right triangles. (Equation $(1)$ can be considered an identity only in the context of the Pythagorean theorem.)

I believe properties are statements about mathematics that are foundational, while theorems are proved from properties

The former is indeed typically the case: "properties"/"laws" tend to be presented as front matter, sometimes misleadingly like axioms or definitions. But make no mistake: they themselves are indeed theorems, not postulates, even when the author obscures this. (To hazard a goosy example: "mathematical law" is to "definition" as "legal law" is to "constitution".)

ryang
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I would consider a (mathematical) law a statement of truth, in which it has been derived from theorems and/or a set of identities. Usually, properties, laws and identities require explanation, and sometimes a rigorous proof, of how they're derived for their audience. For example, someone in the mathematical community may ask, "how is [insert law, property or theorem] derived?" and someone can use a set of mathematical terms to do so.

Also, there are postulates, which are considered to be mathematical statements that are accepted as fact and are most likely not proven (like theorems, laws and identities). Postulates are common in geometry.

bjcolby15
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