A theorem is a mathematical statement that has been proved — shown, step by careful step, to follow from things already accepted as true. The proof is what makes it a theorem.

A theorem定理 is a mathematical statement是一個數學敘述 that has been proved已經被證明——shown, step by careful step經過一步一步仔細地驗證 to follow from things already accepted as true是從已經被接受為真的東西一步步推導出來的。The proof證明 is what makes it a theorem才使它成為定理。

Before the proof, the same statement is only a 'conjecture' (a good guess); after the proof, it is settled forever, and no later discovery can undo it. That is very different from a scientific theory, which fits the evidence we have so far and can be overturned.

Before the proof在證明之前 the same statement同一個敘述 is only a 'conjecture'只是「猜想」 (a good guess)（一個不錯的推測）; after the proof證明之後 it is settled forever它就永遠定案, and no later discovery之後任何新發現 can undo it都無法推翻它。That那 is very different from a scientific theory跟科學上的「理論」很不一樣, which fits the evidence we have so far理論只是符合我們目前所擁有的證據 and can be overturned是可以被推翻的。

Famous ones carry their prover's name: Pythagoras' theorem, Fermat's Last Theorem. The word lives in mathematics, logic, physics and computer science, and it is quite formal.

Famous ones有名的定理 carry their prover's name會用證明者的名字命名: Pythagoras' theorem畢氏定理, Fermat's Last Theorem費馬最後定理。The word這個詞 lives in mathematics, logic, physics and computer science用在數學、邏輯、物理和電腦科學裡, and it is quite formal語氣相當正式。

Outside those subjects people sometimes reach for it playfully to mean 'a rule I am confident about' ('my theorem is that the shortest queue always slows down') — that is a joke borrowing the authority of mathematics, and it works because everyone knows a real theorem is airtight.

Outside those subjects在這些學科之外 people sometimes reach for it playfully人們有時候會半開玩笑地用這個詞 to mean 'a rule I am confident about'來表示「我很有把握的一條規則」 ('my theorem is that the shortest queue always slows down')（像是「我的定理是，最短的那條隊伍一定會變慢」） — that is a joke borrowing the authority of mathematics這是在借用數學的權威開玩笑, and it works因此才有效 because everyone knows因為大家都知道 a real theorem is airtight真正的定理是無懈可擊的。

Countable: a theorem, several theorems. English also has a rare verb 'theorem', meaning to state or treat something as a theorem, but you will almost never meet it, and using the noun is always the safer choice.

Countable可數: a theorem一個定理, several theorems好幾個定理。English英文裡 also has a rare verb 'theorem'也有一個罕見的動詞theorem, meaning to state or treat something as a theorem意思是把某件事陳述或當作定理, but you will almost never meet it但你幾乎不會遇到它, and using the noun is always the safer choice用名詞永遠是比較保險的選擇。