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WORDS IN MATHEMATICS have precise meanings. No surprise, this. And sometimes their etymologies have good tales to tell. Let’s look at “theorem,” and two of its related terms, “corollary” and “lemma.”
Merriam Webster Online is a good starting point, but added precision comes from The Princeton Companion to Mathematics, other mathematical sources, and even Google Translate.
Here are tidbits gleaned from these sources, together with my usual Internet sleuthing.
A Theorem, according to Merriam-Webster, is “1. a formula, proposition, or statement in mathematics or logic deduced or to be deduced from other formulas or propositions. 2. an idea accepted or proposed as a demonstratable truth often as a part of a general theory.” Two other definitions are peripheral to the word’s mathematical use: “stencil” and “a painting produced especially on velvet by use of stencils for each color.”
Utterly peripheral is my own comment that Daughter Suz has Elvis portraits meeting this last definition. At least one has a teardrop; Elvis’s, not hers.
M-W says that theorem’s first known use in English came in 1551, its origin from the Late Latin theorema, from the Greek θεώρημα, theōrēma. The Greek theōrēma shares a root with theōros, spectator; our word “theater” shares its origin from this too.
M-W’s second definition, the one about “demonstratable truth,” probably comes closest to the mathematical one. Its first definition’s muddling of “formula,” “statement,” “mathematics,” “logic,” “deduced,” or “to be deduced” reminds me of Mark Twain’s comment of his wife’s attempt at swearing: “You have the words, dear, but you don’t know the tune.”
In its section “What Do You Find in a Mathematical Paper?,” The Princeton Companion to Mathematics says, “A mathematical statement is established by means of a proof. It is a remarkable feature of mathematics that proofs are possible: that, for example, an argument invented by Euclid over two thousand years ago can still be accepted today and regarded as a completely convincing demonstration.”
“A theorem,” it continues, “is a statement that you regard as intrinsically interesting, a statement that you might think of isolating from the paper and telling other mathematicians about in a seminar, for instance.”
By contrast, “A proposition is a bit like a theorem, but it tends to be slightly ‘boring.’ It may seem odd to want to prove boring results, but they can be important and useful.”
A Corollary, according to Merriam-Webster, is “1. a proposition inferred immediately from a proved proposition with little or no additional proof. 2 a. something that naturally follows. b. something that incidentally or naturally accompanies or parallels.”
M-W’s first meaning coincides with that in The Princeton Companion to Mathematics: “A corollary of a mathematical statement is another statement that follows easily from it. Sometimes the main theorem of a paper is followed by several corollaries, which advertise the strength of the theorem.”
I like M-W’s etymology: “Corollary comes from the Late Latin noun corollarium, which can be translated as ‘a garland given as a reward.’ ” The words are related to the Latin corolla, meaning a “small crown.”
I like the image of a theorem wearing a crown. Indeed, or more than one.
A Lemma, according to Merriam-Webster, has two distinct meanings: First, “1. an auxiliary proposition used in the demonstration of another proposition. 2. the argument or theme of a composition prefixed as a title or introduction. 3. a glossed word or phrase.” Second, and even further from mathematics, “the lower of the two bracts enclosing the flower in the spikelet of grasses.”
M-W says that lemma in the first sense originated in 1570. The word traces to the Greek λεμμά, lēmma, a thing taken, an assumption (in contrast to its later meaning of something requiring proof). The word’s botanical sense arose in 1906.
Of the “auxiliary proposition” concept, The Princeton Companion to Mathematics says, “Some lemmas are difficult to prove and are useful in many different contexts, so the most important lemma can be more important than the least important theorems. However, the general rule is that a result will be called a lemma if the main reason for proving it is to use it as a stepping stone toward the proof of other results.”
I like the German word for lemma: Hilfssatz, “helpful sentence.” ds
© Dennis Simanaitis, SimanaitisSays.com, 2019
I took math all thru high school, and college math thru integral calculus and I never encountered the term “lemma” (this was in the late ‘40s thru mid-50s). Is this a relatively recent term, or is my memory finally giving out on me?
The term has been around for years, but not particularly outside formal mathematics, unlike the terms “theorem” and “corollary.”