What could be good non-mathematical analogy/analogies to explain the difference among the words - theorem, proposition, lemma and corollaries to high school students? I am looking for analogies that would explain their meaning and their usage by mathematicians.
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2$\begingroup$ Once they have the gist, you could try giving them a theorem and asking them to supply the other pieces; I'm thinking something like, "Theorem: Spending more than three hours on facebook every day is a bad idea for a high school student." (Or something else that resonates with your students.) There are many ways of structuring such an argument (and different corollaries) so students could share out their approaches (maybe developed in groups of two or three?) and see whether their presentations could be refined. $\endgroup$– Benjamin DickmanNov 17, 2015 at 22:33
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7$\begingroup$ Surprising that so many answers assume proposition to be another word for conjecture. I don't believe that is standard mathematical usage. $\endgroup$– Ari BrodskyNov 18, 2015 at 15:22
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6$\begingroup$ I agree with @AriBrodsky. A proposition is like a theorem except that the author considers it less important than the things called theorems. In particular, propositions come with proofs (unless they're obviously true). $\endgroup$– Andreas BlassNov 18, 2015 at 19:47
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2$\begingroup$ Why do you want to explain these words to high school students? They all essentially the same thing the only real difference being perceived importance and proof complexity. Both of which are somewhat subjective as can be seen due to the fact that there have been lemmas that have become named and might be much more important then the theorem they were originally created to prove. More over you can have theorems in one book/field which are corollaries in another book/field due to different approaches taken. $\endgroup$– DRFNov 18, 2015 at 21:46
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3$\begingroup$ I have never seen anyone make a clear and useful distinction between proposition and lemma. $\endgroup$– Chris GodsilNov 19, 2015 at 3:32
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12 Answers
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Origami has things like lemmas, theorems and corollaries.
Definition/axiom: The basic folds such as book, mountain, valley etc
Lemma: a folding procedure that is used as part of another, such as the bird base or the water bomb base.
Theorem: a folding procedure that produces something you wanted, such as the traditional crane.
Corollary: a folding procedure that follows on from something you made, such as modifying the traditional crane by expanding the tail.
Proposition: To me this is just the name of the statement you are trying to prove before you've proven it (regardless if whether it's a lemma, a theorem or a corollary), so in that sense it's like the photo of the finished product at the top before the instructions start.
Some use it to mean a result that isn't necessarily in the sequence of things you're trying to prove. So perhaps this would be in the category of "other things you can make from the bird base".
The advantage of origami is that you can bring physical models to illustrate your point!
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Lemma: A wheel.
Proposition: A gear-box.
Theorem: A car.
Corollaries: Photos of the car at interesting destinations.
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From cooking:
Theorem: recipe, something that produces a desired result
Lemma: a technique (e.g. creating a roux, proofing dough, preparing a mirepoix, etc.), a procedure that one uses in many different recipes, but might not produce something interesting/tasty by itself
Corollary: (really stretching here) extra garnish/styling to slightly modify the dish
I'm not going to try for proposition.
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$\begingroup$ A proposition would be a food, and you're wondering what ingredients you need to make it. (A tautology is a food you can spawn from nothing but the power of your mind.) $\endgroup$– PyRulezNov 18, 2015 at 15:14
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2$\begingroup$ I think that cooking is good but your analogies... not to my taste. You could think of a theorem as a meal yes, but a lemma would be something used in a lot of meals but not so big on its own. A corollary would be something that could be easily made from leftovers.... your lemma is somewhat OK but your corollary has gone off alright. $\endgroup$ Nov 18, 2015 at 22:08
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$\begingroup$ A corollary is more of a side-effect, like discovering that raw cookie dough can be added to ice-cream. $\endgroup$– jwgNov 19, 2015 at 13:13
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From Computer Science:
Lemma: Built-In Function (highly technical, but needed)
Proposition: Sub-routine (a good tool to have)
Theorem: Main Program (the main project you're building)
Corollary: A smaller program that calls the Main Program as a sub-routine (the cute things you can use your final project to do)
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2$\begingroup$ i like this analogy a lot but i am not sure if it is accessible to most high school students.. $\endgroup$– celerikoNov 17, 2015 at 17:03
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1$\begingroup$ It's also kind of language-dependent. In C, you might have weird compiler built-ins like
__builtin_clz, but then in other languages, the built-ins might consist mostly of things likesum. Even with someone with programming experience, I don't think these analogies would improve the explanation. $\endgroup$ Nov 17, 2015 at 18:43 -
2$\begingroup$ I'd characterize a "builtin function" more as an "axiom"... $\endgroup$– vonbrandNov 18, 2015 at 1:41
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The distinction is (much more than) somewhat arbitrary. Some people call everything "proposition" (or "result", or even something else). The idea is that in "orderly development" your goal is to prove a theorem. Most of the time this can't be done from cold start, you'd need some auxiliary results, not important in themselves, to get to the goal, lemmata. It often happens that once you have your goal (theorem), you can use it in a simple step/application to derive another important result (or use some intermediate step in the proof as easy starting point), giving rise to a corollary. Then you have conjectures, which are essentially educated guesses. And there are important theorems that are called something else, like the "division algorithm" in elementary number theory.
This is the intent of the author, what happens afterwards is anybody's guess. It is said that the mathematician's wet dream is to become known by a lemma, not a theorem (often lemmas turned out to be much more important than the long forgotten theorems they were stepping stones for, like the various Gauss' lemmata, or Burnside's lemma). And some famous conjectures have their own names, like the Riemann Hypothesis, or the all-time favorite "Last (or great) theorem of Fermat" (which was proved recently).
Trying to find "everyday parallels" is futile, this are technical terms with precise meaning in mathematics (or mathematics writing). You might as well ask for "everyday equivalents" for the distinctions between, say, carving tools.
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Theorem: A business deal that has been signed, sealed, and delivered.
Lemma: A product listing in a catalog, with a known price and availability. A businessman can include this product as part of a larger deal.
Proposition: A proposal. It might or might not work out.
Corollary: An easy follow-up deal, based on a big deal.
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The existence of a Lost City of Atlantis is a proposition. It has not been proven or dis-proven to exist. But if one day, someone discovers a route to get there, we would know it exists. When this happens, its existence becomes a theorem. The route that we take to get there is the proof. Along the way we also discover Davy Jones' Locker. So the existence of said locker is a lemma. When we explore the lost city, we discover it was built by aliens. So the fact that it was built by aliens is a corollary.
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5$\begingroup$ I think you have mis-understood the meaning of 'proposition' in this context. $\endgroup$ Nov 18, 2015 at 15:08
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1$\begingroup$ How so? Here's the definition from Wolfram. I think the analogy fits. $\endgroup$ Nov 18, 2015 at 15:17
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2$\begingroup$ That's certainly the correct definition for "propositional calculus", in which false propositions are very much part of the domain of discourse. On the other hand if you're writing out a proof and labelling things "theorem, lemma, corollary", you're probably not going to write down some false statements and label them "Proposition 26" or whatever ;-) So whether we need to account for false propositions in the analogy probably depends what the analogy is for. $\endgroup$ Nov 19, 2015 at 14:40
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2$\begingroup$ @Steve, that's a very good point. I think it is part of the reason why people have this perception that a proposition is stronger than a conjecture. Most people's experience with the word proposition has been from a math textbook where propositions are always true and always lead to a theorem. But I regard a proposition as weaker than a conjecture. For example Poincare's conjecture was generally thought to be true, but without proof. It wasn't called Poincare's proposition; although propositions are supersets of conjectures. $\endgroup$ Nov 19, 2015 at 15:06
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1$\begingroup$ @uclatommy: A proposition is not always weaker than a conjecture. For example, how would this sound? “Four score and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty, and dedicated to the conjecture that all men are created equal.” $\endgroup$ Dec 7, 2015 at 17:22
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The first thing to bear in mind is that a lemma is a theorem, and usually so is a proposition (although in certain uses in English it might make sense to talk about an "incorrect proposition", which then naturally isn't a theorem, rather it's a failed conjecture). Everything proved is a theorem. But we only give a theorem a name like "Pythagoras' Theorem" if we think it's really important.
Assuming every thing is proved or will be proved by the end of the exercise, though, then:
- A theorem is something you are constructing.
- Propositions and lemmas are components that themselves require construction before incorporating them into the main thing. The difference between the two is usually not important, but the more impressive your proposition is the more likely you are to call it a lemma.
- Corollaries are things that your theorem can be incorporated into the construction of.
quid's answer gives a specific example of this pattern with "car" as the theorem, but you should be able to apply the pattern to any construction process that's complex enough to have recognisable component parts. In particular, one might wish to reflect the fact that some theorems use their lemmas more than once in their proof, by using duplicated parts (wheels for cars, arches for cathedrals, seats for movie theatres, whatever) for the lemma in the analogy.
One of the interesting things about lemmas is that you don't always write your proof such that the lemma is proved before it is used (hence the name, I believe). You can leave the proof until the end if it make the presentation clearer. It's probably useful to draw attention to this fact in the analogy: one might design and even part-construct a car knowing that it's going to have wheels with a certain specification, but actually leave off designing and constructing the wheels until long after the point where we've staked our reputation on them appearing before the end. One cannot complete the car without putting actual wheels in: what we have before that is an incomplete car with missing parts.
Often a conjecture is something that we hoped to use as a lemma, but find we cannot prove even though we've done a lot of work proving that if the conjecture is true, then some useful theorem follows. By (facetious) analogy there might be a fantastic new underwater car that we'll build just as soon as we can figure out how to make waterproof windows. Then these waterproof windows, that nobody knows how to make or if it's even possible, are a conjecture and the underwater car is a consequence (or corollary) of the conjecture being true.
We often use the word "corollary" to indicate a theorem that's very easily proved from our main theorem. "Corollary 1 of Theorem 12", "Corollary 2 of Theorem 12", etc. We're using the term to help organise our thoughts. We also sometimes use it for something more involved: "Theorem 93 is a corollary of Theorem 12, the proof follows ... [12 pages later] QED". What we really mean then, is that Theorem 12 can be used to prove Theorem 93. So in the analogy it would probably be useful to offer more than one corollary, at different "distance" from the theorem.
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- A proof is a path of stepping stones over a creek.
- Every stone is a proposition.
- Whenever you get to a milestone, you get to a lemma.
- The stone right after a milestone is a corollary.
- When you get to the other side, you get to the theorem.
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The first thing that came to mind was food, let's take Cheerios for example (dry breakfast cereal for those not aware).
Theorem - regular old Cheerios
Corollary - Honey Nut Cheerios, Multi-grain Cheerios, Banana Nut Cheerios (any offshoot flavor of cheerios)
Lemma - Oat flour, salt, sugar, flavors (the ingredients in Cheerios)
Proposition - An unreleased, being-tested-by-General-Mills flavor of Cheerios
I know that this isn't perfect but i think it is very accessible for high school students and is appropriate for just being introduced to the concepts. Later on, the definitions can be refined.
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$\begingroup$ I would only suggest that a Proposition is: "definitely true but we don't want to claim it as a proved theorem now." So what kind of cereal is that? Not new Cheerios, but more likely old Cheerios... $\endgroup$ Aug 12, 2018 at 0:53
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If I had to explain common usage to a layperson then I might use an analogy with travelling to beautiful places, as follows.
A theorem is like a beautiful place that you would highlight to anyone exploring the area. Lemmas and propositions are signposts and guides that help one to efficiently reach the beautiful places and better understand their importance. Corollaries are other notewothy things that are easily visible once one reaches the beautiful destination.
Unlike some other analogies, this has the benefit of explicitly highlighting the key role that beauty plays in the judgement of such matters. Of course the judgement of beauty, importance, etc is highly subjective and not everyone may agree on just which results should be elevated to theorems. For example, the eminent algebraist Irving Kaplansky equates them all in his beautiful influential textbook Commutive Rings, with the following explanation:
In the style of Landau, or Hardy and Wright, I have presented the material as an unbroken series of theorems. I prefer this to the n-place decimal system favored by some authors, and I have also grown tired of seeing a barrage of lemmas, propositions, corollaries, and scholia (whatever they are). I admit that this way the lowliest lemma gets elevated to the same eminence as the most awesome theorem. Also, the number of theorems becomes impressive, so impressive that I felt the need to add an index of theorems.
answered Aug 11, 2018 at 15:00
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A love comparative is on the next TED video: https://www.youtube.com/watch?v=jej8qlzlAGw (It's with English subtitles)
The video is a must be seen and the speaker is mathematician who mainly works doing outreach, so it's well explained. I include the explanations.
Theorem:
It's forever
Proposition
If never becomes a theorem, you might divorce, but I think will be forever.
Of my own creation: Lemma
One thing the couple do together
Corollary
the documents you may decide to sign for your wedding in order to organize your marriage (that may be a theorem or a proposition)
