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.
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!
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.
Lemma: A wheel.
Proposition: A gear-box.
Theorem: A car.
Corollaries: Photos of the car at interesting destinations.
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)
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.
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.
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.
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.
- 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.
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.
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.
From a criminal movie:
Proposition (starting point in math problem and in the crime investigation) that the Mr. X murdered Ms. Y will be turned into theorem (proven fact) when the police collects enough evidence (math: proof). A lemma (part of proof) connects him with the murder weapon by matching fingerprints. The corollary (additional statement stemming from the theorem) is that he will be sent to prison by the judge.
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.
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
the documents you may decide to sign for your wedding in order to organize your marriage (that may be a theorem or a proposition)