Are propositions and theorems interchangeable in mathematics?

[jostpuur]

What are propositions? When I read lecture notes, it seems that some of the theorems are randomly called propositions instead.

 

[Hurkyl]

The difference between terms like "axiom", "proposition", "theorem", "lemma", and "definition" are pretty much just a matter of presentation.

(Actually, the term "proposition" is also used to refer to any statement, not just 'true' ones)

 

[jostpuur]

I see... but when text goes like this:

Theorem 1.n. blablablablablabla
Proof. blablablablabla $\square$

Proposition 1.(n+1). blablablablablabla
Proof. blablablablabla $\square$

Theorem 1.(n+2). blablablablablabla
Proof. blablablablabla $\square$

It means that some of the theorems are being called propositions randomly? I wonder if the author has been using dice when writing this.

 

[Hurkyl]

It's not uncommon to have multiple 'types' of statements share the same counter. For example, it's a lot easier to find example 1.5 when it lies between theorem 1.4 and lemma 1.6. ;)

As to why to label a statement as being a theorem and another a proposition, I would guess that the author felt that the theorems are more important statements.

 

[CompuChip]

A lemma, theorem and corollary are all statements that must / should be proven, though there is a sort of intuitive distinction between them. IMO there is a tacit agreement that, for example, Theorems are main results, while Lemmas are often intermediate results needed to prove a theorem (though I have seen cases where the lemma was more general, interesting and important than the theorem whose proof it was needed for); Corollaries are usually short statements which follow (almost) immediately from a given theorem (or they are special cases of those). I often see Propositions following a Definition, in which certain properties of a newly defined object are shown. In my feeling, a proposition is the kind of statement you are usually willing to take for granted unless you are really picky.

So though Hurkyl is right that it's mostly just a matter of presentation, there is a sort of hierarchy as to which is more important (and then which name is given to which result is largely arbitrary and left as a choice to the author, though the idea is that the names are chosen such that they actually add a logical structure in much the same way as divding the text into chapters, sections and paragraphs does)

 

[Hurkyl]

(though I have seen cases where the lemma was more general, interesting and important than the theorem whose proof it was needed for);

Zorn's lemma is a famous example. I've heard he was actually quite annoyed that nobody remembers what he used it to prove! (though that's probably just an urban legend)

 

[jostpuur]

I knew what kind of purpose lemmas and corollaries have in terminology, but for propositions I had no clue.

Zorn's lemma is a famous example. I've heard he was actually quite annoyed that nobody remembers what he used it to prove! (though that's probably just an urban legend)

This must be a Zorn's mistake. He underestimated his own theorem!

 

[yasiru89]

Not all propositions are proven true and some are disproven to illustrate or emphasize a point. It all depends on the motivation of the work and should one look closely certain propositions will be theorems in other works as per their importance in deriving the seminal result therein; Indeed here lies an overlooked part of expositions and the considerations need to be in line of definition to proposition to proof(or disproof). Not definition to theorem to proof.