# Theorems and Propositions

1. Sep 23, 2015

### UncertaintyAjay

So, a proposition is like a theorem but less important. How do you decide whether something is important enough to warrant 'theorem' status because the distinction seems very subjective.

2. Sep 23, 2015

### micromass

Staff Emeritus
It is subjective.

3. Sep 23, 2015

### UncertaintyAjay

So what purpose does the distinction serve?

4. Sep 23, 2015

### micromass

Staff Emeritus
To show the reader what the author thinks is important and what is less important.

5. Sep 23, 2015

### SteamKing

Staff Emeritus
There is a careful distinction made between a proposition and a theorem.

A proposition is a statement or declaration which may be true or false, but not both.

http://mathworld.wolfram.com/Proposition.html

An axiom is a proposition which is accepted to be true.

A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments, other theorems, or axioms. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof.

http://mathworld.wolfram.com/Theorem.html

6. Sep 23, 2015

### SteamKing

Staff Emeritus
There is a careful distinction made between a proposition and a theorem.

A proposition is a statement or declaration which may be true or false, but not both.

http://mathworld.wolfram.com/Proposition.html

An axiom is a proposition which is accepted to be true.

A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments, other theorems, or axioms. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof.

http://mathworld.wolfram.com/Theorem.html

Ranking theorems by 'importance', whatever that means, is immaterial.

7. Sep 23, 2015

### micromass

Staff Emeritus
These are definitions used in mathematical logic texts such as propositional logic. These are not the definitions used in the majority of math texts. In those texts, there is no distinction between a proposition and a theorem.

8. Sep 23, 2015

### SteamKing

Staff Emeritus
That's pretty sloppy. Must be some more of the New, New Math.

I'm still not aware that anyone has ever proven a proposition in the fashion that theorems must be proven to be accepted.

9. Sep 23, 2015

### micromass

Staff Emeritus
Take literally any advanced math book out there, and it does it this way.

10. Sep 23, 2015

### SteamKing

Staff Emeritus
If we start doing mathematics on the basis of majority vote, then Heaven help us.

11. Sep 23, 2015

### SteamKing

Staff Emeritus
It's also not been my experience that the Academy is immune to sloppy thinking. It just takes longer to uncover.

12. Sep 23, 2015

### micromass

Staff Emeritus
So all the mathematicians in the world got it wrong and you got it right? That's basically what you're saying, right?

And yes, basic conventions such as this are decided by majority vote. If the majority decides that the terms proposition and theorem should be used in this or that way, then it is better to adhere to that standard. This is not the same as actually doing mathematics.

13. Sep 23, 2015

### SteamKing

Staff Emeritus
Nope. I'm saying that sloppy thinking is all around us. Just because you read something in a book doesn't necessarily make it true.

Conventions come and go. The ones which are useful stay; the inconvenient ones fall by the wayside.

Personally, I don't see where conflating the concept of a proposition with the concept of a theorem is useful. If it's useful to you, that's a different thing.

14. Sep 23, 2015

### micromass

Staff Emeritus
No, reading it in some book doesn't make it true. But if ALL the math books follow these exact conventions, then this is the convention that we need to follow. If you go ahead and send a math paper to a journal where you label everything a theorem, then you will get as comment that you should rename some theorems to propositions.

15. Sep 23, 2015

### SteamKing

Staff Emeritus
I'm not saying that all propositions should be labeled theorems, or vice versa. In fact, my whole discussion has been that there are certain differences between the two concepts. These differences might be subtle or major, but I have yet to see any authority which states flat out that "proposition" is a synonym for "theorem".

It has been your thesis, and my apologies if I am interpreting it incorrectly, that there really isn't a distinction between the two, at least as far as the "majority of math texts" is concerned, as you yourself said.

16. Sep 23, 2015

### micromass

Staff Emeritus
My thesis is that the distinction is subjective. Propositions are for what the author considers less important statements. Theorems are for the important and crucial results. And then there are lemmas, corollaries, conjectures, remarks.

Your definition, is that a proposition is something which may be false. This is simply incorrect outside of mathematical logic texts. Nobody uses the word proposition for something that is false. In all math texts, propositions are something that is proven to be true (and which mathematical logic calls a theorem). So for mathematical logic, every result which has a proof is a theorem. Outside of mathematical logic, this is no longer true.

17. Sep 23, 2015

### phinds

That has certainly been my experience as well. I was quite surprised by your point of view, Steamking, that the term as used in propositional logic should be applied to all of mathematics.

18. Sep 23, 2015

### SteamKing

Staff Emeritus
Yes, I am aware of the taxonomy of logic devices and proofs.

It's not my definition. I believe I cited a source (not Wikipedia) which defined a proposition as a statement which may be true or false.

I don't believe I said that people do use the word in that context, only that the nature of the truth or falsity of a particular statement may not be clear cut.

I don't think one would get anywhere starting with a statement which is false.

In making certain types of argument, one may initially posit that a certain statement is true, and then by a series of subsequent logical deductions obtain an absurd result, which then suggests that the original statement on which this process was based was, in fact, false.

19. Oct 1, 2015

### aikismos

Not to be a bridge-builder or anything during a perfectly didactic dialectic, but I think both definitions of propositions are in currency depending on the context of discourse. Historically and among anyone with any formal logic education, a proposition is indeed a statement which is essentially provable and generally subject to the law of the excluded middle, and certainly may be shown to be false. https://en.wikipedia.org/wiki/Proposition I don't know of a well-educated mathematician who would deny the interrelation between logic, arithmetic, and set theory, and while they may be different sub-cultures, they generally borrow and share vocabulary.

Now, that being said, micromass is absolutely correct in drawing attention that in general use within publication, propositions are offered and generally proven to be true like theorems (hence the etymological shared origin with the verb 'to propose') as building blocks within the taxonomy of proof. Metaphorically propositions (along with lemma) are the foundations to arrive at theorems and their corollaries.

https://en.wikipedia.org/wiki/Gulliver's_Travels Is it really so difficult to accept that like the word 'statement' (as either a linguistic construction or in a broader sense a text) there are two nuanced, but equally mathematical definitions to the same term? I believe the debate might be seen as little-vs-big-endian.