A syntactically correct expression that is deducible from the given axioms of a deductive system. The difference between Conjecture and Theorem When used as nouns , conjecture means a statement or an idea which is unproven, but is thought to be true, whereas theorem means a mathematical statement of some importance that has been proven to be true.
Conjecture as a noun formal : A statement or an idea which is unproven, but is thought to be true; a guess. Examples: "I explained it, but it is pure conjecture whether he understood, or not.
Examples: "The physicist used his conjecture about subatomic particles to design an experiment. For example, maybe Theorem is a General and Corollary is a Lieutenant. The idea is to use creative writing to explore the nuanced distinctions between similar terms: When is something a Theorem versus a Proposition?
The stories are really fun to read, and sometimes lead to other points. For example, when people give genders to the terms, Theorem is overwhelmingly male and Corollary is almost always female.
Most of our math majors are women, and this observation seems to be rather powerful. Thank you for this info. To begin with, a conjecture is not necessarily a truism. Fundamentally it is a phrase expressing an accepted or an accommodated belief. Does this matter? For what does it mean something to be true?. It means that it can be proven through a logical process of deduction starting with a set of atomic axioms.
Putting aside the issues with A of C for a moment, G1IT shows there are constructs that can not be proven one way or the other and yet still be consistent and factual. That there is no a posteriori way to determine these constructs, but yet we can say they do exist.
Note this is not the same thing as the A of C which is considered to be true a priori, but also not provable. In any case, truth in G1IT is not the same thing as we use it in the arena of axiomatic deduction. Futhermore and back to A of C for a moment, does this mean that the Axiom of Choice, is not an axiom but a postulate, or [gulp] a conjecture?
This segues to my next point. In generally I have taught in the past, definitions are tied together by propositions to enable the creation of axioms. Axioms are the building blocks to construct conjectures and claims, which feed our need to discover, explore and investigate.
If these conjectures and claims are to be accepted a priori they become postulates. And here belies the another issue. Depending on your deductive approach, what one mathematician may call a theorem even if it has been generally accepted for hundreds of years by most of the mathematicians , to another it maybe a lemma, or even a corollary.
In essence, the well known Theorems today, and this goes for Lemmas and Corollaries also, have this tag placed on them for historical reasons, and nothing more. Personally I do consider this to be a dangerous course of endeavor for it forces students to think a certain linear way. The connotations of brainwashing are enliven here. But I digress, and leave this sensitive topic for another time and place. Personally, and this is only a suggestion, for of course, there is much contention on many of the above points I have raised, I believe you should add into your list, the concepts of a priori and a posteriori.
Thus how can you call such things a conjecture, postulate or whatever? Here we are years later, and still people are trying to use words in the same way that Hilbert was trying to build his program for a complete, consistent, sound mathematics. Thanks to Godel, we now know that a Hilbert Mathematics is impossible, so why are we years later pushing a Hilbert Mathematical English? I note lastly that I know that you are trying to give a simple definition to these terms, but as I outlined, there belies the danger.
It just perpetuates the bad mathematical understanding. And mathematics really requires a flexible mind, and not a rigged linear thought process which is how it is, sadly, mostly taught today … Godel should be proof of that! Thank you for your thoughtful response to my post. However, I had to keep my audience in mind—this is a group of first semester freshman or first semester sophomores taking their first proof-writing course. Some of these words were brand new to them.
It would have been completely inappropriate for me to go into the foundations of mathematics with them. Freshman or first years, it matters not. You start teaching false understandings at the beginning of a mathematical education, you end up with just confused and poor mathematicians. Worse yet, you end up with poor engineers and ridiculous physics theories, which is what we have today. Disciplines like Physics are in trouble, this has been so now for the better part of 30 years.
Connect and share knowledge within a single location that is structured and easy to search. Also, when is a mathematical statement a theorem versus a lemma? I've read that a theorem is important while a lemma is not so important and used to prove a theorem. However a theorem is sometimes used to prove some other theorem. This implies that some theorems are also lemmas?
Is it subjective with respect to the author, which statements become a theorem, lemma, etc. I have taken this excerpt out from How to think like a Mathematician.
I think it does a great job of describing what those words mean in a sentence. Later in the chapter, he goes onto describe how we have some conjectures which have been called "Theorems" even though they weren't proven.
For example, Fermat's Last Theorem was referred to as a Theorem even though it hadn't been proven. If you haven't read the book then I highly recommend it if you are a undergraduate in your first two years of math. Theorem vs. Lemma is totally subjective, but typically lemmas are used as components in the proof of a theorem.
Propositions are perhaps even weaker, but again, totally subjective. A conjecture is a statement which requires proof, should be proven, and is not proven. A principle is perhaps the same as a conjecture, but perhaps a statement which is asserted but taken as true even without proof, like an axiom.
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