It is often used like an informal lemma. Paradox — a statement that can be shown, using a given set of axioms and definitions, to be both true and false.
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. There are subtleties that need to be understood, especially now in physics, and these subtleties are being ignored.
There is a fundamental problem how we do theoretical physics and the way we interpret mathematics in physics, that is, how we do mathematical physics. I am of a school where we should be doing physical mathematics — that is, learn from observation and let reality guide your mathematics — this is contrary to mathematical physics, where the math is pushing how we want reality.
That is why I gave the argument above on the interpretation of what is a theory, lemma or corollary. Perspective is illusory, not mathematical fact. As a small but very profound illustration, consider how the Hamiltonian is calculated for a particle in a box, or infinite well it matters not the case. After some effort, the wave functions are determined for the possible states energy, position, momentum, whatever that the particle is allowed.
But is this really true! Consider what the mathematics is really telling us. We have calculated these possible quantized values for the entire system, in situ and a priori refer back to my initial commentary for why this is important.
The mathematics, is insitu! Reality is not. This problem becomes self evident once you get into entanglement, but even there, the mathematics we use breaks down. Let us see how you can explain this to them. Riddle me this, the inner dot product of two vectors is considered to be a scalar, whereas the cross product is vector. Furthermore, the inner dot product is interpreted by every mathematician and physicist in every university in the world today, as a projection of one vector upon the other, and the cross product illustrates the area if we take units of distance of two vectors.
But is this true? Why is it that inner dot product has units of area, like the cross product. And if the inner dot product truly does represent an area, what area is it? A corollary is something that follows trivially from any one of a theorem, lemma, or other corollary. However, when it boils down to it, all of these things are equivalent as they denote the truth of a statement. From a logical point of view, there is no difference between a lemma, proposition, theorem, or corollary - they are all claims waiting to be proved.
However, we use these terms to suggest different levels of importance and difficulty. A lemma is an easily proved claim which is helpful for proving other propositions and theorems, but is usually not particularly interesting in its own right. A corollary is a quick consequence of a proposition or theorem that was proven recently. Here is some information from this link :. Theorem — a mathematical statement that is proved using rigorous mathematical reasoning.
In a mathematical paper, the term theorem is often reserved for the most important results. Lemma — a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. When writing a paper, i usually call lemma a technical result that will be used many times in the remaining part of the paper.
The idea is to avoid repeating a similar argument in different proofs. For that reason a lemma is not always worth remembering in itself and if it is, one can include it in a broader theorem. Proposition: A result which is either true or false. Axioms or postulates: A set of statement without proof which is assumed to be true and used building blocks to prove several mathematical theorems and results.
The "usual" difference is that a lemma is a minor theorem usually towards proving a more significant theorem. Whereas a corollary is an " easy " or " evident " consequence of another theorem or lemma. Axiom or postulate is a statement that is taken as true without proof usually a self-evident or known to hold truth or simply assumed true. A definition , according to a known mathematician, is a special kind of postulate, which introduces new concepts later used in other propositions.
Although that's true, it's misleading. These 3 are all Theorems, in the broader sense of the term, because ultimately Truth is the aim of an axiomatic method in mathematics. However, the truth is Conditional on the Truth of the Axioms, which are merely hypothesized to be so; any axiomatic system, can be changed by replacing an independent axiom with it's Negation. The essential condition of all three is that they be Provable not True , which means that they follow from the Axioms, or other Theorems, by mere Logical Deduction.
That said, Corollaries and Lemmas are Theorems judged within Metamathematics, so to speak, as to how Trivial or Relevant they are to the ultimate Proofs one seeks.
One of these is merely an intermediary Theorem for another Theorem, and the other is a an easy result from a previous Theorem or Axiom. Sign up to join this community.
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