Proposition — a proved and often interesting result, but generally less important than a theorem. Conjecture — a statement that is unproved, but is believed to be true Collatz conjecture, Goldbach conjecture, twin prime conjecture. Claim — an assertion that is then proved. 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.
Ultimately, these are tools to make a paper more readable. I have found it useful to use the terms proposition and lemma to separate different threads in a paper. Lemmas are very much in line with the theorem helping to display progression towards the theorem. I don't agree with Matt E's answer that a lemma should be small and technical, it may be deep and interesting in its own right, but to make it a theorem would distract from the main narrative of the paper.
I agree that propositions may be less aligned, more ad hoc, and hence of independent interest. If propositions are not explicitly used later in the paper, but serve to highlight something important, then perhaps they should be called remarks. The answers and comments in, e. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
Create a free Team What is Teams? Learn more. Ask Question. Asked 10 years, 8 months ago. Active 6 months ago. Viewed 22k times. When to pick Proposition or Theorem?
Sublemma, fine, but Scholium? A famous example is Bayes' Scholium, a well-known result for interpreting observations of a Bernoulli process.
Writing a scholium instead of a remark is like writing a prolegomenon instead of a preface. It's not quite the same thing but has a distinct highbrow smell. Show 1 more comment. Active Oldest Votes. Matt E Matt E k 10 10 gold badges silver badges bronze badges. Add a comment. A Lemma is a useful result that needs to be invoked repeatedly to prove some Theorem or other. Note that sometimes Lemmas can become much more useful than the Theorems they were originally written down to prove.
A Proposition is a technical result that does not need to be invoked as often as a Lemma. Within business and KM, two types of knowledge are usually defined, namely explicit and tacit knowledge. Apart from some specific industry relevant methods, try these general ways to gain more knowledge:. The three most widely accepted contemporary theories of truth are [i] the Correspondence Theory ; [ii] the Semantic Theory of Tarski and Davidson; and [iii] the Deflationary Theory of Frege and Ramsey.
The competing theories are [iv] the Coherence Theory , and [v] the Pragmatic Theory. Begin typing your search term above and press enter to search. Press ESC to cancel. Skip to content Home What is the difference between lemma and proposition? Ben Davis May 31, What is the difference between lemma and proposition?
What does Lemma mean? What is a lemma in philosophy? What is Lemma frequency? What is a false lemma? What is the no false lemmas condition? Is knowledge equal to truth? Is justified true belief sufficient for knowledge? What is the justified true belief theory of knowledge?
What are the three conditions of knowledge? What is the tripartite theory of knowledge? What are the 8 areas of knowledge? What is theory of knowledge in philosophy? What are the 4 types of knowledge? What are sources of knowledge in philosophy? 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? As you can see, if you start off with a poor or misguided understanding in mathematics, it just may take you down a path to paradoxical theories, like Quantum Theory and Relativity. Paradoxes are a sign that it is not your theory that is flawed, but the axioms that the theory is supported upon is in error.
Just imagine, two hundred years of the greatest mathematical minds and greatest physicists in history all missed one important and minor thing, like understanding what the dot product really is. Could this really be true?
Its not like its ever happened before is it? Hmmm, two and half thousand years of believing that the Earth was the center of the known universe, does come to mind.
0コメント