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I have seen all these three terms used quite interchangeably. I've referred to some Logic books, and I couldn't find any clear and sharp distinction between these terms. I found this, but it doesn't say much. In terms of English words, OK, I understand, that verify is to check something; however, in Mathematics context, a lot of books, online courses and etc. are using all these three words with the context of prove. I have just encountered in the online course of Algorithms by Stanford University, instructor saying the phrase "I'm going to mathematically argue, that 5 is prime number".. and then this guy said, at the end, that he proved his claim.

Can anyone explain really clearly difference between these three terms in the Math context? Please also provide the example sentences, so that your answer is not just an opinionated philosophy.

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OK so there are several possible mathematical contexts.

Consider the sentence "I will prove the Orbit-Stabiliser Theorem." Another way of saying the same thing is "I will argue mathematically that the Orbit-Stabiliser Theorem is true." Because a proof is precisely a mathematical argument that something is true. So the Stanford lecturer you mentioned could have said "I will prove that 5 is a prime number", and it would mean the same thing.

So the words are not interchangeable in the sense that you can't say "I will argue the Orbit-Stabiliser Theorem", but if you change the syntax, then you can avoid using the word "proof".

Now the syntax and context really matter, because "argue" is a more versatile word than "prove". For example, we often say "assume X and argue for a contradiction". This means that we will try to prove that X is false. So in general when you see "we will argue..." you should read it as "we will make a series of logical steps", and then you should use the context to see what is being argued.

Now "verify" means something very specific: "Verify Theorem X" never means "prove theorem X" (unless the proof involves checking one particular case, in which case it will be clear what is meant).
Let's say that Theorem X says something about numbers in the set Y. The context is almost always that you have proved Theorem X, and as an example to help your understanding, you will be asked to "Verify theorem X for the case y$\in$Y". And this simply means to check the truth of Theorem X for the number y. So, to reiterate, verifying a theorem almost never proves the theorem.

In summary, you will see all three words used in the context of proofs, but they are not precisely interchangeable.

P.S. გაუმარჯოს შოტლანდიიდან!

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  • $\begingroup$ Hey! the points you've brought confirmed my assumptions.. just couldn't find any assertions anywhere.. I guess my concern is more about proper Math. Vocabulary.. Let me reiterate: I will prove and I will mathematical argue are totally equal expressions. But can a mathematical arguing be the process of Negating a theorem rather than proving it? argue sounds to me like a process.. where you bring your points and prove something is either true or false; Verify theorem X - is to just pick some valid data (e.g. numbers) and check that X is true for them. გაგიმარჯოს ესტონეთიდან! :) $\endgroup$ – Giorgi Tsiklauri Apr 3 at 4:39
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    $\begingroup$ Well, I made several assertions. One is that "argue" should be read as "make a series of logical steps", and I think this is what you already believe. That is correct. In particular, it almost always comes in the phrase "we will argue that...". So you could see "we will argue that X is true" (and therefore prove X) or "we will argue that X is false" (and therefore prove the negation of X). So yes, "mathematical arguing", meaning a series of logical steps, can lead to the proof that something is true or false. $\endgroup$ – Sveti Ivan Rilski Apr 3 at 8:57
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    $\begingroup$ Notice also, that any series of logical steps, ie any "mathematical argument" is a proof for something. Even if you are trying to show the negation, you are then proving the negation. I think you have the right understanding of all three words. $\endgroup$ – Sveti Ivan Rilski Apr 3 at 9:01
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    $\begingroup$ Yes, I think we're on the exact same page, and that's what exactly I was thinking as. Thank you, anyways, very much! $\endgroup$ – Giorgi Tsiklauri Apr 3 at 9:08

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