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Can the terms "Verify" and "Prove" typically be taken as synonymous when reading math texts or in discussion with mathematicians?

If not equivalent, then what are the definitions of "Verify" and "Prove"?

quantif
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  • I like the question, it might be very important to know, especially for beginners which are writing their first papers – Yuriy S Feb 04 '18 at 23:27
  • Similar to https://math.stackexchange.com/questions/42106/difference-between-show-and-prove – lhf Feb 05 '18 at 00:27
  • Goldbach conjecture has been verified and double checked up to $4 \cdot 10^{17}$ but has not been proved, yet – Raffaele Aug 12 '20 at 18:16

4 Answers4

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"Verify" is weaker than "prove", and a verification usually involves checking a finite number of cases or carrying out some routine calculations.

"Verify Goldbachs' conjecture for all even integers less than 1 million" - tedious but straightforward

"Prove Goldbach's conjecture" - much more difficult !

gandalf61
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Verify means "check", used when you need to check some details or whether an argument is true (in for example an already given proof).

Prove means that you need to show something is true by finding the argument yourself.

The Phenotype
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    Aren't proofs by induction, by contradiction and a lot of other methods just verifications of an already stated result, evidently found in some other way? – Yuriy S Feb 04 '18 at 23:15
  • @YuriyS I seriously can't think of a rebuttal to your statement. Should I delete this answer, or turn it into a community wiki so anyone may improve it? – The Phenotype Feb 04 '18 at 23:18
  • I'm not sure, it's your choice. I think the question is hard to answer, because words are never precise enough. I don't think many mathematicians or scientists thing much about it. Proof theory might offer a better insight – Yuriy S Feb 04 '18 at 23:21
  • @YuriyS I'll try the CW then. Hopefully something good will happen :) – The Phenotype Feb 04 '18 at 23:25
  • Does "verify" just mean "check with an arbitrary example?" – quantif Feb 05 '18 at 00:14
  • "because words are never precise enough. I don't think many mathematicians or scientists thing much about it." - but people use the words when writing papers and textbooks. It must have some meaning otherwise the writers are babbling nonsense... – quantif Feb 05 '18 at 00:18
  • "because words are never precise enough. I don't think many mathematicians or scientists thing much about it." - but people use the words when writing papers and textbooks. It must have some meaning otherwise the writers are babbling nonsense... – quantif Feb 05 '18 at 00:18
  • @JosephDasenbrock With verify I mean check with the given data, like no-brainer fill in the details, or just calculating something to conclude a bigger proof. It could happen in examples, in proofs etc. But yeah, "verify" might happen more often in examples than in proofs. – The Phenotype Feb 05 '18 at 00:49
  • Okay, that is helpful – quantif Feb 05 '18 at 01:10
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I believe there is a difference in common use between ''prove" and ''verify". To highlight this, I'd like to give an example.

Example

Prove that if $5x-7$ is odd, then $9x-2$ is even for $x \in \mathbb{Z}$.

Proof

Assume that $5x-7$ is odd. Then $5x-7 = 2n + 1$ for some integer $n$. One may verify \begin{equation*} 9x + 2 = (5x-7) + (4x+9) = 2(n + 2x + 5) \end{equation*}

Because $n + 2x + 5$ is an integer, $9x + 2$ is even.

$\Box$

In this sense, "verifying" $9x+2 = 2(n + 2x + 5)$ is functionally the same as "proving" the result, but it is a matter of algebra. In other words, I would ask a reader to verify something that is well-known but perhaps tedious, whereas a proof typically requires insights at the level of the course or paper.

Note: This slightly edited example was taken from the excellent textbook Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, and Ping Zhang. The comments on 'verify' versus 'prove' are my own.

Gauss
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  • Thanks for the answer @Jackson, so "verify" here is similar to "prove", but "verify" is perhaps more limited in depth, i.e. the verification doesn't handle necessarily handle deeper questions that someone deeply familiar with the topic might raise. – quantif Aug 13 '20 at 19:29
  • the Chartrand text was money and time well spent for me, excellent text, I wasn't a math undergrad student and going from the lower-level math classes to advanced/graduate classes was challenging, the Chartrand text was incredibly helpful. – quantif Aug 13 '20 at 19:31
  • @quantif Yes, I associate 'verify' with brunt force tasks or 'things the reader wouldn't benefit from seeing (but may like to practice)'. There's a 'language' or 'style' with formal math writing, and much of this is preference. And I have a similar story with the Chartrand text - great book. – Gauss Aug 14 '20 at 01:15
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Here, while Maths proving a solution and verifying a solution may be synonyms but are literally different. Verifying means to check as commonly done. Proving means putting down your own opinion in using any other formulae in your way. Now hope you all must have understood. Understood ???

Aryabala Rath
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