# Provide proof or provide a proof (mathematics)

what I want to express is that I want to prove Theorem A.

My supervisor told me that I could use the formulation: We provide a proof of Theorem A.

A friend of mine, who is a Bachelor of Education in english, told me it must be: We provide proof of Theorem A.

Who is correct here? Is this just a typical mathematician thing and nobody else would do it like that?

• books.google.com/ngrams/… shows provide proof is way more used than the other. – Nicholas J. Nov 4 '14 at 18:25
• @JersonZuleta: I believe that this can be partly attributed to the fact that proof is countable mostly in mathematical contexts, and most of Ngrams finding are non-mathematical. – Armen Ծիրունյան Nov 4 '14 at 18:27
• @ArmenԾիրունյան True. Looking up for just mathematical books, it shows the opposite. Provide a proof is more common than the other. However, both are used. – Nicholas J. Nov 4 '14 at 18:33
• In mathematics a "proof" is a specific thing -- words and formulas written on (virtual) paper, starting with an assertion and leading to the conclusion that the assertion has been proven. It is, eg, possible to have 5 different proofs of a single starting assertion, where each is independently sufficient to address the issue -- mathematical proofs are countable. In a non-mathematical sense, "proof" is not usually countable but is a collection of data, to be considered as one mass, which presumably reenforces an assertion. (Then, of course, there are photographer's "proofs".) – Hot Licks Nov 5 '14 at 12:52

In math, you provide a proof of something: a series of logical steps that lead to a conclusion. In the rest of life you provide proof of something: a citation or evidence or other support for a fact.

In the mathematical sense, the proof is somewhat divorced from the theorem it proves, in the sense that a single theorem could have many different proofs. (Most theorems are happy if they have one proof, but some of the more famous ones can be proved many different ways, and each of those ways is a proof.) A mathematical proof is also not a single-step thing: even the simplest proof will have at least three steps (premise→deduction→conclusion).

In the "real world" sense, proof is roughly synonymous with evidence. When you provide proof of insurance, you dig out a single piece of paper from your glove compartment; there's no logical deduction involved.

Bottom line is, your supervisor is correct: you provide a proof of Theorem A. What your English-major friend said isn't really true for math, and means something slightly different in any case.

• The activity of providing evidence for a conjecture is something that happens in mathematics too, although a mathematician would be very unlikely to use the word "proof" to describe it. – user66219 Nov 5 '14 at 9:23

Provide proof of X means provide proof that X exists / obtains:

Often, you will be required to provide proof of the document.

You must provide proof of: your birth or arrival.

How to Provide Proof of Financial Responsibility.

[all internet]

Since you prove the claims of a theorem rather than its existence, the correct wording here is

We provide a proof of Theorem A.

I know of at least four different proofs of Pythagoras's theorem.

I believe both versions are correct, but with slight nuance in meaning. When you say you're going to provide proof of something, it seems to focus on the fact of proving rather than the process of proving. For example

You think these lines intersect. Well, I will now provide proof that these lines are parallel.

When you say you're going to provide a proof, it seems to focus more on the actual process or method of proving

Most of us can prove the Pythagoras's theorem. I will now provide an interesting geometric proof of it which many of you may not have seen.

Of course, this nuance which I may well be imagining is really small. And most of the time both proof and a proof are interchangeable.

Today prof. X will provide proof of the Riemann Hypotheses. //Wow, the hypotheses has been proven

Today prof. X will provide a proof of the Riemann Hypotheses. //Wow, I wonder what the proof is.

• I think you captured the nuances the best. Relatedly, the nuance is the same for the imperative form for homework questions: "Provide a proof of Theorem Z" means "You must provide one proof (series of logical steps) which verifies all claims of Theorem Z to receive full credit." Excluding the article a would turn this straightforward math homework question into an existential crisis. – Patrick M Nov 4 '14 at 21:04

Many online sources use "a proof" instead of just "proof" when used in a sentence. For example, Wikipedia, Wolfram, test prep sites, and college texts all use "a proof." In fact, all of these were the top results for my Google search results of "definition of proof." Whereas in most situations, one would use just "proof," mathematics uses "a proof" - a type of jargon, if you would.

• I didn't know if Wikipedia was a good source here, so I used a whole slew of other sites. – Arradras Nov 4 '14 at 23:12

As noted in previous answers, “We provide a proof of Theorem A” is a more-proper way of saying that you've included a proof of Theorem A than is “We provide proof of Theorem A”.

If the theorem is one that has been proved many different ways, and if your proof is similar to an existing way, append a brief attribution or a description of the proof approach; eg, “following the method of Poisson”, “using methods like those of Euler and Bernoulli”, etc.

If the theorem is new, so that no one else has yet tried to prove it, instead of writing “We provide a proof...”, write “We prove that...”. I think the latter form (directly saying “We prove...”) is rather more common in mathematical writing than is the “We provide a proof...” circumlocution.