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I wanted to ask about the difference between a proof and a plausible argument. What is difference in proving a statement and providing a plausibility argument for it. The example on which I am basing it is:

We know that in numerical analysis, the error of composite trapezoidal rule is $\frac{nh^2}{12}{\times}f^{\prime\prime}(\epsilon)$.Now this can be obtained by using Taylor's series for every division of the the integral $\int_a^bf(x)dx$ and adding the individual errors.

But this is not a proof ( at least I have been told so ). This is ( what they call ) just an argument to give us a formula to work. Then what is a proof?

If anyone of you can think of a better description of this problem, then please edit this post accordingly.

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This isn’t a plausibility argument: it’s a brief outline of a proof, omitting almost all of the details. –  Brian M. Scott Jun 24 '12 at 20:25
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up vote 3 down vote accepted

Before I begin, note that mathematics is an exercise in thought. Mathematics says nothing on what carbon atoms on another sheet of carbon atoms represent - merely how certain ideas like numbers and functions correlate.

Strictly speaking, what mathematicians mean by an actual proof is a formal proof. To quote wikipedia:

A formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language) each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference.

For example, when working with integers, it can be an axiom that $1\in\mathbb N$, and another axiom can be that if $a,b\in\mathbb N$ then $a+b\in\mathbb N$. From this we can deduce that $1+1\in\mathbb N$.

The mathematical field of logic has a lot of tools to notate these sequences formally, and also offers various rules of inference (most notably modus ponens), but ultimately no piece of physical paper can be - ahem - proven to be a proof.

However, hardly any real-life proofs are written in this formal system, because it's quite a hassle. For real analysis, this was done in the famous work Principia mathematica, but the formal proof that $1+1=2$ takes two days, a graduate course in set theory and the sacrifice of a goat to understand.

What mathematicians practically mean by "proof" is then anything which gives a clear hint as to how to write a formal proof. This means that the "usual steps" of simplifications and expansions are abbreviated as one step or skipped altogether. Indeed, there is no fine line between a proof and a proof sketch, because every proof is effectively just a sketch, strictly speaking.

There is no way to distinguish a proof sketch from an invalid argument. Sometimes the best thing you can do is convince the author of the proof sketch that he is incorrect, and mathematicians do that by asking about the details of a proof. "Why can you make this simplification?" "Does this always hold?" Sometimes you know that the theory you're working in, such as real analysis, does not have inconsistencies (ie. everything that is provable is actually true), and in that case you can show a proof is incorrect by giving counterexamples. This is not always as easy as it sounds.

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I can't imagine why would someone downvote this for a reason other than by accident. –  Git Gud Mar 17 '13 at 22:05
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There is no sharp distinction between plausible arguments and proofs. An important point that is not emphasized in mathematical education is that the definition of what constitutes a mathematical proof is ultimately social: it's whatever the mathematicians around you will accept as a proof. Thus it changes over time, it changes between subfields of mathematics, it certainly changes depending on whether the mathematicians around you are actually physicists or not...

I agree with Brian that this is more of a brief outline than a plausible argument. Again this is an ultimately social definition: it means that the mathematicians around you would be able to fill in the details.

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A proof is a general case of an argument. Arguments can be valid or invalid. An argument can be called as a proof only if it is logically valid.

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