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I read a book of logical puzzles once and there was a little story in it (fictional) about some boy who received an F on his geometry test because his professor said his proof that all angles of a triangle add up to 180 degrees was wrong. The boy countered with something like, "You've defined for our class everything about parallel lines, angles, etc., but you haven't defined 'proof'. How am I supposed to know how to prove something if you haven't defined what it means to prove something?".

I thought this was very interesting, so I looked up the Google definition:

Evidence or argument establishing or helping to establish a fact or the truth of a statement.

Merriam-Webster defined it as:

a : the cogency of evidence that compels acceptance by the mind of a truth or a fact

b : the process or an instance of establishing the validity of a statement especially by derivation from other statements in accordance with principles of reasoning

Why don't the majority of teachers concern themselves with defining "proof" for a class?

Furthermore, is it possible to prove that there are a certain number of ways to prove a statement?

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  • For the second question, you'll first have to define when two proofs are the same, which is non-trivial.
    – user2953
    Jun 12 '15 at 17:46
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    A formal definition of what constitutes a mathematical proof would require concepts and abstractions far too demanding for grade school, or even early undergraduate students. The application of sound, deductive reasoning in the formulation of a mathematical proof comes from our mathematical intuition.
    – NWR
    Jun 12 '15 at 18:24
  • The sum of the inner angles of a triangle depends on whether locally parallel straight lines are the same distance from each other everywhere (assumed by e.g. Euclid), diverge (hyperbolic), or cross each other (as on the surface of a sphere). Without taking this into account the attempted proof would necessarily be somewhat lacking. But just tacitly assuming an Euclidian space it could have been accepted as proof for about 2500 years, and only seen as lacking for about 200 years (since ~1830). Jun 13 '15 at 10:37
  • [… continuing previous comment] Thus whether a proof is regarded as valid depends not only on (1) what proof techniques are regarded as valid at the time, but also on (2) what assumptions are regarded as established facts. Regarding proof techniques I like so called "look-see" proofs, e.g. just visually seeing the commutative property of multiplication. These are valid proofs. Sadly, already in the 1990s my students (at college) had not learned such simple things. Not to mention e.g. proof of Pythagora's theorem. :( Jun 13 '15 at 10:46
  • I think that for really understanding the modern concept of proof one would have to start with Gödel's completeness theorem (not his incompleteness theorems, which came later, but his completeness theorem). This is about what can be proved by merely mechanical symbol manipulation, and how more subtle notions of validity and truth maps to that. But then one is into really complex stuff. Jun 13 '15 at 10:53
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Mathematical logic defines in a precise way the concept of "formal" proof and there is a branch of math log, called proof theory dedicated to the study of the mathematical object : proof. [see e.g. Sara Negri & Jan von Plato, Structural Proof Theory (2001) ].

But of course we have an "intuitive" concept of what counts as a proof or as a valid argument, in the same way as we have an intuitive concept of what a natural number (the numbers used for counting) is, prior to any "formal" definition of it, like the Set-theoretic definition of natural numbers.

And our intuition of what counts as a proof has changed in time with the growth of mathematical knowledge.

See : Yuri Manin, A Course in Mathematical Logic for Mathematicians (2010), page 45 :

A proof becomes a proof only after the social act of “accepting it as a proof.” This is as true for mathematics as it is for physics, linguistics, or biology. The evolution of commonly accepted criteria for an argument’s being a proof is an almost untouched theme in the history of science. In any case, the ideal for what constitutes a mathematical demonstration of a “nonobvious truth” has remained unchanged since the time of Euclid: we must arrive at such a truth from “obvious” hypotheses, or assertions that have already been proved, by means of a series of explicitly described, “obviously valid” elementary deductions.

Thus, the method of deduction is a method of mathematics par excellence.

[...] Every proof that is written must be approved and accepted by other mathematicians, sometimes by several generations of mathematicians. In the meantime, both the result and the proof itself are liable to be refined and improved.

Thus, the fact that the current criteria for an "acceptable" proof are quite "uinversally shared" does not contradict the fact that mathematical proofs are human (and social) activities.

We learn them at school, and the "training process" we are subject to at school is the way we elarn how to understand and manage a proof : this is the reason way the mathematical training cannot start simply with the "formal" definition of proof, in the same way as we cannot learn how to count starting from the set-tehoretic definition of natural number ...

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