I am no expert in formal logic, so please forgive me if this question sounds stupid. It's about a common pattern used in many mathematical proofs.
For me it' "obvious" or "trivial" - but I can't prove it.
For a friend of mine it's far from obvious or even wrong - but I don't get his point and I am quite sure that he does not really understand mathematical methods at all ;-)
Let's make a simple example: suppose there's a natural number n and a statement A(n) like "a natural number n is always either even or odd". In many cases there's a proof which does not use a specific property or value of n, so the proof is valid for all n and we conclude that "∀n∈N : A(n)". Now my friend is saying that such a proof is never valid for all numbers n, but only for one single but unspecified number (which I think is nonsense); so he is saying that the "∀n" is wrong (which is think is hogwash).
The common pattern in such proofs is the following: we have a statement A(n) and a proof P which does not use a specific property or value of n. So we could say
"P → ∃n∈N : A(n)" ∧ "P does not use a specific property or value of n" → "∀n∈N : A(n)"
Again: for me it sounds strange; if I have a proof which works for all n then I immediately conclude that "∀n" is right.
Anyway - let me ask the question whether my conclusion is really obvious. Or if one needs a proof for the above mentioned pattern, and how it looks like.
For me it' "obvious" or "trivial" - but I can't prove it.
For a friend of mine it's far from obvious or even wrong - but I don't get his point and I am quite sure that he does not really understand mathematical methods at all ;-)
Let's make a simple example: suppose there's a natural number n and a statement A(n) like "a natural number n is always either even or odd". In many cases there's a proof which does not use a specific property or value of n, so the proof is valid for all n and we conclude that "∀n∈N : A(n)". Now my friend is saying that such a proof is never valid for all numbers n, but only for one single but unspecified number (which I think is nonsense); so he is saying that the "∀n" is wrong (which is think is hogwash).
The common pattern in such proofs is the following: we have a statement A(n) and a proof P which does not use a specific property or value of n. So we could say
"P → ∃n∈N : A(n)" ∧ "P does not use a specific property or value of n" → "∀n∈N : A(n)"
Again: for me it sounds strange; if I have a proof which works for all n then I immediately conclude that "∀n" is right.
Anyway - let me ask the question whether my conclusion is really obvious. Or if one needs a proof for the above mentioned pattern, and how it looks like.
0 commentaires:
Enregistrer un commentaire