AvgStudent
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What's the difference between proof and derivation?
Difference between a proof and derivation?
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What are your thoughts?
Dr.Peterson
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Can you give an example of a problem you have seen that asks for each of these? A specific example will make it easier to talk about.
But generally, I would think a derivation is a kind of proof, and the difference is as much in how the question is asked as in the format or content.
Zermelo
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Well from a logic perspective, every mathematematical statement is considered a theorem, and every way of proving that statement is considered to be proof (except for axioms, which are special theorems that we don’t prove). It’s just the case that some statements are more general than others, this we give them different names so we can do maths more easily. There are extremely general statements like the Banach Fixed Point theorem, (which have rightfully earned the name theorem ?), and statements like 2+2=4.
For example, solving x^2 - 4x + 3 = 0 can be formulated as a theorem and a proof.
Theorem
Let f: R -> R, f(x) =x^2 - 4x + 3 = 0. Then the only roots of f are 1 and 3.
The proof would then be done by using previous knowledge, aka the quadratic formula and the fundamental theorem of algebra.
In the same way, the derivation of the quadratic formula can be considered a proof!
But it’s pretty silly to treat these simple statements as theorems, and if we did, maths would be much more tedious. That’s why we usually call these exercises, and if we only derive some formulas we call them derivations. Blamocur has put the agreed-upon consensus pretty nicely.
So don’t get confused if two people call the same thing a proof and a derivation, in my classes I did some very simple derivations which were considered to be proofs, because of the significance they had for the whole theory.
Also don’t be confused when seeing things like lemmas, corrolaries or consequences, they are all just theorems with different names!
Hope this clears it up, someone correct me if I’m wrong!
For example, solving x^2 - 4x + 3 = 0 can be formulated as a theorem and a proof.
Theorem
Let f: R -> R, f(x) =x^2 - 4x + 3 = 0. Then the only roots of f are 1 and 3.
The proof would then be done by using previous knowledge, aka the quadratic formula and the fundamental theorem of algebra.
In the same way, the derivation of the quadratic formula can be considered a proof!
But it’s pretty silly to treat these simple statements as theorems, and if we did, maths would be much more tedious. That’s why we usually call these exercises, and if we only derive some formulas we call them derivations. Blamocur has put the agreed-upon consensus pretty nicely.
So don’t get confused if two people call the same thing a proof and a derivation, in my classes I did some very simple derivations which were considered to be proofs, because of the significance they had for the whole theory.
Also don’t be confused when seeing things like lemmas, corrolaries or consequences, they are all just theorems with different names!
Hope this clears it up, someone correct me if I’m wrong!