# What's the difference between a proof and a derivation?

I did a BSc in Theoretical Physics, meaning that a lot of my time was spent deriving equations, making hand-wavy arguments, and arriving at solutions with a distinct lack of rigour.

I'm now doing an MSc course which involves maths more in the way of a Maths BSc.

I was explaining to a friend who I did physics with that we didn't really do proper maths in the sense that, despite being very difficult at times, it's really nothing like the maths that you would do an a maths degree, and I proceeded to say that "maths is all about proofs, we never really did proofs", to which he replied: "We derived equations, isn't that a proof?"

I didn't really know how to respond. Is there a formally defined difference between a proof and a derivation? Are they even different?

When I think of derivations I think of the sort of unrigorous maths you get in physics, and when I think of proofs I think of the rigorous maths you get in maths.

I don't know too much about physics. But I do get the following gist:

In physics, you have a bunch of laws, and you come up with some equations and you want to show that they obey these laws. So you need to do derivations and manipulations of these equations until you get to these laws.

Yes, that is a proof in the mathematical sense. But it's not a proof that most mathematicians will consider "a proof", as "proving something" in mathematics is more of an abstract derivation, than verifying an instance of the proof.

When I prove something, I also derive from one statement to the others (and in your case, an equation is a statement). But since a lot of mathematics is not about the actual equations, rather about the properties of objects (which, admittedly, are sometimes phrased in terms of equations), mathematical proofs will more often rely on the axioms/definitions/previous theorems, rather than manipulating equations and showing that they obey some previously agreed upon law.

If you want to be fully formal with your mathematics, however, ultimately, you would need to write your proof in some formal framework (set theory and logic is one, type theory can be another), and use some basic rules of your system to derive from one statement, the other, until you manipulated everything to get to the definition.

So while there is a huge difference between the proofs that you experienced in your two degrees, they are also somewhat the same. So, "same same, but [very] different".

• Feel free to correct me if I'm wrong about the notion of proof when it comes to physics. I'd be happy to delete my answer if it's missing the point. – Asaf Karagila Feb 5 '18 at 17:52
• I really like your point about "mathematics is not about the actual equations". I've noticed too that in physics you would almost always start with an equation, manipulate it/insert other equations, and arrive at a final equation, whereas in maths your using definitions, which are very different, and often finishing with some form of definition or statement. – ODP Feb 5 '18 at 18:21
• I think this is closest of the three answers right now, and don't feel confident enough to write my own. It seems a physics derivation is like a proof, but usually only a proof of certain equations (involving vectors over the complex numbers or similar) using a very narrow selection of algebriac/calculus/linear algebra tools. – Mark S. Feb 5 '18 at 20:05

Typically in physics when someone "derives an equation" I've found this to mean they took other physics equations that they think approximate reality to some extent and then mathematically manipulated them in some way to come up with the equation they "derived".

Now in so far as you accept those original equations as axioms, embedded in whatever formal system allowed them to make those mathematical manipulations or inference rules, then yes its a proof in that system. But this proof says nothing about whether or not what you derived accurately represents reality as that would depend on the accuracy of the original equations, which are either definitions or simply came from looking at a bunch of physical data, intuition etc. and creating some interpolation formula.

Here's an example, let $x$ be your mass, let $y$ be the mass of earth, and let $z$ be the mass of the sun. Now with physics equations $x=y$ and $y=z$ I can derive you have the same mass as the sun. Though this is clearly not reasonable, and yet it is a "proof" in some system if those are part of my axiom schema and where those semantic descriptions of each variable correspond with reality etc. Anyway I'd trust Asaf for any further discussion as he knows more about mathematical logic then someone like me could even comprehend.

Mathematically, Proof is any derivation that validates a Theorem. And Derivation is a mathematical procedure which is performed on the basis of axioms and other known theorems.