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

Question

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.

Answer 1

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".

Answer 2

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". Where by a physics equation I mean a mathematical relationship between physical quantities that scientists have defined (e.g. distance or mass) which can be measured by multiples of units (e.g. meters or kilograms)

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 which I assume was the goal - as that would depend on the accuracy of the original equations.

For example, let $x$ be your mass, let $y$ be the mass of earth, and let $z$ be the mass of the sun. Now with the (clearly inaccurate) physics equations $x=\frac{1}{3}zy^2$ and $y=4z^{1/2}$ I can "prove" or "derive" the new equation $x=\frac{16}{3}z^3>z$ which means you have a larger mass then our sun. Though this is clearly nonsense, and yet it is a "proof" in some deductive system if those equations are part of my axiom schema. Thus its clear that unless the original equations you are making deductions from are good approximations of reality - then there is no evidence to suggest the new equation you derived will be an accurate representation of reality either.

So where do these first equations or "axioms" come from? And why do people trust they are accurate? How do we know they are not largely inaccurate like the ones I just made up?

Well these equations come in two flavors, the first are definitional and are used to describe physical quantities that are needed in later equations. For example the equation $F=MA$. Newton and others invented a parameter called "force" within the context of moving bodies by defining this "force" quantity to be the product of two other quantities. The notion of "force" never had a prior definition before that, so this is not a physics equation we gleaned from studying nature i.e. no one discovered $F=MA$, Newton just defined this new physical quantity called "force" that way. Another example would be velocity e.g. the formula $V=D/T$ - again this was not "discovered" we just realized the ratio of distance units to the ratio of time units was a useful parameter so we gave it a name and labeled it with the variable $V$. However the second flavor of equations is not definitional, an example of one is Newton's law of universal gravitation $F=G\frac{m_1m_2}{r^2}$ where $\small G\approx 6.674×10^{−11}m^{3}(kg)^{−1}s^{−2}$. This equation is a claim that the previous physical quanity of "force" which Newton is using here in the context of gravitational attraction has a mathematical relationship roughly interpolated by the products of two masses, a fixed constant - all taken with the quotient of the square of the length of their given radial distance.

Now where did this formula for "Newton's universal law of gravitation" come from? Well different parts of the equation like the inverse square parameter were already predicted before Newton, while things like the gravitational constant have been repeatedly updated for the past $200$ years. These parameters and constants were predicted by looking at the mathematical relationships between data gathered from either viewing natural phenomena e.g. celestial motion or by artificially creating ideal scenarios in a lab e.g. the famous Cavendish experiment. Basically people eventually noticed from this data that when the masses of the objects acting on each other gravitationally were fixed and the radial distances between them varried there was some form of reciprocal quadratic growth, this allowed the prediction of an inverse square law - likewise by looking at data involving fixed radial distances and varying masses Newton eventually predicted what the numerator sort of looked like, then from here he noted their ratio seemed to oscillate around a scalar amount which he decided to denote by $G$ though he never accurately calculated it beyond estimating $(7\pm 1)×10^{−11}m^{3}(kg)^{−1}s^{−2}$. Later other scientists after him went out and collected even more measurements to verify that his equation was closely approximating the magnitude of gravitational force between objects like Cavendish's as I mentioned earlier - this involved two roughly uniformly dense lead balls and an attempt to measure the force they exerted on each other with a Torison balance. (surveying data is still how modern researchers come up with "physics equations" today - and how they try to verify their results, just we now have computers and software to help analyze data and generate "formulas" like Newtons to help us predict mathematical relationships between quantities, so its easier, we can also tweak these mathematical relationships based on other knowledge we have of relevant context in the area) Thus since for the past few hundred years this relationship has been roughly accurate in its predictions we call it a universal gravitation "law" (again loosely speaking). Whereas because my physics equations I invented do not at all match up with any available data about humans or the earth or our sun, they are largely unknown and useless, resulting in no one calling them "laws" as well as no one trying to teach them in class rooms.

However its important to remember these are all predictions and none of them are exactly accurate in stating the mathematical relationships quantities share in reality. We do not know what exactly effects what, in any given situation the best we can typically do is try to incorporate what the most dominant physical quantities at work seem to be - there is always more fuzz and dependencies we don't know and in fact we never even know for sure if our physical quantities can be represented by the typical real numbers or if they are even quantities at all, for example it could appear some physical quantity we defined is static and measurable simply because its changing or differing values are too small for us to detect. Intuitively you could say we never get to look at the math in the "source code" so to speak of the script which our universe seems to run on - we only see the finished product after its been compiled, thus it could be there is a far more accurate formula which uses many new physical quantities no physicists has yet defined, moreover I can just tell you for a fact that if you let $F'$ be the actual force documented between some associated objects $m_1$ and $m_2$ differing in length by $r$ (that is $F'$ is the actual measurement you record by documenting an experiment like Cavendish's) then the function $f(m_1,m_2,r)=F'-G\frac{m_1m_2}{r^2}$ would display a lot of variance and would never truly be an empty physical quantity (though it would probably get pretty close) despite what Newton's law appears to suggest (you're just going to have to get used to physicists abusing or loosely using mathematical notation, its not worth their time to write approximate and read out possible error estimates/doubts/unqualified dependencies and other disqualifiers every time we want to predict a relationship with these formulas its easier to just write it as a given) in fact if you wanted you could try to develop an even more accurate formula say something like $F=G\frac{m_1m_2}{r^2}+\text{(lower order terms)}$ though we often don't bother as mathematical certainty is often only useful in these contexts up to the degree in which are results stay within some error bounds. Not to mention often times the variance is so great when we try to keep looking at lower and lower order terms that we can't figure out what if any other mathematical structure might be at play here and more importantly what other factors and or unknown physical quantities are effecting this remaining fuzz, another issue is we can't get or just don't have access to enough data to make worth while predictions on lower and lower order terms.

The scientific method - all the inductive reasoning in general which was used to "derive" or "form" these equations, none of this is hard mathematical truth, you can't prove these equations in the mathematical sense, you can tell me based on all available measurements it seems this multivariable formula relating these physical quantities seems like a good approximation, you can say every time you've compared it up with documented data that its been a good estimate, but you can never "prove" this in the mathematical sense. Not anymore then you can prove that the sun will rise again tomorrow or that the universe wont stop existing in $15$ minutes.

However that is not to say the scientific method or all the various forms of heuristics and induction that go into the physical sciences is not useful, it has been very useful in allowing us to make accurate predictions and to get a better approximate grip on what is going on in our surroundings. One should just take care to recognize the varying levels of rigor and certainty when it comes to using terms like "truth" in scientific/philosophical contexts as opposed to mathematical ones. Especially because any attempts at formalizing the processes we use in developing scientific knowledge can never be as set in stone with respect to a formal logistic framework with prior axioms, the distinguishment between inductive and deductive reasoning alone makes them radically different e.g. it might be perfectly okay to claim the earth is $4.54$ billion years old because based on every pre-hl rock we've carbon dated in and around our solar system this seems to be the case, however its not at all okay to claim that because $\gcd((n+1)^{17}+9,n^{17}+9)=1$ for the first one hundred trillion integers $n$ that in general we must have $\gcd((n+1)^{17}+9,n^{17}+9)=1$ for all integers $n$ - in fact this fails to be true for some integer $10^{50}<n<10^{60}$. So in short roughly speaking, mathematicians require pure deductive reasoning from agreed upon axioms before accepting claims as truth whereas scientists are permitted to use inductive reasoning and generalizations in their notion of truth as well as deductive reasoning (for example as we did earlier we defined some quantities like "force" via $F=MA$ then based on the inductive reasoning of Newton and others we accepted the equation $F=G\frac{m_1m_2}{r^2}$ which we could then combine if we wanted with either more equations conjured through inductive reasoning or through mathematical manipulations using some degree of deductive reasoning as described in your theoretical physics class).

Answer 3

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.