When a calculation or formula is made from first principles, what exactly does this mean? What are the first principles? How many are there?
There's probably a more mathsy explanation, but from my point of view doing something from first principles is taking the fundamental definitions and working from them. If you aren't working from FP's there are rules you can follow that are derived from FP's for a general class of problems. For example: calculating a derivative. From first principles \(\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0} \frac{y(x + \delta x) - y(x)}{\delta x}\). This can be pretty cumbersome in practice so you use rules, for example \(\frac{d}{dx} (x^n) = n x ^{n-1}\) which is derived from first principles.
But is there an agreed set of first principles? How many are there? Are most of the formulae in high school physics books generally the first principles?
Equations like \(F=ma\) were considered first principles up 'til the 20th century. Same with laws like Coulomb's law for static electric forces. From experimental justifications, you can treat these laws as first principles and mathematically deduce how more complicated systems will behave when you combine various individual pieces that each separately obey these laws. Of course, with quantum mechanics and relativity, classical equations that were once considered first principles are now known to be mere approximations that work nearly perfectly at the everyday level. The fundamental postulates of physics have yet to be determined, that's one of the loftier goals String theorists are hoping to achieve. So as an example, you might be given some physics equations and told to assume them as given. In high school, you would typically be asked to plug some numbers into these equations and then put the resulting info together to calculate something, like the time it takes a wheel to rotate or whatever. Often you'll be given equations that in fact can be deduced from simpler equations, but you won't be told how these things are deduced, they'll just quote it as if it were written in stone. When you get to the university level, there's a lot more mathematical focus on deducing complicated equations for complicated systems based on the more simple rules that govern their components, there's less magic and handwaving.
Shorter: From first principles means that the calculation proceeds directly from the most fundamental scientific theory currently known to cover an experiment and not from a phenomenological fit to some experimental data which in principle could have also been explained from basic theory. Hydrogen spectra and the spectra of ions with just one electron are treated in most textbooks from first principles. Helium's spectrum is treated from first principles in scientific papers and perhaps a few cutting edge textbooks. While the consensus is that a calculation from first principles of iron's spectrum is likely to closely match with experiment, the details of the calculation are so complex that I don't believe that they have yet been made. So most of what we know about the details of iron's spectrum is from experimental data and only a small part of it is from first principles. And yet what we know is compatible with first principles, as iron has a well-defined spectrum with the lines where they should be according to crude approximations which are based on first principles.
In mathematics the first principles are either the axioms of the particular area of maths you're considering or about as close as you wish to get without getting bogged down in over the top details. For instance, Prom's example of derivatives. The derivation of \(\frac{d}{dx}(x^{n}) = n x^{n-1}\) from first principles involves using the definition \(\frac{dy}{dx} = \lim_{dx\to 0} \frac{y(x+dx)-y(x)}{dx}\) but you don't need to do such things as prove the well defined nature of limits or derive the concept of multiplication, they can be assumed. \(\frac{d}{dx}(x^{n}) = n x^{n-1}\) is a result in calculus so you can start from the founding principles of calculus, not algebra itself. In physics axiomatic constructions are less rigorously defined and sometimes do not have a set of well agreed first principles. This is because we have the ability to look at Nature and obtain results by observation, rather than in mathematics where everything must be derived from more fundamental concepts. There's probably hundreds, if not thousands, of axiomatic definitions in each area of mathematics. No. If anything it works in reverse, the older you get (and thus the further into education to get) the closer and closer you get to the first principles. In primary school you're told everything is made up of atoms. In high school you learn about elements. In 6th form (aged 16~18 here in the UK) you learn about electron shells and how they define chemical properties. In university you learn how the derive the electron shell structure. In grad school you learn how to describe how single electrons and photons interact. As you learn more you can grasp more and more of the fundamental details because you understand the 'level up'. Things get simpler when you ignore the internal components, the fine details. In mathematics its even worse. People can spent their entire academic careers working on a tiny area of mathematics, trying to prove particular results.
