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what mathematical subjects do I have to learn in order to create my own formula about a topic? For example, I am curious how Einstein could find his final formulas. Is it all about logic or did he have to know some deep math rules and subjects in order to get his first formula?

I was also wondering how an equation can appear from nothing, and what would be its start.

What subjects should I study to create my own formula and test to see if if its results are plausible?

thanks in advance

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closed as not a real question by Jonas Meyer, Chandru1, Aryabhata, Qiaochu Yuan, Mariano Suárez-Alvarez Oct 23 '10 at 21:29

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Um... I don't really think there's any way to answer this question. There are formulas in all areas of mathematics and science, and people discover them by working long and hard on understanding what's already been done. In this sense, they're never "from scratch." And most mathematical results don't include formulas, though this has no bearing on how important they are. Einstein found his formulas through a deep understanding of physics and differential geometry. I can't tell you where to go for such a broad goal -- just work hard and learn lots of math. –  Paul VanKoughnett Oct 23 '10 at 20:23
Great question. (except for the spelling, which I fixed) –  anon Oct 23 '10 at 20:26
"Creating a formula" does not mean anything, really. The only way to rescue a real question from this is to ask "how do I become a mathematician/mathematical physicist/physicist?" The best thing you can do, from the point of view of math, is to read about what math is and what mathematicians do. Parts of the book The Princeton Companion to Mathematics, for example, and I Imagine there are similar things about physics. –  Mariano Suárez-Alvarez Oct 23 '10 at 21:33
I'm hoping that this question can be re-opened, meta.math.stackexchange.com/questions/960/… –  anon Oct 23 '10 at 21:48
It's a pity this was closed. Formulas (and theorems) ultimately express patterns or regularities in observable data. It could be experimental data from science or calculable patterns within mathematics. This does not mean that the derivation of formulas is by staring at a mass of data until enlightenment dawns, but the primary ingredient is the accumulation of concrete knowledge and experience specific to the subject matter. This is the material on which insight, intelligence, luck and brute force labor operate to -- sometimes, on a good day -- produce useful formulas and theories. –  T.. Oct 24 '10 at 18:24

2 Answers 2

up vote 3 down vote accepted

Just consider doing an experiment: Let's take a bowling ball up a scaffold, the higher we drop it off the building the harder a dent it will put into the soft earth below. Suppose we did a whole series of experiments and measured each case - when plotting the results we might get a graph that curves up quickly - since graphs are and functions are essentially the same thing - we have a formula too. This is how Willem Gravesande found the famous formula for kinetic energy $\frac{1}{2}m v^2$. Another fun example of this is Rydberg formula, data had been measured and he just tried out lots of different equations until finding one which fit the data nicely. This is the experimental derivation of a physical formula.

It is not always necessary to do experiments to discover physical formulae though. Newtons theory of mechanics gives the formula $F = ma$. Using calculus (i.e. purely mathematical reasoning) we can derive the formula $E_k= \frac{1}{2}m v^2$. Here is the explicit derivation but the point is just that it is possible to derive new formula as consequences of the mathematical model. This is the mathematical derivation of a physical formula.

Another approach (you can follow this approach to derive the equations for relativity) is to define some physical principles that you want to model - define a physical theory as a set of postulates. For example "The speed of light is constant for all observers", etc.. then construct mathematical models of these ideas. Once you have the model you can derive consequences from it and build experiments to check them. Another example: Niels Bohr devised a model of the atom from which he was able to derive the Rydberg formula. I think this is called Mathematical Physics, or maybe just Physics.

Anyway, to do this sort of work you need to be a very strong mathematician - but also have a good intuition about reality. If you just study basic physics (like watching the Caltech lectures) you can practice the skills needed.

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If there had been a simple rule/algorithm that would allow us to derive new things out of the air, everybody would have been an Einstein now.

The background of such formulas is usually a simple idea that the scientist plays with in his/her head and expands it to a more and more complex theory. I personally think that the formulas are the last thing - just a formal formulation of the ideas behind.

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