The original question was:
How does a scientist/physicist map ideas and concepts into/onto mathematics? Are there a sequence of steps? If one starts with a mathematical model I can sort of see how using deduction, induction etc….
And what about the case where there is no mathematical model? How does ideas concepts get from mind to math/formulas? What is the “dirty work” that goes on behind the scenes?
Physicist: It’s hard to imagine most of the goings on in the world being turned into math. Generally when “math” is brought up, most of us think of arithmetic, which really doesn’t do a whole lot on its own. The majority of the interesting stuff in the world involves stuff much more fancy. There’s already a lot of mathematical structures out there (calculus and topology and all kinds of stuff) so, when you can, you find one that’s familiar and see if it fits. Otherwise you just create new maths.
Here’s how it starts: you tinker about with something until you notice a pattern. Then you make up symbols that allow you to write down those patterns. Then you see what the repercussions are.
For example: “***”, “ggg”, and “@@@”, all have something in common. So, make up a symbol: “3”. Make up some more symbols for other similar patterns and you’ve got the natural numbers: 1, 2, 3, 4, …
Now you can start making up symbols to describe all the reasonable things you can do with numbers: “ “. Boom! Now you’re cooking with arithmetic!
Here’s where the abstraction starts. New symbols come as part of a package that includes rules. Generally by staring at the situation you can pull the rules out. For example, by looking at the picture above you’ll notice that it doesn’t matter which group on the left side is first, so: 2+3=3+2. Holy crap! Commutativity!
Now you can start poking the rules to see how far they bend. If they bend too far, you may find yourself in need of new symbols and rules. For example: What happens if you subtract a larger number from a smaller number? What if you add three or more numbers together instead of just two? Keep up this line of questions and pretty soon you get parentheses, primes, rational numbers, all kinds of stuff.
As often as not, new ideas, symbols, and rules are created because they are needed, as opposed to being found (like the picture above). For example: you can get from the natural numbers (1,2,3,..) to the rational numbers (2/3, 17/5, …) by merely asking some very reasonable questions about division. But the rational numbers have “gaps” (like and ). So now you need a number system with no gaps. What do you do? Make it up! Call it the “real numbers”, .
What’s surprising, from a philosophical point of view, is that the rules that apply in one place apply in others. Which is really nice, because then the rules and symbols that have been found in one area can immediately be applied in another.
Calculus, which only works on continuous things (like the real numbers), applies very well to describing fluid flow, despite the fact that fluids aren’t continuous (they’re made of atoms). Even weirder; most quantum mechanical systems share a lot of properties in common with complex numbers. So as QM was first being fleshed out it already had a couple hundred years of complex analysis to build on.
Don’t be deceived by the notion that the field of mathematics is dominated by numbers. Keeping with the idea of 1) find a pattern, 2) make up a symbol and some rules, and 3) run with it: here are some other fields of math that have very little to do with numbers.
Geometry: That’s right, geometry. It doesn’t really need numbers, it’s just taught that way. You start with “two points make a line” and out pops about a thousand years of Greek history.
Algebra (group theory): This is the entirely abstract study of “there’s stuff, and things can happen to that stuff”.
To be more rigorous, and to actually get stuff done, it primarily considers “groups“, which have more structure than just “there’s stuff”.
Graph theory: The study of dots, and the lines that connect them. If you’ve spent any time doodling, there’s a good chance you accidentally did some graph theory. One simple result is: Any graph where every “vertex” has an even number of “edges” connected to it can be drawn completely without ever picking up your pen or repeating a line.
This theorem is just another example of what happens when you start writing down patterns and rules, and following where they lead.
Topology: You usually hear about this in the context of “coffee mugs and donuts”. This covers stuff like Mobius strips, and tori, to really weird stuff like projective planes. Try this: take two Mobius strips and glue their edges to each other. If you did it right you should have a Klein bottle. Also, you can’t do it right because we don’t live in 4 or more dimensions. Another result from the wide and weird world of algebraic topology!
Logic: This includes stuff like: “If P, then Q” is equivalent to “If not Q, then not P”. Surprisingly useful stuff, and it started with someone sitting down and making up rules and symbols (specifically: George Boole). By the way, he would have written this last statement, “ “.
Knot theory: Knot logic is logic nonetheless (suck on that, logicians!). This is seriously alien math that takes the frustration of tangled string, and kicks it up a notch. Knot theory also borrows some of the machinery from polynomials (despite having nothing to do with polynomials themselves) to form the various forms of knot polynomials.
So the point is: “turning something into math” just means writing down the patterns you see, and then pondering, for a while, the consequences of those patterns. If there’s no readily available notation or symbols, make something up. Let creativity be .
When “there’s no mathematical model” you either make one up, or you’re screwed. But the rules you come up with really need to be solid. For example, you could say “Let the quantity of love be L. Love grows with time, therefore ” (the time derivative of love is positive). But this isn’t a well defined quantity, nor is this a hard and fast rule (You taught me well, Becky!). You could create an entire mathematical system based on your lovsumptions (assumptions about love), but it won’t have anything to do with reality, or indeed love, if the rules aren’t solid.