When VCRs were originally introduced, they had a counter which was based on the number of revolutions made by the reels. A friend of mine with an extensive tape collection bought 6 hour tapes and would record several movies, TV shows, etc. on each tape, and then write down the counter number of the beginning and end of each show. At some point in the early 90s, the machines began to use counters which were based on the amount of tape that passed over the heads, i.e., on the amount of time. So, my friend was having to guess where to wind the tape to a particular program on his older tapes. He was not a happy camper.
I was able to use a bit of differential calculus and linear regression to give him a very accurate conversion chart, which he thought was just magical!
a very common construction in algebra is called "conjugation":
C = ABA^{-1}
we say C is the conjugate of B by A.
i tend to think of things in the real world as "processes". in this context, ABA^{-1} means: do A, then do B, then undo A (this is the reverse of the usual left-to-right composition, so if you insist on the two matching, use A^{-1}BA instead).
here is how i apply this in the real world:
suppose C is something you would like to do, but there's an obstacle, of some sort (difficulty, lack of materials, whatever). suppose further that B is not problem, it can be done easily.
if there is a way to "transfer the problem to another arena", perhaps one in which the obstacle no longer exists, (this is what A does), we do that instead, and transfer back. i call this "by-passing the problem".
a simple example:
you want to travel 2 blocks north in your car, but there's an accident, blocking the road.
solution: travel one block EAST, then go two blocks north, then go one block west. problem solved!
this is how most people do "difficult integrals"....they "change variables" (make a substitution), now the "difficult integral" becomes an "easier one", which they then solve. finally, they "change variables back" obtaining an easy answer to a difficult problem.
the key is: the "change arena" process must be reversible.
put another way: problems can often be reduced to certain kinds of canonical forms. we can manipulate these canonical forms in a very general way. the "by-pass" is just ONE canonical form i use, others include:
look for symmetry
filter out irrelevant details
replace long sequences of repetition by a single operation
do things backwards
it is a rare occurrence that none of these methods is of no help whatsoever. math can be "symbolic manipulation for its own sake", but often, there is "some real meaning" attached to what we're doing. i see the process as:
D = ABC
where D = "real world problem"
A = "abstract to mathematical formulation"
B = "mathematical manipulation"
C = "specify real-world instance"
the ideal situation is where "A" is reversible, that is our abstraction captures "enough" of our problem to be "faithful", so we don't get "nonsense" when we do C. most of the history of mathematics (and science) consists of "enlarging" our interpretation of B (which is what C essentially is) to increase the reversibility of A.
there are limits to this view of things, if B is just as complicated as D, we haven't gained anything. to me, on a philosophical level, this means: if math can solve any (real-world) problem, it's useless. the whole point is to "forget stuff we don't care about". if we are adding apples, or adding oranges to get a total, the fact that we are adding apples or oranges doesn't affect the total. if we have to take "everything into account" in our abstraction, it's not a very useful abstraction, we may as well just use the original formulation of the problem. turning this interpretation on its head: there should be real-world problems that don't HAVE a mathematical solution.
fortunately for us, this is often not the case. for example, when considering financial transactions, often the musical tastes of the people making the transactions are irrelelvant. your bank doesn't ask you if you like AC/DC before they compute the interest you've earned on your account.