What is the mathematical definition of Resistance? I've heard contortions like "measure of the potential difference that must be impressed across the object to cause unit current to flow through it", but these don't provide real meaning to me, and they certainly don't provide much help with understanding.

Brad

I forgot to also ask, if R is defined as V/I, in what sense is Ohms law meaningful?

I think that you can define R=V/I and then Ohms law says that R is constant. The odd contortion you gave is just another way of saying R=V/I.

The formal definitions is (of an ohm):"The ohm is the electric resistance between two points of a conductor when a constant difference of potentional of 1 volt, applied between these two points, produces in this conductor a current of 1 ampere, this conductor not being the sourceof any electromotive force."
Ohm's law states that: V=IR
The resistance could be looked as the ratio between the difference of potential energy of two points and their current.
This ratio was called resistance.
Some electrical units don't have a liniar connection between the difference of potential energy and the current (Like a lightbulb => its resistance changes, because the temp. changes).
Ohm's Law is meaningful to electrical units that do obey this liniar ratio (Like resistors). When all the units in a circuit obey ohm's law, so does the circuit himself obey ohm's law.

I hope i made it a little clearer, if you have any questions, do ask.

Yatir

Brad,
i am not exactly sure what do you mean by "mathematical definition of resistance" though if you need to know what resistance is then....

when potential is applied to ends of a conductor,then there is flow of current(due to motion of electrons)......however these conductors have stationary ions present in them which tend to block the passage of flow of electrons(i am not going much into detail here).....in other words these conductors offer some resistance to flow of electrons.....

now it is found that in such conductors potential applied at the ends bears a constant ratio to the currrent flowing through it(or you can say that V-I graph is linear...this line is very important).....this ratio is called the resistance......(V/I=R...OHM'S LAW)

this gives the measure of resistance of a given conductor.....the less the value of R the more is the conductance of the conductor......

onto a little more info....
now in a more general terms..........
linear element-it is an element which shows linear V-I graph i.e they obey ohm's law
example:resistor

linear circuit-it is a circuit containing such linear elements.on a whole you can say that linear circuits obey ohm's law.....

when considering complex electrical networks....this simple looking law becomes a powerful tool for solving complex problems....especially theorems like Superposition theorem,Thevenin's theorem and Norton's theorem which rely on ohm's law.....help in solving many problems in an electrical network.....

hope this is of any help!!
love arun

Kerwin