What a continuous symmetry is
Mathematical counterpart: One-parameter groups and their generators.
A snowflake looks the same if you turn it by sixty degrees. A dinner plate looks the same if you turn it by any angle you like.
Both are symmetries. Only the second is the kind Noether's theorems use.
Continuous means adjustable
The snowflake gives you six choices. The plate gives you a dial: turn it by any amount, and every amount works.
That dial is what "continuous" means. There is a knob, it turns smoothly, and every setting is as good as any other.
A symmetry can have more than one knob. Move an object in space and you have three: left-right, forward-back, up-down. Turn it and you have three more. Each knob is called a parameter, and counting them is the first thing either theorem asks you to do, because the count of knobs is the count of results you get back.
The smallest possible turn
Set every knob to zero and nothing happens. That is the setting where the transformation does nothing at all, and you can always arrange the knobs so that zero means "do nothing".
Now turn one knob by the tiniest amount. What you get is an infinitesimal transformation — the smallest recognisable version of the change.
This turns out to be the only version you need. Turn the knob by a tiny amount, then again, then again, and you can build up any amount you like. So the whole symmetry is recoverable from its smallest step, and both proofs work entirely with the smallest step. Physicists call the smallest step the generator, because it generates everything else.
Two kinds of knob, and the difference is not size
Here is the distinction the whole volume turns on.
One knob for the whole thing. Rotate the plate. You choose one angle and that angle applies everywhere on the plate at once. Turn the knob and the entire object moves together.
One knob at every point. Now imagine a change you can dial differently here than there — a little here, a lot over there, nothing at all in the corner. You are not setting a number. You are choosing a whole pattern of numbers, one for each place.
The first kind gives you Noether's first theorem. The second gives you her second theorem. They give different answers, and the difference is not that the second has more knobs. It is that the second does not have knobs at all in the ordinary sense — it has a shape you can draw freely.
Adding more knobs to the first kind never turns it into the second. A hundred knobs is still a hundred numbers. A free pattern is a different sort of thing. See Symmetry that can differ from place to place.
Why the count is exact
Both theorems return one result per knob, or one per free pattern. Not roughly one. Exactly one, and the results are independent of each other.
That exactness is doing real work. If you find three conserved quantities and the symmetry has three knobs, there are no more to find. If you find a fourth, you have missed a knob.
Symmetries that give nothing
Not every symmetry is the useful kind, and it is worth knowing which are not.
Reflect a system in a mirror. The physics may well be unchanged. But there is no dial: you either reflect or you do not, and there is no such thing as reflecting by thirty per cent.
Run time backwards. Same situation — a switch, not a knob.
Swap two identical particles. Again a switch.
These are all genuine symmetries and they matter enormously in physics. Mirror symmetry, for instance, is what tells you certain reactions cannot happen. But neither of Noether's theorems has anything to say about them, and no conserved quantity of the kind she describes comes out of any of them.
The reason is the smallest step. Both proofs work by taking a symmetry, shrinking it until it barely does anything, and reading off what that tiny change costs. A switch has no tiny version. There is nothing to shrink.
So the theorems are silent on a whole class of real symmetries, and the silence is structural rather than an oversight. Physics does get conservation laws out of switch-like symmetries — parity is conserved by most interactions — but those laws are found another way and obey different rules. They are conserved as labels, not as amounts, and you cannot have half of one.
No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.