ProngHorn's statements are both incorrect and correct. He's just using it the wrong context.
How ProngHorn is wrong:
There is no free lunch. To move an object at a given speed takes a certain amount of energy per second which is called "power", measured in watts or horsepower. Other that slight transmission losses, it doesn't matter what you do to get that power, but that power doesn't change and you need it to move the vehicle at that speed.
Power = Volts * Amps
So, if you cut the volts in half, you have to double the amps to get the same power... and you need the same power to move at the same speed because there is no free lunch.
Volts are something you control or is constant. Amps are usually the result of what you try to force something to do.
For example, if you spin a motor up with no load and measure the amps, they will be small, because the only current is the losses in motor wires and the air resistance of the spinning motor guts against the air. But if you grab onto the shaft and try to slow it down, you'll find very little (if any) change in speed. Instead what happens is the harder you squeeze, the more amps the motor draws to maintain the same speed with the voltage it's given.
Well in the case of a bike it's not your hand slowing down the shaft, it's the tire not wanting to spin because of the bike's air and rolling resistance slowing it down.
So the result of using half the voltage is that current drawn by the motor to maintain speed and hit the power requirement doubles.
That's how ProngHorn is wrong.
Here's how ProngHorn is right:
Amps = Volts / Resistance ("Ohm's Law")
Let's plug in a made-up number for resistance of "12 ohms", the value isn't important, just the relationship. And pick any voltage, let's use the real ones our OP has mentioned: 48V and 24V.
Amps = 48 volts / 12 ohms
Amps = 4
Amps = 24 volts / 12 ohms
Amps = 2
Again, the exact numbers don't matter, just the relationship. The formula is true for any numbers you pick.
So, you see, you control amps by controlling voltage. When you cut voltage by half, you also drop the amps by half. The "Power Law" still applies: Power = Volts x Amps. So if volts are 1/2 as large, they make amps 1/2 as large, and thus power is 1/4 what it originally was.
So ProngHorn was right.
So why are these results incompatible?
Here's how ProngHorn's context was wrong:
Pronghorn assumed the resistance stayed the same in both circuits. And with the back tire lifted up, he would be correct, his results would hold true.
But unlike most electrical circuits, we don't have constant resistance, we have varying electrical resistance based on the fixed power demand of traveling a certain vehicle at a certain speed because there is no free lunch.
When you cut the voltage by half but insist on demanding the same power, the electrical resistance gets cut to 1/2 which spikes the amps by double.
This might seem counter-intuitive in common language. You're thinking "But this makes it harder on the motor, wouldn't the resistance go up?" Yes, in common language. But electrical resistance is a specific term that means something different. Any voltage will cause infinite current (amps) unless you have some resistance. So for example, the filament of a 100 watt light bulb has half the resistance of the filament on a 50 watt light bulb. A high load is one that *doesn't* limit the current to be lower.
This might be too many new concepts all at once, but to relate it back to motors... you can think of jammed motor like short circuit. Like if you jammed a paperclip into a socket. And in fact, if you give a seized motor a normal amount of voltage, it will melt. A motor is in the constant process of pushing the short circuit away from itself, like a solenoid or railgun. Only we cheat and make it a circle so when the motor tries to push part of itself away, it just circles right back around and draws more current again. The more hard we make it for the motor to push the rotating part away from the non-rotating part, the more time it stays in the low-resistance close-contact area and the more time the current is "allowed" to spike.
This is how a motor's resistance would be non-constant depending on the voltage given, and how that resistance drops with less voltage and causes the average amps to rise when you have a set power requirement.
That's dumbed down a bit (and parts of my explanation are close to being outright false, to limit the number of new concepts to cover), but conceptually it all holds true.
Any of that make any sense?
The real-world consequences may or may not be okay. You get away with half as high of a battery voltage, but double the battery current. The controller and wiring will also need to support double the current, but only half the voltage.
In terms of wiring, voltage rating = how thick the insulation is. Almost all wire insulation is rated for 120+V (if not 300+) before electricity will jump through it so that's moot, no saving there.
In terms of controller voltage rating, that depends on the the way the innards of the component are designed, but generally at modest voltages like these they're all in the same ballpark anyway. So, probably no savings.
Higher current on the other hand, wire will require double the cross sectional area (50% increase in diameter) and weight (negligible). And for controller, generally devices have a max current (internal wire and magic stuff thickness) so there is a penalty. You might need components twice as big, or if bundled, twice as many of them side by side. It depends how your controller is built.
If your controller had excess capacity by double, there's no penalty in going this route.
Back at batteries... unless you're limited by pack size (and can't pick smaller cells), technically there'd be no savings here either, either way.
For example, if you had imaginary tiny 2V cells, perhaps you had 4 in parallel (for capacity) by 24 of those in series. So, 96 total batteries at 48V. To go down to 24V, you could just cut the pack in half and have 4x12, for 48 batteries at 24V. But then you got rid of half your battery pack and half your range. Maybe you want this, it's half the price and an option.
To maintain your old range, you need to put 8 in parallel, 8x12, still the same 96 batteries. For equivalent range, your battery neither shrinks in size nor weight nor cost nor anything. Just changes in wiring.
The other concern is to make sure your batteries can supply double the current. This usually isn't an issue unless your cells are truly tiny. Most batteries anyone would even consider for riding can spit out plenty more current than the motor could take anyway. It's a side-effect of picking high enough capacity batteries to have useful range anyway.
One final caveat... suppose you wanted to do the reverse. You wanted to go from 24V to 48V, using the same batteries with different wiring. That *MAY* not be possible, if your batteries are too big. For example if you have 2V batteries and only 1x12 to get 24V... you can't just take a pair of scissors and cut single batteries in half to get half the capacity, then chain up 1x24. For LIPOs, this may be an actual limitation since individual cells are often ordered quite large, and it's why you see people often doubling their range when they double their voltage.. because there's no rearrangement of cells possible.
Data Dump Complete. Confused yet? Probably too much at once, but, if you learned at least some of it, some is better than none.