It is not a law that u have to use your clutch on changing up through the gears. It is commonly called a "motocross change" when u keep the engine pinned and just snap the gears thru with a little flick of the clutch while doing it. Thats how motocross riders do it and while it does put extra stresses on the gearbox it is not the worst thing in the world.
A motorbike gearbox uses dogs to engage the gears rather than being a direct mesh from gear to gear. This means that one gears face has a male dog on the side of it, the mating gear has a female hole on the side of it, when you shift the gear, the shifter forks move the gears on the gear cluster and it is the dogs that engage together rather than the teeth on the gears. What happens when u use the clutch is the engines gear (which spins at the same speed as the engine) gets disconnected from the gear cluster which stops the driving force of the bike, when u shift the gear and release the clutch the engine gear re-engages to the new ratio selected and away u go. When not using the clutch to change, the engine gear is still driving the first ratio (say second gear) when u hit the shifter the fork moves the gears to engage the third gear which is spinning at a different speed hence the dogs have to undergo more stresses in order to engage (and it is harder for them to do so).
Therefore, using the rationalle, when changing down gears (say from third to second) your engine revs increase because second gear has a taller ratio than third gear. When u try and do this without using the clutch the gearbox has to try and engage the dogs of a slower gear with a faster gear while the engine is still spinning and driving the bike forward. This WILL increase the wear on the dogs on the gears and it will increase the stresses the dogs incur to a certain degree however it is not impossible to do it.
I realise this is very long winded and probably doesn't make a great deal of sense so i'll try and sum it up pretty easily.
On a well made japanese, austrian or italian bike, the gearbox can change up without the clutch all day every day. On a cheaper chinese made engine, it is still capable of doing it however the quality of the manufacturing/materials is not quite as high so the likelyhood of your gearbox breaking is increased. I always on anybike use the clutch to change down gears whilst giving the throttle a "blip". This Blip attempts to match the engine revs to the revs the engine will produce when the lower gear is engaged, this makes the dogs mesh a little easier and minimises wear to the gearbox. this is also used twofold to stop compression lockups of the wear wheel when u realease the clutch ina lower gear (as someone previously mentioned).
Perhaps someone could add a slipper clutch to their mini to stop this problem of the rear wheel locking up after a change down! HAHAAHAHHHA.
I'm sorry this is so long, its a bit hard to try and explain without visual aids! If anyone has any more queries feel free to PM me and i'll try and explain it further (probably doesn't help that i didn't sleep real well last night and am not really thinking 100% clearly!!) HAHAAHHHA