Assume that a magician has the ability to create both flames and ice. His Muggle (for the lack of a better word) friend is wounded and is bleeding profusely. Traditionally, in most fantasy books where modern technology does not exist, cauterization would be used to seal that wound. However, I know that realistically cauterization is a contentious method of stanching blood flow. Since this magician can manipulate ice as well, would sealing a wound with a sheet of ice over the wound be preferable to searing the wound with fire? Assume that the magician can reinforce the ice sheet at any time. I should also mention that I do not mean for the ice to be a permanent solution—just a method to prevent the friend from bleeding out on the spot.
Both are useful in different ways.
Ice: Ice will not staunch the bleeding in any meaningful way unless you were to actually freeze the tissue/blood.
But this is a terrible idea as the cells in your body are primarily...water. Water does what when it freezes? Right, it expands. When your body freezes the cells literally burst. This is what we of course call 'frostbite.' Once they thaw you have simply exacerbated the wound and as you can't heal while frozen there is no way to remedy the situation.
So in short, trying to use ice to staunch a wound with either do nothing or make things worse.
On the positive side ice can be used to clean the wound before it is sealed (although it may not be totally necessary...which Ill get to on fire). It can also be used to treat swelling after the fact which even though very simple could help save a fair number of lives.
Fire: Its going to hurt like crazy and it is less ideal than...you know stitches, but of the two options, fire is the only thing that will allow you staunch the wound in your scenario.
In short you burn flesh to seal things up. We know from history that it works and was used for a long long time before medicine became a practice and came up with better options.
One piece of misinformation is the idea that the process of cauterizing a wound sterilized it and kept infection at bay. While partially true, as fire will certainly kill bacteria, the process actually makes the tissue more prone and friendly to bacteria.
Use fire to cauterize the wound. You could optionally clean it with ice water before hand...this might help...but frankly...if you are cauterizing to save a life...
Once the wound is cauterized wrap it and keep it as clean as possible
Ice the wound to keep swelling down. You should generally (with the exception of run away fever) never apply ice directly to the wound. Putting cloth or something in between is best. Ice can potentially save limbs...things can burst if swelling gets out of hand. Also never underestimate the value of clean drinking water when healing.
Fire, most definitely. Cauterization works because the vessels are literally fused shut. The blood has a barrier to stop against. Tissue is destroyed, but will heal again in most cases; even if by painful skin grafts.
Ice would constrict the vessels and slow the bleeding but it would not stop. The warm blood against the ice would just constantly melt the ice and you'd end up with a wet bloody mess. In a cold environment, this would end up killing the "patient" faster.
Medially speaking, neither is enough on their own, even together.
Fire can cauterise but it has to be targeted. This "profuse" bleeding sounds deep. Unless you can target it, your magician is just going to toast the patient. The bleeding thing needs to be found and fixed, wound packed, held together and covered.
Cold ice can help with vasoconstriction. You're not trying to freeze the wound, just make it cold so all the blood vessels in the immediate area limit the amount of blood they're sending through. Won't help with a profuse bleed (certainly not if arterial pressure) but will help with surface wounds and bruises.
But it's probably just going to kill them quicker if they're already in circulatory shock. Which they probably are. Ice when they're stable, maybe.
With some medical training and sterile equipment your wizard might be able to save a life. Without that they can just make ice cream and hotdogs (so they're not totally useless).
like most posts above I would have said Fire would be best but then remembered that using modern medicine that ICE might actually be better:
This allows doctors to cool down patients to around 50 degrees Fahrenheit, putting them in a coma for 60 or 90 minutes, when they may have bled out in minutes otherwise. Although I don't know the particulars (like how much transfusion blood this requires), it seems there could be more value in very select situations where said mage would be better off using some very controlled ice magic over the quicker burn it now approach.
Why not both? I'm going to take some liberties here. You magician's abilities have progressed at the same pace, instead of one developing later on his/her magical journey of discovery. With that in mind, your magician would be proficient to the same degree in both schools of magic. Now, in your 'wounded muggle' situation, depending on where the muggle was injured he could approach its in different way. Personally, 'I' would use 'my' ice ability to lower the core temp of the muggle, not in a dangerous way, but as the body is chilled the blood loss should slow. Then depending on where the injury is 'I' would use my fire ability to cauterize the wound. The other approach would be if your magician had medical training as well. He/she could use his/her magical abilities to speed up medical treatment. He/she could channel a block of ice, then use fire to shape the ice into implements. Obviously the medical tools are made of ice and they would numb the area in question. With the tools he/she would be able to perform field triage and save the Muggle's life. The ability to move around without medical implements then make them when they are needed.