My bathroom sink is not connected to any hot water source, so I always shave with cold water. I COULD shower first in hot water, or use the shower's hot water heater to heat up a wet towel and wrap it first, but just can't be bothered. I'm used to cold-water shaving, and get a very close, comfortable shave with it.
That said, I've never thought of it as superior. I think I get a great shave because of the newer shaving gels and mutli-blade razors (Trac III or Proglide Fusion are the best). I've long assumed that a hot-water shave would be even better, but I guess I think that just because it is "accepted wisdom" and not from any careful experimentation.
Hot water should "soften" the beard more. That said, cold water closes up the pores and makes the hairs stand out straighter, which might actually help with a closer cut. Which one is more effective in the long run, I'm not sure.
Cold water will also make any nicks bleed less; hot water increases circulation to an area, and will therefore increase bleeding. That's one reason people committing suicide are advised to slit their wrists while in a tub of warm water; the water helps prevent coagulation, and the warmth encourages bleed-out. [Note; I do NOT, NOT, NOT advocate this procedure in any way, shape or form!

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If I were using a traditional single-edged/stropped razor, I'd assume a hot/wet approach would be preferable, but that's only because that's the way I've always experienced it when done professionally in a barber place. Somehow it seems "right", but again that is just accepted tradition, and not based on any careful experimentation.
What I DO STRONGLY RECOMMEND is cold rinsing at the end of showers. I used to like hot, steamy showers, but about 16 years ago I was reading in some old Natural Hygiene books about "water cures", and learned about alternate hot-and-cold baths. I adapted this for my showers, and have never looked back, although now I do a simpler technique, with just a cold final rinse.
At first, I started with just cold-rinsing my feet and ankles. In the beginning, even that seemed uncomfortable, but over a few weeks I moved on to cold-rinsing the legs, then legs and arms, then legs, arms and torso, and finally the whole body, including the head.
For a while, I was filling up a large bucket with cold water before my shower, then at the end of the shower I'd pick up the bucket and pour it all over me (over my head, letting it gush down all over me). A bit like "cannonballing" into an icy river in winter. Now I don't bother with that (limited room in the bathroom for a bucket!), but still do the cold rinse thing.
For a while, I did the back-and-forth thing, alternating hot water with cold, over and over again. One time I was at a spa (in a 5-star hotel in China) and went back and forth between the hot water pool and the cold water pool, thirty or forty times over a half hour or more. The staff were staring, open-mouthed, at this crazy white-skinned fool; they'd never seen anyone do that before, but I walked out of there feeling completely "detoxed" or "cleaned" and refreshed.
[CAUTION: I tried this once at another place that had heavily chlorinated water. The hot water opens up the pores, letting the chlorine in, and the cold water then closes the pores, trapping the chlorine in... the net effect is BAD; don't do this with chlorinated pools, only with clean, non-chlorinated water!]
Now most days I can't be bothered with the alternating hot/cold stuff. I shower with lukewarm water, and rinse with cold. (Even in winter, when the water is icy-cold.) What was initially uncomfortable, I now enjoy. The tingly, refreshing feeling of a cold rinse actually makes you feel more "alive" and more focused (because the extra blood that is no longer filling the extremities is still feeding the brain, actually increasing brain oxygenation).
Also, when the weather is cold, if you dry off after a cold rinse and put on clothes reasonably quickly (the sooner the better), you feel warm for hours afterward, even when others all around you are shivering. Cold winter weather is actually the BEST time to do cold showering!
I believe you "train" the circulation --the tiny muscles and control systems that help expand/contract the blood vessels-- by doing this, and make your body more efficient at adjusting to temperature fluctuations. You can feel the increased ability to adjust to the temperature changes during the first few weeks of doing it. I know that I'm a lot less sensitive to cold temperatures (or even hot, but that might be because I grew up mostly in the desert) than I used to be, and apparently less sensitive than many/most people, since I've been doing this. I've repeatedly had the experience of teaching groups in rooms with the air-conditioning on, and having people ask me after a while if the air-con could be turned down, because they are "freezing", sometimes even in sweaters, while I haven't even noticed anything while in shirt-sleeves.
I had a chance once to discuss this with a cardiologist and a couple of his nurses, and they all said that this kind of cold rinsing is actually good for the heart, too --it strengthens the whole system.
So while I'm not entirely convinced about cold-water shaving, I'm certain that cold-water shower rinses are a very good thing. I plan to continue doing them for the rest of my life.
[NOTE: I do not advocate cold showers when you are seriously ill, running a fever, or are extremely weak, although beginning with small parts (just the hands or just the feet) and gradually increasing the cold-water coverage can strengthen a weak (but not sick) body.]
For fevers and various infectious diseases, there is a very effective cold-water treatment, but it requires others' help and a serious commitment to healing, to endure it. One plunges into a tub of ice water, and is then wrapped up with layer upon layer of cloth soaked in the ice water, the inner layers of cotton and the outer layers of wool. The entire body is wrapped, leaving only the face open (mostly for breathing, and because you don't want to chill your eyeballs!).
One is then left in this state, wrapped like an egyptian mummy, at least overnight (8-12 hours), until the wrappings are warmed and nearly dried, This chills the body (mild hypothermia) at first, then over a number of hours of shivering, the body's heat gradually warms up the wrappings, until a feverish state (hyperthermia) exists at the end of the process. (That is why many layers of wrapping are needed; otherwise you only chill the body but don't heat it.)
This transition through the whole range of temperatures will kill off most harmful micro-organisms, and also allows each of the body's enzyme systems to pass through its most active stage. This cures multiple diseases, especially infectious ones. It may kill minor or "focal" infections that one wasn't even aware of. One may feel drained a bit initially, but once the immediate recovery is achieved, the patient may feel significantly "renewed".
The procedure I just described has been all but forgotten in our modern age of antibiotics. Taking a pill or getting a shot takes less courage and fortitude, and less labor (it helps to have several strong doctors and/or nurses to wrap a person in layer upon layer of icy-cold cloths!), etc... It was a well-accepted treatment in the victorian era when labor was cheaper and modern drugs hadn't yet been invented. Forgotten and largely abandoned today, BUT it is a very effective treatment, used for many years before antibiotics were discovered.
Sorry, I just did another L-O-N-G post, didn't I? If you made it this far, I hope at least that you learned something useful!
