one can buy LED bulbs with defined color temperature.
why cool white = many kelvins (= high temperature?)
why warm white = few kelvins (= low temperature?)
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Any and every object naturally gives off light that is to a good approximation that of a blackbody. The amount of each wavelength given off is dependent on the object's temperature.
For cool enough objects, most of this emitted light is too far into the infrared to be seen by our eyes. Once something gets to be hot enough, a significant part of its emission will be visible. Our eyes take in all the visible wavelengths and interpret their sum (weighted by how much of each wavelength is emitted) as a single color. As you vary the temperature, this average color varies.
That linked article contains a diagram to this effect:
As you can see, low (but still rather hot) temperatures lead to red, then the color of blackbody light shifts through orange and yellow to white and whitish-blue. When it comes to blackbodies, red is cooler and blue is hotter. This applies to any object that is glowing because of its heat, whether it be a piece of hot metal or a star. (Yes, if a star and a newly-forged sword have the same exact color, that is because they have the same exact temperature; composition and solid/gas/plasma phase don't matter at all.)
So why do we think of blue as "cool"? This is just a coincidence of evolution/environmental exposure. Much of the visible light we see on a day-to-day basis is not pure blackbody emission; it comes from more complicated physics. Water tends to reflect the sky, which tends to be blue because blues are scattered by particles in the air better than reds. And natural fires tend not to be hot enough to make anything glow white-hot, much less blueish-white hot.
The designation of different colors of light as warm and cool, in contradiction to their actual color temperatures, has to do with the items that we experience in ordinary life.
Most objects hat we encounter are not warm enough to emit detectable levels of visible light. As they warm up, they begin to emit more visible light, principally at the red end of the spectrum. We associate this appearance of red light with dangerous heat: stove elements, candle flames, red-hot pokers... We seldom encounter items hot enough to emit predominantly blue light...
On the other hand, the presence of snow and ice can give a decidedly blue tint to natural light, especially if it is reflecting a lot of blue sky light. So we associate this blue color with the coldness of the climate.
I remember on an eclipse cruise a long time ago, coming out of a lecture on Cool Red Giants and Hot White Dwarves, and staring in confusion at the red and blue color coded taps in the washroom! Add to this the fact that the sink was several meters below sea level. There is such a thing as knowing too much physics...