A set of primary colors is, most tangibly, a set of real colorants or colored lights that can be combined in varying amounts to produce a gamut of colors. This is the essential method used in applications that are intended to elicit the perception of diverse sets of color, e.g. electronic displays, color printing, and paintings. Perceptions associated with a given combination of primary colors are predicted by applying the appropriate mixing model (additive, subtractive, additive averaging, etc.) that embodies the underlying physics of how light interacts with the media and ultimately the retina.
Primary colors can also be conceptual, either as additive mathematical elements of a color space or as irreducible phenomenological categories in domains such as psychology and philosophy. Color-space primaries are precisely defined and empirically rooted in psychophysical color matching experiments which are foundational for understanding the color vision. Primaries of some color spaces are complete (that is, all visible colors are described in terms of their weighted sums with nonnegative weights) but necessarily imaginary (that is, there is no plausible way that those primary colors could be represented physically, or perceived). Phenomenological accounts of primary colors, such as the psychological primaries, have been used as the conceptual basis for practical color applications even though they are not a quantitative description in and of themselves.
All sets of real and color-space primaries are arbitrary, in the sense that there is no one set of primaries that can be considered the canonical set. Primary pigments or light sources selected for a given application on the basis of subjective preferences as well as practical factors such as cost, stability, availability, etc. Color-space primaries can be subjected to meaningful one-to-one transformations so that the transformed space is still complete and each color is specified with a unique sum.
Elementary art education materials, dictionaries, and electronic search engines often define primary colors effectively as conceptual colors (generally magenta, yellow, and cyan; or red, green, and blue) that can be used to mix "all" other colors and often go further and suggest that these conceptual colors correspond to specific hues and precise wavelengths. Such sources do not present a coherent, consistent definition of primary colors since real primaries cannot be complete.