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Just like other phenomena -- through experimentation and build theories to explain the results (light is probable one of these phenomena). Other times, we have a theory first and go on to do experiments to prove it (for example, theories of relativity).

Light is generally agreed to have two natures -- wave and particle. Scientists found light can result in an interference pattern with double slots, hence a wave. They also found that its energy is quantized (photon energies are in discrete values, not continuous, as you think in a wave) -- you can count number of photons! Another puzzle about light is that a photon has its associated energy but no mass (not that we can find).

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