One of the goals of mathematical rigor is to create formal structure that are somewhat consistent with our intuitions. The usual notions of randomness do not capture the common intuition that 1000000000000 is less random than 101010100101. Kolmogorov complexity is a formalization of this intuition, and it can be shown to have many relationships to regular randomness (for example: a string drawn from the uniform distribution over strings is very likely to have a Kolmogorov complexity close to the string's length).