Evolution cannot occur without at least some traits being heritable. Heritability is a measure of parent-offspring resemblance; so a trait with high heritability has a high genetic component to it, right?
This is one strange and paradoxical thing about heredity studies: heritability is defined as something specific to a certain population in certain circumstances. It is neither a property of an individual nor a property of a trait.
In parent-offspring correlation studies, heritability is simply the slope that measures the resemblance between the average trait value of the two parents and the trait value of the offspring. In twin studies, heritability is the correlation between the trait values of the twins.
From my reading about scientific and philosophical articles on heritability, I figured that two assumptions are implied in those studies without even being mentioned:
(1) Parents and offspring always resemble, instead of differ from, each other. (In other word, the slope of heritability on the parent-offspring graph is always positive.)
(2) Heritability can never equal to zero. This is because of how the math works out. Heritability is a correlation: a covariance over the product of two standard deviations. Covariances and standard deviations are always positive numbers.
Think about it: pick any measurable trait, its heritability will be a non-zero positive number, even if in reality it does not have genetic component. So what does heritability really tell us?
What would our evolution looks like if parents and offspring always tend to differ, not resemble, each other? Would we evolve backwards? Within one generation, the evolutionary response in the offspring will shift in the opposite direction as the selective pressure; however, the response will be cancelled in the grand-offspring generation under the same selective pressure. Just thinking about this weird scenario gave me a giggle fit in my philosophy-biology class.