"The squares marked A and B are the same shade of gray, yet they appear different."
It is in front of us: as our eyes so brazenly fool us, the visual information (based on data from the eyes that might be misinterpreted by the brain) should never be taken for granted; we should always use information from other sources (e.g., other experiments) to verify what our eyes are trying to tell us about the world.
In the second edition of my book "From Illusions to Reality: Time, Spacetime and the Nature of Reality" I will add a special section on the checkershadow illusion above. What is remarkable is that even Edward H. Adelson, who discovered it, wrote "I still don't believe it. There must be something sneaky going on in those images."
For years I have been wondering at what point we will start awakening from our millennia-long "three-dimensional" dream and will stop fooling ourselves by blindly trusting our senses, particularly our eyes:
We see three-dimensional objects only at a single moment of time ("now") and take it as self-evident that reality is a three-dimensional world which itself exists only at the moment "now" (for the "obvious" reason that we see it at that moment; most of us even forget another illusion - that what we see at "now" is all past, because light needs some time to bring the light reflected from what we see at "now" to our eyes).
More than a hundred years ago Hermann Minkowski (Einstein's mathematics professor) showed that the three-dimensional world we see is merely an illusion and that the true (higher) reality is a four-dimensional world in which all moments of time exist "at once" since they form the fourth dimension of the true world. However, the physical meaning of Minkowski's discovery has apparently been conveniently ignored because of its huge implications for everything in our life; only Minkowski's four-dimensional mathematics is used because it cannot be ignored in physics and because it does not bite... What is more disturbing is that a fact (that no one has been able to question scientifically) has been ignored - none of the experiments which confirmed the kinematic relativistic effects (the twin "paradox" and the two manifestations of relativity of simultaneity - time dilation and length contraction) would be possible if the world were NOT four-dimensional.
Of course, no one can impose on us a given (even scientific) view - we are all entitled to our own views. However, we should never forget that there is a small problem - Nature does not care at all about our personal opinions.