I’ve observed that those who are the biggest cheerleaders of Science are those who have only a shallow understanding of it. There are many, many things for which Science is not the appropriate tool and it’s only by understanding where Science should not be used that one can learn to wield it appropriately.
Science cannot comment on Qualia. That is, the actual experience of perceiving something. Science can tell you that Neuron X fired in your brain and that this correlates to what other people report when their Neuron X fires but it cannot tell you that the flavor of strawberries that you perceive is of the same perceptual type as other people’s taste of strawberries. Some people use this as evidence that Qualia either does not exist or is an ill-defined concept.
Popperian Science cannot comment on counter-factuals. If you take the strictly Popperian view of Science as producing claims which are falsifiable, then the entire space of counter-factuals are outside of the realm of science. Counter-factuals are claims of the nature “If X had happened, then Y would occur”. “If Hitler had got accepted into Art School, WWII would have never happened”, “If you had gone to college, you would be more successful”, “If the ball had gone left, it would have fallen down the ramp”. All claims of this nature are strictly unfalsifiable since we can never construct an experiment that occurs in the past. Counter-factuals represent a HUGE part of human knowledge and some people use this as an argument that the Popperian definition of Science is flawed.
Science cannot function when the deductive principle is relaxed. Science fundamentally rests on the deductive principle. We observe that X has happened under Y conditions in the past and we assume that it will continue to do so in the future. However, we already know of areas where the deductive principle is provably not true. We call these situations black swans. Under these conditions, applying the principles of Science will actively damage us, as the most recent financial crisis has demonstrated.
Science cannot comment on positive claims. That is, about what should happen. Science cannot comment on whether the death penalty is a morally just course of action or whether we should save the environment. It can point out the consequences of various courses of actions but it cannot guide the direction.
Science cannot comment on aesthetic claims. It cannot tell you whether something is or is not beautiful. It can sit people in front of something and measure circuits corresponding to beauty light up in their brain but it cannot measure the intrinsic beauty of an object. Some people claim this is evidence that the concept of intrinsic beauty is flawed but others, notably religious scholars, claim that acts such as genital mutilation or forced sex are intrinsically ugly acts whose wrongness lies outside of human preference.
Science cannot comment on fictional worlds. Imagine if you were a character in a novel who possessed all the sufficient Qualia to believe that you actually existed. Science would be completely useless to you. You might think that Science was a valid system but this is just a convenient conspiracy by the novel author. He could write on the next page “and then he walked into a room and saw, on the table, a triangle with edge lengths 1, 1 & 8” and your entire system of not only science but logic breaks down (to see why, try drawing one to scale right now). Now, from a metaphysical perspective, it is impossible to determine whether you currently existing in a fictional universe, believing yourself to be real or not.
Science cannot comment on empirically identical claims. I call this the “surprise birthday party problem”. Say you suspect your friends are planning to hold a surprise birthday party for you. You want to use science to determine whether this is empirically true or not. Well, the answer is, if you’re friends are any good, you can’t. Your friends will act in a way that is absolutely identical to as if they were not planning a birthday party at all, that’s why it’s a surprise. The only way to find out is to wait until your birthday and see if a party is held. A surprise birthday party is essentially a conspiracy theory and so this finding holds true across all conspiracy theories. Any good conspiracist will structure the world such that it would behave as if such a conspiracy doesn’t exist, until when it becomes time to strike. You cannot use science to solve this problem.