Limitations of science
While the scientific method has shown itself to be very powerful in investigating the physical world, it cannot directly address the questions posed by our experience of consciousness. This strongly suggests that there is more to the Universe than science can ever describe.
The scientific method requires measurements and observations which can be made available to public scrutiny and which can be repeated and verified by other experimentalists. Not everything can be observed or measured and some things, particularly conscious experience, are not available for public observation. [Strahler 1992]
Science cannot explain conscious experience or how it is that we all possess it and it cannot deal with things like appreciation of beauty. While science can tell us how the brain works, sometimes in exquisite detail, science cannot tell us why an experience feels the way it does. Why is the experience of green like it is and not like something else, or like nothing at all.
It is the fact that we have conscious experiences, rather than just being robots, which is the strongest indicator that there has to be more to reality than the physical Universe.
In addition science cannot tell us about:
Origins
While science can make models of the Universe which are valid further and further back in time, it can never find a starting point. However far back in time we go, there will always have to be something which caused what we find.
Free will (Agency)
Science has little to say about “free will” or “agency”. In fact physics, working from initial assumptions, tells us that the Universe is deterministic with the addition of random quantum fluctuations.
Such a scenario is incompatible with free will or agency, there is no possible freedom here any more than in a clockwork mechanism or a computer. And yet freedom is our experience. Either free will is an illusion or there is more to the Universe than physics can tell us about and/or the assumptions that physicalists have made are false.
Morality
Science cannot answer questions of morality. Although science can tell us how to make a nuclear bomb, it cannot tell us when, or if, it would be right to use it. Neither can it tell us whether objective morality exists at all even though most people instinctively believe that it does.
Purpose
Scientific investigation can tell us a great deal about how the world works but is not able to tell us how or why it came to exist or whether it has a purpose. A scientist coming to Earth from another world and seeing a motor car, or a human being, could tell in great detail how it works but not who put it there or why. Lennox gives this example in [Lennox 2009,page 45].
It is conceivable that someone from a remote part of the world, who was seeing one for the first time and who knew nothing about modern engineering, might imagine that there is a god (Mr. Ford) inside the engine, making it go. … Of course, if he were subsequently to study engineering and take the engine to pieces, he would discover that there is no Mr. Ford inside it. Neither would it take much intelligence for him to see that he did not need to introduce Mr. Ford as an explanation for its working. … So far so good. But if he then decided that his understanding of the principles of how the engine works made it impossible to believe in the existence of a Mr. Ford who designed the engine in the first place, this would be patently false – in philosophical terms he would be committing a category mistake. Had there never been a Mr. Ford to design the mechanisms, none would exist for him to understand.
It is likewise a category mistake to suppose that our understanding of the impersonal principles according to which the Universe works makes it either unnecessary or impossible to believe in the existence of a personal Creator who designed, made and upholds the Universe.
If we are to find out anything about these things, we need something other than science.
The limitations of science are affirmed by many eminent scientists. For instance:
“The limits of science are however made clear by its inability to answer childlike elementary questions such as: “How did everything begin?”, “What are we here for?”, “What is the point of existence?”
“I am very astonished that the scientific picture of the real world around me is very deficient. It gives a lot of factual information, puts all of our experience in a magnificently consistent order, but is ghastly silent about all and sundry that is really near to our heart that really matters to us. It cannot tell us a word about red or blue, bitter and sweet, physical pain and physical delight; it knows nothing of beautiful and ugly, good or bad, God and eternity. Science sometimes pretends to answer questions in these domains but the answers are often so silly that we are not inclined to take them seriously”
The usual approach of science of constructing a mathematical model cannot answer the questions of why there is a Universe for the model to describe. Why does the Universe go to all the bother of existing? Is the unified theory so compelling that it brings about its own existence? Or does it need a Creator, and, if so, does he have any other effect on the Universe?
There are some, however, who would assert that the scientific method is the only way we can know anything. This view is known as Scientism or Physicalism.
Other refutations of physicalism can be found, for instance here or here.