Have you ever had the feeling that you're alone in the world, that nothing exists beyond you? Perhaps you have, especially if you've just seen The Matrix and its kin. The idea that nothing exists besides you and your thoughts is called "solipsism". While it may make for an interesting science fiction film (leaving aside crappy sequels), it doesn't do well when you're living in the real world. However, like it or not, solipsism is actually valid. Why, then, are none of us actually solipsists? Well, someone once pointed out that whenever a solipsist stubs his toe, it means he must be subconsciously into self-injury.
More seriously, the answer lies in recognizing the difference between between two different, but very similar, ways of saying something exists. The first is, "it exists"; the second is, "it probably exists". When they're written like that, you probably don't see how they can be easily confused. However, I assure you, you do it all the time. In fact, your brain is wired to treat the second statement ("it probably exists") as the first ("it exists"). This is why people who hallucinate treat their hallucinations as reality, a tendency used to great effect in the film A Beautiful Mind. John Nash believed his college roommate and his employer were real precisely because his mind obeyed the rule of thumb "If it very likely exists, then it does exist." This is the generalization science is based on; indeed, it's the rule upon which we base all of our lives.
You see, solipsism points out that the rule of thumb is not, in of itself, logically valid. Given an arbitrarily high, but not absolute, certainty of the truth of a statement, there's no reason for you to say it's actually true. Solipsism points this out, and then stops, saying that you can't know for certain if your body exists, that you're not a brain floating in a tank somewhere, or hooked up to a computer simulation a la The Matrix, or that you're insane locked up in a sanitarium in eighteenth century England, or dreaming, or that the world and you and everything you remember all sprang into existence thirty seconds ago, or any other highly implausible yet perfectly possible situation you could care to invent. Solipsism is a philosophical dead end: if you subscribe to it, then you can't get anywhere beyond "I think, therefore I am".
The key to getting past solipsism lies in the words "for certain". You can't know for certain, so who cares? You certainly don't, I guarantee, since you're reading this right now. You operate on the assumption that your senses are not lying to you: in effect, you operate on the assumption that if something very probably exists, then it actually does exist. To give a rather exotic example, suppose you're sitting at your computer, and you feel a nudge on your shoulder. You turn around and, lo and behold, there is a unicorn standing behind you. It's even pink. Your first instinct is probably disbelief, so you reach out and touch it. It feels real enough (its pink hair is short and coarse). You can smell the horsey sweat on its body, and you can hear it snuffling as it sniffs your hand. It seems real enough; it's certainly not a mirage that you can see but neither hear nor smell nor feel. So, you're on the verge of concluding it's not an illusion, but just to be sure, you yell for your wife. She enters. Her first reaction? "Oh my gosh. Is that a unicorn?"
What was the line of reasoning you used there? Well, a unicorn standing behind you is wildly implausible, to say the least. Your instinct was that there's a chance your eyes are being deceived, as in the case of a mirage, so you checked it with all your other senses. Because the probability that all of your senses are being deceived simultaneously is exponentially smaller than the probability that one sense is being deceived, that confirms that there seemed to be a real, live unicorn standing there. A unicorn is still way, way out there, so you called your wife. What's the probability that both you and she are being deceived by some optical illusion? Astronomically small. She sees it, too. Congratulations! You've jacked up the probability that there actually is a unicorn standing in your room to levels that appear to you to be very near certainty.
For another example, turn to A Beautiful Mind. When the representative of the Nobel Committee comes to see John Nash, he turns to a student just exiting the classroom and asks, "Do you see this man?" He's just double-checking and making sure that his senses are not, in fact, deceiving him.
So, no human can lay claim to absolute knowledge regarding existence outside that of his own mind. He can only be certain to within an incredibly tiny probability of error that he's right. Science simply formalizes and quantifies the methods that we humans use to figure out the world we live in, using a process called methodological naturalism- i.e., the scientific method. It's a process in five steps:
This process is, essentially, what you went through when you discovered the unicorn behind you in your room. You saw the unicorn, hypothesized that it really did exist, tested your hypothesis, and noted that your hypothesis had been confirmed. Science also requires repeated testing of a hypothesis if it is initially successful; this permits many people to confirm your hypothesis. There is one rather large detail this summary of the scientific method glosses over. Science requires that scientific hypotheses be falsifiabile -- i.e., the hypothesis must be able to be proven false. This permits it to employ the logic described above: the probability of passing the test while still being incorrect becomes impossibly small after subsequent passed tests.
This is probably unclear, so let me illustrate. As we've discussed above, it's impossible to determine if any statement about the world is absolutely true, as solipsism acknowledges. Therefore, science must make do with trying to determine how probable a given proposition is. This is why it requires falsifiability: the testing in the scientific method does not look for confirmation, but rather seeks to prove the hypothesis wrong. Confirmation, in of itself, tells us nothing if it's not possible to prove the hypothesis wrong. Going back to our pink unicorn hypothesis, is it possible to prove that there's actually no pink unicorn standing behind you? Sure; you could turn around and not see it, or you could see it but not feel it, or you could see it and feel it but not hear it or smell it, or your wife maybe can't see it.
Again, to reiterate, falsifiability is important because when we test our hypothesis, we assume that it is false and evaluate our results in the light of that assumption. If the results are what we expect if it's false, we consider the original hypothesis disproven; if they're wildly improbable in light of our assumption of falsehood, we do not consider the original hypothesis proven, but rather that the test has provided evidence for it. We test it again, then, to see if it passes; then we test again, and again. The probability that it passes all the tests but is false is like the probability that you see and hear and feel the unicorn, but it's actually not there.
If we can't prove a hypothesis false, then, like solipsism, it tells us nothing. What good is something that we can't prove wrong? It doesn't tell us anything because we can't test it and let the probability that it isn't wrong build up until we're pretty certain that it's correct. If there's no chance we can find a result that shows a hypothesis is incorrect, then we can't crank it through our scientific method to get a result. For an example of an unfalsifiable hypothesis, check out Carl Sagan's invisible dragon in the garage.
Once it's been repeatedly confirmed, giving us reasonable confidence that it's true, a scientific hypothesis may fall into one of three categories:
A fact is simply an observation; for example, it is fact that objects dropped at the surface of the earth accelerate downward at about 9.8 meters per second per second (m/s/s). This can be confirmed through testing. A law is a description that generalizes a phenomenon to describe many observations. For example, Newton's law of gravitation predicts an object will fall at 9.8 m/s/s when it's dropped on the Earth's surface. Note that it makes predictions that can be falsified; for example, if objects consistently fell at 13.2 m/s/s, Newton's law would, of course, be wrong. Finally, a theory is a hypothesis that gives a mechanism to generate predictions. For example, Einstein's theory of relativity postulates a mechanism (curved spacetime) to describe gravity. From this mechanism, it generates predictions, which can be tested against observations and falsified. It hasn't failed so far, which is why relativity is so widely accepted in the scientific community. Do note that the scientific definition of theory is different from the colloquial definition, although the two definitions are often confused; a scientific theory is anything but a casual guess.
So, to recap, science evades the problem of absolute truth by refusing to lay claim to it. Since no truth is absolute beyond your own existence, science doesn't seek absolute truth, but rather to establish that a particular model or approximation is very likely correct. This is formal and mathematical, but in principle exactly the same as the way in which you would convince yourself of the existence of a startling occurrence, such as a pink unicorn standing behind you. The scientific method (also known as methodological naturalism) formalizes the process you use to discover knowledge. It requires hypotheses to generate predictions which can be tested and proven wrong, and in doing so will weed out inaccurate descriptions of the world. Hypotheses that are tested, retested, etc., and pass eventually upon becoming accepted fall into three categories: fact, law, and theory. This descriptive power, the ability to weed out hypotheses which are inaccurate and useless, leads science to generate useful descriptions and approximations of the universe around us. These descriptions, in turn, have led to knowledge, technological advancement, greater wealth, and, in general, the affluence, comfort, and luxury we as a society enjoy.