(Edit: this answer is now split into two parts, thanks to a lengthy discussion with Rex Kerr. I made my original answer on a very specific reading of the scientific method. He had a very different reading, which came to a different but very related outcome. I've tried to capture that in the first part. The second part is my original answer, for those who wish to use the stricter reading)
There are at least two extremes as to how one can define the scientific method. One is a process, one is more of a set of principles and a goal. The process is well defined as:
- Observe something interesting
- Formulate a hypothesis that you think would model this interesting thing better than existing model.
- Run a series of independent tests of the hypothesis.
- Statistically demonstrate that the original model (often called the null-hypothesis) predicts the outcomes of the tests to be highly unlikely.
- Reject the null hypothesis (assuming the data backs your claim)
- Demonstrate that your new model does a better job of predicting the statistical results.
This is what I was taught the scientific method was in high school. If that is the version you are after, skip ahead to the second part, which explicitly targets that reading.
However, there is another more fluid reading which also exists. The statistical requirement is relaxed, because it can cause trouble. However, there is a focus on both the elimination of hypotheses through testing and the preference towards hypotheses which are testable. This reading of the scientific method is a very general direction, so the alternatives are equally general.
Science is a very deductive learning approach. It depends upon one writing a hypothesis in highly objective terms and then testing it. There are many situations where deductive learning does not work. Procedural learning is often viewed as an alternative approach. Consider the case of an athlete. They collect large amounts of information from scientific approaches, but the final bit that takes them from a "good athlete" to a "great athlete" is all "feel." There may be no written hypotheses. There may be no statistical testing. Yet, the mind absolutely learns in this way. Thus, procedural learning like this would be a valid alternative method. In fact, many Chinese martial arts focus almost entirely on procedural learning because it is so hard to learn deductively.
Which reading of the scientific method you want to use is up to you. What follows is written entirely from the perspective of a strict statistically valid approach to the scientific method. Much of what has made science great is its ability to build upon previous hypotheses. While statistical rigor is a nicety for rejecting hypotheses, it becomes essential for building hypothesis which can support others.
Finding an alternative to the scientific method depends on you deciding what you want out of a method. You will never find a better tool than the scientific method at it's game. However, if that game is not what you really want it to be, there are alternatives.
The most visible example of this I have seen is western medicine compared to Traditional Chinese Medicine. They developed very different approaches, and yet both appear to yield results. TCM actually does work on line along your questioning: it is not fundamentally built off of rejecting hypothesis.
Let's look at the scientific method, and see if we can make some headway. There are two major features of the scientific method which stand out as "interesting" for this line of thinking:
- The scientific method is highly steeped in the language of statistics.
- The scientific method seeks objective theories.
- The scientific method tests theories.
These are traditionally seen as strengths. However, they can also be seen as weaknesses (like all good superheros, their strength is their weakness.. that's what makes them interesting).
The scientific method is completely and utterly useless without statistics. This means any singular event is completely beyond its reasoning. It cannot provide answers to topics such as "the purpose of your life" because there is only 1 you, and N=1 means there is no statistics.
Related to this, the scientific method strives to be objective. It always tries to remove the observer from the picture. This is very valuable, because it ensures that your discoveries are applicable to others. However, it also proves to be tricky in many situations. Social studies in particular have great difficulties with the scientific method because it is so very difficult to make good tests that keep the observer out of the loop. As an example, TCM claims that acupuncture works. Those who have tried it, claim it works with uncanny success. However, science has had fits trying to find any effect of acupuncture beyond the infamous cop-out "the placebo effect." The issue is that it is almost impossible to develop an effective control to measure against because the acupuncture practitioner knows if they are doing it right or not. Whether you believe acupuncture works or not depends heavily on whether you accept results which lack a solid control to ensure objectivity.
Finally, science tests its theories. This sounds absurd, because it seems so obvious that you should test them. However, a theory is not accepted at all until it is tested. The result is that anyone with a theory must expend the resources to do the testing before science will do anything with it. Other approaches get away with a different style: you use a theory once you have it, and you test it when you get an opportunity to do so. The tests can also be dangerous. (Edit: I had a reference to the LHC and potential to create black holes here, but it was too contentious. Instead, it has been replaced with a hypothetical example) Consider a hypothetical particle physics experiment. The scientist is rather confident that their theory is correct. They begin experimenting, after calculating that they would like 100 samples to do statistics on. Generally speaking, they are finding their theory holds out for test after test. However, on tests which disagree with their hypothesis (which happens in the scientific method due to noise), the observer notices a burst of energy from the test apparatus. That burst becomes stronger and more dangerous with every data point that disagrees with their hypothesis. At some point, the scientist decides to cut the experiment short, because they are uncomfortable putting their life at risk to finish the test. By the strictest reading of the scientific method, that data cannot be analyzed because it is tainted with the scientist's choice to cut the tests off early. This might induce biases because the scientist is more likely to cut them off faster if the results look good for their theory. Other methodologies are capable of using this data (including the intuition of that scientist, who will not try the exact same experiment again).
Seeing that the strengths and weaknesses of science are so confounded, it is up to each individual to decide if those are ideal for them. There are many others, none so visibly different from the scientific method than that of TCM. As described to me in a lecture, the difference is in the approach towards healing the human body:
- Western medicine tears the body apart into components, develops hypotheses about these components, then builds them up. At each step, it develops testable hypotheses, and tests them. From there, it finds things which may provide results, and tests those.
- TCM starts with the body as a whole, finds things that cause good results, then develops testable theories about why the results occurred.
The end result is that much of TCM is doctor-centric. A doctor finds out what works well for them, and suggests it to others. The focus is less on rejecting bad hypotheses, and more on finding new good hypotheses. TCM relies more on natural attrition to weed out the hypothesis, rather than actively trying to disprove them.
Can I claim one is better than the other? I'm not sure if I can. However, I do feel comfortable claiming that they are different, and that a remarkably large number of individuals consider one better than the other in both directions. It's simply another way to approach things.