The flaw in the Scientific Method is that there might be half truths or lesser truths. If you are framing experiments around a single hypothesis you are only testing that specific idea, rather than all ideas.
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When you hypothesize first and create an experiment around that hypothesis your experiment is fallacious because you are deliberately framing your experiment around whatever you are trying to prove. You might find a half-truth or misdirection. Finding the truth of the matter has nothing to do with the Scientific Method. With the Scientific Method you are attempting to prove your idea (hypothesis) true. In the Scientific Method you're also told to stop experimenting as soon as you get a successful result.
Experimenting without hypothesis = Zetetic Method
Experimenting around a specific hypothesis = Scientific Method
Zeteticism (Empericism) blows the Scientific Method out of the water. While it might not always be the most practical option to test all possibilities before coming to a conclusion, it is certainly the method which will bring the experimenter the closest to truth. The Scientific Method fails because it is based on creating and testing a hypothesis, rather than testing all competing possibilities. Multiple hypothesis' might very well be correct in a particular subject, but the experimenter wouldn't know that, as he was taught in school to publish his result and declare victory as soon as his hypothesis achieves a positive result.
This is what Zeteticism is. It is not "you have to see it to believe it," or any other such nonsense. The text does not say that at all. I am appalled at the poor reading comprehension on this forum.
You misunderstand the scientific method. First, I want to point out that it shouldn't make much difference whether or not the hypothesis is constructed before or after the experiment, and that in practice many hypotheses are kept in mind both before and after the experiment is conducted. Robotham's experiment ,accurate or not, can easily be described in the language of the scientific method. Robotham had two hypotheses in mind: if the Earth is flat, I must see
x. If the Earth is round, I must see
y. It hardly matters whether he forms the hypothesis before or after the experiment. He is using the scientific method.
More importantly, you ignore a crucial element of the scientific method, that experiments are hardly framed around a single hypothesis, and that the results of an experiment can be simultaneously compared to a multiplicity of hypotheses. This is what Robotham is attempting to do, like it or not. It's also what other scientists do, such as the
Geuger-Marsden experiment. This experiment illustrates how Rutherford actually had several hypotheses in mind: If atoms are entirely empty, one must observe
x; if atoms are partially solid, one must observe
y; if atoms are entirely solid, one must observe
z.
This experiment is also an illustration of the willingness of scientists to abandon their working hypotheses and theories when the data clearly reject those theories. The modification of hypotheses based on observation is an integral part of the scientific method.
You also portray the scientific method as an inquiry with a terminus. It is not. The scientific method does not suggest that you should perform a single experiment, come to a conclusion based on that experiment, and then be finished with your inquiry. If anything, that describes the way FEers treat the Robotham experiments. The scientific method assumes that your inquiry is never complete, if it assumes anything at all. After analyzing the results of an experiment, one should modify any remaining hypotheses and begin a new experiment. Verify past results. Compare findings to the findings of others. Have your work reviewed. Conduct more experiments. Find anomalies. Conduct more experiments. Modify hypotheses along the way. This is the practice of the scientific method.
Finally, it is worth noting that the scientific method uses experiments to rule out hypotheses as much or more than it uses them to validate them. I don't think the same can be said for zeteticism. Zeteticism seems to be based almost exclusively on saying "I have a achieved the confirming result I was looking for. No further inquiry is necessary. All negative results must be false."