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As a result of my recent interest in the field of epistemology, I read that there is no such thing as a unique scientific method.
However, during my medical studies I had been told that every scientific inquiry follows the following five steps:
-Make an observation
-Formulate a hypothesis
-Make a prediction based on the hypothesis
-Test the prediction
-Conclude about the hypothesis.
Of course these steps are vague and it seems difficult to understand precisely how they are or should be carried. Anyway I cannot imagine any scientific inquiry which would not use these steps, explicitly or not and in their various possible forms.
But as we usually say that there is no scientific method, such inquiries must exist. Can someone give me an example of such scientific inquiry not using the five step method ? Many thanks in advance.

Edit:
Rejection of the idea that there is a unique scientific method can for example be found in the SEP, "Scientific method" (https://plato.stanford.edu/entries/scientific-method/):
"One of the settings in which the legend of a single, universal scientific method has been particularly strong is science education [...]. Often, ‘the scientific method’ is presented in textbooks and educational web pages as a fixed four or five step procedure [...]."
In chapter 1 of Lee McIntyre's "The Scientific Attitude", you can also find the following statement: "If there is one thing that most people think is special about science, it is that it follows a distinctive “scientific method.” If there is one thing that the majority of philosophers of science agree on, it is the idea that there is no such thing as “scientific method.”".

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    @BrianZ this is namely mentioned here: plato.stanford.edu/entries/scientific-method/#OveOrgThe
    – user47679
    Aug 7, 2020 at 15:59
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    Mathematicians often do not make observations and predictions or conduct tests. In many cases in biology, sociology or psychology field studies aim not to test hypotheses, because there are none available, but to aggregate and classify data. Conducting tests in psychology, sociology and economics is often impossible for practical or ethical reasons, so hypotheses are "tested" against pre-existing data. Sometimes no testing is possible at all, as with string theory, so hypotheses are sorted based on properties like simplicity and coherence, internal and with other existing theories.
    – Conifold
    Aug 8, 2020 at 7:54
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    @BrianZ please see my edit.
    – user47679
    Aug 8, 2020 at 13:23
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    Wikipedia's article on it refers you to sources that call it the demarcation of science.
    – J D
    Aug 8, 2020 at 14:24

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One example of a four-step process was Einstein's formulation of special relativity. According to Einstein and the historians who have studied his work, his hypothesis formulation step was not motivated by, say, collecting a sheaf of celestial observations but by a thought experiment. You might consider this to be a four-and-a-half step process ;-) but it did not start with observations.

It could similarly be argued that Weinberg's unification of the weak and electromagnetic forces was not motivated by observational data either, but started with a mathematical hypothesis. It was not until several years after he wrote down his model that data started to come in which furnished its validation.

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