What is the set of conditions under which a given knowledge can or cannot be acquired without testing (statistics) hypotheses?
Thoughts on different fields
Below is a list and definitions (and selected sentences of interest) of different field of knowledge and my opinion on whether it appears to me that the field in question is testing hypotheses to gather more knowledge.
Natural science is a branch of science concerned with the description, prediction, and understanding of natural phenomena, based on observational and empirical evidence. Validity, accuracy, and social mechanisms ensuring quality control, such as peer review and repeatability of findings, are amongst the criteria and methods used for this purpose. (wiki)
In physics, chemistry, biology, geology (and other fields of natural sciences), it seems to me that acquiring knowledge can only be achieved through the statistical test of hypotheses.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),2 structure,3 space,2 and change.45 There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. (wiki)
In Mathematics, knowledge is not acquired through testing hypotheses. I will note however, that any knowledge in mathematics is dependent on axioms that are assumed to be true. As natural sciences largely (or completely) depends on mathematics and of course, more generally speaking on logic, I think I could say that natural sciences also depend on axioms that are assumed to be true.
Psychology is the study of mind and behavior.1 It is an academic discipline and an applied science which seeks to understand individuals and groups by establishing general principles and researching specific cases.
Psychologists employ empirical methods to infer causal and correlational relationships between psychosocial variables. In addition, or in opposition, to employing empirical and deductive methods, some—especially clinical and counseling psychologists—at times rely upon symbolic interpretation and other inductive techniques. Psychology has been described as a "hub science",5 with psychological findings linking to research and perspectives from the social sciences, natural sciences, medicine, humanities, and philosophy.
In psychology, both hypothesis testing and other methods seem to be involved.
Applied psychology is the use of psychological methods and findings of scientific psychology to solve practical problems of human and animal behavior and experience.
simply specifies the locations of features and populations (yahoo answer)
The only way I can think of descriptive geography as a field in which knowledge is acquired through testing hypothesis would be if we consider any possible set of characteristics of a given river as a given hypothesis in which, only the set that is observed is accepted. However, there is no statistical testing involved except for the evaluation of the characteristics of the river such as the measurement of the volumetric flow rate.
History (from Greek ἱστορία, historia, meaning "inquiry, knowledge acquired by investigation")2 is the study of the past, particularly how it relates to humans.3 It is an umbrella term that relates to past events as well as the memory, discovery, collection, organization, presentation, and interpretation of information about these events
History can also refer to the academic discipline which uses a narrative to examine and analyse a sequence of past events, and objectively determine the patterns of cause and effect that determine them (wiki)
There is a whole (and long) wikipedia article on the methods of inquiry in history: Historical method.
similarly to descriptive geography, I see history as a very descriptive science, for which I fail to think of it as a testing hypotheses. However I do recognize the use of statistical methods in history as described in the above wiki article (Historical method).
Ethics or moral philosophy is the branch of philosophy that involves systematizing, defending, and recommending concepts of right and wrong conduct (wiki)
I would doubt that much statistical testing would be involved in here.