For instance, if you plan to investigate the 2010 population of Emperor Penguin colonies it is of no consequence that you presuppose global warming as the cause for missing nesting grounds, i.e. an axiom/postulate.
If however in the conduct of that investigation you excluded selective colonies from the count on the presumption that those colonies will not be there due to global warming, then your investigation is worthless with reference to your stated objectives.
In this example, it would be objective to the stated investigation to footmark the colonies you expect to be lost due to the global warming presupposition.
Maybe we're getting stuck on semantics. You say they should approach it that way mathematicians do, but then use terms like axiom and postulate as if they are more or less interchangeable in the example.
In mathematics these have very different meanings, and are not interchangeable.
Postulates can be proven or disproven. The postulate is what is to be tested (it is not assumed to be either true or false when it is declared) and the methodology provides the framework for that test.
Axioms are assumed to be true, and provide the framework for the methodology.