Do positivists believe you can prove a hypothesis? The ontology states that there is one reality, so I would like to find out the definitive positivist stance when it comes to proving hypothesis....
The positivists you are looking for are called logical positivists. Here I offer a brief intellectual history that explains how the theories of hypothesis testing were developed by logical positivists. Since your statements could cause objections (your talk of ontology), let’s first begin with the following doctrines.
3 doctrines of logical positivists
- Verifiability principle
What is scientific and meaningful is what is verifiable. Rudolf Carnap proposed this ideology in Pseudo Problems in Philosophy (1928). Ethics and metaphysics deal with unverifiable stuff, so is unscientific, and so is meaningless.
- Logical atomism
There is a one-to-one correspondence between the structure of the language and the structure of the world, and logic is the key to marry the two. Moved by Wittgenstein’s Tractatus, Rudolf Carnap proposed the ideology in The Logical Structure of the World (1928)
- Reductionism (unification of science)
The Frege-Russell logicism project, ulteriorly, was to reduce biology to chemistry, chemistry to physics, physics to math, and finally, math to logic.
Theories of hypothesis testing
Your question on hypothesis testing is concerned with the first ideology of logical positivism: verifiability principle. The following are theories that aimed to materialize the principle.
Phase 1: Vindicationism (1920's)
Initially, positivists thought that sentences must be vindicated to be meaningful and scientific. Soon they realized that vindication was impossible even at the level of scientific observations (Many claims of Einstein could not be vindicated at the time). Hypotheses in particular cannot be vindicated since it is a universal statement. ‘All ravens are black’ can never be vindicated by 1000’s observations of black ravens. By 1935, Carnap himself gave up on vindicationism which he proposed a decade ago.
Phase 2: Confirmationism (1930's)
Instead of vindication, Carnap now offers confirmation as the way to understand the truth of a hypothesis. If there are 1000’s observations of black ravens, then the hypothesis ‘All ravens are black; is very much confirmed. So the hypothesis approximates the truth. Carnap thus asserts that what matters is degrees of confirmation, not absolute veridication. His Logical Foundations of Probability (1950) is designed to realize his idea that probability is concerned with the logical relationship between a hypothesis and evidence.
Phase 3: Falsificationism (1950's)
Karl Popper, in his The Logic of Scientific Discovery (first published in German 1934: introduced to the English speaking world in 1959), having criticized confirmation as a logically invalid form of inference (the fallacy of affirming the consequent), offered falsification as the logically correct form of hypothesis testing. (I have a post on Popper’s falsificationism somewhere in Stackexchange).
Phase 4: Crises (1960’s)
Logical positivists were at the brink of being squeezed out due to internal and external pressures. The external pressure is the advance of a new paradigm, ushered by Thomas Kuhn. The internal pressure began with the Duhem-Quine thesis. Confirmation theorists themselves start to find paradoxes of confirmation (e.g., Hempel’s raven paradox and Goodman’s grue paradox). Does the observation of a white shoe confirm the hypothesis ‘All ravens are black’? (Hempel's Studies in the Logic of Confirmation (1965)). Does the observation of a green emerald confirm the hypothesis ‘All emeralds are green’ or ‘All emeralds are grue’? (Goodman’s The Structure of Appearance (1951) These crises incentivised logical positivists to change their name to logical empiricists.
Phase 5: Bayesian confirmation (1990's)
A Bayesian method became the last hope for logical empiricists. Bayesians found that the paradoxes of confirmation are solvable within the Bayesian framework. New ideas, of course, generate new (pseudo) problems (like the problems of old evidence and of new hypothesis). John Earman examined the Bayesian method in his Bayes or Bust (1992). (I know this sh** because the Bayesian methodology was my dissertation topic in my previous academic life.)
Phase 6: Present
The doctrines of logical empiricism are fatally attractive. The building of an empire: i.e., the unification of science! If defection is not an option, as the wise Earman once said, one should continue the imperial march of empiricism since the alternative is to get busted.
For those thinking of defection, these are some options: revolutionism (“Scientific progress is made by revolution, not by approximation.”: Thomas Kuhn), historicism (“Science is what scientists do.”: Peter Galison), Aristotelianism (“I hunted for causality all my life, and I now realize that Aristotle captured it best.”: Nancy Cartwright), and pluralism (“Biology is no slave to physics. We sciences are all equal.”: John Dupré).