Scientific method assumes the principle of induction by simple enumeration (IBSE); but the principle itself cannot be proved empirically.
IBSE is fallacious, but is simply assumed in science: Any theory that does not contradict observations is valid. Given a finite number observed occurrences, infinite number of theories are valid; with each additional observation, infinite number of theories are invalidated, but at the same time, infinite more become possible. This is why scientific outlook must be tentative, subject to revision based on new evidence.
Among infinite number of theories, it is Occam's razor that helps us select one out of many. But what exactly is called simple is rather vague. In the theoretical community, people are actually making do: Everything is tentative. For a given set of observations and a sufficient theory, neither the theory nor "the entities" implied by it are necessary. The only reason that makes a theory acceptable is that the theory does not contradict observations. And infinite number of theories are acceptable for a given set of observations: this is a feature of inductive reasoning.
Philosophical scrutiny only increases doubt.
In Human Knowledge: its scope and limits, Bertrand Russell scrutinized IBSE and traced a priori assumptions further back. He suggested that "the postulates required to validate scientific method may be reduced to five." They are:
a. The postulate of quasi-permanence.
b. The postulate of separable causal lines.
c. The postulate of spatio-temporal continuity in causal lines.
d. The postulate of the common causal origin of similar structures ranged about a center, or, more simply, the structural postulate.
e. The postulate of analogy.
Source: Russell, Bertrand. Human Knowledge, Its Scope and Limits. New York: Simon and Schuster, 1948