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Such behaviors are effective when some verifiable functional state or outcome can be reached regardless of initial conditions. In these cases, systems repeatedly undergo distinct and abortive trajectories; the process only ends when the system finds \u201cright\u201d outcomes. This dominance of the terminal condition (and indifference to the initial state) provokes a very different perspective than the conventional \u201cdynamical systems\u201d framework that has been a centerpiece of the quantitative sciences for centuries. We hypothesize that many of these problems defy the initial-condition driven or gradients on landscapes so useful in problems ranging from mechanics to electrodynamics to chemical kinetics to mass and heat transport. We examine several mathematical frameworks that capture and unify key aspects of exploratory dynamics. One powerful way of thinking of such processes is the geometric distribution, where repeated failures are punctuated by a successful trajectory. We develop intuitions for how random walks with resets accelerate search processes. We highlight fresh and surprising behaviors of search under drift, cues, and checkpoints. Last, appreciating the probability of trajectories conditioned on satisfying macroscopic final outcomes reveals a language for the potency of variation sculpted by selection in their broadest forms. This can give the fictitious appearance of the future making itself known in the present, but we view this as conceptually similar to the way \u201cfictitious forces\u201d arise in non-inertial reference frames. 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