"I was introduced to the expression by a mathematician who was an expert in the many hierarchies of mathematical logic, typically infinite sequences of types of sets definable by some class of formulas. Each step is defined by some critical bump in complexity or definitional power which can't be achieved at a lower level, and then one looks at what it takes to get beyond the whole sequence. One of the prof's PhD students, working in this area, punned on the Latin by titling his thesis Ad Astra Per Aspera.
Once you go high enough into one of these hierarchies, called the projective sets (of real numbers, or subsets of higher dimensional R^n), there are all kinds of interactions with the highest infinities. Assume bigger infinites and you get more structure and organization "down below". There are a bunch of mathematicians for whom this is the holy grail, to figure out how far out to go into the infinite, based on these more "concrete" consequences.
Others think this is pure moonshine, kind of a mathematical ideology. The originator of this line of thought, though, was Kurt Godel, who was a kook and believer in the reality of the mathematical infinite (he also starved himself to death in Princeton after his wife died, he was paranoid about people poisoning him I think). So the research programme has the sanctification of genius, and that goes a long way in math."