The discipline informed by science is computer engineering. They have to worry about, e.g., things melting. I'm critical of computability as discussed here to

The discipline informed by science is computer engineering. They have to worry about, e.g., things melting. I'm critical of computability as discussed here to some degree, in that many of the algorithms in effect do not compute anything but instead melt a computer, or the earth, or would otherwise deplete the environment of negentropy without answering the question. In standard mathematical analysis there is no behavior "at" infinity, rather we may have some decent picture of what happens at each n (a series' remainder, truncation error, a bound on the value of certain terms, etc.), and talk about what happens to it for greater and greater n, as you mention. This is that other science, numerical methods and analysis. Is it fair to compare the two? If I recall this right every register machine is associated with a system of diophantine equations, with the number of variables about equal to the register count, and the size of the system equal to the length of the program. This is a precise representation of the machine at any step n for a given starting state. To do computer science really analytically you have to solve huge systems of dio equations or something equivalent. Beyond sounding hard, this was Hilbert's 10th problem and has long been answered as not possible. We've been puzzled by these systems all along since antiquity. I am pretty ambivalent towards thinking of BB numbers as scientific results, for the basic assumption of every working model we have (that erasure of a bit is possible) would break before they could compute a result. But at the same time, we are not the first constructivists -- the situation is really puzzling still. I think we no longer realize how shocking some early-mid 20th century results were to mathematicians and logicians of the time. This still hasn't been resolved, nor on the "other" side in physics. We'd love to have the tools we have in applied math in computer science, but that's what all the fuss is about.