Laplace, de Moivre, and their immediate intellectual heirs developed most of the basic components of probability theory in the eighteenth and early nineteenth c

Laplace, de Moivre, and their immediate intellectual heirs developed most of the basic components of probability theory in the eighteenth and early nineteenth centuries, and more or less all of the algebraic properties of probability spaces were developed before Kolmogorov formulated his axioms in the twentieth century. Indeed, Kolmogorov's formulation of probability theory, and measure theory generally, is mostly relevant only as a formalism that brought probability into step with the formal developments in analysis to the time. Yes, Kolomogorov's foundation is far and away the dominant flavor of formal probability theory today, but it is possible to develop substantially all of the field, formally or informally, using finite spaces and the basic body of knowledge about them that existed even before Kolmogorov was born [0]. You are certainly right that the advent of the computer age has enormously enhanced the practical utility of probability theory, but again I think you're mistaken in believing that computers were very necessary for the conception or basic use of techniques like Monte Carlo simulations. Twentieth-century mathematicians and engineers of the analogue age had books of random numbers (generated by physical techniques, like rolling dice) and knowledge of practical techniques for generating random-ish sequences that made many probabilistic approaches possible before computing time was cheap and readily available; for example, in his Mathematical Theory of Communication, Claude Shannon describes how he generates Markov chains modeling the English language by (1) flipping through books to find a sequence of character or words, (2) noting the sequence that follows it, and then (3) flipping ahead to find that sequence's next occurrence in the text, before repeating the process to determine the next part of the sequence. (These sorts of pre-computational techniques also help to build intuition about probabilistic processes that seem to be missing in many people who turn to computers right away; I've found some mid-century papers and books on probabilistic topics to be particularly enjoyable and insightful because they present things in this pre-computational setting.) 0. To do so formally requires nonstandard analysis for many applications, but the nonstandard approach only rigorously develops techniques that were known, albeit without formal grounding, to probabilists of previous eras. One needs Kolmogorov's axioms and modern measure theory only to do highly abstract probabilistic sorts of things on uncountable spaces that cannot be sufficiently well approximated by nonstandard or informal models of "very large" spaces. I have never seen an application of probability outside of a pure maths setting that cannot be modeled by a nonstandard finite space, and for good reason: our world is finite, if not actually, then in our experience of it.