The foundations of mathematics are so simple. Seriously. The fact that they even phrase the question as such is hysterical. The reason mathematics is so powerful a tool is precisely because its foundations are so trivial. Like discourse on property in ethics and law it is a word game because no one establishes sufficient limits under which the general term obscures a change in state.
Math very simple. Correspondence (what remains and what does not), Types, operations, grammar, syntax. Generally we use mathematics for the purpose of scale independence. in other words, we remove the property of scale from the set of correspondences. But we might also pass from physical dimensions to logical dimensions (there are only so many possible physical dimensions). So now we leave dimensional correspondence. In mathematics we remove time correspondence by default, and only add it in when we specifically want to make use of it. In sets we remove temporal and causal correspondence … at least in most cases. So we can add and remove many different correspondences, and work only with reciprocal (self referencing) correspondence (constant relations). But there is nothing magic here at all except for the fields (results) that can be produced by these different definitions as we use them to describe the consequences of using different values in different orders.
But if you say “I want to study the parsimony, limits, and full accounting, of this set of types using this set of operations, with the common grammar and syntax” that is pretty much what someone means when they say ‘foundations’. Most of the time. Sometimes they have no clue.
There is nothing much more difficult here in the ‘foundations’ so to speak. What’s hard in mathematics is holding operations, grammar and syntax constant, what happens as we use different correspondences (dimensions), types, and values in combination with others and yet others, to produce these various kinds of patterns that represent phenomenon that we want to describe. And what mathematicians find beautiful is that there is a bizaare set of regularities (that they call symmetries or some variation thereof), that emerge once you becomes skilled in these models, just like some games become predictable if you see a certain pattern.
But really, math is interesting because by describing regular patterns that produce complex phenomenon, we are able to describe things very accurately that we cannot ‘see’ without math to help us find it.
Its seems mystical. It isn’t. Its just the adult version of mommy saying ‘boo’ to the toddler and the joy he gets from the stimulation. There is nothing magical here. it’s creative, and interesting, but it’s just engineering with cheaper tools at lower risk: paper, pencil, and time.