Dear Bertrand Russell...what kind of "Foundations of Mathematics" were you looking for?

I was thinking about how set theory and what Russell et al call the "Foundations of Mathematics" is actually the most abstract type of math. It devolves into questions about infinite collections. Why would that be foundational? Did we really need transfinite arithmetic to understand the use of small whole numbers? Of course we didn't. But they were not really looking for foundations so much as gleaming golden spires, that could stand high above all other math as its parent in logic,  dependent on nothing more "basic" - rather - nothing more lofty.

There appears to have been a recent blossoming of homotopy theory around revisions of Russell's failed "Type Theory" attempt at defining foundations. Good luck with that! Rather than providing a deeper understanding of any sort of "foundation" these guys are off to the races doing a kind of math (Category Theory) that we called "abstract nonsense" in graduate school. As far as I am concerned, if you are going to skip ahead to the "For all"s, "There exists"s, "Not"s, etc., then you have already jumped the shark.

Rather than looking in the most abstract direction to find the foundations, why not look in the opposite direction with the most concrete realities possible, namely psychology and human behavior?

I think they were still (maybe also to this day) striving to understand Plato's idealized world. I am sure mathematicians are most comfortable believing that their work stems from something eternal and perfect, rather than something messy like Piaget's stages of cognitive development. I have no doubt whatsoever that mathematicians have no interest in psychology. 

But, as it turns out, psychologists [me] have an interest in mathematics. For what it is worth, I think you need to start with the name relation, persistence of meaning of letter and other symbols. And move from their into the definitions of "topic" data structures, thesaurus's, and ledgers. That leads to a kind of math that I remain interested in - the theory of patterns. But I do agree with those better mathematicians (than me) that it is important to get to a place where 1-1 or 1-many correspondences can be discussed.