Each mathematics field defines a language, a set of terms each defined in terms of some basic truths (axioms) and a way of building sentences, often with adjunction of the language of logic. Any sentence you speak in this language is Always True, because logic often uses universal quantification. It's possible to make truths which hold in specific cases using the idea of existential quantification, but they are less interesting/celebrated because they are specific applications.
However, there is also the pitfall - the power can only be wielded on problems which can be described in the language of that mathematical field. Here is where it's useful to have and understand multiple fields of mathematics - if you can't easily describe your problem in the language of one field, perhaps there is another field which has a language in which it is easier to describe your problem.
So it becomes useful to study the truths which hold across all fields of mathematics. This requires defining a language in which you can re-define any and all other mathematical fields. I think this is what is called a "univalent foundations of mathematics". For a long time people thought set theory was that thing, but it's always felt like "a hack" to use the language of set theory as such a foundation, because it feels too concrete, too much based on counting.
The current hotness, I believe, is category theory, which is like a slightly more abstracted idea of set theory. Where set theory defines things as sets and subsets, category theory only defines things as how they relate to other things. It's so beautiful! A relation is like a single fact, it's so much like the theory of the mind that analytic philosophers invented in the past century or two.
Now, to see if this conception of mine about the form and function of mathematics is correct. Now I need to study a category theory book to see what the theorems mathematicians in that field are creating and if they can serve as tools useful in many situations or as the basis for other mathematical fields, more specific/concrete fields.
@chexxor The other hotness is Type Theory and HOTT in particular.
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