Disclaimer: I am a self taught pleb, not a mathematician.
The pdf blows up my phone but the first equations look like basic calculus, and the rest seem to be in the form of logical judgments where above the horizontal line are called the premises of the rule, and the judgment below the line is called its conclusion. I've seen Godel's T (System T) used in this style, and it seems this pdf is using set theory (again, my shit phone explodes and can't load half the pages) though I would imagine these would be done in category theory these days.
To "understand" math, (scare quotes because again, I'm just a pleb), you really need to know what field, or set of axioms, or universe you're working in. The rational numbers are the easiest imho, when you get into the reals you're dealing with mathematical objects that may not even be possible to be represented as numbers (as I understand it) so if you can understand symbolic programming ala Sussman Lisp/Scheme you can understand math just you're unfamiliar with the notation, and unfamilar with whatever the rules of this particular abstraction are, that's all. By axioms I mean somebody has merely stated "I created this universe, here are the axioms that rule this abstract universe" meaning they often will not be congruent with any kind of other fields/axioms, in my experience as a plebmatician.
If you want to understand there's a guy from here who wrote a book https://pimbook.org/ and to get into Per Martin-Löf you'll probably need some logic courses, professor Robert Harper or Frank Pfenning both have 'intuitionistic' constructivist math resources you can read/learn from if you google.
The pdf blows up my phone but the first equations look like basic calculus, and the rest seem to be in the form of logical judgments where above the horizontal line are called the premises of the rule, and the judgment below the line is called its conclusion. I've seen Godel's T (System T) used in this style, and it seems this pdf is using set theory (again, my shit phone explodes and can't load half the pages) though I would imagine these would be done in category theory these days.
To "understand" math, (scare quotes because again, I'm just a pleb), you really need to know what field, or set of axioms, or universe you're working in. The rational numbers are the easiest imho, when you get into the reals you're dealing with mathematical objects that may not even be possible to be represented as numbers (as I understand it) so if you can understand symbolic programming ala Sussman Lisp/Scheme you can understand math just you're unfamiliar with the notation, and unfamilar with whatever the rules of this particular abstraction are, that's all. By axioms I mean somebody has merely stated "I created this universe, here are the axioms that rule this abstract universe" meaning they often will not be congruent with any kind of other fields/axioms, in my experience as a plebmatician.
If you want to understand there's a guy from here who wrote a book https://pimbook.org/ and to get into Per Martin-Löf you'll probably need some logic courses, professor Robert Harper or Frank Pfenning both have 'intuitionistic' constructivist math resources you can read/learn from if you google.