Mathematics is the study of structures themselves. These are necessary in Science and in Art, as both require the invention of structures to either explain (and thus understand) the world, or for any other purpose (in the case of Art).
Mathematics, however, doesn't concern itself with the purposes or details of particular structures; rather, it commits itself with the abstract properties common among many structures.
Math is just what happens when one decides to think logically. It is the study of the logical consequences of arbitrary axioms.
It is both the Art and Science of the structure of structures. It studies many structures in the world, and creates an abstract structure to understand them. In this sense it is a Science. It also creates new unobserved abstract structures, often, generalizing observed ones. In this sense it is an Art.
Maps of mathematics
map pic – https://minireference.com/static/tutorials/conceptmap.pdf – another map
Mathematics is sometimes called formal sciences.
lecture with nice comments about math being about classifying objects at begninning
Useful resources and tools
http://www.dudamath.com/action.html
3blue1brown,
https://en.wikipedia.org/wiki/Category:Mathematics_portals
http://www.msri.org/web/msri/online-videos
https://www.youtube.com/watch?v=glearwgR1Ls
Books
People
http://math.ucr.edu/home/baez/
http://euler.nmt.edu/~jstarret/
Other links:
http://mathgl.sourceforge.net/doc_en/Main.html
http://www.theshapeofmath.com/princeton/dynsys
https://www0.maths.ox.ac.uk/courses
From http://bactra.org/thesis/single-spaced-thesis.pdf :
Formalizing intuitions (Quine 1961) insist, the goal [of formalizing some notion] is that the formal notion match the intuitive one in all theeasy cases; resolve the hard ones in ways which don't make us boggle; and let us frame simple and fruitfulgeneralizations.
http://us.metamath.org/mpegif/mmset.html#overview
Classification of mathematics: http://www.ams.org/mathscinet/msc/msc2010.html?t=53Axx&btn=Current
Journal of humanistic mathematics
Journal of mathematics and the arts
https://www.risc.jku.at/research/theorema/software/
https://mir.fi.muni.cz/webmias-ntcir-12-100/ps
Solve common equations: https://www.fxsolver.com/
"in mathematics you don't understand things. You just get used to them." ~ Von Neumann
Physically, math is just people playing with thoughts (neuronal activations), in the same way that we play with wooden blocks and other things. So math is playing with yourself.. Feynman was right.. "math is to physics what masturbation is to sex"
For each stop—each timbre, or type of sound, that the organ could make (viz. blockflöte, trumpet, piccolo)—there was a separate row of pipes, arranged in a line from long to short. Long pipes made low notes, short high. The tops of the pipes defined a graph: not a straight line but an upward-tending curve. The organist/math teacher sat down with a few loose pipes, a pencil, and paper, and helped Lawrence figure out why. When Lawrence understood, it was as if the math teacher had suddenly played the good part of Bach’s Fantasia and Fugue in G Minor on a pipe organ the size of the Spiral Nebula in Andromeda—the part where Uncle Johann dissects the architecture of the Universe in one merciless descending ever-mutating chord, as if his foot is thrusting through skidding layers of garbage until it finally strikes bedrock. In particular, the final steps of the organist’s explanation were like a falcon’s dive through layer after layer of pretense and illusion, thrilling or sickening or confusing depending on what you were. The heavens were riven open. Lawrence glimpsed choirs of angels ranking off into geometrical infinity. (cryptonomicon)
The basic problem for Lawrence was that he was lazy. He had figured out that everything was much simpler if, like Superman with his X-ray vision, you just stared through the cosmetic distractions and saw the underlying mathematical skeleton. Once you found the math in a thing, you knew everything about it, and you could manipulate it to your heart’s content with nothing more than a pencil and a napkin. He saw it in the curve of the silver bars on his glockenspiel, saw it in the catenary arch of a bridge and in the capacitor-studded drum of Atanasoff and Berry’s computing machine. (cryptonomicon)