I'm sorry to be the one to have to bring this up. But not all mathematics can be represented visually. Maths breaks into a number of main branches, among them principally algebra, analysis and geometry.
Usually only the third can be represented visually, though often mathematicians develop diagrams which help internalise the complex notions involved in analysis and algebra too.
For example, in algebraic geometry, the more familiar notions of geometric objects such as curves and surfaces are replaced with purely algebraic notions, such as schemes. This is because of various categorical equivalences between geometric objects (on the geometric side) and various algebraic objects (on the algebraic side). But on the algebraic side, schemes are a very expansive generalisation of things that actually correspond to geometric (and visualisable) objects.
I once went to a teaching seminar on the use of a package called GeoGebra for the teaching of mathematics. None of us mathematicians could bring ourselves to put up our hands and ask how one might represent a complex of modules over a noetherian ring pictorially in GeoGebra. There's this fundamental misunderstanding amongst educators that symbolic mathematics is not essential to understanding maths.
This is an important insight when it comes to computer programs though. The same thing happens in computer science. You get splits between things that are geometric, symbolic and purely computational.
Often I get really annoyed at people showing off their latest concurrent programming paradigm by implementing a GUI or event loop for some graphical or network application. They forget that many things simply don't fit into that paradigm.
I equally get annoyed at computer scientists for forgetting that the number of integers is not about 10. Sometimes us mathematicians really want to do things with matrices of ten thousand by ten thousand entries.
Usually only the third can be represented visually, though often mathematicians develop diagrams which help internalise the complex notions involved in analysis and algebra too.
For example, in algebraic geometry, the more familiar notions of geometric objects such as curves and surfaces are replaced with purely algebraic notions, such as schemes. This is because of various categorical equivalences between geometric objects (on the geometric side) and various algebraic objects (on the algebraic side). But on the algebraic side, schemes are a very expansive generalisation of things that actually correspond to geometric (and visualisable) objects.
I once went to a teaching seminar on the use of a package called GeoGebra for the teaching of mathematics. None of us mathematicians could bring ourselves to put up our hands and ask how one might represent a complex of modules over a noetherian ring pictorially in GeoGebra. There's this fundamental misunderstanding amongst educators that symbolic mathematics is not essential to understanding maths.
This is an important insight when it comes to computer programs though. The same thing happens in computer science. You get splits between things that are geometric, symbolic and purely computational.
Often I get really annoyed at people showing off their latest concurrent programming paradigm by implementing a GUI or event loop for some graphical or network application. They forget that many things simply don't fit into that paradigm.
I equally get annoyed at computer scientists for forgetting that the number of integers is not about 10. Sometimes us mathematicians really want to do things with matrices of ten thousand by ten thousand entries.