I want to overview and understand all of math. I'm taking two approaches.
I'm working on this at my page on the Azimuth project.
Areas of math
Ieškau matematikos pagrindų. Apžvelgiu matematikos sritis ir jas išdėstau pagal tai, kaip viena nuo kitos priklauso.
Thank you for your advice!
I ended up making my maps with yEd, which is available for free at http://www.yworks.com It's a well rounded tool.
Here's a flash viewer of one map of mathematical areas: http://www.ms.lt/derlius/Math/MathWays2/mathways2.html At the bottom I've placed the math areas which are starting points for math such as "logic" and "geometry" seem to be. Then each arrow leads to a type of math that requires a bit more structure or knowledge. At the very top is "number theory" which seems to pull together absolutely every kind of math. You can zoom into the image using the "zoom" scale at top. Then you can move around the image using your browser's scroll bars on the right and on the bottom.
I've colored coded:
So this is a further development of this earlier map...
The new map has twice as many nodes but it hasn't made things clearer for me. However, the last map was not scalable, which is to say, I couldn't make it any bigger. Whereas this new map I could probably grow to include 10,000 nodes, or simply a node for every math page in Wikipedia. So I can play around with this new map and I think within a year I will find helpful ways of organizing the big picture in math.
Kirby, yes it could be advantageous to put it on a sphere or tetrahedron, etc. However, if there is some deep pattern lurking here, it may be 6 dimensional or 600 dimensional for all we know. So on the one hand a 2-dimensional spherical surface isn't that much of an advance over a 2-dimensional flat surface. In any event, what really matters to me is to come up with a good mental model and then visualize accordingly. For example, it seems there may be a single "starting point" (foundations) and a single "ending point" (number theory). In that case I have a globe with a specified south pole and north pole. Then it's not a sphere where any point could be the axis. Instead, there is only one axis for the globe, and so basically it is a cylinder like the usual map of the world, where, as usual, the right hand side is adjacent to the left hand side. There may be a handful of "starting points", it's still not clear.
I also made a second map based on the system of ways of figuring out which I've uncovered: http://www.ms.lt/sodas/Mintys/MatematikosR%C5%ABmai Here's that system linked together with the areas in math: http://www.ms.lt/derlius/Math/MathWays/math.html
These maps use "organic" layout in yEd, which is most compact. Other possible layouts include: hierarchical, orthogonal, circular, tree, radial, series parallel.
Here is a third map based on the circular view: http://www.ms.lt/derlius/Math/MathWays3/mathways3.html You can zoom in. This view was very helpful for seeing how the nodes group together by subject. There do seem to be some general patterns in terms of content. I tried to pick a node from each group and make a large node so that it would stand out. The groups are I think more arbitrary than they may seem, however. Anyways, this was helpful.
Everybody is welcome to download the data and try it out in yEd. http://www.ms.lt/derlius/Math/MathWays3/mathways3.graphml Perhaps somebody can put it on a 3-D surface.
This circular view also reminds me (superficially?) of circle folding. Bradford, I look forward to folding more circles and sharing how that goes.
Hi Kirby, Joseph and all,
Kirby, thank you for mentioning Synergetics. I will look into that and add it to my map of math areas. http://www.azimuthproject.org/azimuth/show/Andrius+Kulikauskas I hope to work on that tomorrow. Also, I want to highlight some areas that I think relate to my philosophy.
Kirby, thank you for encouraging me regarding my map of the big picture. I will keep working on that. I like your idea of adding a time of discovery, thank you! I now made number theory more central...
It's getting messy. I wonder if anybody knows of a diagramming/visualizaing tool that I might try to use. I'm currently using DIA. I'm thinking of trying out TouchGraph which I've used before. There's a free version: https://sourceforge.net/projects/touchgraph/
Algebra and analysis