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Epistemology - m a t h 4 w i s d o m - g m a i l
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See: Math collaboration May 4, 2016 Math Future Worlds Kirby, Bradford, Joseph, Ted and all, I'm grateful for your letters. Kirby wrote about "worlds" in math education. I will try to build on his concept, talk about my own world and relate it to other worlds. First of all, math is so rich that we can develop our own personal worlds. Each of our worlds thrives on themes that we find especially evocative. In this Math Future forum, we get to write about our worlds and their themes, and that helps us understand and inspire each other. Some themes are: - the tetrahedron and other Platonic solids in Kirby's World
- circle folding in Bradford's World
- spreadsheets in Joseph's World
- patternmatics (teaching math, computer science, and logic together) in Ted's World
- the map of areas in math in Andrius's World.
Aside from these personal worlds, we also have social worlds: - the Mandated World, the math which one is required to learn or teach in K-12, organized sequentially
- the Academic World, which is divided up into highly specialized areas
More generally, we can talk about the worlds of jazz, chess, rowing, dance, cooking, ship building, and so on. Related to math we have worlds of physics, engineering, computer science and other disciplines. Aristotle thought in terms of "techne" (know-how) and Heidegger simply called these "worlds". I like to think of each of these worlds in terms of their "ways of figuring things out". My main interest is in Andrius's Philosophy World, which I'm writing a book about. But I want to show that I can usefully systematize any of these worlds in terms of their "ways of figuring things out". I think I can show that an instance of the same system of 24 ways is at work in each case. I would like to describe that for math and physics and produce some results that would at least inspire people to get interested. In physics, that amounts to surveying the kinds of experiments that have been done. Fields medal winner Terrence Tao presents a wonderful collection in his video lecture "The Cosmic Distance Ladder": Most of these can be understood using proportions or trigonometry. But, in general, I need to learn lots of math to truly understand quantum field theory and general relativity. I realized that I might as well try to master the "big picture" in math.I am actually not interested in my own world, but I would very much like to know or at least imagine God's World. By that I mean the "big picture", the absolute truth about absolutely everything. That's the goal of Andrius's Philosophy World. But I also believe that there can be at least one Big Picture World in math and likewise at least one Big Picture World in physics. Quite a few physicists are working hard on the latter. But since David Hilbert (1862-1943) and Jules Henri Poincaré (1854-1912) there's no mathematician who would openly dream of knowing a Big Picture World in math or even wanting to. An exception might have been Alexander Grothendieck (1928-2014). His work was highly abstract but apparently most influential. Now with the Internet, through math blogs, wikis, forums and video lectures, it is possible to find people like Terrence Tao, John Baez, Urs Schreiber and others who are not ashamed to be interested in a very wide range of math. My reason for writing about these worlds is to consider how they might be related. My particular interest with Andrius's Math World is to develop what might let me comprehend a Big Picture Math World. It's not critical for now that people understand me. Rather, it's important for me to understand others. I would like to understand and make sense of whatever they find insightful in their own personal Math Worlds, especially their ways of figuring things out. Also, I'm glad to learn what they may know of the Big Picture Math World, or simply, what might help me learn more math. Terrence Tao wrote his thoughts on "What is Good Mathematics?" https://arxiv.org/pdf/math/0702396v1.pdf This is a notion from the Academic World that is rarely discussed explicitly. He lists 21 meanings for "good math". But a general theme seems to be that good math relates different Math Worlds. I was thrilled to watch John Baez's video lectures on his "favorite numbers": 5, 8 and 24. http://math.ucr.edu/home/baez/numbers/ (Numbers 8 and 24 happen to be big in Andrius's Philosophy World.) His lecture on the number 5 discusses the golden mean and continued fractions. He explains how the golden mean 1.618... (related to the square root of 5) can be considered the "most irrational" number by noting that it has the simplest continued fraction 1/(1 + 1/(1 + 1/(1 + ... ... forever .... ) ) ) and so gives the most unsatisfactory approximations by rational numbers. He relates this to getting good rational approximations for pi (explaining why 333 radians is close to 0, it's because pi can be approximated by 1/(3 + 1/(7 + 1/15)) = 333/106 where 15 is quite larger than 1. Such talks encourage me to explore the connections he points out. His talk on 24 drew on all kinds of math such as Lie groups and Lie algebras that recur in the Academic World and seem should be part of the Big Picture World which I wish to learn. However, he's able to say that it ultimately may be simply that 24 = 6x4 where 6 represents the equilateral triangle world and 4 represents the square world. So that's very much a Kirby's World issue, I think. And as I learn how the the Lie groups are classified based on their Lie algebras, which are classified based on the Dynkin diagrams of their root systems, well, the latter are just the kind of coordinate systems for generating crystallographic lattices that Kirby thinks in terms of. So there's something relevant here for Andrius's Math World. What I think makes these root spaces so special and rare is that they balance two very different ways of defining a space: top-down and bottom-up. Suppose you want to describe a 3 dimensional space. In the bottom-up view, you would start with 0 dimensions, then add a first dimension (a first basis vector), and then a second dimension (a second basis vector), and finally a third dimension (a third basis vector). In the top-down view, you start with 3 dimensions, then remove a basis vector (a dimension), and be left with the 2 dimensional space that is perpendicular to the vector you removed. Then remove another basis vector to be left with a 1 dimensional space, and finally, remove a remaining basis vector to be left with 0 dimensional space. The space that you get when you remove 1 dimension is called a "hyperplane". And you can use the removed vector to define two sides of the hyperplane, positive (front side) and negative (back side). And so you can also define a "reflection" that takes a vector (its component in the direction of the removed vector) and reflect it over to the opposite side of the hyperplane. What makes the root systems special is that each system is a handful of vectors which keep to themselves both bottom-up and top-down. Bottom-up, you can generate all of the roots as integer sums of fundamental roots (basis vectors). Top-down, each root defines reflections such that all of the other roots are reflected into roots on the other side. So it is a system that is not only highly symmetric, but also finds a very special balance between the bottom-up (addition) and top-down (reflection) views of what a vector can mean. I think that this bottom-up and top-down "duality" is at the heart of "tensors". I keep trying to figure that out and I hope to write more what I'm learning as I do. "Duality" is another key idea in Andrius's Math World that I'm trying to master. I intend to go through this list of dualities at Wikipedia: https://en.wikipedia.org/wiki/List_of_dualities In Andrius's Philosophy World there is a key distinction between "marked opposites" and "unmarked opposites". In math, we might say that 1 and -1 are marked opposites. In the sense of addition, 1 and -1 are symmetric (they are unmarked opposites). But in the sense of multiplication, 1 is more basic than -1 because 1x1 = 1 but -1 x -1 = 1. This becomes important with the complex numbers. I write i and neglect -i. But actually, -i is just as fundamental as i. Multiplying by i takes you counterclockwise and multiplying by -i takes you clockwise. Both are square roots of -1. There is absolutely no reason for prefering one over the other. So the "marking" -i is highly misleading. I didn't realize this until a week ago. Now I suddenly understand why complex conjugation is so important, going back and forth between x+yi and x-yi. It simply says that it should never make a fundamental difference when you switch between the two, +i and -i. Sometimes it would be more helpful to call them i and j. This is a basic idea about complex numbers and what makes them different from real numbers. And then I realized that this relates to "circle folding". Because laying down this coordinate system is simply choosing a way to draw a line across the circle, which is to say, fold it. So those are some of the links between worlds that I'm finding as I build Andrius's Math World to try to understand the Big Picture Math World as best I can. I add below some questions related to the Big Picture Math World. Andrius Several big questions provide new angles on the big picture of mathematics as an endeavor: - discovery: What are the ways of figuring things out in mathematics? We can study mathematics as an activity by which we create and solve mathematical problems. These techniques are much more limited than the mathematical output which they generate.
- beauty: Mathematicians are guided by a sense of beauty. What is meant by beauty? What principles determine it? How does beauty lead to mathematical insight?
- organization: How can mathematics be organized so as to survey the most basic concepts from which it arises and understand it in terms of its most fundamental divisions and how they relate to each other?
- education: What resources are available to mathematicians that would help them most effectively learn mathematics so as to try to understand it as a whole? How might mathematicians collaborate effectively in trying to understand the big picture?
- insight: What are the most fruitful insights in trying to understand mathematics? How can such insights best be stated?
- premathematics: What concepts express intuitions that are prior to explicit mathematics and make it possible?
- history: How can the history of mathematical discovery inform frameworks for the future development of mathematics?
- humanity: What parts or aspects of mathematics are specific to the human mind, body, culture, society, and what might be more broadly meaningful to other species in the universe?
May 9, 2016 Kirby, Christian, 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. - There are a variety of ways to import data from Excel spreadsheets. I imported a list of 100 nodes that way.
- But the graphical user interface is also just right for creating notes and edges. I quickly created another 100 nodes and 300 edges.
- And there is a variety of export options including SVG, HTML Image Map and HTML Flash Viewer. I will show some examples of the latter.
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: - yellow nodes are areas of theoretical math
- orange nodes are areas of applied math
- blue nodes are math structures known for their beauty
- green nodes are math structures that I think would be helpful to be familiar with
- purple nodes are for ways of figuring things out which I'm systematizing (they appear in the second map)
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. May 22, 2016 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. Andrius 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/ |

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