Andrius Kulikauskas

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See: Math

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?" 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. (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: 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.


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?


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