Everything is connected... but how?

    Mathematical reasoning is not exclusively grounded in the use of numbers, and it does not need to be inaccessible to people who lack a knack for the numbers. At the same time numbers can be very handy for describing different kinds of relationships, and people who don't completely understand how the math works behind the scenes can still benefit from the information provided by logically structured and quantified arguments.

Collaborative Knowledge Distribution and Problem Solving Networks

    The modern world is faced with some truly astronomical problems, having to do with the environment, the distribution of resources among its residents, and of course the ability our species has developed for self-destruction. Fortunately, we are also equipped with some unbelievably powerful tools. Cloud computing and social media are still blooming technological industries whose full potentials are as yet no where near being realized. Lets go over what there is though.

Google

    The PageRank algorithm itself is a great example of math being used behind the scenes to everyone's great benefit. This brilliant application of Markov Chains was one of the most recent great leaps towards human beings attaining the ability to make decisions based on the collective knowledge of our entire species. One could almost call it the best thing since written language (Note Baidu is about the same age, those Chinese, and who knows what else is out there). Google of course didn't stop there though. Google Drive, Doc, Calendar, and of course Gmail (obviously they didn't invent email but it needs to be mentioned, might as well be here), are all services that revolutionize our workflow, letting us with people from miles away (we might as well throw Google Hangouts in here, though yes Skype came first), and access our documents anywhere with an internet connection. Google's search is still probably the most influential however, connecting people with information, people, and businesses they would otherwise never know about, enjoy, or make use of.

#KnowledgeDistribution

Wikipedia

    Next up comes the online encyclopedia of everything ever that anyone has ever talked about, or so it attempts to be. It seems to do a pretty wonderful job of it though, and the partnership between Google and Wikipedia is one of the few things that gives me hope for the future of humankind. Wikipedia is the current paradigm for crowd sourced knowledge aggregation. The way the encyclopedia is interconnected with links supports the way psychologists believe knowledge is structured in the human mind (Not brain, mind. We are talking about behavioral data here). A main point of support from the realm of psychology has to do purely with motivation. People tend to start digging around on Wikipedia out of real interest. Studies of attention tend to show that as we might expect, people are better able to recall information if they were genuinely interested in what they were reading in the first place. This comes back to Google, and especially to the Google-Wikipedia alliance. For the readers at home - never stop Googling things when they interest you, never tell yourself you are wasting time on Wikipedia. I'm not entirely sure there really is such a thing.

#KnowledgeDistribution

#KnowledgeRepresentation

Reddit

    After discussing that enormous body of information that is simply there for the reading, we should probably discuss the one that lives and breathes. Well, one of the digital ones that is. Up-voting, down-voting, link karma, and comment karma are all ideas that fill me with heady thoughts of distributed cognition. We will have to come back to this after we discuss logical structures and the current under-use of the idea of isomorphism in social media.

#KnowledgeDistribution

Facebook and Twitter

    While we are on the topic of social media, lets talk about the sites everyone actually uses, or that most people use. This is how a lot of people, if not the majority of my generation at least, get their news. I personally think that it is much more efficient than televised news at least. Newspapers do tend to have more investigative reporting and less cat videos, but even these are more systematically biased than the aggregation of the things everyone you know personally finds important. The Facebook activist can be seen as a lazy bystander sitting at his or her computer instead of standing in a street protest somewhere, but is that really necessary? The first point of activism is almost always to spread awareness about something that may be wrong with our society, or about something that could be improved upon. This we can do through Facebook and Twitter, and it really can be meaningful. That isn't to mention the ease with which petitions can be created and signed over the internet. I have yet to discuss Twitter specifically, but don't worry. It'll come again up soon.

#KnowledgeDistribution

HashTagging

   This phenomenon is actually very mathematical to me. It conveys a very simple, widely interpretable "is related to" relationship between some concept or media object and the label of some other concept or media object. These links can be treated as edges in a graph where the concepts (network files; the Graphs) and objects (text, links, images, video, applications) are vertices.

#KnowledgeRepresentation

#Ok,enoughhashtagging. 

Wolfram

   The world's first large attempt at a computational knowledge engine that I know of, WolframAlpha is impressive, and Mathematica is so useful for so many things. I don't understand why they don't work with Wikipedia, but whatever. Their natural language programming project will be useful as well. It is young, and seems to have a lot of potential. They already have libraries of code to deal with different types of social network data. While we are talking about problem solving networks though,

Yahoo Answers and Cha-Cha

    From every text book physics problem to relationship advice, these human powered question answering services answer a lot of questions. They are also faster than Wolfram's buggy free version at giving you useful information, especially from a phone. The idea of forums in general, and this goes back to Reddit, is incredibly powerful and still underutilized in the political domain. Domain experts have valuable, targeted information to offer in response to reasonably well formed linguistic queries, and once some questions have been answered once, that answer can be easily reused. The challenge is to store them in a sensible way so that they may be retrieved when necessary. These types of systems could be incorporated with knowledge computation systems like the ones Wolfram is working on to great effect. It may even become reasonable to approach larger complex problems from this direction. However, new was to structure information and represent knowledge might be helpful.

Mind and Concept Mapping

    Mind maps are more and more frequently being used by professionals to structure their thought process and workflow, and to create demonstrations and present information. Concept mapping on the other hand seems to interest more psychologists and linguists.

CMAPS and IHMC

   Researchers at the Florida Institute for Human and Machine Cognition head a large collaborative online concept mapping project, and are connected with the engineering of the Semantic Web, as it is called. They do other interesting things too, like teaching robots to make use of spoken natural language (http://www.ihmc.us/Research/human_machine.php), and using our understanding of human cognition to most advantageously integrate human interaction into complex automated decision-making processes (http://www.ihmc.us/Research/knowledge_discovery.php). The CMaps interface comes along with great features, like the ability for multiple users to simultaneously edit a concept map, and a built in similarity test with several options that compares any two concept maps.

XMind, FreeMind, MindMeister, etc.

   The mind mapping game seems to always have been more about creating visual displays of information that they were about attempting to encode complex contextual information in way that can be retrieved by natural language queries. The structures that can be made in CMaps can also be made in programs like XMind, and I think that these mind mapping companies have done a good job of making the interface flexible and pretty.

The 3rdMind

    When I originally had this idea I had never heard of mind mapping software or CMaps, or the semantic web. What it amounts to though is combination of those and a lot of the ideas I have already been describing, with one added little feature. The ability to send messages between any two nodes. An object-oriented database would be formed that could be populated through crowd-sourced knowledge, pulled right from Yahoo Answers and Wikipedia themselves, preferably. Users would be nodes themselves, or indexed by nodes anyway. This approach would allow for personal knowledge databases to be compiled and accessed by others. 

Local Solutions

    One truth that I think new technologies could exploit, is that local problems often have local answers that can be addressed locally but informed by large, complex data that regards systems more or less holistically, on different levels one could say. Systems for rewarding and making use of people with good ideas could be more prevalent than they are now on a casual level. This is getting unfocused though, and I think I will pick up here sometime later.

The Koch-Sierpinski Ninja Star

Fractals

    A fractal is, in general, a mathematical set that displays self-similar patterns. Most people recognize these geometrically, as a set of points plotted in two or three dimensions. We think of them by the step, or iteration, going on towards infinity. Two of the first-to-be-discussed, simplest, and most recognized fractals are known as Koch’s Snowflake and Sierpinski’s Triangle. A few years ago I had an idea for how to draw a fractal that involves some similar generative functions, but is indeed quite distinct from both and possesses some surprisingly unique properties. I’ve been looking into it on and off ever since.

Koch’s Snowflake

    As Wikipedia explains: 

The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows:

1) divide the line segment into three segments of equal length.

2) draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.

3) remove the line segment that is the base of the triangle from step 2.

Like so:

   What I find most interesting about Koch’s Snowflake, is that its perimeter has infinite length. It will be interesting to see if the Ninja Star does as well, because its perimeter grows similarly to but slower than that of Koch's Snowflake.

Sierpinski’s Triangle

The cool thing about this guy’s fractal is that it has connections to other foundational patterns in mathematics, like Pascal’s Triangle and the Towers of Hanoi puzzle. It doesn’t have really cool limits in it like Koch’s Snowflake or the Ninja Star, but there are cool limits having to do with other things (like Pascal’s Triangle) that actually converge to it, which is also cool.

The Koch-Sierpinski Ninja Star

    What I did originally was to literally just start drawing Sierpinski's triangle inside of Koch's snowflake, which you can do. However, though it looks kind of cool there isn't much that is interesting about it that isn't interesting about the first two. I was interested in the area ratio of dark to light triangles, so continuing the Sierpinski pattern in the white triangle didn't appeal to me for long. I then took a liking to the way the corners of the inner triangles touched each other at a point, and decided to limit generation of new triangle to include only those. Notice the difference that emerges between the outer bounds of this first version and Koch's Snowflake ( You can see it in the fourth iteration ).

It wasn't until this year that I realized the whole snowflake can be inscribed into each blank triangle that composes it. That realization was how it came to look like this.

The area question was actually really easy before, though somehow I wasn't convinced at the time - it will stay 1/4 colored the whole time as it iterates on toward infinity. However, that changes entirely when the white area in one iteration starts to be filled with more Ninja Stars.

Notation

    After going through a few different versions of tables trying to count the triangles in each iteration of this thing, I settled on keeping track of the triangles by level using a sub-scripted variable TL sub k of n. This variable tracks how many new full triangles ( 3 blank surrounding one colored ) are created at level k, where a level k is defined as one less than how many triangles the base of the star that is sprouting the triangles is directly and indirectly inscribed into. This way we are are able to count the number of a given size of triangle by the n value at which they are produced, while reusing the original recursive generating function. This is necessary because the way it goes, 3 triangles sprout off of the initial one, but then only 2 sprout off of each of those and so on. Notice in the GIF that we add all of the triangles that there will ever be of a given size to the design in each iteration. Then a recursive formula for the number of triangles produced at each level can be:

**There is a typo here! The second to last line should read,

TL_k (n) = 3 (SUM from i = 1 to k - 2 of: TL_i (k - 1)) TL_1 (n - k) 

Basically we are just reusing the base sequence, which is a binary tree of valency 3.

Remember however that this only counts the number of new triangles at a particular level. A formula for the number of new triangles produced in a given iteration can be:

Then the total number of triangles is given by

The sequences given by both of these last two formulas look like this:

T(n): 1,3,9,30,96,309,996,3207,...

TT(n): 1,4,13,43,139,448,1444,4651,...

Neither of these two sequences are present in the Online Encyclopedia of Integer Sequences (OEIS), so that's sort of cool.

The Area Question 

    Now that we can track how many triangles are created in each iteration under the original assumption that all triangles of a given size are created in a single iteration, we can start to really pursue the question: What is the ratio of colored to blank space in the whole design as n approached infinity?

First we could define another sequence, this one an ordinary convergent geometric series. We count the area in terms of the original triangle at n=1 having an area of 4, so that each triangle composing it has area 1. Then each new triangle made in iteration n has an area of 4(1/4)^n. We plug this expression in to the sum given by TL sub 1 of n. This will calculate the area of the entire design as n approaches infinity. The limit we are interested is given below, followed by a simplification of the function we are trying to evaluate as n approaches infinity. The way we represent it initially includes recursive actions that we can't deal with in a limit.

    The limit of A(n) is now easy to calculate as n approaches infinity. It is equal to 10. This is interesting, because we can observe that the equilateral triangle that the design could be inscribed into has a area of 16 by our convention, and the hexagon within that triangle that clearly bounds the outward growth of the design has an area of 13 by that same convention. Now I will leave you with a limit that I have yet to solve, as it also includes recursive functions. We can define a function CA(n) to represent the total area of colored triangles in the design as well. We do so by starting at one fourth the area of the whole pattern at n=1, so 1. This is the area of the first colored triangle. We then add (1/4)^n of the number of the total number of triangles created at each iteration to the total colored area in each iteration. It looks like this,

Since T(n) is recursively dependent on the weird TL_k(n), it is tricky to get a closed formula for. TL_1(n) is actually in the OEIS, and a closed formula for it was not so hard to find. This on the other hand stumps me, and the corresponding limit is thus out of my reach as of now. The final relationship of interest is precisely the following; the ratio of colored area to total, 

In the spreadsheet where I came up with these recursive definitions I approximated this R(n) function up to the 17th iteration, and it actually seems that it may be approaching 1/2. I find that very interesting personally, but as of yet we cannot know for sure what the value is. We know visually that the limit must converge to some number between 0 and 1, because there are some white spaces which we know will never be filled, as given by the difference between 10 and 16, the area of the design and the equilateral triangle it is inscribed into.

Check out the spreadsheet. Just make a copy if you want to mess with it.

https://docs.google.com/spreadsheet/ccc?key=0AkvjjjAi1kyWdE9YcW1XWXlCeFJURUxBTFBHS3ZRSnc&usp=sharing

GIF of triangles by level.

Geogebra drawings all courtesy of Professor John Golden. Your beautiful pictures inspire me to my best :)

Formal Systems and Applications Thereof (continued)

    The ideas expressed in Hofstadter's first masterpiece, Gödel, Escher, Bach: An Eternal Golden Braid, are powerful, if confusing. However I think he makes them about as accessible as can be expected. He makes humorous use of dialogue in illustrating some of his more far-fetched points. He weaves concepts from mathematics, psychology, philosophy, computer science, and even biology together in a way that draws on one's understandings of some in explaining the others. The beautiful thing, is that the fact he can do that mirrors what the book seeks to explain.

Form and Content

    Central to how any formal system is utilized in a real world setting is the distinction between how it operates and how it is interpreted. How it operates has to do with the axioms upon which it is based and the rules governing the creation of theorems from axioms and other theorems. How it is interpreted has to do with another key word.

Isomorphisms

    What makes a formal system useful, is a powerful isomorphism between the axioms and theorems of the system and some feature of reality. Abstract Algebra is useful in one instance, because if in the real world you have one apple, and someone gives you another, you have two apples. Abstract Algebra does not rely on that fact itself to produce the true statement 2 + 2 = 2, yet in that case and in any similar cases it will produce a theorem that corresponds accurately to our observable reality, given that we interpret it reasonably. The system was specifically developed to so, and is as such an extremely powerful tool in executing all kinds of tasks.

Meta-Mathematics

    Kurt Gödel understood this very well, and so took the path of using a formal mathematical system (number theory) to study itself. By "stepping outside of the system" he used the system itself to prove that it (number theory) was either inconsistent or incomplete.  The more wide reaching implication of Gödel's Theorem is, by the nature of his argument, which can be imported into any sufficiently powerful formal system, that every formal system is by necessarily either inconsistent or incomplete. That is to say it either contradicts itself, or it is not capable of producing every theorem of the system that reality dictates should be true.

The Future of Mathematics in Everyday Life

    We live in an age where technology is developing more rapidly, one might worry, than anyone not directly involved in the development is ready for. The key to maintaining stability, fairness, and equality among our fellow human beings in the coming years will lie in utilizing new technologies reasonably and responsibly. Mathematics plays a critical role in all of this, as every day we find more efficient ways to meaningfully quantify and make use of dimensions associated with economics, engineering, medicine, and psychology, while more and more frequently mathematicians and physicists are found knocking at even philosophy's doorstep. But what of it? Has it not always been the case that scientists are pushing the boundaries of our knowledge to places we have never seen before?

    Absolutely, but lets go over a specific example of a technological area that could become harmful or unsustainable left unchecked, and some that have potential to serve as checks on those. The details tend to hold the truth of a matter, and herein I believe is evidence that we face challenges today that are, by any quantitative measures, beyond the magnitude of almost any we have ever faced as a species. The way I see it, systematic decisions about the distribution of resources in a nation should be made according to scientific realities with the goals of maintaining the health, well-being, and productivity of its citizens and guests. At present no comprehensive system for effectively collecting and aggregating professional opinions in such a way seems to exist.

The Food and Pharmaceutical Industries

    We start with one of the larger problems in the United States as it stands. The free market is almost by definition not supposed to have entirely benevolent intentions towards their customers; companies and corporations are expected to compete for profit above all else. However, a successful free market must be regulated by government bodies that do have benevolent intentions towards their residents, not towards the profits of their corporations. It is not conspiracy that the FDA is controlled by if not entirely run by former executives of the very institutions they are charged with regulating. It is fact. 

Corn

    I introduced this whole topic with the idea of scientific realities. Lets generalize that to quantitatively significant statements. At the expensive of seeming less than scientifically professional, I'm going to use some hash tags to categorize statements without getting any more excessive in my use of headings and subheadings. I am not an expert in many of the areas I tag, but nonetheless I believe my impressions to be accurate-ish. It would be nice to have a system that let me easily find and cite legitimately peer-reviewed sources of information a little more intuitively than Google does now (which is already leagues ahead of the previously best system, the library).

Corn is among the most subsidized agricultural products in the United States #policy

Cows are often fed a stock of almost all corn, #foodIndustry

regardless of the fact that cows are evolved to eat grass, not corn #biology

I'm fairly certain growing nothing but corn is bad for the dirt itself, fertility-wise #agriculture

Corn contains very simple carbohydrates,  #chemistry

and is most commonly refined into pure glucose #foodIndustry

This corn syrup is in almost everything. #theGroceryStore

The human body is not evolved to process anywhere near the levels of pure glucose an average American consumes #biology

The risk of diabetes for an average American is and has been growing #publicHealth

Obesity is a leading cause of death and a driving force behind rising health care costs in the US #publicHealth

Those health risks are most prevalent among minorities and low-income families, who presumably eat more of the products that government subsidies render inexpensive. #I could go on, but I hope I have made a point by now.

    What I mean is this: if the people in charge (of this specific area and others besides) were taking into account the actual numbers, that correspond to actual reality, they would not make the policy decisions they make. It's easy to blame financial interests and greed in this situation. It is more difficult to move people to do anything about it. First, someone has to make them listen, then understand the problem. Even then we are left the arduous task of convincing them that they can do anything to help, and then guiding them through the process of actually doing what they can to help, i.e. nominating and then voting for officials who will enact the necessary policy changes. Scientists are busy people. They need help with this stuff. This corn example is simply meant to illustrate one area in which rigorous discussion could be started from general tagged comments, if there were a system designed to link quantitative results to those tagged statements ad hoc.

Genetically Modified Organisms

    I believe that the scientifically literate opinion on GMO's in general is simple: they are a profound agricultural breakthrough that should be exploited energetically, but carefully. Furthermore they should be created with altruistic, not profit-based goals in mind. There is an enormous problem when it comes to the way genetic modification is being conducted. We are killing all of the bees! The enormity of that quasi-fact is more than present enough to warrant treating it like a fact when it comes to corn that is genetically modified to naturally contain pesticides. But I digress.

Social Media

    A never-before-seen power for solidarity and free distribution of critical information is present among the hoards of kitten videos and relationship advice columns we find on sites like twitter and facebook. However, the machinery is not doing all it can do yet, and won't be until it is purposefully nudged in the right direction. 

Big Data

    Scientists have been making use of the huge data pools that come out of social media sites for a while now, but at this point advertising agencies seem to have more fun with it. I posit that this can change, and moreover that it needs to change. I believe that the scientifically, mathematically, and statistically literate can make a push for expanding that literacy through social media, to the benefit of all. I believe that people need to be aware that they are, in addition to being a human being, a data point. It feels dehumanizing, and to some extent it is. However, it is the reality we live in and to make the best of it, we could for instance maybe find ways to make racist data points see themselves in the context of the whole spectrum of racial awareness.

Quantitatively Transparent Media

    The whole point of my corn rambling was to illustrate a portion of government policy that I find reprehensibly idiotic.  I think that we as human beings are at a point in our evolution where we can make clear statistically at least when something reprehensibly idiotic is in fact, reprehensibly idiotic. If only there was a way to broadcast the truth of such things to everyone in the world in a way that all could understand and believe... it seems no one tries because they don't think it possible. I think all we need are much more interactive systems that allow people to record their thoughts and beliefs relationally to see how they integrate with each other and with scientific evidence.

Where are the p-values?

    I know that most people don't understand what a p-value is, but that doesn't mean they shouldn't have some awareness of what one is. The news of the future is customizable to the viewers knowledge and interests. There must be ways to make it clear that if people want the truth they need to see the p-values. This is shooting a little high, I know, but any movement in that direction would be good movement. I started this blog with a discussion of how numbers may or may not lie. Well, they have a lot less motivation than human beings to do so. That much is certain. The key to defeating corruption in government policy is not to make lying more difficult. It is to make telling the truth easier. To do so will require advancements in education and user interface design, all of which are well within reach.

MoKai's Ninja Star

I have been thinking about this triangle / star shape for years now, and this semester it is hopefully going to really come together. I made these images with Geogebra, and they start by first portraying just the growth outward.

 Then on the last one I show what it looks like if we inscribe this whole thing scaled down into one of the empty triangles.

Before I started trying to work with GeoGebra I did some drawings by hand, in which I let the design grow inwards as well as outwards. The rule I used was to add all of the triangles of a particular size to the drawing where ever they fit.

 Notice that when I drew these ones I did them all inscribed into an equilateral triangle, as the design would be when placed inside of itself.

 On the last iteration that I drew I realized that the whole design could be inscribed into the irregular hexagon you see below. This will end up being crucial to calculating the areas involved in the design as it iterates towards infinity. The triangular pieces left over when the hexagon is drawn inside a triangle of corresponding size will be the only untouched spaces in a way.

Formal Systems and Applications Thereof

A common reason people who find mathematics fun and interesting tend to give for their interest in it can be summarized in four words: The numbers don't lie. I personally think the saying has a ring to it, but perhaps the most important question for a mathematician to ask is this: do they? Do the numbers lie? The answer is not exactly simple. It has a lot to do with how one is to define the term "lie", and at least as much to do with how that definition applies to personified symbolic representations of abstract concepts. Anyone who studies statistics will tell you that it is very easy for people to lie to themselves about the numbers, but is that really the numbers fault? When it comes down to it I think a decent, simple answer is that numbers and mathematical reasoning in general are worth precisely as much as their interpretations are.

Gödel, Escher, Bach: an Eternal Golden Braid

I am not the first person to discuss questions like these. A man named Douglas Hofstadter has already done quite a bit of very good work in elucidating the kinds of points I want to make, and as I read his book I will be using some of his language to discuss relationships that exist  between thought, reality, and mathematical reasoning.

Strange Loops

A central theme and something that Hofstadter discusses early in his book is one of self-referential statements he calls strange loops. Such loops are depicted visually in paintings by the artist M.C. Escher, as in his painting of two hands that appear to be simultaneously drawing each other, and auditorily in the structure of some musical peices written by Sebastion Bach. Most importantly though, Kurt Gödel used one to prove mathematically that no consistent axiomatic system could possibly produce all of the rules of number theory. This result was an important step in the history of human beings discovering the meaninglessness of absolute truth, a path with roots as deep as those found in the Daodejing and Zhuangzi of ancient China. Great minds like Schrödinger, Heisenberg, and Einstein have since unleashed general and special relativity as well as quantum field theory upon the public. What makes Gödel's Theorem so interesting and beautiful is its direct connection to one of these strange loops, because those strange loops seem to be very connected to what makes us what we are. Another mathematician by the name of René Descartes is known for having said "I think, therefore I am," a foundational philosophical statement. This statement could be called a strange loop itself, If you assume that someone has to be in order to think. Are we not ourselves simply complex self-referential statements?