The Collatz Conjecture: A Dance of Simplicity and Complexity
This portfolio is a reflective outlook on my final project, “The Collatz Conjecture: A Dance of Simplicity and Complexity“, a book that aims to bridge the disconnect between the number of people who like mathematics and those who actually pursue it by inspiring young mathematically inclined students and giving them confidence in their abilities. In this portfolio I also ponder about the things I have learned and incorporated into my project this semester.
First, I learnt about positionality and its important for both research and writing. Being aware of one’s biases and thoughts only enhances one’s understanding of their audience and allows them to appeal to their audience in a more sophisticated manner. Talking about the meaning of the Swastika in class, and the differing perspectives people have about it made me cognizant of the differences in opinions that I would have to be aware of before writing my book. It was especially important for me because, as I stated in my project proposal, “my perspectives are colored by the belief that mathematics is the language of the universe and of logic and that mathematics underlies the simplest and the most complex of things.” However, I understand that my whole audience would not share the same belief and while interested in mathematics, they could be of the opinion that mathematics is not a language of the universe but is a human language invented to describe the universe. Therefore, I revised my positionality statement and in my final project, I took a balanced viewpoint that mathematics in its current form is a human invention but it describes what is the language of the universe. This statement from the first chapter reflects this: “I believe that the normal world behaves like the Platonic world, only that we have yet to gain knowledge of what differentiates a number
like 0.99999999999……… and 1.”
Secondly, it was important to balance facts and opinions. While I speculated that this was a completely fact-based and impersonal book, it turned out to be a combination of both. I learnt from the Kurbin, et.al. reading that personal experiences can be very convincing. I noted that this could be applicable in the philosophical chapters of my book, including the first chapter. I explore the nature of my topic in the first chapter and talk about things including Wigner’s enigma (is mathematics an invention or discovery) and the connection of personal experiences and objectivity. While giving examples of algorithms and formulae, representative of facts, I also talked about anecdotes from the lives of personalities like Kurt Godel. For example, this excerpt from my book combines both facts and anecdotes: “In fact, these theorems may also have been a product of Gödel’s cultural environment. In the early 1900s, Gödel joined a group of mathematicians called the Vienna Circle. Gödel’s beliefs corresponded to those of Platonists, a school of thought believing in the abstract nature of mathematics. However, the Vienna Circle consisted of empiricists and a group of anti-Platonists known as logical positivists. Gödel’s characteristic paranoia led him to oppose them further and come up with his theories.” In fact I anticipated the technical nature of my topic to make this book completely impersonal and talked about this as a challenge to convey my message well to my readers in the introductory video. I think working with anecdotes and experiences of other people really improved my book and would appeal to the audience.
In the introductory video, I thought using multimodality would be a problem. However, as I learnt from reading research and articles and making annotated notes, the topic I worked on can be a very visual one too. I read about things that could be very useful for creating diagrams and illustrations for my book so that I can explain the problem better to readers and can make the content more interesting. The diagrams I talked about also give context to explanations about the ‘passive uses of mathematics’ and its presence in nature given the natural coral-like looking representation:
I also used references too and images from shows and movies like the Simpson’s in order to appeal more to the audience. The language I use is easy to understand and uses jargon but only after explaining it properly. This was a key element I learnt from my presentation, where I used topic-specific words assuming I did not need to explain them to my audience. For example, I introduced hailstone sequences as follows:
“Take any number. For your convenience, keep the number small. Now calculate a sequence of numbers using the following:
- If the number is even, divide by 2.
- If the number is odd, multiply by 3 and add 1. Keep repeating the process until you feel the numbers
are repeating.…Did you get the following repeating numbers? 1, 4, 2, 1, … Now take any other number and try the same.”
Therefore, I believe I have used multimodality appropriately.
Artefacts
Now, to give a brief introduction to my artefacts:
Artefact 0
Artefact 0 is my introductory video. It provides information about expected challenges. I wanted to present the video as a news channel, with the challenge being a forecast. I tried to bring out my preference to use formal and non-fictional writing through this video and also wanted to showcase a bit of creativity.
Artefact 1
A Research Proposal explaining aim, target audience, table of contents and chapter introductions and plans.
Artefact 2
Research Notebook documenting my notes, annotated references and research.
Artefact 3
Final Project Chapter 1: Introduction to mathematics and unsolved problems in general, about how mathematics represents a logical understanding of the world and the universe and how it involves not just numbers and calculations, but patterns and challenges that intrigue the human mind.
Artefact 4
Final Project Chapter 2: I go on to give a number of examples (Take a number greater than 1. If it cannot be divided by 2, multiply by 3 and add 1, otherwise divide by 2. Repeating this gives 1 in the end.) This party trick is what the Collatz Conjecture assumes to be true. I explain the topic in detail.
