Mathematical structures: Foundations & explorations

Overview

Excerpts

Structuralism, as a philosophy of mathematics, is a particular set of beliefs regarding the ontology and epistemology of mathematical objects. It is in many ways a response to the contemporary Platonist view, which treats mathematical objects as individuals that exist independent from one another, e.g. the number 3 exists whether or not the rest of the numbers exist. For Gottlob Frege, a number is a concept. When one says ‘there are two authors of Philosophica Mathematica’ or ‘there are no moons of Venus,’ one evokes the concepts of 2 or 0. These statements may or may not be true but the numbers themselves, as concepts, are ontologically independent of the statements. “That a statement of number should express something factual independent of our way of regarding things can surprise only those who think a concept is something subjective like an idea. But this is a mistaken view.” Thus, according to Frege, concepts are objective. Whether statements regarding concepts are true or not the concept exists independently from our regarding them. The structuralist rejects this view on the grounds that the existence of a number is dependent upon the existence of the other numbers.

The focus of this paper is on a particular type of sequence of numbers known as a graceful permutation and the enumeration of them with respect to their length. In the literature so far, the best lower bound on their enumeration is given by Michal Adamaszek. We use an alternative method from Adamaszek’s which involves reducing infinite trees to finite directed graphs and using techniques from linear algebra to compute the largest eigenvalue of that directed graph’s adjacency matrix. This gives us a higher lower bound than has been found so far. We also find the exact growth rate for graceful permutations with particular properties.

Reflections

I did not come to Marlboro with the intention of focusing on mathematics but I got hooked on the first several math courses I took. I ended up spending a summer at Marlboro to do mathematical research with Matt which is when my plan really started to take shape. At this point all of my philosophical thinking had shifted towards thinking about mathematics. How do we acquire mathematical knowledge? What exactly are these abstract objects I have spent so long studying? These are the questions I asked myself over and over again that summer which became the early writings for my philosophy paper. The campus is absolutely gorgeous in the summer so I would take long walks on the various trails around campus contemplating mathematical problems. That summer I internalized the famous quote of Friedrich Nieztsche: “All truly great thoughts are conceived by walking.”

My research project is the most important part for me because it is genuinely new work in the field of combinatorics. The results Matt and I came up with were new improvements on previous results in the field. I will be continuing work on this project over the summer because I believe I can get it published in the Journal of Combinatorics or a similarly prestigious academic journal.

I hope to go to graduate school for a master’s degree in teaching mathematics. Before I went to Marlboro, mathematics was a dry and boring subject. It was as if there was nothing new to it: just follow the formulas and apply them on the tests. The math courses at Marlboro taught me that mathematics can be fun and creative. In high school, mathematics is a source of intense anxiety for many students. I hope to apply the techniques I have learned at Marlboro to make mathematics a subject that is fun and approachable for everyone.