Chapter 1: The Nature of Mathematics

Is mathematics entirely self-referential? The straightforward answer is no, despite the common misconception.

Understanding True vs. Is True

Why do I argue that mathematics cannot be simplified to a self-referential system? In other words, can mathematics merely be reduced to examining foundational principles (axioms) and rules of deduction (implications)? For instance, when a mathematician utilizes proof-checking software, this is what she is effectively doing.

When stating "if X is true, then Y" based on axioms and deduction rules, one creates a closed system that may not yield practical insights. The error lies in confusing the representation of mathematical domains—symbolic axioms and deduction rules—with the mathematics itself. An essential preliminary step is often overlooked.

However, delving into mathematics reveals that the most challenging aspect is crafting definitions. With modern notation and concepts, I can articulate ideas that were unformulated 200 years ago. Significant advancements in mathematics stem from precise definitions and clearer statements.

The confusion arises from conflating two distinct aspects. Once definitions are established, we can represent them as a set of axioms and deductions. Thus, once we articulate what we aim to prove, computers can assist us in solving it, albeit they are not particularly adept at this yet.

Nevertheless, mathematics encompasses more than mere deduction. It involves classifying concepts into meaningful categories, formalizing them into axiom sets and rules of implication, and ultimately deriving valid conclusions—proofs. It is noteworthy that mathematical fields typically begin with problems, proceed to solutions, and finally formalize methods.

Thus, mathematics claims to be more than an "if true" framework; it aspires to be an "is true" or perhaps an "is meaningful" system. A prime example of meaningful categorization is the equality sign.

Group Theory and the Rubik's Cube

To model and resolve a Rubik's Cube using group theory, several elements are necessary. First, one must conceptually link the abstract principles of group theory to the physical object of the Rubik's Cube and its manipulation. This process is successful only because we have developed meaningful categories.

(For more on group theory and the Rubik's Cube, see here.)

Bridging the Conceptual Gap

A core issue is that pure mathematics increasingly engages with abstract concepts. Through notation, we can simplify it to a spatial problem; however, these manipulations are fundamentally conceptual. Without a solid conceptual grasp, the ability to apply techniques becomes limited, as one may not fully understand the underlying principles.

When faced with complex conceptual manipulations, we often feel overwhelmed. Examples serve as crucial clarifiers. After all, a professor cannot always point to the conceptual object; you must discover or create the concept independently, particularly if you align with Brouwer's intuitionist perspective.

If I were to rearrange several objects on a page, it would be unreasonable to deem it meaningless. The assertion that it lacks meaning is, in itself, absurd.

As I navigate through abstract manipulations, the temptation arises to label it as self-referential, partly due to the computability issues discussed, but largely because concepts often seem enigmatic. I may feel that I grasp my mathematical reasoning, yet there is a limit to how much I can elucidate my thought process. Ultimately, I rely on others to infer my intentions, rather than articulating them explicitly.

This notion parallels our understanding of physical objects. Just because I cannot articulate the concept of "greenness" does not strip it of meaning. A green object may be present, yet explaining it to someone blind is challenging. I can point it out to a sighted person, assuming they will perceive what I see.

The Essence of Emotions and Literature

Similarly, I cannot adequately convey the experience of pain without referencing my own understanding. Yet, it would be absurd to deny its existence. At thirteen, I mistakenly concluded that emotions were nonexistent—a naïve thought that ultimately led to considerable distress.

Attempting to explain how we comprehend a sentence reveals the complexity of understanding. You grasped the previous sentence but may find it challenging to articulate how and why.

The categories within mathematics are as mysterious and significant as any other aspect of our existence. I hope this discussion sheds light on why mathematics holds a unique position, akin to every other element of our somewhat mystical reality.

Epilogue I: The Question of Self-Referentiality

If mathematics were truly self-referential, how do we comprehend the implications of a self-referential system? Are we expected to grasp what an implication signifies devoid of context? How can we understand an abstract statement, perhaps represented by P?

If these necessary abstractions or conceptual manipulations are achievable, why not others?

Epilogue II: Inductive vs. Deductive Learning

One epistemological challenge arises when we convert meaningful categories into valid axiom sets and deduction rules. For example, Euclid’s geometric principles were primarily justified through visual intuition. However, special relativity indicates that these axioms do not universally apply.

Similarly, when formalizing set theory, some axioms may come across as somewhat artificial. This contrived nature emerges from the need to prevent contradictions.

Mathematics requires a blend of deductive and inductive learning. The inductive aspect appears in two areas. Firstly, in practical applications, Euclid’s laws may not hold precisely, but they suffice for our purposes. Secondly, in refining our axioms and deduction rules. In a previous article, I extensively discussed the complexities of establishing appropriate logical frameworks.

This highlights the importance of precise definitions, as articulating relevant concepts, axioms, and deductions is no simple task. For those interested in my ongoing exploration of this subject, I refer you to an article I wrote a year ago. I would like to express my gratitude to Frédéric Prost, whose recent response inspired my late-night reflections.

Chapter 2: Exploring Mathematical Concepts

In this video titled "Sets, Logic and Computability | Math History | NJ Wildberger," the presenter discusses the foundational aspects of sets, logic, and computability in mathematics, offering insights into their historical development.

In the video "13-1 Pure Mathematics Set Theory, Logic, Recursion, Computability, Model Theory," the speaker delves into the intricacies of set theory, logic, and related concepts, providing a comprehensive overview of these essential mathematical frameworks.