Chapter 1: The Linguistic Nature of Mathematics

Mathematics fundamentally serves as a language. Like all languages, it is a construct of humans, rich in expression yet fraught with implicit knowledge and ambiguous meanings. This piece delves into how the expression of mathematical concepts intertwines with programming and technology.

Mathematicians often claim that mathematical notation is clear-cut. However, consider this:

Let’s denote Figure 1 as the notation for a mixed number.

And Figure 2 as the notation for an algebraic expression.

Figure 1 comprises the elements {3, 1/3}, while Figure 2 consists of {3, x}. Despite both notations aligning closely, how do we determine the operations between these elements? If you interpret Figure 1 as "addition" and Figure 2 as "multiplication," how did you arrive at such differing evaluations?

At its core, mathematics is a language, and like any language, it is inherently complex, shaped by its historical context. Various fields—such as probability, statistical mechanics, computer science, economics, and machine learning—cultivate their own specialized dialects. This results in not only different terminologies for mathematical objects but also unique representations of these objects.

Metalinguistic. Definition: Language that discusses language itself.

For instance, the term "one-hot," as in "one-hot encoded variable," originates from digital circuit design and signifies a group of bits where only one bit is active (1) while the others are inactive (0). "One-hot" serves as a metalinguistic term for representations like "00000001", "00000010", and "00010000".

However, ambiguity in mathematical language is not limited to metalanguage alone. Consider the example of double factorials and semifactorials, denoted as n!!. While these notations may seem equivalent, n!! is not identical to (n!)!, and the term "double" actually implies a greater magnitude than "semi-".

Double factorial. Definition: The product of all integers up to n that share the same parity (even or odd) as n. Also known as the semifactorial.

Example: The semifactorial for even, odd, and zero values.

In the realm of mathematical study, many students remain oblivious to the fact that they are learning a foreign language. Mathematics has developed alongside human language for millennia across various cultures—from Asia and India to Mesopotamia, Europe, Africa, and Mesoamerica. This linguistic evolution is as intricate and layered as any established language.

The lack of awareness regarding this linguistic challenge exacerbates math anxiety, particularly among underrepresented groups, including women and minorities, who are significantly fewer in STEM fields. High-paying jobs in technology remain unfilled, not due to a lack of interest, but rather because many individuals do not speak the language of mathematics.

Among those proficient in mathematics, some may prefer to maintain exclusivity within the community, using mathematical notation as a shibboleth to reinforce their status and professional reputation.

Shibboleth. Definition: A custom or phrase that identifies individuals within a group.

Example: "If you're on any tech support call, saying 'shibboleet' will connect you to someone knowledgeable in multiple programming languages."

Mathematics often acts as a shibboleth, leading to contempt for those who struggle to interpret its notation. This phenomenon is not unique to data scientists and programmers; for centuries, lawyers in England have employed complex legal language to safeguard their profession's elite status.

Unlike Law French, mathematics serves as a universal language. It is the foundation of algorithms that power our programs and the scientific principles that explain our reality. Math embodies creativity, playfulness, art, and humor, continually captivating humanity throughout history. As Francis Su argues, mathematics is fundamentally tied to human flourishing.

The importance of language in fostering human development is undeniable. Consider the story of a deaf man who, raised by hearing parents unable to teach him sign language, did not comprehend the concept of language until he turned twenty-seven. A woman demonstrated both sides of a conversation, allowing him to grasp the connection between symbols and their meanings. His emotional response upon discovering this connection was profound, illustrating the essential role of shared symbols in human interaction.

The absence of language renders life inhuman. We have an intrinsic need to nurture our curiosity, assign significance to concepts, and communicate with others through shared symbols. Mathematical language emerges as a means to convey these ideas.

In the 1930s, linguistic anthropologists Sapir and Whorf introduced a theory positing that language constrains our cognitive abilities. In its most extreme form—linguistic determinism—it suggests that lacking a word for a concept means one cannot understand that concept. This viewpoint has been used by some to assert superiority over others.

Linguistic determinism. Definition: The belief that language limits human cognition and understanding.

Example: The Hopi language lacks tense markers for verbs, implying that its speakers do not conceive of linear time.

Claiming that certain groups are "just not good at math" because they have not yet mastered its vocabulary serves as a gatekeeping mechanism across disciplines reliant on mathematics. This narrative suggests that those lacking fluency are unwelcome to learn.

Conversely, languages are human inventions designed to meet our needs. The mathematical notation we create contradicts the Sapir-Whorf hypothesis, demonstrating that it is thought that shapes language, leading us to develop innovative writing systems and visual representations to communicate logical progressions.

While there is a weaker form of the Sapir-Whorf hypothesis that suggests language influences cognition, it does not impose strict limitations. This perspective acknowledges that language can shape our understanding of information.

Linguistic influence. Definition: The notion that language affects cognitive processes without determining them.

Example: Individuals shown a racially ambiguous photograph and asked to draw the person based on differing racial cues produced varying representations.

Kenneth E. Iverson's Turing Award Lecture, "Notation as a Tool of Thought," examines this idea of linguistic influence, asserting that mathematical notation is a powerful cognitive tool. However, he notes its limitations, such as its lack of universality and the need for contextual interpretation.

We employ mathematical notation to formulate concepts and reframe old challenges, creating symbolic bridges that connect theorems to proofs. Yet, as Iverson points out, notation varies by topic, author, and context, which may explain why many find programming more approachable before encountering the underlying mathematics.

No one is born fluent in mathematics; we all acquire it through exposure and practice. Achieving proficiency in any language necessitates interaction with both the language and its speakers. Engaging with others is crucial for mastering mathematical concepts, as self-study alone often fails to yield fluency.

Language is fundamentally a tool for communication, meant to be inclusive rather than exclusive. We have a responsibility to cultivate a welcoming environment in mathematics, clarify our own notation, and teach one another the language central to our lives and careers.

Every human possesses the potential to master any language, including mathematics.

Chapter 2: The Role of Language in STEM Inclusion