# Understanding Mathematical Language in Machine Learning

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Mathematics is fundamentally a language, rich in expression yet laden with implicit knowledge and ambiguous meanings. This piece delves into how mathematical notation relates to programming and technology.

Many mathematicians assert that mathematical notation is clear and without ambiguity. However, consider the following example.

Let’s examine Figure 1:

Now, let’s look at Figure 2:

In Figure 1, the components are {3, 1/3}, while Figure 2 consists of {3, x}. In both cases, the elements are presented side by side. What operation is implied between the elements of these expressions?

If you interpret Figure 1 as “addition” and Figure 2 as “multiplication,” how did you discern the different evaluations when the notation appears identical?

At its core, mathematics serves as a language. Like all languages crafted by humans, it is expressive yet often requires a shared understanding of context and meaning.

Various disciplines—such as probability, statistical mechanics, computer science, economics, and machine learning—develop unique dialects and terminologies. The terminology for mathematical concepts not only differs across fields, but so do the forms of those concepts.

**Metalinguistic. Definition.** Language that discusses language.

**Example:** The phrase “one-hot,” as in “one-hot encoded variable,” originates from digital circuit design, denoting a set of bits where only one bit is high (1) while the others are low (0). “One-hot” serves as a metalinguistic label for representations like “00000001,” “00000010,” and “00010000.”

Yet, ambiguity in mathematical language arises not only from its metalanguage. For instance, the notation for a double factorial, represented as n!!, can lead to confusion, as it is distinct from (n!)! while the prefix “double” suggests a multiplication of two.

**Double Factorial. Definition.** The product of all integers from 1 to n that match the parity (odd or even) of n, also referred to as the semifactorial.

**Example:** The semifactorial for even numbers, odd numbers, and 0:

In different notation:

If you've engaged with any math-based discipline, you may not have realized you were learning a foreign language. Mathematics has evolved alongside human communication over millennia, shaped by mathematicians across continents and cultures. Its complexity rivals that of any ancient language, from Sanskrit to Arabic.

The lack of awareness regarding this linguistic aspect complicates the learning process, exacerbates math anxiety, and deters individuals from entering a growing number of science and technology sectors. Women and minorities are notably underrepresented in STEM fields, which offer significantly higher wages across educational levels. The unfilled STEM positions stem not from a lack of interest but from an absence of individuals proficient in this mathematical language.

Some who are fluent in mathematics seem to prefer maintaining a limited and exclusive community. For certain groups, mathematical notation acts as a shibboleth, a means to uphold their elite standing and professional reputation.

**Shibboleth. Definition.** A word, phrase, or custom that serves as a marker of identity or exclusivity.

**Example:** “During any tech support call, saying the phrase ‘shibboleet’ will ensure you are connected to someone knowledgeable in at least two programming languages.”

Mathematics often serves as a shibboleth, casting disdain on those who struggle to grasp its notation. This behavior isn't unique to data scientists and programmers; historically, English lawyers have used complex legal language to protect their profession's elite status.

Unlike Law French, mathematics is the universal language of the cosmos. It underpins the algorithms that govern our technologies and the scientific principles that explain our world. Mathematics embodies creativity, playfulness, artistry, and humor, continuously captivating humanity throughout history. As Francis Su argues, mathematics fosters human flourishing.

Language is integral to human well-being. Consider the story of a deaf man who, raised by hearing parents without exposure to sign language, only recognized the concept of language at age twenty-seven when a woman demonstrated a conversation for him. He then began pointing to items around him, seeking their names and symbols:

And then he started—this was one of the most emotional moments of my life. He began to point at everything in the room. Suddenly, this twenty-seven-year-old man, who had previously seen walls, doors, and windows, wanted the symbol for "table," the name for "window."

The expression on his face indicated he was encountering these objects anew, as if they had transformed with the introduction of shared symbols. He could now communicate “window” to someone he hadn't met, illustrating the magic that occurs when humans share symbols.

That was his first “Aha!” moment; he became ecstatic, pointing to everything and mirroring my signs. Then he broke down, overwhelmed by the realization of what he had missed for twenty-seven years.

A life devoid of language lacks humanity. We have an inherent need to explore, assign meaning, and share ideas through common symbols. Language is the medium of exchange, and when those ideas are mathematical, we create a mathematical language to facilitate communication.

In the 1930s, linguistic anthropologists Sapir and Whorf posited that one's mental capacity is restricted by the language they speak. This strong view, known as linguistic determinism, suggests that lacking a term in one’s language limits one’s ability to comprehend that concept.

**Linguistic Determinism. Definition.** The notion that language and its structures constrain human thought and knowledge processes.

**Example:** The Hopi language, devoid of tense markers for verbs, implies that Hopi speakers may not possess a concept of linear time.

Claiming certain groups are “just not good at math” due to their unfamiliarity with its syntax and vocabulary is a similar gatekeeping mechanism that excludes underrepresented demographics from fields where mathematics serves as the common language.

In contrast, language is a human invention, crafted to meet our needs. The mathematical notation we develop contradicts the Sapir-Whorf hypothesis: it is thought that inspires language, leading to innovative applications of existing systems, drawings, colors, and substitutions to illustrate logical connections from hypothesis to conclusion.

**Rainbow Proof Shows Graphs Have Uniform Parts.** On January 8, three mathematicians submitted a proof addressing a nearly sixty-year-old problem in combinatorics known as Ringel's...

While the Sapir-Whorf hypothesis has a weaker version, linguistic influence, which posits that language does not confine cognition but can shape it.

**Linguistic Influence. Definition.** The idea that language can affect human thought processes and knowledge.

**Example:** A study showed a racially ambiguous image to participants, with half informed it depicted a “Black man” and the other half a “White man.” The group perceiving the image as a Black man drew more exaggerated features.

Kenneth E. Iverson's Turing Award Lecture, “Notation as a Tool of Thought,” elaborates on this weaker linguistic relativity:

- “Mathematical notation exemplifies how language can be consciously used as a tool for thought.”
- “However, mathematical notation has significant limitations, lacking universality and requiring contextual interpretation.”
- “Programming languages, designed for directing computers, serve as superior thought tools, being universal and executable.”
- “Executability allows for extensive experimentation with ideas expressed through programming languages, while clarity facilitates precise thought experiments.”

Mathematical notation is a means of developing ideas and constructing fresh perspectives on existing challenges. We create symbolic bridges that guide our reasoning from theorems to proofs. However, as Iverson points out, notation differs based on the topic, author, and context. This variability may explain why many find it easier to begin with programming, only to feel overwhelmed by the mathematics underlying the code.

No one is born fluent in mathematics; we acquire it through exposure and repetition. All of us started as novices at some point.

Learners need access to both the language and its speakers for effective mastery. If you have ever attempted to learn French solely from a textbook, you know that self-study often falls short. Silent consonants are challenging in French, but the complexities of mathematical symbols across disciplines can be daunting to navigate without guidance.

Communication is the essence of language. It is meant to be inclusive, not exclusive. We have a responsibility to foster a welcoming environment in mathematics, ensuring clarity in our notation and teaching each other to communicate using the fundamental language that shapes our lives and careers.

Every individual is inherently capable of learning any human language. There is no reason to believe we are less able to master mathematics.