by Nina Ryalls, Mathematics
There’s a paradox at the heart of mathematical writing. Any communication between humans relies on connotations, inference, and assumptions; a communal well of ideas filled over centuries. Unfortunately, these are all things which mathematics hates. Because math is purely conceptual, imprecise concepts “break” math, and very quickly cause contradictions and chaos. Math needs to be clear, clean, un-connotation-y. Math needs to be sterilized of nonlogical inferences. Math needs us to rid ourselves of assumptions. That is, while humans naturally communicate with every word tied to a thousand ideas with thousands of fibers, making them fuzzy with meaning, math asks us to run around and shave the words bald.
While humans naturally communicate with every word tied to a thousand ideas with thousands of fibers, making them fuzzy with meaning, math asks us to run around and shave the words bald.
However, any attempt to communicate more than the most basic math without doing the work of translating it from bald logic-ese to furry human-ese will fail at making itself understood—as any translator knows, connotations are vital to communicating meaning. Below is a section from the Principia Mathematica, an attempt by philosophers Bertrand Russell and Alfred North Whitehead to communicate math and math alone. Notice the equation they were trying to describe at the bottom.
This is incredibly good math, but is unreadable, even to most mathematicians. It is historically important in that the attempt to describe math with total formality was made, but the only thing communicated clearly is that the concepts are not clear. Totally correct math writing that starts from scratch, so to speak, instead of using the intuition the reader already has, fails to communicate. Math needs no intuition, mathematicians do; the essential problem of mathematical writing is how to take the concepts of math, clean and slippery like a bar of soap, and make them graspable but not dirtied.
This soap should not be dirtied, but it can be carved – mathematicians have developed a way to effectively communicate with words (logos) the meaning of logic (also logos). Consider the following convention: x is a variable, and f is a function. This is widely and consistently used by mathematicians, although the choice of name for mathematical objects technically does not matter at all, as they are simply symbols. However, this consistency allows deviations from the norm of names to be notable – and to “pull along” meaning. For example, if x were used as a function instead of a variable, the experienced reader might intuitively guess that the function is not fixed, and is in fact still varying. Whether this is a correct assumption does not impact the integrity of the math itself; the fuzziness is attached to the name and not the named. For familiar objects, mathematicians can depart from the conventional name in order to suggest which particular aspect of the object they want the reader to understand.
Another use of symbols is to draw analogies. While the operation of addition is well known for numbers, the same meaning simply doesn’t exist for other mathematical objects, such as sets or surfaces. However, sets have other operations, which are also named addition. While using the same word for entirely different operations may seem like “bad math,” it is done in order to implicitly make an analogy between the familiar—addition of numbers—and the unfamiliar—a new operation on sets. This analogy is not strictly speaking mathematical; naming it “addition” does not force it to have any similarity to the familiar addition of numbers, but rather it was a choice by mathematicians to evoke similar already-present knowledge of the reader in order to build intuition about the entirely new object they describe. The symbols themselves can also be altered—the symbol for direct sum, another operation on sets, is a plus sign inside a circle. The plus sign evokes addition, the roundness of the circle puts it in the same category as other set operations, like union and intersection. For new objects, mathematicians use old symbols in order to draw parallels and analogies with the old ideas.
These are simple examples; in fact all mathematical symbology is dense with common usage, inferences, and connotations, which mathematicians have attached to the symbols of math and not math itself. We use symbols in math not because mathematics are inherently symbolic, but because we can drench each symbol with inferences, connotations, and other dark, secret, alogical meanings. As long as the symbols remain only symbols, the exactness and precision of the math represented by those symbols is not affected. Just as the artist needs that shared well of ideas and uses a medium to evoke those ideas, mathematicians rely on the medium of symbols to evoke mathematical ideas.
Citation
Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica. Page 397. University of Michigan Historical Math Collection. Source.