You may be surprised to learn that the symbolic language that we see in mathematics today did not come to light until about the sixteenth century (Mazur, 2014). Even then it was minimal in the symbolic usage. The first use of mathematical jargon that you may have seen is the use of the plus, minus, and equals signs. Though these signs are very simple and easy to use, if you think back far enough, you may realize that this has not always been the case. If you have any doubts, ask a first grader.
The first appearance of mathematical symbolism was in 1557 when Robert Recorde wrote The Whetstone of Witte. Up to this point, all of mathematics was written long hand in whatever language you were speaking in the day and would take an extreme amount of time to do. We as humans tend to make arduous and strenuous tasks simpler, and that is exactly what Recorde did. The first symbol that Recorde introduced was the equality sign. “=” (Thackeray, 1994). Recorde states “I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe [twin] lines of one length, thus: =, bicause noe 2 thynges, can be moare equalle” (Jones et al, 2006). Translation: "I will set, as I often do in work, a pair of parallel or Gemowe lines of one length, thus: =, because no two things can be more equal." There are many times in mathematics or statistics that we wish to show that two entities are of equal value. For example, $4+4=8$, shows the sum of four and four is eight. We could also write that $2x+3= 2\left(x+\frac{3}{2}\right)$ which is also a true and equal statement.
More symbols like the addition sign "+"and the subtraction sign "—" were also introduced around the same time as the equality sign. John Widmann first introduced the addition and subtraction signs in his book Arithmetic, but the symbols were not made widely known until Recorde used them in his Whetstone of Witte (Thackeray, 1994). Recorde wrote: "There be other 2 signes in often use of which the first is made thus -+- and betokeneth more: the other is made thus --- and betokeneth lesse"(Thackeray, 1994). Translation: There are to other signs that are often used. The first is made -+- and it gives more. The second is made --- and it gives less (or takes away). Thomas Hariot, who worked with Recorde on some of his works, came up with our inequality signs "<" and ">" (Thackeray, 1994). A few more well know symbols were introduced between 1542 – 1631 by William Oughtred. Oughtred introduced the common multiplication sign $\times$ as well as the trigonometric terms sine, cosine, and tangent (Thackeray, 1994). The division sign ÷ was not introduced until 1659 by J.H. Rahn and then later in England by John Pell in 1668 (Thackeray, 1994).
Much of the notation that we now see in mathematics was written out in plain text in whatever language you were reading in. Imagine that you are in a math class and instead of seeing "$f(x) = x^2 + 2x + 1$" you see "The function of f containing the variable x is equal to the square term of x in addition to two times the term of x in addition to one." Not only did the full English sentence take up twice the amount of room, but it was also harder to piece together exactly what the sentence was trying to tell you. We can now note that even though the mathematical expression looks more daunting than its English counterpart, pairing the two side by side helps us to see that mathematics is a language and can be written in complete sentences using only notation and symbols. Mazur states that "mathematics uses symbols to express its content with precision" (2014). With the use of mathematical notation, mathematics can be deemed easier, though this was not always the case.
As mathematical notation and jargon emerged, it took about twice the amount of time to understand the meaning of what the author was trying to convey. In the eighteenth century, the language of mathematics had become so symbolized that in order to interpret what a mathematician was trying to discuss, you had to have an excessive amount of "preliminary tutoring" (Mazur, 2014). Maybe that is not much different than the mathematics that we see today, but at least you know what some of the primary symbols mean. Mazur continues, the "quantity [of the symbols] was not the problem; rather, it was that the novice had to learn a new visual language while trying to comprehend new material. Understanding such a language either took a very special expertise or enormously intense work persistence" (2014).
In the beginning, mathematical jargon was not widely accepted, but it continued to develop and become more commonly used. A step in that development was the use of logic sentences or what we think of today as Set Theory. Gottfried Wilhelm Leibnitz (1646-1716) is credited with being one of the first mathematicians to begin creating his own mathematical language. Leibnitz believed that all of mathematics could be written in a logical format, as was first shown by Aristotle (Knott, 1977). An example of the logic used by Aristotle can be showing by using "syllogism" (Knott, 1977), which typically has two premises and a conclusion, such as:
Some A is B, Every B is C, Some A is C.Leibnitz worked on his language multiple times throughout his life. In his second attempt, he began to use the notation $AB$ and $A+B$ for the conjunction of concepts (Knott, 1977). Note that a conjunction of concepts means that we have two concepts put together. Typically, this means "and".
Though there is evidence that Leibnitz was the first to create a mathematical language, many accept George Boole (1815-1864) as the father of symbolic logic (Knott, 1977). Boole began his work by naming events in relation to probability. Boole used letters like $x, y,$ and $z$ as "appellative or descriptive signs, expressing either the name of a thing, or some quality or circumstance belonging to it" (Knott, 1977). As his worked progressed, he continued to create much of the jargon that we see in statistics courses today like: $xy$ to denote an intersection, $x+y$ as a union of disjoint classes, 1 was the universal set, 0 was the empty set, and $1 – x$ was the complement of a set (Knott, 1977). Many of these symbols are different as "$\cap$" is now representative of an intersection of sets, "$\cup$" is the union, and "$\emptyset$" is the representation for the empty set ("$\emptyset$" was first introduced in 1939 by N. Bourbaki in his book Éléments de mathématique [Lankham et al, 2007]).
Boole’s work was quickly picked up and extended by two mathematicians by the names of C. S. Pierce (1839-1914) and Ernst Schröder (1841-1902). They continued the work based on logical statements. At this point in the development of mathematical jargon, most of the mathematics being worked on were far too developed for the symbols that were being created. In fact, Schröder said that "in symbolic logic we have elaborated an instrument and nothing for it to do" (Knott, 1977). Though the symbolic language was not yet up to speed with the mathematics, Schröder still continued to work on the mathematical language and is given credit for the symbol that means "is included" (usually in a set) "$\subset$" (Lankham et al, 2007).
The next mathematician to continue the work of Boole was Giuiseppe Peano (1858-1932). Peano, and many mathematicians before him, were very concerned with the lack of rigor within analysis (i.e., there was not enough of it). Analysis is a branch of study within mathematics that began with calculus. It involves the study of differentiation, integration, sequences and series, and functions and how those different portions of calculus behave in different sets of numbers. Such as the real (usually symbolized as "ℝ") or integer numbers (usually symbolized as "ℤ").
Peano began to continue the work of creating, or adding onto, the developing language of mathematics. He first published a book called Calcolo Geometrico, where he introduced his own set of notations including the intersection and union symbols that we now use today in probability (Knott, 1977). Peano’s next work, the Arithmetices Principia nova methods exposita, published in 1889, was the first work to introduce a symbol that is used often throughout set theory and mathematical jargon, "$\in$" (Knott, 1977). The symbol "$\in$" means "is an element of" and is usually followed by a set (e.g., $x \in ℝ$, where "ℝ" is the set of real numbers).
Leibnitz, Boole, Pierce, Schröder, and Peano laid the groundwork for what we see in the language of mathematics. Many mathematicians continued their work and added their own symbolic language like $\pi$ for the ratio of a circle’s circumference and its diameter, first introduced by William Jones in 1706 and then popularized by Leonhard Euler in 1748 (Lankham et al, 2007). Another well-known symbol is i for the notation of an imaginary number (previously notated as $\sqrt{-1}$ ) first used by Leonhard Euler in 1777 (Lankham et al, 2007).
Though these symbols can be hard to grasp at first, knowing what they mean will greatly increase your knowledge in mathematics and one’s ability to write precisely and formally in mathematics or statistics.