Quantifiably Enigmatic Operations
Math’s precision is so exacting that it does not leave any room to explain what it is telling you. The measurements that come out of an equation are abstract to such a high degree that without any objects attached to the value, there is no telling what has been said. For some, this makes math a chore best avoided. There is no denying that the operations are meaningless without objects present, but there is something at the basic level that bears examination in order to best view how the abstraction makes math unable to engage language in a meaningful dialogue.
Unlike language, math has several properties that make it universal and fungible. This does not mean the values or quantities can be moved around with willful disregard. Rater, it means that eight apples and eight oranges are quantifiably equal even if the two fruits are not. Even if one uses different symbols to represent numbers (e.g. Roman and Arabic numerals), their value remains unaffected. Thus, the measures of each value remain consistent and immutable even while their representation or objects vary. The power of math is tied to its ability to transmit information past linguistic or symbolic barriers. This is evinced in mathematical patterns.
Certain sequences are found everywhere in nature. The Golden Ratio is one such sequence. It appears in nautiloid structures like snail and sea shells. It also appears in weather patterns and ocean waves. Even plants make use of this pattern. Sunflowers are an example of this along with the placement of leaves along the stalks of plants as well as the placement of limbs on trees. And these are just examples from the natural world we can observe. Physics does the same in such notations as the gravitation constant of 9.8m/s2 for falling objects on Earth.
Math follows syntactical rules like language, but the rules are driven towards outcome. Language works towards an outcome as well, but it is not driven; it meanders compared to math. With math it is all about operations more so than it is nuance. To be sure, the complex higher orders of math are nuanced, but not to the degree of language. That is because even then the subtleties are in the employment of the operating symbols. Math lacks the ability to create metaphors; which, having a part of catachresis to some extent, defy nature. We can equate a trait of an animal to a human and have the point be understood with no issues for the validity of the quality. If you tried to make 2 + 2 = 5 in math without invoking complex rules or synergy, you will be challenged and found at fault for violating the laws of nature.
The basic structural grammar of math uses the acronym of PEMDAS to describe the order of importance for the main operations in any equation. Languages also have a strict word order for understanding to be achieved. In English, the basic structure is SVX (subject, verb, and everything else). For math, the operations are parentheses, exponents, multiplication, division, addition, and subtraction. Notice that there are no rules governing where the information subjected to the operands go, save that they must balance out when an equal sign appears. This lack of “word order” is what makes math operationally-driven.
Consider the following statements: 16 ÷ 2, 2 x 4, 23, and 12 – 4. All of them have the same resolution. They are all expressions of the value of 8. None of these expressions are conceptual. Given enough objects to see this, you can physically see the answer. In fact, with most mathematical equations, you can directly observe the results. Some might require very specific knowledge sets, but observers with said knowledge can see the math in action. Language does not work this way. As described in the previous section, words generally have multiple meanings, leaving the sentence open to interpretation based on the concepts bound in the word and the mental image needed to construct the sentence’s intended meaning.
But, one might point out, I have left out mathematical formulae and variables. They are not basic math, for one. The use of variables as stand-ins for math problems represents the values that will eventually go into an equation, but they are not concepts in math. Variables and equations are concepts of language of how the world works based on experience and then rendered as equations. The equation F = ma does not measure force; it tells the observer how to measure force. Force equals mass times acceleration. What about the formula d = rt? This tells us how to measure distance by multiplying the rate of speed by the time of movement. Let us replace the variables with numbers: 8 = 4 x 2. Without the equation, is that a measure of distance or force? Both formulae use the same mathematical operations. The famous E = mc2 equation is not much different once you reduce c2 to the speed of light squared. This means once you plug in the number for mass, you can carry out the multiplication just like with F = ma and d = rt. The concept is not in the operation, it is in what values you select for and the measurement you are making.
The last point bears focus as it is operational in nature. The act of data selection to parse through an equation’s operands is not a concept, it is information objectively observed. We assign meaning to measurements so that they have conceptual weight. To prove this point, let us refer back to the numeric symbol of 8. Outside of the expressions used thus far, what does it mean? What does it quantify on its own? Outside the concepts informing the equations, the result of 8 has no attachments. For our purposes of understanding the syntax and outcomes of math, the symbol of 8 represents a quantified unknown. I can express the symbol as the word “eight” and still be no closer to the truth of what is being signified. The meaning is missing. In fact, it cannot be found in the word’s definition.
According to the Oxford English Dictionary, the word “eight” is an adjective, and thus, by grammatical rules of the language, requires something substantive (a noun) to follow its use. The OED also defines the word as a substantive and goes on to define it as “the abstract number eight.” Other definitions are listed for the adjectival form of the word, but the root of many of the problems in gaming can be seen in the definition of the number as an abstract. Language does not do abstracts well. Why? Because we rely on our eyes to gather the greatest amount of information. Abstract ideas lack physicality and sensory details we can readily experience or imagine experiencing. Math cannot tell us what is happening, only the mechanics underlying why it is. Thus, math can give an exacting answer, but it cannot contextualize it.
And here we find what language does well: contextualizing raw data we observe and experience. Here is an example: five apples plus three oranges equals eight fruit. I mixed objects, but I still have a mathematical function that occurs despite categorical changes (actual objects in apples and oranges to abstract concepts: fruit). But wait, you say, you just said language does not do abstracts well. Yes, which is true, but what is a fruit? Tomatoes and avocados are fruit, but most people do not consider anything not sweet a fruit even if that is its scientific category. But while I am arguing with myself rhetorically, 5 + 3 still equals 8 and the math did not care one whit that words got appended to the equation. That is because the values parsed are universal; the rest was a linguistic argument.
What about the symbols used in higher levels of math? They appear conceptual at first glance, but they are not. A capital sigma (Σ) is used for summations. Now that symbol may have annotations subscripted, superscripted, and to either side, but the sigma is shorthand for the procedures behind the summation function. The same applies to the lower case sigma (σ) used to denote the math that governs the standard deviation function of a bell curve distribution. What we have to do is unpack the math behind the symbols to understand what they mean. I might have to think of the concept that describes the math, but it does not impact the shortcut the symbol represents, only your understanding of how it works. The concept is within your linguistic capabilities of describing what the math is used for and why you are doing it.
Even those who feel more comfortable doing math than describing the world through words hang their calculations on observable concepts. We cannot see gravity, for instance, but we can measure its effects. So, when a physicist uses 9.8m/s2, he knows it is a constant referring to the rate of speed of falling objects on Earth. If a chemist needs to measure molecular mass for a mixture, she knows that 1 mol of water has 18.02 grams of weight. A mol, like a meter or a second, is a concept that has had a specific meaning attached to it so the math can be understood as something concrete. Remove the units of measure and it looks like the chemist is saying 1 = 18.02.
Units of measure are one of the linguistic concepts that underpin the way we interact with math to make use of it. It is through our assigned units that the numbers acquire meaning. Otherwise, complex math has no meaning beyond numbers that cannot tell us much. It makes the difference between 2″ x 4″ x 8′ and 2″ x 4″ x 8′ as the marks tell a carpenter one thing and others that these marks are exponents. That is how math is so exacting as to be vague. Just like language, it is the context of the observer’s frame of reference that defines the content.
Previous:
Anatomy of Game Design: An Unbridgeable Divide, Part 3
Next:
Anatomy of Game Design: An Unbridgeable Divide, Part 5