“Abstract” is exactly right. The fact that the paragraph is a useful summation/generalization of the contents of the article supports the primal point of the article itself. This is very similar to how a meme is a quick generalization of a large urban concept. Mathematics is the study of concepts. We use mathematics without even realizing it.

## Introduction

We would like to propose that the following conjecture could be defined:

*The concept of memes is closer to the fundamentals of mathematics compared to how mathematically fundamental high school arithmetic is.*

Grounds will be given by first establishing formal definitions and concepts for abstraction, memes, and an estimate of high school arithmetic. The relation between the concept of a meme and abstraction will be elaborated. Furthermore, we will discuss the mathematical fundamentality of abstraction and use all these baselines to make an argument that informally defines the conjecture stated above facetiously calling it the Becker-Du Plessis Conjecture. We believe this conjecture to be profound because if the “conjecture” is true, most people (especially the youth) “use mathematics” without even realizing it.

## Comic

## Definitions and concepts

### (A) A formal definition of mathematical abstraction

“If S_{1} and S_{2} are statements such that S_{2 }refines S_{1} then we say S_{1 }is more abstract than S_{2 }if and only if S_{1 }is shorter than S_{2}.” (Ward, 1996, p. 3). Abstraction is also described as the process of generalization, eliminating inessential detail, or removing inessential information. Abstraction allows for potential implementations, moving to a lower-level means restricting the number of potential implementations. (Ward, 1996, p. 1)

### Examples: Compare:

One:

- Calculate the product of
**a**and**b**and store the result in**c**. - Calculate the product of
**a**and**b**using only addition and store the result in

Two

- A certain value
**x**is larger than another value**y** - A certain integer value
**x**is equal to another integer value**y**after adding 1 to it

Ward argues that (a) is the most abstract (Ward, 1996, p. 2)

### (B) A formal definition of memes

“(Internet memes) often take the form of pictures, videos, or other media containing cultural information”. (*Meme | Definition, Meaning, History, & Facts | Britannica*, n.d.)Internet memes can be thought of as a means of conveying a concept, specifically in the realm of culture, without explicitly stating it. For example, consider a picture of a frowning cat in Figure 2, which people might use to convey faux anger. Nothing about the image makes that particularly obvious, but when one person sends the picture to another, the recipient understands it is not meant to just be a random picture to look at or that the sender is not legitimately angry with them, and the sender does not need to state it explicitly. This idea is understood, largely thanks to cultural context.

### (C) High School arithmetic

Number patterns are a topic that one can find in most high school textbooks. (Urban, 2008, p. 54). For the purposes of this report, we will consider this as typical high school arithmetic as, slaving through these exercises, is what our student’s minds when thinking about high school mathematics.

Sets and set theory is considered quite fundamental. A set is considered determined if a formula with a free variable is used to define it. A vector space is a type of determined set. A vector space is a set with 2 operations and a designated zero vector. Specific rules to be followed. The Vector space must be closed under addition and multiplication and must have a designated zero vector. There are 9 different rules that must be followed to even be considered a vector space. There are several different vector spaces, of which the cartesian vector space is merely one example.

This vector space has restrictions which include those needed for number patterns. A typical reader who hasn’t studied any true mathematics yet might not even understand what it all means and that it is our point. (David Poole, 2011, p. 445)

### Relation between the concept of a meme and abstraction

For similar reasons as stated in **2. b** Figure 5 can also be used to convey the same meaning. Two different pictures and yet we understand them to mean the same thing. These two and a seemingly infinite number of pictures of frowning cats can be used to convey the same thing. As discussed above, you can easily draw parallels between these internet memes and abstraction. The idea, in this case, faux anger, can be seen as an abstraction, and all the possible pictures are specific implementations.

### Mathematical fundamentality of abstraction

Abstraction can be seen as a way of hiding inessential details. This is crucial in mathematics, especially considering modern mathematics complexity level. What if mathematicians had to prove associativity, commutativity, etc. every single time they wanted to use one of these properties? One can only imagine how much this would slow down mathematical progress. In saving time, something which hides unnecessary detail is most definitely useful. It’s also useful for generalizing certain truths. Instead of explaining why commutativity works in a specific case, every time, we can explain why it works in general, then the specific implementations follow.

### Argument for the informal conjecture out of what is conceptualized

From what we can glean the very idea of abstraction has to be established before even initializing working with sets. We generalize ideas into concepts so that we don’t have to repetitively prove or elaborate redundantly. This is like what a meme does in the way explained in **4**

Figure 1 provides a visual representation of what we are trying to convey. We presented several different sets. The lowest level is all considered sets. While all the sets with only zero elements can be examples of the zero vector. All the comics on the left-hand bottom corner are examples of rage comics. They are also considered to be memes on their own. Similarly, the specific grumpy cat memes are on the same level of abstraction as the comics.

We hope the following can be gleaned: Making memes and abstraction are very similar. Abstraction is more fundamental to mathematics than high school arithmetic is. That means memes are more mathematically fundamental compared to high school arithmetic.** **

### How the informal conjecture means math is for everyone

If the conjecture is true that means that most people, especially the youth use maths without even realizing it. Most of today’s youth often interact with memes. They take ideas like “faux anger” and make memes out of it so that transfer of information can be optimized. The study of concepts i.e., mathematics is for everyone. Abstraction of ideas could be seen as so essential to communication that the concept took form automatically in memes.

### Conclusion

Our initial “conjecture” was that memes are more closely related to the fundamentals of mathematics than simple arithmetic. If we consider especially abstraction as the mathematical fundamental in question, We believe we have made a strong case for the close correlation between internet memes and abstraction. Furthermore, we have shown how this relates to the topic of Mathematics for Everyone.

### Darius Becker

Computer Science (Second Year) Student at Stellenbosch University

### Pierre Du Plessis

Mathematical Sciences (Second Year) Student at Stellenbosch University

## References

David Poole. (2011).

*EXPLORATIONS AND VIGNETTES*.*Meme | Definition, Meaning, History, & Facts | Britannica*. (n.d.). Retrieved March 9, 2023, from https://www.britannica.com/topic/memeUrban, Paul. (2008).

Ward, M. (1996).*Mathematics for the international students : ib dipolma hl core.*Haese & Harris Publ.*A Definition of Abstraction*.