In this essay, I want to mention three mathematical perspectives that formed my thought about mathematics. These perspectives have existed for a long time, and they have a significant influence on the development of mathematics in general. They are Formalism, Formalization, and Abstraction. These names probably will not be familiar to everyone, nevertheless they are important to how people think about mathematics.

Formalism:

Formalism is a philosophy perspective of mathematics. The key idea of formalism is there are no mathematical objects in nature. This perspective is totally counter to another famous philosophy perspective: Platonism – “mathematical objects are real, they are immutable” (David & Hersch, 1981, p.318). For Platonists, humans only play the role of mathematical discovery, not the inventors. 

According to formalism, all mathematical objects, such as numbers, expressions, sets, points, and polygons, were all created by humans. While exploring mathematics, humans abstract their idea to mathematical objects and denote them by symbols. So, mathematics can be understood as symbols. Moreover, mathematicians also create rules to work with symbols. These rules are used for manipulating symbols, developing new formulas, and even proving them. Indeed, David & Hersh (1981) also said, “Mathematics just consists of axioms, definitions, and theorems – in other words, formulas” (p.319). However, when symbols or formulas (the string of symbols) stand-alone, they make no meaning. They need to be placed in a specific situation and to be interpreted to have mathematical meaning. For example, “=” is concrete symbol. Without the abstract meaning as equal or put in the context of comparing numbers, it is meaningless. 

The early formalism discussed above was initiated by two German mathematicians, H. E. Heine and Johannes Thomae (Weir, 2020). In the mid-twentieth century, formalism was the primary philosophy perspective of mathematics. It was widely used in textbooks and mathematical official writing (David & Hersh, 1981, p.339). Then this view became less popular when Hilbert introduced his formalism. Unlike the early perspective: mathematics is symbols, Hilbert split mathematics into finite part (finite mathematics) and infinite part (ideal elements). The finite part can easily be understood and decided as true or false, such as 1 < 2 is true. In contrast, the infinite part is meaningless and lacking real value, such as x < 2. It can be true or false based on the value of x. He also applied this idea to geometry and complex number and infinity sets. In contrast to early formalism, Hilbert’s idea about formalism is to “study these symbols mathematically” (Brown, 2010, p.73). He thought mathematics was a logical deduction game (David & Hersh, 1981, p.339). Humans write down some rules or axioms and then build all mathematics from that. It is really like when you give other people chessmen and chess rules, they can play chess. If you give them another objects and rules, they play another game.

I believe in Formalism because, with all of my experience in learning math, I think mathematics is a human creation. First, there are many apples on the tree. But if humans do not think about the idea of counting them, numbers will not exist. Similarly, many things have the circle shape in nature. But if humans do not think about the idea of calculating the circumference and area of the circle to use it in construction, π number does not exist. Second, all the mathematics formulas are only can be understood in the context that humans gave for them. Without the explanation that humans place on each symbol, we cannot understand it. If symbols and formulas are natural objects, they should be understood without explanation. Indeed, mountains and sea are natural objects and when we mention them, everybody will think about the same things without ambiguity and assumption. Finally, mathematics also includes rules to work with symbols. When we talk about rules, of course, they are the product of human creation. For example, when adding 17 and 18, we will add the unit first. 7 + 8 = 15 but we cannot write down number 15 and then move tens. The rule here is we just write down the unit number which is 5 and then we add one to the tens. So, we have 1 (from 17) + 1 (from 18) + 1 (from 15) = 3. The final result should be 35. Nature does not have a rule like this. Humans build the rules and build mathematics from that. In conclusion, I believe that mathematical objects are not natural objects, they were created by humans to serve human purposes.

Formalization:

Based on the formalization perspective, whole mathematics can be represented by formalized text such as symbols and formulas. It does not contain normal text. The strength of this mathematical view is it emphasizes precision in mathematics. With signs, we only can understand a formula in one way. There is no need to wonder about the meaning of words, especially if they are not your native language. This mathematics perspective has also reached a new level because of this clarity: it can be used in mechanical processing.

Peano and Frege created formalization in the 19th century (David & Hersch, 1981, p.137). As its name suggests, they wanted to make the mathematical proof more convincing and coherent by removing normal text that was supposed to carry multiple layers of meaning in some cases. However, this idea was not supported by the mathematicians at that time. Most mathematical works were written for human reading, but humans were not familiar with this new language. Thus, this sign language caused a lot of difficulties. For example, Russel and Whitehead’s Principia Mathematica was called an “unreadable masterpiece” (David & Hersch, p.138). Besides, there is one more disadvantage to this perspective. It relies on one specific foundation (Iancu & Rabe, 2011). Indeed, mathematics has a long history of development with many different perspectives. In many cases, two points of view, such as Platonism and Fallibilism, were incompatible. So, trying to convert every statement of all perspectives into only one theory set that covers all aspects was an almost impossible challenge. 

After World War II, the formal language was gradually recognized with the development of electronic computers. David & Hersh said, “For the machine, nothing must be left unstated, nothing must be left to the imagination. For the human reader, nothing should be included which is so obvious, so mechanical that it would distract from the ideas being communicated” (p.140). This is the big difference between humans and machines. To take advantage of the machine’s capability, humans must change themselves. Then, “texts written in formal language have become one of the characteristic artifacts of our culture” (David & Hersh, 1981, p. 139). When humans overcome the preconception about formal language and get used to it, mathematics will have a new look, more accurate, and avoid implicit assumptions about obvious knowledge. Math is simply a set of rules that can always be followed and replicated. With a standardized notation, everybody can understand, communicate, and reproduce deductive results.

I used to work as a software engineer, and now I am teaching coding for children, so my closest example of a formal language is programming languages. Programming languages belong to computer science rather than mathematics. However, they were built on the formalization perspective of mathematics, so I think they should be reasonable to discuss in here.

Transistors inside a computer use two states: on (usually represented by 1) and off (usually represented by 0) to describe all information. To make computers understand what humans want them to do, we need to convert our ideas into 0 and 1. For example, when converting the “A” character into computer language, it turns into “01000001.” It takes eight numbers to represent just a letter, so humans cannot write our requests in binary numbers. This is the reason why we create programming languages. A Programming language is a bridge between humans and computers because it is user-friendly and close to English. However, behind programming languages, we always have a compiler. The compiler will convert the commands that are written in a programming language into binary numbers. In short, we need translators to communicate with computers. 

Let’s imagine you have a dictionary to convert English into machine language. What will happen if you give the machine instructions with the words that are not included in the dictionary? Of course, computers cannot understand your instructions. Human language is too diverse, complicated with many layers of meaning. It is not suitable for machines. Formal language, in this case programming language, is the solution to help the communication between humans and computers more straightforward and more precise. 

The first time I learned to use programming language, I was struggling with it. There are too many rules in the programming language. Everything such as the words that you use, space and semicolon, must be accurate to run a program. I really understand why formalization was not supported the first time it was public. However, when I get used to using programming language, I realized the power of it. It is very clear and consistent. Everybody has a common understanding of a command. And sometimes, there is only one way to write about a problem. People also wonder whether different programming languages make general programming language ambiguous. In practice, there are some differences between programming languages. But the difference is only in writing conventions. The basic ideas behind most programming languages are similar. It will take time to get used to a new programming language, but it is hard to misunderstand the commands. 

Abstraction:

Abstraction is the process by which humans convert concrete things in the real-world into mathematics theory. There is a one-to-one correspondence between an object and a math theoretical counterpart. According to David & Hersh (1981), there are two ways to create abstraction (p.126). First, abstraction is a result of the idealization process. Objects in the real-world and objects in a math problem are not precisely the same. “In the idealized version, all the accidentals and imperfections of the concrete instance have been miraculously eliminated” (David & Hersh, 1981, p.126). Nevertheless, that does not mean mathematics has lost its precision. There is no concrete link between a real object and a mathematical object. For example, we cannot make a perfect comparison between a triangle shape in real life and a drawn triangle on a paper. The comparison here is just the estimation; small errors can be accepted and ignored. Therefore, we keep using imagination to abstract the real-world objects. Without abstraction and imagination, we cannot explore mathematical elements in nature and develop mathematics as we do today. 

Second, abstraction is the process of extraction. When we extract the core thing from a big problem, it is the abstraction. David & Hersh (1981) said when a mathematician takes the role of abstractor, he needs to answer, “What is at the heart of the matter? What makes the process tick? What gives it its characteristic aspect?” (p.115). When mathematicians find all of the answers, they can look at the problem in a very different way. The big question will turn into smaller questions, and mathematicians can solve them separately. This way of solving problems is right for all problems, not just math. Additionally, when we extract a problem into mathematical variables and formulas, it is also abstraction. For example, when you have five books and five cups of water. You can think of it as having five objects and ignore the difference between books and water. To summarize, abstraction can be understood as an idealization or extraction process. It will help us gain the overview and pull out the object or problem’s key feature instead of looking at its detail.

As a mathematics learner, I believe most of us have worked with abstraction. When I was in elementary school, I worked with arithmetic, and I did the simple form of math like addition, subtraction, multiplication, and division with real numbers. Even when the exercises became more difficult, instead of asking what is 5+ 3, the teacher gave me a problem. However, it still was a problem with specific numbers. For example, Tom has five candies. His mom gives him three more. How many candies does Tom have in total? I just needed to add 5 and 3 to solve this problem. Things changed when I moved to secondary school. Arithmetic turned into algebra. Specific numbers turned into abstract variables and formulas. The exercises not only simply added or subtracted numbers in the problem. The exercises required me to create equations to find the solution. For example, this is a typical math problem in grade 8: One person goes from A to B at a 20-mph speed and then returns with 15-mph speed. The return time is 10 minutes more than the travel time. Calculate the distance from A to B. In this problem, I needed to call x as the distance from A to B. In that case, the travel time would be x/12 because speed multiple with time will be equal to distance. Similarly, the return time would be x/10. Because the return time is 10 minutes more than the travel time, so I had the equation: x/15 – x/20 = 10/60. Then x = 10 miles. Creating variable x and the equations were abstraction processes. I did not need a real number to work with. I could treat a variable as a specific number and calculate with it. Throughout this kind of exercise, I had learned and used abstraction as a powerful tool to solve a real-life problem.

I have another experiment with abstraction when I teach coding for children. When children want to create an animation of fireworks flying into the sky for their game, this is a challenge about abstraction. The first question that helps students find out the key concept is when fireworks fly into the sky, do they move horizontally or vertically? Mostly, they move vertically. The second question is, if we bring it into a plane, will the vertical equal to x-axis or y-axis? The process of imagining and converting a real firework into an object in a plane is an abstraction. Also, the process that translates the natural movement into the movement in a 2-D plane is abstraction. Two fireworks are not precisely the same, but when we ignore the details such as spatial factors and just focus on the fireworks, we still can illustrate them effectively. In this case, if we do not use abstraction, we cannot create fireworks animation on a screen.

The relationship among three perspectives:

Formalism, formalization, and abstraction have their own answer for the question: What is mathematics? For people who believe in formalism, there are no natural mathematic objects, they were created by humans. For people who believe in formalization, mathematics can be represented by formalized text such as symbols and formulas. And under the abstraction perspective, humans can convert concrete things into mathematic theory. Despite the differences in understanding mathematics, there is a relationship among them. Formalism and formalization have the same idea about symbols. Nowadays, when we talk about mathematics, the first thought would be numeric symbols, operations, equations. According to that way of thinking, symbols, or more broadly formulas, are an essential part of mathematics. The creation, usage, and research of symbols help us develop mathematics, which is a precision science, in a sustainable way. Furthermore, abstraction also can be seen in both formalism and formalization. Mathematicians use abstraction as a tool to create symbols to explain their ideas about mathematics. Humans give concrete things an abstract meaning and turn them into mathematical elements. Therefore, without abstraction, we could not have mathematical objects in formalism and symbols in formalization. In conclusion, these three perspectives on mathematics are different, but they still have a strong connection with each other.

References:

Davis, P. J., Hersh, R., & Hersh, R. (1981). The mathematical experience. Boston: Birkhäuser.

Weir, A. (2020, Spring). Formalism in the Philosophy of Mathematics (E. N. Zalta, Ed.). The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/archives/spr2020/entries/formalism-mathematics/

Brown, J. R. (2010). Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures (2nd ed.). Routledge.

Iancu, M., & Rabe, F. (2011). Formalizing foundations of mathematics. Mathematical Structures in Computer Science, 21(4), 883-911. DOI:10.1017/S0960129511000144