## Nonexistence of a Universal Algorithm for Investment Problems in Constructive Mathematics

Student: Jiahong Sun
Table: MATH7
Experimentation location: Home
Regulated Research (Form 1c): No
Project continuation (Form 7): No

#### Bibliography/Citations:

[1] Constructive Mathematics (Stanford Encyclopedia of Philosophy). https://plato.stanford.edu/entries/mathematics-constructive/.

[2] Shen, Alexander, and Nikolai Konstantinovich Vereshchagin. Computable Functions. American Mathematical Society, 2003.

[3] Turing: Computer Pioneer, Code-Breaker, Gay Icon | Live Science. https://www.livescience.com/29483-alan-turing.html.

[4] Bishop, Errett, and Douglas S. Bridges. Constructive Analysis. Springer, 1985.

[5] Constructive Analysis - Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Constructive\_analysis.

[6] Kushner, B. A. Lectures on Constructive Mathematical Analysis. American Mathematical Society, 1985.

[7] Geuvers, Herman, Milad Niqui, Bas Spitters, and Freek Wiedijk. n.d. “Constructive Analysis, Types and Exact Real Numbers.”

[8] Bishop, E., &amp; Bridges, D. S. (1985). Constructive analysis. Springer-Verlag.

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#### Research Plan:

a. Developing / Initiating the purpose of the research

I did the literature review in order to understand Constructive Mathematics. This simple observation leads to the creation of Constructive Mathematics, where the only numbers one can and has to consider are those given by a computer program in some language (Turing machine). All rational numbers and the famous real numbers such as π, e, and others are constructive, but as I mentioned above most real numbers in the sense of the traditional real analysis are not constructive.

Specifically, the Cantor diagonal construction indicates that the set of real numbers is not countable. Therefore, most real numbers cannot be described in any Mathematical sense. Therefore, this consideration leads to the creation of Constructive Mathematics. In addition, Mathematics plays one of the central roles in economics. I notice that investment options cannot be described in traditional mathematics. Thus, I introduce Constructive Mathematics into economics. In this research project, I explore how the existence of solutions to investment change by introducing Constructive Real Numbers as the profits. However, I do not deny the cases that when the profits are ordinary real numbers or rational numbers cannot be easily compared, because the solutions to these conditions are understood and well-known.

b. Designing the procedures

To complete this project, I needed to deeply learn and understand Constructive Mathematics. And then, I started to investigate whether there is an algorithm that can obtain the optimal way for investment with large profits by providing two investment options. Therefore, I need to establish basic settings for my research project, including the assumed non-extendable algorithm and investment profits of each company represented by Constructive Real Numbers(CRN). Subsequently, I use the concept of a non-extendable algorithm and Constructive Real Numbers to study the existence of solutions. After I get the result, I will generalize the result into several investment options. Eventually, I can establish a conclusion for my research project.

c. Implementing the procedure (including special techniques or the use of special equipment)

My mentor provided me with 6 lectures for me in order to deeply understand Constructive Mathematics. The most essential knowledge and techniques are Constructive Real Numbers and Non-extendable algorithms. I used what I have learned to construct the basic setting for my research. I designed a non-extendable partially-defined algorithm and provided its definition. And then, I used the concept of Constructive Real Numbers to assume two companies’ profits. Later, I used the non-extendable property of my algorithm to prove the contradiction and obtain the conclusion. Moreover, I generalize two investment options to several investment options by using the conclusion of my proof of two investment options.

d. Gathering / Recording data

My research is only based on theoretical Mathematics, so I do not need to gather or record data. I would like to provide my basic setting for this project here.

Every constructive number x is given as a Cauchy sequence xn of rational numbers generated by a computer algorithm after n steps. The standard condition in Constructive Mathematics is that this sequence of rational numbers is equipped with the convergence regulator that basically is the requirement that for all natural numbers N and all m, n > N we have that |xm − xn | < 2-N. I assume the profit of each investment option is one of the constructive real numbers. Moreover, I also illustrate that there is a non-extendable partially-defined program that transforms positive integers into 0 and 1.

e. Formulating conclusions

In this project, I work on the optimal investment of money between different possible options. The expected incomes were constructive real numbers CRNs. The main result of the project is that it is impossible to have a computer algorithm that always chooses these investments in an optimal way.

1. What was the major objective of your project and what was your plan to achieve it?

The introduction of Constructive Mathematics can help people to construct these kinds of real numbers. Thus, I believe that I can apply Constructive Mathematics to economic problems. I mainly study whether this is an algorithm that can determine the optimal investment of money between several options. The expected incomes were constructive real numbers. The main result of the project is that there is no computer algorithm that can always choose the investment in an optimal way. The main tool was that there are programs that are partially defined and that can not be possibly extended to be defined on all positive integers.

a. Was that goal the result of any specific situation, experience, or problem you encountered?

When I did more literature review, I noticed that scholars barely connect Constructive Mathematics with the economical problem or even the application of Constructive Mathematics. People are more focused on theoretical breakthroughs and theorems. However, Constructive Mathematics can somehow be represented in that daily life. For instance, when we have a box that is theoretically 1m × 1m × 1m, measuring it up to a centimeter may get 0.99m × 1.01m × 1.00m. However, if we measure the box in millimeters, the result might be 0.991m × 1.009m × 1.001m. Therefore, when we get more precise measurements, the result will be slight changes. Consequently, we cannot describe the exact size of the box. Similarly, the investments cannot tell the precise answer in some situations, which traditional math cannot describe. Thus, I am determined to bring Constructive Mathematics into the economy, specifically investment scenarios. In the economy, people are always eager to obtain investment options to maximize profits. Therefore, I want to test if the optimal investment option always exists. I can use the property of Constructive Real Number to describe the profit by getting it closer and closer by doing a sequence of successive measurements represented by the Cauchy Sequence and an algorithm. Finally, I obtain a method to describe the investment option and figure out if we can obtain the algorithm that can discover the optimal investment option among several possible options with Constructive Real Number profits by applying Constructive Mathematics.

b. Were you trying to solve a problem, answer a question, or test a hypothesis?

My project is to try to test if there is an algorithm that can obtain the optimal solution for a few given options of investment in constructive mathematics. Therefore, I am testing a hypothesis and answering a question.

To complete this project, I learned Constructive Mathematics. And then I established basic settings for my research. I connected the abstract concept of a constructive real number with the investment problem. Finally, I use the contradiction about non-extendable algorithms to prove the non-existence of an algorithm to obtain the optimal solution for a few given options of investment.

a. For teams, describe what each member worked on.

I worked as an individual.

a. Is there some aspect of your project's objective, or how you achieved it that you haven't done before?

The entire project is new to me. Thus, I learned Constructive Mathematics which is an advanced and innovative field of Math. The basic settings of my research is established while I was learning constructive Mathematics. I approximately spent 10 to 15 hours per week and a total of 150 hours. In the first six weeks, Professor Dr. Chernov gave me 3 hours of lectures per week. In the last four weeks, we work on a research project in Constructive Mathematical Economics. We met and discussed 2 hours per hour through Zoom. In addition, I worked more than 10 hours individually on the research project. After August, Dr. Chernov and I used a few hours per week to revamp my paper

b. Is your project's objective, or the way you implemented it, different from anything you have seen?

When I was doing the literature review, I discovered that most people are focused on the proof of theory about Constructive Mathematics. Therefore, my research is probably the first project that applied Constructive Mathematics to economic problems, or even the first research that bring Constructive Mathematics to reality. Because Constructive Mathematics is a quite theoretical and cutting-edged mathematical field. Therefore, I think my application of Constructive Mathematics in economics is a breakthrough. Although my research only proves the non-existence of the algorithm that can choose the optimal solution from several investment options, it provides a new perspective to define lots of economic circumstances that we cannot describe by traditional Mathematics. Therefore, this new description of economics can lead to the emergence of much more research. For instance, people probably can use this approach to prove why can’t people find the pattern of the stock market through neural networks and machine learning. Constructive Mathematics can be useful in this field. If we can use Constructive Mathematics to prove the non-existence of the algorithm, it has a large impact that can save people time and money

c. If you believe your work to be unique in some way, what research have you done to confirm that it is?

I reviewed the past paper about their research topics. I observed that there is not a paper that associated Constructive Mathematics with economic problems, especially investment options. In addition, I asked my mentor about his insights into Constructive Mathematics and Economics. He believed that this application has not been researched by scholars. Therefore, I believe this is a unique project.

4. What was the most challenging part of completing your project?

First, learning Constructive Mathematics was the most challenging part of my project. I needed to understand abstract concepts through professional mathematical notions. However, I cannot understand Constructive Mathematics in the traditional way, since the logic is different from traditional mathematics, and the field is hard to understand by giving some specific examples. For example, many surprising facts that are clearly false in the traditional versions of the subjects are true in the constructive world. For example, Ceitin (Tzeitin) showed that every function defined on (constructive) real numbers is continuous, and in particular the functions similar to the Heaviside function do not exist in the world of Constructive Mathematics. In fact, he showed more: that given ε from the definition of the continuous function one can always algorithmically find δ. However, surprisingly not all functions on the constructive interval [0,1] are uniformly continuous even though in the classical Real Analysis all continuous functions on a closed unit interval are uniformly continuous.

a. What problems did you encounter, and how did you overcome them?

As I mentioned before, it is pretty hard to understand the concepts of Constructive Mathematics. Therefore, I read a book which is called “Lectures on constructive Mathematical Analysis”, written by Kusher. Moreover, putting the concept of Constructive Real Numbers into the investment options is also a challenging part to complete my project. Since Constructive Mathematics is an abstract concept, it is challenging to relate the concepts with investment options. I comprehend the concepts more times to understand them.

b. What did you learn from overcoming these problems?

Throughout my research, I learned a different perspective to comprehend mathematics. To be more specific, I used to think that most of the values in the reality can be represented by a certain number. However, after I learned Constructive Mathematics, I discover that there are actually plentiful situations that cannot be described by certain numbers. When we make more precise measurements and calculations, the result keeps changing. Therefore, Constructive Mathematics can be an effective tool to describe it.

My curiosity and creativity in this research can demonstrate my affinity and aptitude for being a good scientist. I creatively connect theoretical Constructive Mathematics with the economic problem. This innovative idea leads to my research. In addition, my resilience to overcome the difficulties in learning Constructive Mathematics can also help me to be a better scientist. In Constructive Mathematics, there are many surprising facts that are clearly false in the traditional versions of the subjects are true in the constructive world. I use more than 10 hours to read the relevant book and Dr.Chernov’s notes and understand such abstract concepts. Therefore, creativity and resilience are essential parts of me. After this research, I became more convinced of my interest in applied mathematics. I enjoy the exploration between theoretical mathematics and reality. In this way, as a scientist, I can more directly to contribute to the society or world. Moreover, Constructive Mathematics is not only relevant to Mathematics but is also related to Computer Science. The concept is based on the logic and algorithm of Computer Science because it is the future trend. The interdisciplinary research also makes me excited.

5. If you were going to do this project again, are there any things you would you do differently the next time?

I will start earlier in the background reading. A deeper and broader understanding of Constructive Mathematics can provide more paths to solving a certain problem.

6. Did working on this project give you any ideas for other projects?

In the future, I would like to proceed in the following direction. We have shown the non-existence of an optimal solution for a few given options of constructive real number profits, but our proof doesn’t show whether we can acquire an optimal solution for the option of real number profits and the option of constructive real number profits. Furthermore, since the application of constructive real numbers is abundant in reality, I will extend my research to discuss more problems such as employees' salaries and stock markets. Recently, people are using machine learning or neural networks to predict the stock market. Maybe I can use constructive mathematics to prove that it is impossible to predict the stock market.

Moreover, I believe that the guarantee of the final result(Eg. the non-existence of the algorithm) will be essential for human beings. Specifically, Computer Science and Artificial Intelligence are quickly developing. More and more scholars spend large amounts of time inventing and researching innovative things. Thus, most of the time they don’t know whether what they are doing will be futile or a breakthrough. However, the introduction of Constructive Mathematics can describe lots of circumstances and logic in Computer Science. Thus, Constructive Mathematics can be a tool to briefly prove the existence or non-existence of some algorithms. This process will greatly save time, money, and resources for the researchers.

7. How did COVID-19 affect the completion of your project?

Because my research project doesn’t involve any real-world experiment, it doesn’t affect my research a lot. However, COVID-19 Pandemic prevent my communication with other scholars and Scientists. Specifically, I participated in a Conference and discussed my paper at the Conference. In addition, I noticed that there are lots of other people who discussed their interesting findings at the Conference. However, COVID-19 makes the conference online which blocks my direct communication with other scholars. Thus, I think this is the only defect that COVID-19 affects me.