## Non-existence of the Algorithm that can Obtain the Optimal Solution for a Few Given Options of Investment in Constructive Mathematics

Student: Jiahong Sun
Table: MATH3
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.

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Presentation files:
Research paper:
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Project files

#### Research Plan:

Rationale

When we have a box that is 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. But Constructive Mathematics can be an effective method to explain it by getting the profit closer and closer by doing a sequence of successive measurements. Thus, we can describe the investment option in Constructive Mathematics. As a result, I generate an idea to prove the non-existence of a computer program that can always choose the optimal solution from a few investment options when the profits are constructive real numbers.

Questions:

Can we find an algorithm that can always choose the most profitable investment?

Goals:

Using Constructive Mathematics to prove that no algorithm can choose the optimal solutions from a few investment options.

Procedures:

1. Learning Constructive Mathematics understand some concepts
2. Establishing Basic Settings
1. The entire project stems from a partially defined algorithm that is non-extendable to all the inputs
2. Connecting the investment options with the concepts of Constructive Real Numbers(CRN)
3. Active Research
4. Using Constructive Real Numbers, the algorithm, and contradiction to prove my hypothesis.

Bibliography:

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.

Safety and Risk:

There is no safety risk in my project.

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

The major objective of my project is to prove the non-existence of the algorithm that can obtain the optimal solution for a few given options of investment in Constructive Mathematics. To achieve it, first, I learned about Constructive Mathematics. Then, I used a partially defined non-extendable algorithm as the primary tool of my project. Using the concept of constructive real numbers and contradiction to achieve my goal.

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

People are always eager to obtain the investment option to maximize profits in the economy. Therefore, I want to test if the optimal investment option always exists. When I explored investment options, I discovered that there is a situation that we cannot tell the exact answer, but we can get closer and closer to it by doing a sequence of successive measurements. This kind of process is hard to be described by traditional mathematics. However, this condition is highly associated with defining a constructive real number. Therefore, I use the definition of Constructive Real Numbers and the non-extendable of the algorithm to prove that there is no optimal solution for several investment options.

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

My project is to test if there is an algorithm that can obtain the optimal solution for a few given investment options 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. Next, 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 investment options.

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

N/A

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, an advanced and innovative field in Math. While I was learning constructive Mathematics, I established the basic settings of my research.

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. However, no research uses Constructive Mathematics as a tool to prove economic problems. Thus, my research is a novel project.

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 no research associated Constructive Mathematics with economic problems, especially investment options.

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 traditionally understand Constructive Mathematics since the logic is different from traditional mathematics, and the field is hard to understand by giving some specific examples. Thus, I spent ample time learning Constructive Mathematics.

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

As I mentioned before, it is difficult to understand the concepts of Constructive Mathematics. Therefore, I read a book called Lectures on Constructive Mathematical Analysis, written by Boris Abramovich Kusher. Moreover, putting the idea of Constructive Real Numbers into the investment options is also a challenging part of completing my project. Since Constructive Mathematics is an abstract concept, it is challenging to relate the concepts of investment options. Therefore, I reviewed the concepts frequently for better understanding.

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 reality, can be represented by a certain number. However, after learning Constructive Mathematics, I discovered that there are plenty situations that specific numbers cannot describe. When we make more precise measurements and calculations, the result keeps changing. Therefore, Constructive Mathematics can be an effective tool to describe the situation.

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 with the background reading so I could read more articles. A deeper and broader understanding of Constructive Mathematics can provide more paths to solve 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 optimal solutions for a few given options of constructive real number profits. Still, our proof doesn't indicate whether we can acquire an optimal solution for the possibility of real number profits and constructive real number profits. Furthermore, since the application of constructive real numbers is abundant, I will extend my research to discuss more problems such as employees' salaries and stock markets. Recently, people have been 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.

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

COVID-19  made my entire research experience to be virtual. However, it didn't significantly impact my research because mathematics research does not involve real-world experiments.