Tag: math

As a transition from education to profession, college is the place where students have gone through lots of changes, and interpersonal relationships are definitely included. For example, let’s say a young and innocent college kid, Bob, has this feeling that although he is always getting to know new people, it is becoming more difficult for him to keep up with existing connections and friends. Of course, he has some “close friends” whom he keeps in touch with every day, but besides that, how close he is to “non-close friends” completely depends on how lucky they are to bump into each other on the way to Ferris Booth.


Graphic courtesy of Leadership Close Up

In our society, interpersonal relationships can be characterized as weak ties and strong ties. When you have a strong tie with someone, you keep up with him or her frequently and there are ways that you two meet or connect frequently — just think of someone in your housing group. When you have a weak tie with someone, one the other hand, you might still share a lot of common interests with that person, but for some reason you are connecting with him or her less frequently, and the cause of such infrequency could be unintentional — it might be because you two aren’t in the same class, or because you are living in Wien but your friend is living in Harmony and you don’t meet each other.

However, such infrequent connection could also, if not more likely, be caused by human calculation, and therefore we can actually draw a parallel between friendship and the prisoner’s dilemma in game theory. Yes, friendships can indeed be a game.

Let’s suppose we have two friends, Bob and Jim, and they know each other. One of the assumptions that we apply here is that Bob is a friend to Jim because Bob believes Jim will make him happy, and vice versa for Jim (I’m guessing you don’t to be friends with someone who makes you unhappy!). They both want to be very close to each other, but they are also aware that it takes some cost to keep up, for example if they hang out for an hour they lose an hour of study time, and therefore the more they want to connect, the higher the cost they have to pay to sustain a high level of connection.

In economics, we believe all happiness can be quantified, so we can do a simple experiment here to see how the outcome is derived. Suppose both Bob and Jim have two choices: either to connect the other one frequently or not. There is a special case: let’s say Bob chooses to connect with Jim frequently but Jim chooses to connect with Bob less frequently, what does each other gain out of this relationship? Remember since there is a cost of connecting, Jim will be better off choosing to connect frequently, because he is getting the same care and attention from Bob with a smaller cost. But, on the other hand, it will be hurtful, mostly emotionally, for Bob when he sees Jim is not giving him same attention and caring he deserves. We can characterize the situation in the following payoff matrix:

If you have taken economic classes you can see what I mean by this matrix, and you know what Bob and Jim will choose to do.  If you are not familiar with economics, the basic idea behind this is that, because Bob knows he will be better off if when Jim contacts him frequently, he contacts Jim less frequently, Bob has an incentive to choose to connect Jim less frequently. But Jim faces the same situation and he will have an incentive to choose to connect less frequently and as a result, they will end up in a weak tie with each other.

It may seem disappointing in the beginning, but such incentives disappear as long as the game of friendship is being played repeatedly for a long time, and for a long time the real benefit of keeping a close relationship will outweigh the advantage of not returning to your friend once, and in such case a strong tie can be sustained. In other words, your relationship ends when you realize that there is an end of it, and upon such calculation from pure reasoning, there is a sentiment of friendship that we cherish.


Meet Mathew Pregasen. Mathew is a Columbia junior studying computer science who founded a startup with Anuke Ganegoda (CC ’18), Sahir Jaggi (SEAS ’17) and Rikhav Shah (MIT ’19). Named Parsegon Inc, the company implements a new method of transcribing English descriptions of math into mathematical script. For example, Parsegon’s technology could take a sentence “integral from 0 to 10 in region D of 2x squared + 3x cubed – the square root of x” and convert it into visual, textbook-formatted math.

How did you come up with the idea of Parsegon? What experience made you want to start your own business?

The way it started was pretty accidental. It was first a small project that we had no intention of turning into a company, but as it developed we realized it had more potential. Soon, we started to think of this project in a business context. We did Almaworks, raised some funding, hired some people for the summer, and further developed our business. In the ending, it is a technology project.

How did Almaworks facilitate your business development process?

I think the most beneficial part is that it connects you with incredibly helpful mentors. At first, you might not know too much about design, planning, or the law associated with a startup business, but as long as you get close to a mentor, you will get proper advice on business direction, project development, and especially important legal services.

What’s the current entrepreneurial environment at Columbia like? How does it compare to other schools?

I think in the last two years, there has been some significant changes, where the administration—especially entrepreneurship administration—has been putting a lot of resources into the entrepreneurship community. They raised the amount of provided grants and have organized the Columbia Entrepreneurship Competition for the last four years.  Alongside that, you have clubs like CORE (Columbia Organization of Rising Entrepreneurs) and ADI (Application Development Initiative) that push this culture. I think ultimately the culture should be self-accelerating instead of accessory, but you need to have some initial velocity at the beginning.


Mathew Pregasen

Image via Mathew Pregasen

So back to Parsegon. It seems to be designed for people who are not fast at mathematical typing. How do you attract people who are already proficient at mathematical expression in typing packages such as LaTeX?

We are not competing with LaTeX and we don’t expect people to write papers in Parsegon. That being said, we do have a very user-friendly environment that reduces time and difficulty in typing. Parsegon is also educational in the sense that it makes teaching more accessible to students and enables the entire classroom to engage in interactive math.


You have been trying to integrate Parsegon into classrooms. What is the feedback from teachers and students?

We primarily focus on high schools, and we’ve been having very strong feedback.

What do you think is the biggest challenge for Parsegon?

I think the greatest challenge for us is to make a technology that provides a number of services for very diverse classroom environments. Some people might not be familiar with computer typing and some do prefer a very traditional and structured typing style, so although we are making it more accessible to people, it is still a big challenge to build the technology that accommodates the needs of everyone and strikes a proper balance between accessibility and formality.

Are there any computer science classes at Columbia that have helped you in this process?

Namely Operating Systems (W4118) with Jason Nieh. I also took a class called Computer Theory with Alfred Aho which was useful for the theoretical angle.

What do you think is the future of Parsegon?

We want to build the best tool for educational practices in the America. We believe that there is a big gap between the technology side of users and the technology provided for educational professionals, and we believe that our implementation will not only complement the traditional learning method, but also improve it. The importance of Parsegon is that it teaches students to understand the language of math. If you can understand the language of math, you usually also understand the theory of math much more coherently. And we believe that is the best way Parsegon could improve the learning process of math on a more cognitive level.

A year is long enough for surprises to happen, especially this past year. People suddenly found out that the world they live in has gone through a course that no one could ever have predicted. One year ago, hardly anyone foresaw the European Union losing its most important player and perhaps in the future losing a second one; hardly anyone could say affirmatively that a billionaire without any political experience could become the president of US. People are shocked, fearful, and puzzled by these facts. We call this series of events a wave of anti-establishment—not only anti-establishing the social system that we have relied on so much, but, more importantly, challenging the ways we look at and interpret the world.

While people are asking, “What is wrong with our world?” it is equally important to be introspective and ask what is “Wrong with my own thinking that causes so much disillusionment with what really happened?” It is not an easy question, and people can have diverse answers for it, but besides the debates of ideology, social norms and political correctness, maybe we can focus on something that is less paid attention to, something that seems to be irrelevant to politics: math.

Throughout human civilization, people have used reason to understand the world, and after thousands of years of development, almost every field of study has become dependent on the use of rationality. Usually, we tend to call such rational tool “model.” People use models to capture the factors of the issue being studied and use logical representations to depict the fundamental laws that govern the behavior of these factors. The most widely used model among people is the mathematical model, where the logical representations are necessarily mathematical expressions. Such a use of math has been adapted in economics, political science, sociology, and even psychology. Investors use math to make investment strategy, economists use math to understand the behavior of economics, politicians use math to predict the patterns of voters, and policymakers use math to structure the best policy for the country.

For a long time, math has successfully captured the behavior of the world and did a pretty good job in assisting people with their applications in the real world, and people have been more and more dependent on math to solve the complex situations they face. But because of these successes, people also ignore the shortcomings of mathematical models in social sciences, and such ignorance could cause problems.

One weakness of math models is that in order to achieve more accurate depiction of the scenario, comprehensiveness is sacrificed. The first thing to do when constructing a math model is to make assumptions and simplify the situation to a bunch of factors that are representable by math expressions. But in this simplification some important factors are tossed away because they cannot be clearly quantitated. One clear example is people’s sentiment, which in some cases dictates the situation, but because it is too hard to be modeled, it is usually left out or excessively simplified. With an absence of sentimental factors, the model can sometimes interpret facts incorrectly.

Another shortcoming of math is not a shortcoming in nature, but could hurt people when they are too dependent on a math model. The nature of math assumes a deterministic model. That is to say, with given conditions and given principles, the outcome can be well defined. Usually the models taught in economic classes and political science classes don’t assume any stochastic scenarios, and they don’t talk about things that are not solvable (otherwise what is the point of studying it?) But neither of the preconditions are always true. Sometimes we cannot grasp the condition correctly due to our limited ability to observe the complete picture, and sometimes we just simply don’t have the correct principles on which this world functions. In either case, model thinking could fail.

The use of math modeling in our daily life is not essentially problematic. It is the overdependence on it that causes some misleading in perceiving the world and interpreting it. If we don’t fully grasp the pros and cons of using mathematical reasoning in social science fields, we will constantly encounter conflicts between our assumptive beliefs and real facts.