A Mathematical View of Disagreement

Opinions, Debate, Psychology, and Convolution

This newsletter offers insights at the intersection of data science and social justice. In today’s entry, we veer slightly, substituting mathematics for data science and world peace for social justice. Though it will be a bit more personal in tone, I assure you that we are still touching the edges of that intersection. Our focus for today is the exploration of a mathematical concept known as convolution, and we will use it to think about opinions and debates.

My Previous Post

A few days ago, I wrote a post called “Palestine, Israel, and Logical Fallacies: Mathematics for Shared Humanity.” The conceit of the post, as stated, was to teach people about fallacies in argumentation through the lens of mathematical logic, all in the context of Middle East conflict, and all without “taking a side,” other than that I am opposed to oppression, displacement, terrorism, and violence in general. “Logic can be a tool for world peace,” I wrote. “It fosters clear, rational thinking-essential for navigating complex issues.” I tried to show how “thinking mathematically can help us recognize the intrinsic worth of every human,” even though I admitted that it “isn’t going to end war today.”

A Lesson Gone Awry

It did not go well. I got some strong pushback and in the end, I unpublished the post. Why? Not because I can’t handle pushback (to do social justice work is to receive constant pushback) but because my mission in writing the post was to reduce the amount of pain in the world, and it was clear from some responses I got that I simply wasn’t accomplishing that goal.

A Range of Reactions

Here are some things that readers told me:

• My post was biased towards Palestine.
• My post was biased towards Israel.
• I am a pro-Palestinian terrorist anti-Semitic jerk.
• I am a settler colonialist pro-Israel racist piece of excrement.
• I was engaging in both-sides-ism and am morally bankrupt.
• I was engaging in neither-side-ism and am morally bankrupt.
• I wrote a helpful, insightful piece that helped some people feel grounded.

After receiving these responses, I joked to a friend that I am the greatest Rorschach test of all time. I am all things, depending on who you are.

I found myself wondering if there were a mathematical explanation for everything that had happened. I couldn’t stop myself: I often turn to mathematics for clarity and understanding in a complex world. But before we get to mathematics, we need to begin with…

The Psychological Angle

As I mentioned, I got pushback from all corners of the Middle East conflict opinion space, and I was curious about why people were engaging with me rather than with people much more ideologically opposed (though I shouldn’t assume — perhaps they did that as well). I’m not an expert in psychology, so I want to tread cautiously and humbly here, but I did some reading up and learned about some possible mechanisms/explanations.

Similarity-Attraction Effect. People are generally more attracted to others who are similar to them. If Person A feels more similar to Person Z than they do to person B, they might be more motivated to engage in a conversation, even if it’s a disagreement.

In-Group Bias. This is the tendency for individuals to favor their own group. Person A might see Person Z as part of their “in-group” because of their relatively similar beliefs and may focus their energy on trying to persuade them.

Cognitive Dissonance. People want to resolve contradictory beliefs, ideas, or values in interactions. If Person A encounters Person Z and their beliefs have some dissonance, Person A may try to resolve it by arguing with Person Z.

The Boomerang Effect. Attempts to persuade someone to change their beliefs can often backfire, causing them to adopt an opposing position. It might be that Person A feels that trying to persuade Person B is futile, and the attempt might strengthen Person B’s opposing viewpoint.

Selective Exposure Theory. People tend to favor information which reinforces their pre-existing views while avoiding contradictory information. If Person A feels that they can’t change Person B’s perspective, they might avoid engaging them entirely in favor of Person Z.

Confirmation Bias. Person A might engage more with Person Z because, being relatively more aligned, there’s a higher chance of having some of their own views confirmed, even in a debate.

What Does This Have to Do With Math

I’ll steer clear of using specific terms and examples associated with the Middle East conflict and, instead, adopt an abstract approach to express my thoughts. It’s crucial to note that this abstraction won’t and can’t perfectly mirror my experience, especially as I don’t view my stance — a rejection of war, oppression, displacement, terrorism, and violence — as a “middle ground” between other viewpoints.

Nonetheless, imagine a graph where the horizontal axis abstractly represents different viewpoints, and the vertical axis indicates the popularity of each viewpoint.

Picture group A, a cluster of individuals with similar beliefs, represented by a bell curve on this graph (in blue). The core belief is the peak but there is of course some variation.

Now, introduce group B, another cluster with a shared belief, distinct from group A, visualized by another (red) bell curve situated some distance away, with minimal overlap with A’s curve.

Enter individual Z (the vertical orange line) positioned between the two groups on the belief spectrum. Who, among groups A and B, would likely confront Z? For group A, individuals with beliefs closely aligned with Z might see no reason for confrontation, owing to minimal disagreement. Those with beliefs significantly diverging might also refrain, perceiving Z as an outgroup or deeming the discussion futile.

However, those whose beliefs are at an intermediate distance from Z’s might engage, finding just enough common ground for dialogue and enough disagreement to spark debate. Take the example person (in purple) towards the left of group A, whose debate zone (also in purple) includes Z.

A similar pattern of interaction to all of this applies to group B.

So, how many confrontations can Z expect? To determine this, we examine each belief within group A’s distribution. We note the group members for whom their debate zone, at intermediate distance, contains Z. The process is replicated for group B, and the counts are summed.

The total count hinges on numerous factors: the central beliefs of groups A and B, the spread of beliefs within these groups, the specific range of beliefs considered “intermediate,” and Z’s exact position on the spectrum. The key takeaway? Under certain conditions, Z could face a significant number of confrontations.

An Even More Mathematical View

There’s a mathematical concept for all of this: convolution. Convolution is a process where two mathematical functions combine to form a third function, showcasing how one modifies the other. In the context of ideological confrontations, one function represents the belief distribution of a group and the other is a kernel — a function that encapsulates a range of beliefs likely to incite debate.

The kernel is meant to measure a distance from a particular reference belief. Visualize the kernel gliding across the belief spectrum of group A, with each shift unveiling varying degrees of overlap, calculated through an integral, a fundamental concept in calculus.

What emerges is a plot of potential confrontations, dependent on the interplay of the ideological distributions A and B and the precise positioning of individual Z. The convolution’s value, indicative of the amount of confrontations, can, under specific conditions, be large.

Conclusion

My initial post on the Middle East, anchored in my rejection of violence and my confidence in the constructive role of mathematical logic, elicited a range of responses. I wrote it because I felt the need to voice my thoughts during a tense period. My core beliefs remain unchanged.

I value each response I received. Every comment, whether critical or supportive, is important to me. They offer insights that broaden my understanding and perspective, and I am not ignoring or discounting them.

In this follow-up post, I have tried to add another layer to the discussion. My writing today has been an attempt to explore the dynamics of our interactions and responses in a different light. It isn’t meant to invalidate any reactions, but rather, to consider the broader context in which they occur.

I appreciate everyone who took the time to respond. The diversity of perspectives enriches the dialogue and underscores the complexity of the issues at hand. Thank you for your engagement and for contributing to the conversation.

Your neighbor,

Chad