Math in Social Sciences

Mathematics is often associated with numbers, equations, and formulas, but its application extends far beyond the realms of pure mathematics and science. In social sciences, math plays a crucial role in analyzing and understanding human behavior, societal trends, and the relationships between different variables. This article explores how math is used in social sciences, supported by theories, specific formulas, and coding examples. We will also discuss how these mathematical tools are applied in understanding social intersectionality, political processes, and how they influence voting behaviors and other institutions like sports.

Counting and Measuring: The Foundation

At the heart of social sciences lies the need to measure and quantify human behavior and societal trends. This is where basic mathematical operations like counting and measuring come into play.

Example 1: Survey Data Analysis

Imagine a survey conducted to understand people’s preferences for different types of music. The data collected can be represented as follows:

Pop: 50 votes
Rock: 30 votes
Jazz: 20 votes

Using basic math, we can calculate the total number of votes:

$$
\text{Total Votes} = 50 + 30 + 20 = 100
$$

To find the percentage of people who prefer each genre:

$$
\text{Percentage of Pop} = \left(\frac{50}{100}\right) \times 100 = 50%
$$

Coding Example: Calculating Percentages in Python

votes = {'Pop': 50, 'Rock': 30, 'Jazz': 20}
total_votes = sum(votes.values())

percentages = {genre: (count / total_votes) * 100 for genre, count in votes.items()}
print(percentages)

Output:

1	{'Pop': 50.0%, 'Rock': 30.0%, 'Jazz': 20.0%}

Social intersectionality is a framework for understanding how various social identities—such as race, gender, class, and sexuality—intersect to create unique experiences of oppression and privilege. In the context of politics, this concept helps analyze how these identities influence voting behavior, political representation, and access to resources.

Application to Voting Behavior

Consider a voter who is both a working-class woman and a person of color. Social intersectionality would examine how these overlapping identities might influence her voting choices. She may prioritize policies that address economic inequality, gender equity, and racial justice. Understanding these intersections allows political campaigns to tailor their messages more effectively and policymakers to address the needs of diverse communities.

Example 2: Income and Education Level

Let’s say we want to explore the relationship between a person’s education level (measured in years) and their income. We can use linear regression to find the relationship.

Theoretical Formula

The linear regression equation is:

$$
y = mx + b
$$

Where:

(y) is the dependent variable (income)
(x) is the independent variable (years of education)
(m) is the slope of the line (the change in income for each additional year of education)
(b) is the y-intercept (the income level when years of education is zero)

Coding Example: Linear Regression in Python

import numpy as np
from sklearn.linear_model import LinearRegression

# Example data
education_years = np.array([10, 12, 14, 16, 18]).reshape(-1, 1)
income = np.array([30000, 35000, 40000, 45000, 50000])

# Create a linear regression model
model = LinearRegression()
model.fit(education_years, income)

# Predict income based on education years
predicted_income = model.predict([[20]])  # Predict income for 20 years of education
print(f"Predicted Income: ${predicted_income[0]:,.2f}")

Output:

1	Predicted Income: $55,000.00

Measuring Political Subjectivity

Political subjectivity refers to how individuals perceive and interpret political information based on their personal beliefs, values, and experiences. Measuring political subjectivity is essential for understanding how people form opinions about candidates, policies, and political events.

Mathematical Approaches

Quantitative methods like sentiment analysis, survey data analysis, and statistical modeling can measure political subjectivity. For instance, researchers might analyze social media posts to gauge public sentiment toward a candidate or policy. By applying mathematical models, they can predict how these sentiments might influence voting behavior.

Example: Predicting Voting Behavior

A logistic regression model could be used to predict the likelihood of someone voting for a particular candidate based on their expressed sentiments and demographic factors.

from sklearn.linear_model import LogisticRegression

# Example data
sentiments = [0.1, 0.4, 0.6, 0.8]  # sentiment scores
votes = [0, 0, 1, 1]  # 0 = not voting for candidate, 1 = voting for candidate

# Create and fit the model
model = LogisticRegression()
model.fit([[s] for s in sentiments], votes)

# Predicting the likelihood of voting based on sentiment
prediction = model.predict_proba([[0.7]])[0][1]  # sentiment score of 0.7
print(f"Predicted likelihood of voting for the candidate: {prediction:.2f}")

Utility Theory and Choices

The individual will choose the product with the higher utility. If we substitute the values for (x) and (y):

For Product A: (x = 2), (y = 3)
$$
U_A = 2(2) + 3(3) = 4 + 9 = 13
$$
For Product B: (x = 2), (y = 3)
$$
U_B = 3(2) + 1(3) = 6 + 3 = 9
$$

The individual would choose Product A because it provides higher utility (13 vs. 9).

Example 3: Choosing Between Two Products

Suppose an individual is choosing between two products, A and B. The utility derived from each product can be represented by a utility function:

$$
U_A = 2x + 3y
$$
$$
U_B = 3x + y
$$

Where (x) and (y) are attributes of the products (e.g., quality and price).

Coding Example: Utility Calculation in Python

def calculate_utility(x, y, coefficients):
    return coefficients[0] * x + coefficients[1] * y

# Example coefficients for U_A and U_B
coefficients_A = [2, 3]
coefficients_B = [3, 1]

# Values for x and y
x = 2
y = 3

utility_A = calculate_utility(x, y, coefficients_A)
utility_B = calculate_utility(x, y, coefficients_B)

chosen_product = 'A' if utility_A > utility_B else 'B'
print(f"Chosen Product: Product {chosen_product}")

Output:

1	Chosen Product: Product A

Mathematics in social sciences, particularly in politics, provides the tools necessary for analyzing voting patterns, electoral outcomes, and the effectiveness of political campaigns. Techniques such as regression analysis, game theory, and statistical modeling are often used to make sense of complex political landscapes.

Game Theory Example: Strategic Interactions in Elections

In an election, game theory can model the strategic interactions between different political parties. Each party’s strategies might include forming coalitions, focusing on particular issues, or targeting specific voter demographics.

import nashpy as nash
import numpy as np

# Example game matrix where rows are strategies of party A and columns are strategies of party B
A = np.array([[3, 1], [0, 2]])  # Party A's payoffs
B = np.array([[3, 1], [2, 0]])  # Party B's payoffs

game = nash.Game(A, B)
equilibria = game.support_enumeration()
for eq in equilibria:
    print("Equilibrium strategies:", eq)

Facilitating Voting Processes and Institutional Management

By applying mathematical techniques to voting behaviors, political analysts can forecast election outcomes, understand voter turnout, and identify key issues that drive electoral participation. Additionally, these tools can be used to design fair and efficient voting systems, ensuring that every vote counts equally.

In institutions like sports, math can be used to analyze performance data, optimize team strategies, and predict game outcomes. For example, statistical models can be applied to player performance metrics to determine the most effective team formations or to assess the impact of specific training programs.

Predicting the Future: Time Series Analysis

Time series analysis is used to analyze data points collected or recorded at specific time intervals. In social sciences, it can be used to forecast future trends based on historical data.

Example 4: Predicting Unemployment Rates

If we have historical data on unemployment rates, we can use time series analysis to predict future rates.

Coding Example: Time Series Forecasting in Python

import pandas as pd
from statsmodels.tsa.arima_model import ARIMA

Example data (Year, Unemployment Rate)

data = {
    'Year': [2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019],
    'Rate': [8.0, 7.5, 7.0, 6.5, 6.0, 5.5, 5.3, 5.0, 4.8, 4.7]
}

df = pd.DataFrame(data)
df.set_index('Year', inplace=True)

# Fit the model
model = ARIMA(df['Rate'], order=(1, 1, 1))
model_fit = model.fit(disp=False)

# Forecast the next year's rate
forecast = model_fit.forecast(steps=1)[0]
print(f"Predicted Unemployment Rate for next year: {forecast[0]:.2f}%")

Conclusion

Mathematics is an invaluable tool in social sciences, helping researchers and analysts quantify behavior, discover patterns, predict outcomes, and understand the complex choices individuals and groups make. From basic counting and measuring to advanced techniques like regression analysis, game theory, and time series forecasting, math provides the foundation for much of the work done in social sciences.

The integration of mathematical principles with social sciences, particularly in areas like social intersectionality, political processes, and institutional management, allows for a more systematic and precise exploration of social phenomena. These tools enable us to turn data into insights and actions, making informed decisions that shape our society.

By applying these concepts through coding and mathematical models, we can further enhance our understanding and application of social sciences, whether it’s forecasting unemployment rates, analyzing voting behavior, or optimizing strategies in various institutions like sports and politics. This holistic approach demonstrates how math, theory, and technology converge to deepen our understanding of society, making it an essential component of social sciences.

POLITICS DocsMath in Social Sciences 2024-08-21