Self reflection
I began my Computer Science journey pretty early – almost around my entrance to secondary education. Hence, I am able to grasp a solid understanding of what CS is trying to do and what the ultimate goals are for CS. However, the same does not hold for mathematics and statistics: I can describe my learning in these two subjects as “stuffing a duck” – I digest knowledge without knowing why we need to study it (not the shallow application of each formula or rule I learned, but the ultimate purpose of what mathematics (the subject) wants to achieve. The same holds for statistics as well.) That’s why, after having three semesters in university, I feel increasingly bored by mathematics and statistics subjects and even question myself why I need to study them (I don’t want any answers involving real-life applications e.g., “To calculate areas for home renovation; predict future income;” but what I want is something philosophical and not influenced by society (aka not sticking to the “values” society wants us to have) – Why did we create mathematics in the first place?).
That’s why you see this post. It is a summary of what I have discussed with AI (Particularly, Google Gemini 2.0 Flash Thinking Experimental from https://aistudio.google.com) and I want to share with all the people who found lost in their life meaning while they enter university.
This post is inspired by ThisIsXXZ posts:
- 我的第一次反向传播
- He talked about the study method transition after graduated from secondary school and entered university, which I pretty much agree with him.
- Side note: He mentioned the new way of taking notes which I found pretty inspirational.
- Meanings of Life
- Literally, meaning of life. I have fallen into the “meaning of life” the society fabricated many times - and is not really what I want.
Mathematics: A Top-Down View (The Grand System Chart)
Imagine mathematics as a vast, incredibly powerful, and deeply interconnected intellectual system. Our goal is to understand its overall architecture and purpose.
Level 1: The Ultimate Purpose of Mathematics (The “Why” of Math’s Existence)
- Understanding and Modeling Reality (Both Abstract and Concrete): At the very highest level, mathematics is fundamentally about understanding the underlying structures and patterns of reality, both in the physical world and in the realm of abstract ideas. It’s about building precise, logical models to describe, explain, and predict phenomena.
- Think of it as: Creating a universal language and toolkit for describing and reasoning about anything that can be quantified, structured, or logically analyzed.
- Analogy: Like architecture, mathematics is about designing the framework upon which other disciplines (science, engineering, even parts of philosophy and economics) are built.
Level 2: Major Branches of Mathematical Activity (The “2nd Top” - Different Ways Math is Done)
To achieve this ultimate purpose, mathematical activity can be broadly categorized into these interconnected branches:
- Pure Mathematics (The Exploratory and Foundation-Building Branch):
- Focus: Exploring abstract mathematical structures, patterns, and relationships for their own sake, driven by curiosity, internal logic, and aesthetic principles. This is where the “self-highing” feeling sometimes comes from, but it’s actually about deep exploration of the mathematical universe.
- Motivation: Intrinsic beauty, intellectual challenge, uncovering fundamental truths within mathematics itself. Often, pure math discoveries later find unexpected applications (sometimes decades or centuries later!).
- Examples: Number Theory, Abstract Algebra, Topology, Real and Complex Analysis, Set Theory, Logic.
- Analogy: Like fundamental research in physics – exploring the deepest laws of nature, even without immediate practical applications, but these discoveries often become the basis for future technologies.
- Applied Mathematics (The Problem-Solving and Connection-Making Branch):
- Focus: Using mathematical tools and techniques developed in pure math to solve real-world problems in science, engineering, technology, finance, social sciences, and many other fields. This is where math becomes directly useful and impactful.
- Motivation: Practical problem-solving, creating new technologies, improving efficiency, understanding and predicting real-world phenomena.
- Examples: Mathematical Modeling, Numerical Analysis, Optimization, Statistics and Probability, Differential Equations, Operations Research, Financial Mathematics, Cryptography.
- Analogy: Like engineering – taking the fundamental principles of physics and applying them to design bridges, airplanes, computers, etc.
- Mathematical Foundations (The Rigorous and Logical Bedrock):
- Focus: Establishing the logical foundations, rigor, and consistency of mathematics itself. Ensuring that mathematical reasoning is sound and free from contradictions.
- Motivation: Certainty, precision, avoiding logical fallacies, understanding the nature of mathematical truth.
- Examples: Mathematical Logic, Set Theory, Proof Theory, Category Theory.
- Analogy: Like the foundations of a building – unseen but essential for the stability and integrity of the entire structure.
Level 3: Core Mathematical Areas within Each Branch (Specific Fields of Study)
Within each of these branches, we have specific areas of mathematical study (this is where you find Calculus, Linear Algebra, Optimization, etc.):
- From Pure Math:
- Analysis (Calculus and its extensions): Study of continuous change, limits, derivatives, integrals, functions – fundamental for modeling continuous phenomena. Multivariable Calculus fits here.
- Algebra (Abstract Structures): Study of groups, rings, fields, vector spaces, matrices – provides powerful tools for abstraction and structure. Linear Algebra fits here.
- Geometry and Topology: Study of shapes, spaces, and their properties, both in familiar and abstract settings.
- Number Theory: Study of integers and their properties, surprisingly deep and applicable to cryptography.
- From Applied Math:
- Statistics and Probability: Dealing with uncertainty, data analysis, inference, prediction – essential for data-driven fields like AI.
- Numerical Analysis: Developing algorithms for approximating solutions to mathematical problems, especially those that can’t be solved exactly (crucial for computer simulations and calculations).
- Optimization: Finding the best solutions within constraints – central to machine learning, operations research, engineering design. Introduction to Optimization fits here.
- Differential Equations: Modeling rates of change and dynamic systems – used everywhere in science and engineering.
- From Mathematical Foundations:
- Mathematical Logic: Formalizing reasoning and proof.
- Set Theory: The language for describing collections and relationships.
Within each core area, you learn specific techniques, theorems, formulas, and algorithms. For example:
- Calculus: Derivatives, integrals, limits, series, partial derivatives, multiple integrals.
- Linear Algebra: Vectors, matrices, matrix operations, determinants, eigenvalues, eigenvectors, linear transformations.
- Optimization: Gradient descent, linear programming, convex optimization, dynamic programming.
- Statistics: Hypothesis testing, regression, probability distributions, confidence intervals, machine learning algorithms.
How Multivariable Calculus, Linear Algebra, and Optimization Fit In:
- Multivariable Calculus: Provides the tools to analyze and model systems with multiple interacting factors, essential for understanding complex real-world phenomena and for optimization in higher dimensions.
- Linear Algebra: Provides the language and methods for working with data in a structured way (vectors, matrices), for transformations, and for solving systems of equations – fundamental for data analysis, machine learning, computer graphics, and many other areas.
- Introduction to Optimization: Gives you the fundamental techniques for finding “best” solutions, which is at the heart of machine learning, AI, operations research, and engineering design.
The Interconnectedness is Key:
It’s crucial to see that these branches and areas of math are not isolated. They are deeply interconnected and build upon each other. Linear algebra is used extensively in calculus, statistics, and optimization. Calculus is used in probability, statistics, optimization, and differential equations. And so on.
Think of it like a vast toolkit: Pure math develops the tools, applied math uses them to solve problems, and mathematical foundations ensure the toolkit is reliable and consistent. You are learning to use these tools!
How to Make Math Feel Less Abstract and More Meaningful:
- Focus on the “Why” and “What For”: Always ask: Why is this concept important? What problems does it help solve?
- Seek Applications: Actively look for real-world applications and examples for every mathematical concept you learn, especially in your field of Applied AI.
- See the Interconnections: Try to see how different mathematical areas relate to each other.
- Practice Problem Solving: Working through problems, especially applied problems, helps solidify your understanding and see the utility of the math.
- Be Patient and Persistent: Mathematical understanding often comes gradually. It’s okay to feel confused or frustrated at times. Keep practicing, keep asking questions, and gradually the pieces will start to fit together.
By adopting this “top-down” perspective, you can start to see mathematics not as a collection of random symbols and abstract concepts, but as a vast, powerful, and deeply interconnected system for understanding and shaping the world. Your current courses (Multivariable Calculus, Linear Algebra, Optimization) are essential parts of this system, equipping you with fundamental tools for your future in Applied AI and beyond. Keep exploring – the deeper you go, the more you’ll appreciate the beauty and power of mathematics!
You’ve hit on a really important point! It’s completely valid that you can visualize the “top-down” structure of CS, but statistics feels much more nebulous. And you’re right, my previous explanations about STAT3600 being “linear regression and useful” are helpful for motivation but don’t give you that “big picture” structural understanding you crave.
Let’s build a “top-down diagram” or mental map for statistics, explicitly addressing your feeling of it being continuous or discrete and helping you see the overall structure.
Statistics: A Top-Down Mental Map (Like a System Design Chart for Stats!)
Think of statistics as a system for understanding and learning from data. Just like a software system has different layers and components, statistics has different branches and tools that work together.
Level 1: The Overarching Goal (The “Top” - What is Statistics REALLY for?)
- Understanding the World Through Data: At its highest level, statistics is about using data to answer questions about the world, make informed decisions, and gain insights. This is the ultimate “why.”
- Examples:
- Understanding customer behavior to improve business strategies.
- Analyzing medical data to find effective treatments.
- Predicting climate change impacts.
- Building AI models that learn from data.
Level 2: Major Branches (The “2nd Top” - Broad Categories of Statistical Work)
To achieve the overarching goal, statistics is broadly divided into these major areas:
- Descriptive Statistics: Describing and summarizing data. This is about getting a handle on the data you have.
- Tools: Mean, median, mode, standard deviation, variance, histograms, boxplots, summary tables.
- Purpose: To get a clear picture of the data’s main features: central tendency, spread, distribution shape. Think of it as the “data exploration” phase.
- Inferential Statistics: Drawing conclusions and making generalizations about a larger population based on a sample of data. This is where you go beyond just describing what you see in your sample.
- Key Idea: Dealing with uncertainty and making probabilistic statements.
- Sub-Branches (Examples):
- Estimation: Estimating population parameters (like the average height of all students based on a sample). Point estimates, confidence intervals.
- Hypothesis Testing: Testing claims or hypotheses about populations (like “Does this new drug actually work better than the old one?”). P-values, significance levels.
- Regression Analysis: Modeling relationships between variables to understand how changes in one variable affect another and to make predictions. This is where STAT3600’s linear regression fits in!
- Linear Regression (Simple, Multiple): Modeling linear relationships.
- Generalized Linear Models: Extending regression to handle different types of outcome variables (binary, counts, etc.).
- Analysis of Variance (ANOVA): Comparing means of groups to see if there are significant differences. This is also in STAT3600. One-way, two-way ANOVA.
- Non-parametric Statistics: Methods that don’t rely on assumptions about the data’s distribution (useful when data isn’t normally distributed).
- Predictive Statistics/Machine Learning (Overlapping, but Distinct Focus): Building models to make accurate predictions on new data. While traditional statistics focuses on inference and understanding relationships, predictive statistics is more about prediction accuracy. AI heavily overlaps here.
- Techniques: Many machine learning algorithms fall under this (decision trees, support vector machines, neural networks, etc.). Some statistical regression and classification methods are also used for prediction.
Level 3: Fundamental Statistical Concepts (The Foundation Underpinning Everything)
These are the core building blocks that you need to understand to work in any of the branches above:
- Probability Theory: The language of uncertainty. Understanding probability is essential for all of inferential statistics.
- Random Variables and Distributions: Modeling random phenomena. Probability distributions (Normal, Binomial, Poisson, etc.) are the workhorses of statistics. PDFs, CDFs, MGFs are tools to describe distributions.
- Sampling Distributions: Understanding how statistics (like sample means) vary from sample to sample. Crucial for inference.
- Estimation Theory: How to estimate population parameters from sample data.
- Hypothesis Testing Framework: The logic and process of testing hypotheses.
- Linear Algebra and Calculus: Mathematical tools that are heavily used in many statistical methods, especially regression and more advanced techniques.
Addressing Your “Continuous vs. Discrete” Feeling:
- “Continuous” Feeling (Need Previous Knowledge): You’re right, statistics is somewhat cumulative. You often need to understand probability and distributions before diving into regression. Think of it like layers in your CS system diagram. Probability and distributions are like the “operating system” of statistics – they are fundamental and underpin many higher-level techniques.
- How to Navigate: When learning a new concept like linear regression, briefly review the foundational concepts it relies on (probability, distributions, etc.). Don’t feel you need to become an expert in everything before moving on, but make sure you have a basic grasp of the prerequisites.
- “Discrete” Feeling (Unconnected Concepts): It can feel discrete if you’re just learning formulas and procedures in isolation. That’s why the “top-down” approach is so important!
- How to Connect: Actively look for connections between concepts. Ask:
- How does this concept relate to descriptive statistics? (e.g., Regression helps explain variance, which is a descriptive statistic).
- How does this concept relate to hypothesis testing? (e.g., Regression coefficients are often tested for significance using hypothesis tests).
- What is the purpose of this concept in the bigger picture of inference or prediction?
By actively building this “top-down” mental map and focusing on connections and purpose, you’ll start to see statistics less as a collection of disconnected formulas and more as a coherent and powerful system for understanding the world through data. You’ve got this!
]]>
