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Venn diagrams to choose good math books/articles

Venn diagram of great mathematicians Venn diagram of great math explainers Read books from those who are at the intersection of above two venn diagrams

Halmos, Courant, Davenport, Hilbert,

Those books that I know of are Courant's What is Mathematics? (praised by Einstein), David Hilbert's Geometry and Imagination, and W. W. Sawyer's Math Books which are highly praised by Paul Graham. Additionally, Davenport's Higher Arithmetics and The Princeton Companion to Mathematics (it is more like an encyclopedia from many different authors. Each author is a great mathematicians, and some of them are also great explainers, while some others not)

Universal truth with books. Most books are bad. Nearly all textbooks are bad finding good books and learning material is extremely important

Below explanation of Paul R. Halmos in his linear algebra book summarizes what I expect from a good math book.

The complimentary book - Linear Algebra Problem Book (book) by Halmos

Halmos clearly had fun writing his books, that is apparent when reading them. Note that Halmos himself is a martian.

Dolciani Mathematical Expositions is a good place to find other such books.

Halmos method

Paul R. Halmos in his Hilbert Space problem book (preface, page vii)

#### Historical context What did the mathematician know at the time he discovered this or proved something? What was his reasoning process? Which previous or contemporary mathematicians he talked to. What is the landscape of the problem and what are consequences. After knowing the surrounding context of a discovery, it becomes much more clear how he was able to found it. It's not magic. Genius is lamer than what would those who didn't know the context think.

Great math explainers explain within the context. Context of history, what the mathematician knew about others, process of reasoning, etc.

More generally, a canon of knowledge become consistent and meaningful only when thought within its historical development. Because all knowledge carries inconsistencies caused by its non-linear historical development. For example, polynomial means many-named/termed in latin. But, an equation involving factorials are not considered polynomial. This is not elegant. But it is what it is because of the historical development of the term. Another thing, which is not merely incorrect, but non-elegant is the fact that the term quadratic represents x^2. The word quadratic comes from quadratus in latin, meaning 'made square', hence involve 'quad' that means number four in latin. It is kind of confusing because it involves 'quad (four)' as it makes me think of x^4. Not elegant.

Problems with current mathematical education

See rant of Terence Tao on at his Lex Fridman interview.

current math education is very bad. we have one-size fits all approach to teaching now.
it is designed for classrooms of 30-40, but each actually needs an individual tutor
math can be much more accessible
evolution didn't give us a math center to do math directly. instead of we have other centers, like vision, language and others. how to do math is repurposing any of the other centers to do math. A mathematician can be very quick with one center and slow on others. Terence Tao himself admits to be bad with visual thinking

Tree/graph of proofs

Math dependency graph e.g.

[THEOREM] Euler’s Totient Theorem
└── If a ∈ ℤ, gcd(a, n) = 1 ⇒ a^φ(n) ≡ 1 mod n
    |
    ├── [DEFINITION] Euler’s Totient Function φ(n)
    │   └── φ(n) = count of integers 1 ≤ k < n such that gcd(k, n) = 1
    │       ├── [DEFINITION] Coprimality: gcd(a, b) = 1
    │       │   └── [ALGORITHM] Euclidean Algorithm (to compute gcd)
    │       │       └── [AXIOM] Division Algorithm
    │       └── [ALGORITHM] Compute φ(n) using prime factorization
    │           └── φ(n) = n × ∏(1 - 1/p), for p | n
    │               └── [DEFINITION] Prime Factorization
    │                   └── [THEOREM] Fundamental Theorem of Arithmetic
    │                       └── Every integer > 1 is uniquely a product of primes
    │
    ├── [LEMMA] Multiplicative Property of φ(n)
    │   └── If gcd(m, n) = 1 ⇒ φ(mn) = φ(m)·φ(n)
    │       └── [PROOF] Uses Chinese Remainder Theorem
    │           └── [THEOREM] Chinese Remainder Theorem (CRT)
    │               └── [CONSTRUCTION] Isomorphism of ℤ/mnℤ ≅ ℤ/mℤ × ℤ/nℤ
    │
    ├── [LEMMA] Units Modulo n form a Group under Multiplication
    │   └── [CONSTRUCTION] (ℤ/nℤ)* = {a < n | gcd(a, n) = 1}
    │       ├── [DEFINITION] Group (Algebraic structure)
    │       └── [AXIOM] Associativity, identity, inverse, closure
    │
    ├── [THEOREM] Lagrange’s Theorem (Group Theory)
    │   └── In finite group G, order of any element divides |G|
    │       └── [PROOF] Coset partitioning argument
    │
    ├── [THEOREM] Fermat’s Little Theorem (special case of Euler’s)
    │   └── If p prime, a not divisible by p ⇒ a^(p−1) ≡ 1 mod p
    │       ├── [DEFINITION] Prime Number
    │       ├── [PROOF] Uses multiplicative group modulo p
    │       │   └── [LEMMA] (ℤ/pℤ)* is cyclic for prime p
    │       └── [COROLLARY] Euler’s Theorem generalizes Fermat’s
    │
    ├── [DEFINITION] Modular Arithmetic
    │   ├── [DEFINITION] Congruence: a ≡ b mod n ⇔ n | (a − b)
    │   ├── [DEFINITION] Modular Exponentiation: a^k mod n
    │   ├── [THEOREM] Properties of Congruences
    │   │   └── e.g., a ≡ b ⇒ a+c ≡ b+c and ac ≡ bc
    │   └── [ALGORITHM] Fast Modular Exponentiation
    │
    └── [EXAMPLE] Euler’s Theorem with n = 9, a = 2
        ├── φ(9) = 6 ⇒ 2^6 ≡ 1 mod 9
        └── (2^6 = 64; 64 mod 9 = 1 ✔️)

See how I generate dependency graphs at https://chatgpt.com/share/688c4b20-9050-8010-8070-52c2f85b0fac

Looking at this graph, one can divide the problem into sub-pieces. If you can prove/understand each node, then you can construct the whole proof leading to the theorem at hand. An ambitious project would be to generate an enormous map for the whole math where each node has its own document of proof.

Relevantly, Timothy Gowers works on an automatic theorem proving project.

Take multiple perspectives

Learn to Think in Multiple Ways — The Anthology of Balaji (archived)

verbal, visual, algebraic, numerical, computational, historical

Sorted by importance (by Baris)

historical, context, story [HOW? WHY?]
computational (in python code), constructive [HOW?]
algebraic [WHAT?]
numerical / visual (depending on the topic) [WHAT?]
verbal [WHAT? WHY?]

Relevant: humans learn through stories because they consist how and why

Visual perspective is especially helpful especially when remembering, for an overview.

e.g.
- ]

Another very helpful is code representation, as in Python.

e.g. RSA algorithm

Nuggets from the best math books

How To Solve It (book) by Polya

One-page summary from the book

Open: Pasted image 20250810171024.png

NotebookLM Summary

George Pólya's "How to Solve It" presents a comprehensive approach to problem-solving, particularly in mathematics, emphasizing a heuristic method rather than rigid formal rules. The book organizes problem-solving into a four-phase framework and details numerous specific techniques within these phases.

Overarching Problem-Solving Framework: Four Phases

The core of Pólya's method is divided into four phases:

1. Understanding the Problem: This initial phase involves clearly identifying and comprehending all aspects of the problem. Key steps include determining what the unknown is, what the data are, and what the condition is. It also involves assessing if the condition is sufficient, insufficient, redundant, or contradictory, and using tools like drawing a figure and introducing suitable notation.
2. Devising a Plan: In this phase, the solver seeks to establish a connection between the given data and the unknown, ultimately formulating a strategy for the solution. This often involves asking if a related problem has been seen before, considering theorems, and contemplating the introduction of auxiliary elements. If the proposed problem cannot be solved, one might attempt to solve a simpler, more general, more special, or analogous problem, or try to restate the problem.
3. Carrying Out the Plan: This phase focuses on meticulously executing the plan devised in the previous step. It requires checking each step for clarity and correctness, and providing proof for each action.
4. Looking Back: The final phase involves reviewing the solution obtained to deepen understanding and consolidate knowledge. This includes checking the result and the argument, considering if the result could be derived differently, and exploring whether the solution or its method can be used for some other problem.

Specific Problem-Solving Techniques (Heuristics)

Beyond the four phases, the book outlines various heuristic techniques, many detailed in the "Short Dictionary of Heuristic":

Analogy: A method that leverages similarity in relations between parts of different objects or problems. It suggests that if two problems share structural resemblances, the solution method for one might be applicable or adaptable to the other, often by identifying a simpler analogous problem.
Auxiliary Elements: The strategic introduction of new components into a problem (e.g., auxiliary lines in geometry, or an auxiliary unknown in algebra) to create new connections, simplify the problem, or make the path to a solution more apparent.
Auxiliary Problem: A simpler, related problem whose solution helps clarify or simplify the original, more complex problem. This is a sub-problem that, once solved, assists in resolving the main problem.
Bright Idea: Refers to a sudden, insightful leap or "seeing the light" that reveals a crucial connection or path to the solution. It's often linked to Aristotle's concept of "sagacity" – hitting upon the essential connection by intuition.
Can you check the result?: A verification technique to build confidence in the solution, either by numerical estimation, common sense, or formal logical checks of the argument.
Can you derive the result differently?: Encourages finding alternative methods or proofs for a solution. This practice enhances understanding, broadens problem-solving perspectives, and strengthens conviction in the result.
Can you use the result?: Promotes exploring the broader utility of a solution or method, prompting reflection on how it might apply to or inspire the solution of other problems.
Condition: Emphasizes analyzing the given conditions of a problem. This involves understanding the relationships between the unknown and the data, and determining if the conditions are sufficient, insufficient, redundant, or contradictory for a solution.
Could you derive something useful from the data?: A technique for extracting preliminary insights by systematically exploring the given data and identifying any useful information or relationships, even before a complete solution plan is formed.
Could you restate the problem?: Involves rephrasing the problem in different terms or from various perspectives to gain clarity, uncover hidden implications, or suggest new approaches that were not obvious in the original phrasing.
Decomposing and Recombining: The strategy of breaking a complex problem down into smaller, more manageable parts, solving these individual sub-problems, and then synthesizing their solutions to construct the complete solution to the original problem. This simplifies complexity and helps focus on essential elements.
Did you use all the data?: A crucial check to ensure that all given information has been fully incorporated into the solution. Failure to use all data can indicate an incomplete understanding or an erroneous approach.
Do you know a related problem?: A technique that prompts the solver to recall past experiences with similar problems, drawing upon previously acquired knowledge and successful methods to inform the current problem's solution.
Draw a figure: A visual aid, particularly useful in geometry, where sketching the problem helps visualize the relationships between elements, clarify the unknown and data, and often stimulates new ideas for the solution.
Examine your guess: Encourages testing initial hypotheses or partial solutions. Even if a guess is incorrect, analyzing why it fails can provide significant insights into the problem's nature and guide towards a correct solution.
Generalization: A strategy of solving a broader or more general problem that encompasses the original problem as a special case. This can sometimes make the solution process simpler or reveal deeper underlying principles.
Have you seen it before? / Here is a problem related to yours and solved before: These inquiries prompt the solver to access their existing knowledge base and past problem-solving experiences, mobilizing "dormant knowledge" to find relevant patterns or solutions.
Heuristic Reasoning: A type of reasoning that is provisional and plausible, designed for the discovery of a solution, as opposed to strict, formal proof. It often relies on induction or analogy and serves as a method of inquiry and invention.
If you cannot solve the proposed problem: A meta-strategy that suggests modifying the problem if the original proves intractable. This can involve attempting a related, more accessible, more general, more special, or analogous problem, or simplifying the conditions.
Induction and Mathematical Induction: Induction is the process of inferring general laws from specific observations. Mathematical induction is a formal proof method used to establish the truth of propositions across an infinite set of cases, often building upon patterns observed through induction.
Inventor's Paradox: The surprising observation that sometimes a more ambitious, general plan or problem might be easier to solve than a more modest, specific one. This can occur because a broader scope might reveal simpler underlying structures or connections.
Is it possible to satisfy the condition?: A diagnostic question to check the consistency and solvability of the problem. It helps determine if the given conditions are contradictory or if a solution is even feasible.
Pappus's Analysis and Synthesis:
- Analysis: A method of discovery that involves starting from what is sought (the unknown) and tracing backward through its logical consequences until a known or previously established fact is reached. This is akin to Working Backwards.
- Synthesis: A method of presentation, often used for formal proofs, which starts from known or given facts and proceeds forward through logical steps to arrive at the desired conclusion.
Reductio ad Absurdum and Indirect Proof:
- Reductio ad Absurdum: A proof technique that demonstrates the falsity of a proposition by showing that assuming it to be true leads to a logical absurdity or contradiction.
- Indirect Proof: A broader category of proof where the truth of a statement is established by proving that its negation leads to a contradiction.
Teacher's Method of Questioning: A pedagogical approach where the teacher guides the student towards the solution by asking a series of carefully chosen questions, prompting the student to discover the steps themselves rather than simply being told the answer. This cultivates the student's independent mental habits.
Variation of the problem: Systematically altering aspects of the problem (e.g., parameters, conditions, or the unknown) to explore different versions, which can sometimes reveal new insights or simplify the original problem.
Working Backwards: A problem-solving strategy where the solver begins with the desired outcome or unknown and determines the necessary preceding steps to reach it from the given information. This technique is closely related to Pappus's analysis.

How I Learn Math