Math From Scratch

For a while now (at the very least since 2019, at most since 2015), I've been wanting to prove all the mathematical knowledge (theorems and algorithms) that I've learned. If you were to learn Mathematics from the very beginning, how would you go about it? Rather than blindly trusting a source, you should verify a claim by tracing it back to the axioms it's built on, the elementary blocks which you assume to be true.

This is my attempt at this challenge. I would like to rigorously derive all my current mathematical toolset and make my way up to my favorite topics, like Game Theory.


Will Nuno follow the footsteps of Euclid, Whitehead and Russell and accomplish Hilbert's dream?

Find out in the next episode of Nuno X Math From Scratch!

To be continued...

(Spoiler alert: nope. Then again, the goal of Math From Scratch isn't the same as Hilbert's, so who cares?)


Table of Contents

  1. The Very Beginning
  2. Syntax
  3. Propositional Logic
    1. Language
    2. Semantics
      1. Definitions
      2. Truth Functions
    3. Propositional Calculus: \(P_2\)
    4. Useful Theorems, Lemmas and Metatheorems
    5. Deduction Metatheorem \(\rightarrow\) still working on it
    6. Soundness and Consistency
    7. Completeness and Decidability
    8. Independence Proof
    9. Other Propositional Calculi
      1. Łukasiewicz System
      2. Walsh's System with \(A_1\)
      3. Nicod's System
      4. Natural Deduction


Special thanks to Charlie Park (박찬솔) and Orfeas Liossatos for checking for typos and improving the text.


List of main books/papers that seem useful:

  • Analysis:
    • A Course in Mathematical AnalysisD. J. H. Garling
    • Analyse I Théorie des Ensembles et TopologieLaurent Schwartz
    • Analyse II Calcul Différentiel et Équations DifférentiellesLaurent Schwartz
    • Analyse III Calcul IntégralLaurent Schwartz
    • Analyse IV Applications à la Théorie de la MesureLaurent Schwartz
    • Analysis ITerence Tao
    • Analysis IITerence Tao
  • Game Theory:
    • A Course In Game TheoryThomas S. Ferguson
    • Introducing Game Theory and Its ApplicationsElliott Mendelson
  • Geometry:
    • Elementary Geometry from an Advanced StandpointEdwin E. Moise
    • Why is it that the Ratio of Any Circle's Circumference to its Diameter is a Constant?F. M. S. Lima & P. G. F. Jordão
  • Mathematical Logic and Set Theory:
    • A Primer for Logic and ProofHolly P. Hirst & Jeffry L. Hirst
    • Answers to Some Open Questions of Ulrich & MeredithMatthew Walsh & Branden Fitelson
    • Axiomatization of Propositional Calculus With Sheffer FunctorsThomas W. Scharle
    • Differential geometry reconstructedAlan U. Kennington
    • Elements of Mathematical LogicJan Łukasiewicz
    • Foundations of MathematicsMohammad Safdari
    • Friendly Introduction to Mathematical LogicChristopher Leary & Lars Kristiansen
    • Gödel's Theorems and Zermelo's AxiomsLorenz Halbeisen & Regula Krapf
    • Introduction to Axiomatic Set TheoryGaisi Takeuti & Wilson M. Zaring
    • Introduction to Mathematical LogicEliott Mendelson
    • Logic for MathematiciansJ. Barkley Rosser
    • Logics for Computer ScienceAnita Wasilewska
    • Metalogic, An Introduction to the Metatheory of Standard First Order LogicGeoffrey Hunter
    • Models and GamesJouko Väänänen

Books that have useful formal proofs:

  • A Primer for Logic and ProofHolly P. Hirst & Jeffry L. Hirst
  • Differential geometry reconstructedAlan U. Kennington
  • Elements of Mathematical LogicJan Łukasiewicz
  • Foundations of MathematicsMohammad Safdari
  • Gödel's Theorems and Zermelo's AxiomsLorenz Halbeisen & Regula Krapf
  • Introduction to Mathematical LogicEliott Mendelson
  • Metalogic, An Introduction to the Metatheory of Standard First Order LogicGeoffrey Hunter

Books that appear to have useful formal proofs (need to check):

  • A Mathematical Introduction to LogicHerbert B. Enderton
  • Computability and LogicGeorge S. Boolos & John P. Burgess & Richard C. Jeffrey
  • Foundations of MathematicsMohammad Safdari
  • Introduction to Axiomatic Set TheoryGaisi Takeuti & Wilson M. Zaring
  • Introduction to MetamathematicsStephen Cole Kleene
  • Logic for MathematiciansJ. Barkley Rosser
  • Mathematical LogicJoseph R. Shoenfield
  • Mathematical LogicStephen Cole Kleene
  • Mathematical LogicW. V. Quine
  • Principia MathamaticaAlfred North Whitehead & Bertrand Russel

Posted 18/08/2024 | Last edited 03/04/2025