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Subjects
STEM
Mathematics
Logic & Foundations
Foundations of Mathematics

Free Online

Foundations of Mathematics Courses and Certifications

Build strong mathematical foundations from elementary arithmetic through pre-algebra concepts and mathematical reasoning. Access grade-level curricula on Study.com and explore advanced problem-solving on Coursera and edX, preparing students for higher mathematics and real-world applications.

32 courses

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Mathematical Proofs
Set Theory
Propositional Logic
Category Theory
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Math Foundation

Build mathematical maturity through logic, sets, functions, relations, proof techniques, geometric intuition, and mathematical writing.
- Swayam
- Available until Oct 9, 2026
- Free Online Course
Infinity

Expand your mind with the paradoxes and beauties of infinity.
- Brilliant
- 4 hours
- Self-Paced
- Paid Course
Insights into Mathematics
Problems With the Calculus - Math History - NJ Wildberger

A historical examination of the unresolved conceptual problems behind calculus, from Newton and Leibniz’s infinitesimals to limits, complex numbers, and probability.
- YouTube
- 49 minutes
- Self-Paced
- Free Video
Insights into Mathematics
The Rational Number Line and Irrationalities - Famous Math Problems 19b

A geometric account of the rational number line as a model of the continuum, leading to projective constructions and the introduction of irrationalities.
- YouTube
- 38 minutes
- Self-Paced
- Free Video
Insights into Mathematics
Towards a New Mathematics and Types versus Sets - Sociology and Pure Maths

Explore Box Arithmetic, a revolutionary approach to mathematics foundations using nested empty boxes instead of infinite sets, distinguishing types from sets.
- YouTube
- 14 minutes
- Self-Paced
- Free Video
Insights into Mathematics
The Operation of Caret - Exponentiation via Multisets - Math Foundations

Explore a novel arithmetic on multisets, where a recursively defined caret operation extends addition and multiplication into a new form of exponentiation.
- YouTube
- 35 minutes
- Self-Paced
- Free Video
Insights into Mathematics
A Multiset Approach to Arithmetic - Math Foundations

Explore arithmetic and algebra through multisets, defining natural numbers, polynumbers, and multinumbers as nested unordered collections.
- YouTube
- 39 minutes
- Self-Paced
- Free Video
Insights into Mathematics
Is Pure Mathematics Logically Viable? Five Challenges - Sociology and Pure Maths

Five challenges question the logical foundations and professional assumptions of pure mathematics.
- YouTube
- 22 minutes
- Self-Paced
- Free Video
Insights into Mathematics
Emile Borel - Real Number Enthusiast or Skeptic? - Sociology and Pure Mathematics

Explore Borel’s skepticism about real numbers through infinite decimals, normal numbers, and his famous know-it-all number.
- YouTube
- 24 minutes
- Self-Paced
- Free Video
Insights into Mathematics
The Realm of Natural Numbers - Data Structures in Mathematics

Explore natural numbers through multiset-inspired addition, union, and intersection, proving their laws and a surprising distributive relation linked to tropical calculus.
- YouTube
- 25 minutes
- Self-Paced
- Free Video
Insights into Mathematics
Introducing Box Arithmetic - Math Foundations 237

An overview of Box Arithmetic, using multisets and a box/anti-box duality to rethink arithmetic and its foundations.
- YouTube
- 23 minutes
- Self-Paced
- Free Video
Insights into Mathematics
Arithmetic with Negative Multisets and Integral Polynumbers - Math Foundations 232

Using anti-msets to define negative numbers, this lesson develops arithmetic for integer polynumbers and their Laurent forms.
- YouTube
- 40 minutes
- Self-Paced
- Free Video
Insights into Mathematics
Negative Numbers from Antimultisets - Math Foundations 231

Particle-antiparticle annihilation and creation offer a surprising route to deriving negative numbers in arithmetic built from multisets.
- YouTube
- 39 minutes
- Self-Paced
- Free Video
Insights into Mathematics
The Successor-Limit Hierarchy in Higher Arithmetic Operations

Builds a hierarchy of ever-larger arithmetic operations using successors and diagonal limits, then works through a detailed higher-operation calculation.
- YouTube
- 43 minutes
- Self-Paced
- Free Video
Insights into Mathematics
Cracking the Riemann Hypothesis - A Computational Approach

A computational critique of infinite operations reframes Lagarias’ equivalent form of the Riemann Hypothesis as a finite calculation.
- YouTube
- 28 minutes
- Self-Paced
- Free Video

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