The Compass Philosophy - Foundations & Mission
MATH-CS COMPASS is designed to bridge the gap between the rigorous logic of pure mathematics and the practical innovation of computer science. Its common thread is verification: knowing when an answer can be trusted.
In the modern view, mathematics is the study of structure. Pure mathematicians tend to value formal rigor, and computer scientists value what works in practice, but understanding the structure underneath a system rewards both. This site offers an accessible yet principled look at the mathematical blueprints behind modern technology. Its topics are chosen so that a tool built for one purpose turns out to matter somewhere unexpected. An idea introduced in one section is often not paid off until another, sometimes far away: what looks at first like an aside becomes, pages later, the key to something you could not have solved without it. That return of ideas is a large part of what this curriculum is.
Calling mathematics the study of structure used to sound like a philosophical position. It is now a practical one. Machines write code, proofs, and predictions faster than anyone can read them, so the hard part is no longer producing an answer. It is knowing whether to trust it. An answer that looks right and an answer that is right can be very different things, and telling them apart is what mathematics has always done.
That is what we mean by verification, and every section of the Compass contributes to it. Algebra gives us invariants, which can show that two objects are different, and it lets a network respect a symmetry by design. Analysis gives us completeness, which guarantees that approximations which settle down have something real to settle on. Probability tells us what finite data can and cannot support. The theory of computation tells us which questions can be decided at all, and how a machine can check a proof step by step. Machine learning is where these guarantees are put to work, and where their absence shows first.
The Two Worlds of the Compass
Everything here is anchored by Section I (Linear Algebra to Algebraic Foundations) — the Core, where the common language of the whole site is established. It runs from the vectors and matrices of linear algebra, through the abstract structures of groups, rings, and fields, and up to the mathematics of continuous symmetry. Every later section speaks this language. From the Core, two great domains open up:
- The Discrete World — the realm of things that can be counted and steps that can be carried out one at a time. Section IV (Discrete Mathematics & Algorithms) studies graphs, logic, and the theory of computation, including its sharpest question: what a machine can and cannot decide at all. Three long arcs grow from it. Category theory is the language that names the single pattern behind every structure-preserving map. Cryptography is the field whose public-key half is threatened by a quantum algorithm and is being rebuilt on new mathematics believed to resist it. Formal methods treat a proof as a program that a machine can check.
- The Continuous World — the realm of smooth change and of quantities approached in the limit. Section II (Calculus to Optimization & Analysis) carries calculus into rigorous analysis and optimization. Its longer half then rebuilds those tools on curved spaces: the surface of a sphere, the space of all rotations, the shapes data traces out in high dimensions. Its unifying thread is completeness, the guarantee that a space has no holes, and the ground on which proofs of convergence stand.
The bridge between these two worlds is Section III (Probability & Statistics). There the counting of the discrete meets the integration of the continuous, and the two turn out to be one idea. It is the mathematics of reasoning when the answer is uncertain, and the direct foundation of how machines learn from data.
All of this converges in Section V (Machine Learning). But this is less a final destination than a high vantage point. From each topic, the tools built earlier can be seen working together, and the deeper mathematics still ahead comes into view. That is why the curriculum so often circles back, meeting an idea first as an application and only later building it properly from below. The clearest example is the geometry of learning itself, the principle that a good architecture respects the symmetries of its data. That path can now be followed end to end. Other views, such as a compositional account of learning drawn from category theory, are only beginning to take shape.
Specific tools, libraries, and frameworks age quickly; the mathematical foundations beneath them do not. The aim of this site is to help you understand, adapt, and even create new approaches, not just use today's technology as a black box. That understanding lasts long after the libraries and techniques of any current curriculum have been replaced.
If you have any suggestions, requests, or inquiries, please feel free to use the contact form below. Your input is invaluable in helping improve this site for everyone.
I would like to express my deep appreciation to the students and faculty of the Western Washington University Mathematics and Computer Science Departments. Their passion and inspiration were the primary motivation for creating this platform following my graduation.
Thank you for visiting, and enjoy your exploration!
(*Please note that the content on this site is a continuous work in progress. More resources are frequently added as the Compass expands.)
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Explore Topics
I - Linear Algebra to Algebraic Foundations
Explore foundations of modern mathematics & computer science.
II - Calculus to Optimization & Analysis
Explore optimization techniques and mathematical analysis.
III - Probability & Statistics
Explore probability theory and statistical methods.
IV - Discrete Mathematics & Algorithms
Explore graph theory, combinatorics, the theory of computation, and algorithms.
V - Machine Learning
Application viewpoints — from core ML to equivariant and topological deep learning. (Most of the underlying mathematics lives in Sections I–IV.)