Whenever you spot a disconnect between what your gut is telling you and what is supposed to be rational, it’s an important opportunity to understand something new. And then you can start this game of back-and-forth. Can you articulate your gut instinct and place it within a rational discussion? If there’s still a disconnect, can you visualize why? As you play that game, your imagination will gradually reconfigure. And in the end, if you’re persistent, your instinct and your reason will align, and you will be smarter. This is mathematical thinking.

Gödel proved in 1931 that mathematics is not decidable, an earth-shattering result. He proved that there are statements in mathematics, which are true but not provable within the system. Worse yet, it turns out that you can’t build a more powerful mathematical system. Once a system becomes sufficiently complex, there will always be statements which are undecidable. You’re left with a choice: either have weak system of mathematics or accept that there will always be theorems out of reach. A rough analogy to incompleteness Heisenberg’s Uncertainty Principle, which shows that physics makes it impossible to determine both the position and velocity of a particle with exact precision.

A third major theme of the book is isomorphism, which is unique to Hofstadter’s vernacular. In formal …

Visualise matrix multiplication

<> A Programmer's Introduction to Mathematics uses your familiarity with ideas from programming and software to teach mathematics.

You'll learn about the central objects and theorems of mathematics, covering graphs, calculus, linear algebra, eigenvalues, optimization, and more. You'll also be immersed in the often unspoken cultural attitudes of mathematics, learning both how to read and write proofs while understanding why mathematics is the way it is. Between each technical chapter is an essay describing a different aspect of mathematical culture, and discussions of the insights and meta-insights that constitute mathematical intuition.

As you learn, we'll use new mathematical ideas to create wondrous programs, from cryptographic schemes to neural networks to hyperbolic tessellations. Each chapter also contains a set of exercises that have you actively explore …

Pre-course resources to refresh knowledge of math, programming, and neuroscience.

Free Calculus book from MIT

Khan Academy's Linear Algebra videos

Math explained visually. Accessible math lessons. I watched the linear algebra videos.

<<<This notebook will explore these concepts in a concrete way using Python code. The code is meant to be succint and explicit, and fast enough to handle sample spaces with millions of outcomes. If you need to handle trillions, you'll want a more efficient implementation. I also have another notebook that covers paradoxes in Probability Theory.

Cognicull is a learning site that pursues "ease of understanding" and "comfort". It is designed so that knowledge of mathematics, natural science, and engineering can be gained easily.

[img[https://upload.wikimedia.org/wikipedia/commons/e/e5/Gospers_glider_gun.gif]]

Biography of the exceptional mathematician who invented the Game of Life.

books, resources and tips for anyone who wants to learn mathematics

31.4159265 North, 31.4159265 East is near Gamasa. (The real Pi 3.145, 3.145 is in the sea)