Foundations of geometry

Definitions, specification and interpretation | Arithmetic and Geometry Math Foundations 44

We discuss important meta-issues regarding definitions and specification in mathematics. We also introduce the idea that mathematical definitions, expressions, formulas or theorems may support a variety of possible interpretations. Examples use our previous definitions from elementary ge

From playlist Math Foundations

The basic framework for geometry (II) | Arithmetic and Geometry Math Foundations 24 | N J Wildberger

We discuss parallel and perpendicular lines, and basic notions relating to triangles, including the notion of a side and a vertex of a triangle. This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discus

From playlist Math Foundations

The essential dichotomy underlying mathematics | Data Structures Math Foundations 186

What lies at the very core of mathematics? What is mathematics ultimately about, once we strip away all the hoopla and complexity? In this video I give you my answer to this intriguing question. Surprisingly, it is not really the natural numbers: they are fundamental, but not the most fund

From playlist Math Foundations

An introduction to algebraic curves | Arithmetic and Geometry Math Foundations 76 | N J Wildberger

This is a gentle introduction to curves and more specifically algebraic curves. We look at historical aspects of curves, going back to the ancient Greeks, then on the 17th century work of Descartes. We point out some of the difficulties with Jordan's notion of curve, and move to the polynu

From playlist Math Foundations

What is a number? | Arithmetic and Geometry Math Foundations 1 | N J Wildberger

The first of a series that will discuss foundations of mathematics. Contains a general introduction to the series, and then the beginnings of arithmetic with natural numbers. This series will methodically develop a lot of basic mathematics, starting with arithmetic, then geometry, then alg

From playlist Math Foundations

The basic framework for geometry (I) | Arithmetic and Geometry Math Foundations 23 | N J Wildberger

This video begins to lay out proper foundations for planar Euclidean geometry, based on arithmetic. We follow Descartes and Fermat in working in a coordinate plane, but a novel feature is that we use only rational numbers. Points and lines are the basic objects which need to be defined. T

From playlist Math Foundations

The deep structure of the rational numbers | Real numbers and limits Math Foundations 95

The rational numbers deserve a lot of attention, as they are the heart of mathematics. I am hopeful that modern mathematics will (slowly) swing around to the crucial realization that a lot of things which are currently framed in terms of "real numbers" are more properly understood in terms

From playlist Math Foundations

What is the Fundamental theorem of Algebra, really? | Abstract Algebra Math Foundations 217

Here we give restatements of the Fundamental theorems of Algebra (I) and (II) that we critiqued in our last video, so that they are now at least meaningful and correct statements, at least to the best of our knowledge. The key is to abstain from any prior assumptions about our understandin

From playlist Math Foundations

Mathematical space and a basic duality in geometry | Rational Geometry Math Foundations 122

In this video we introduce some basic orientation to the problem of how we represent, and think about, space in mathematics. One key idea is the fundamental duality between the affine and projective views: two sides of the same coin. We explain how the Cartesian revolution of the 17th ce

From playlist Math Foundations

Why it took 379 pages to prove 1+1=2

Sign up to Brilliant to receive a 20% discount with this link! https://brilliant.org/upandatom/ Hi! I'm Jade. If you'd like to consider supporting Up and Atom, head over to my Patreon page :) https://www.patreon.com/upandatom Visit the Up and Atom store https://store.nebula.app/collecti

From playlist Math

Geometry Course – Chapter 1 (Foundations) Let’s Start!

Learn Geometry - chapter 1 full Geometry course, Foundations to Geometry. For more in-depth math help check out my catalog of courses. Every course includes over 275 videos of easy to follow and understand math instruction, with fully explained practice problems and printable worksheets

From playlist GED Prep Videos

Payel Das - Design and Evaluation of Foundation Models and Generative AI in Molecular Space

Recorded 23 January 2023. Payel Das of IBM Research presents "Design and Evaluation of Foundation Models and Generative AI in Molecular Space" at IPAM's Learning and Emergence in Molecular Systems Workshop. Abstract: Learning effective representations of molecules is critical in a variety

From playlist 2023 Learning and Emergence in Molecular Systems

Geometry | Arithmetic and Geometry Math Foundations 18 | N J Wildberger

How to begin geometry? What is the correct framework? How to define point, line, circle etc etc? These are some of the issues we will be addressing in this first look at the logical foundations of geometry. This lecture is part of the MathFoundations series, which tries to lay out proper

From playlist Math Foundations

Michael Atiyah: Poincaré conjecture, Hodge conjecture, Yang-Mills, Navier-Stokes [2000]

Millennium Meeting These videos document the Institute's landmark Paris millennium event which took place on May 24-25, 2000, at the Collège de France. On this occasion, CMI unveiled the "Millennium Prize Problems," seven mathematical quandaries that have long resisted solution. The announ

From playlist Number Theory

Euclid's Books VI--XIII | Arithmetic and Geometry Math Foundations 21 | N J Wildberger

A very brief outline of the contents of the later books in Euclid's Elements dealing with geometry. This includes the work on three dimensional, or solid, geometry, culminating in the construction of the five Platonic solids. This lecture is part of the MathFoundations series, which tries

From playlist Math Foundations

Difficulties with Euclid | Arithmetic and Geometry Math Foundations 22 | N J Wildberger

There are logical ambiguities with Euclid's Elements, despite its being the most important mathematical work of all time. Here we discuss some of these, as well as Hilbert's attempt at an alternative formulation. We prepare the ground for a new and more modern approach to the foundations o

From playlist Math Foundations

Euclid Book 1 Props VI-VIII - a foundation for geometry | Sociology and Pure Maths | N J Wildberger

We look at Propositions VI to VIII of Book 1 of Euclid's Elements, perhaps the first place where proofs by contradiction arise in mathematics. The proofs are not entirely transparent however, and a reasonable question arises as to the suitability of Euclid as a foundation for modern geomet

From playlist Sociology and Pure Mathematics

The basic framework for geometry (IV) | Arithmetic and Geometry Math Foundations 26 | N J Wildberger

Angles don't make sense in the rational number system. The proper notion of the separation of two lines is the `spread' between them, which is a purely algebraic quantity and can be calculated easily using rational arithmetic only. This video highlights some of the advantages in replacing

From playlist Math Foundations

The three-fold symmetry of chromogeometry | Rational Geometry Math Foundations 141 | NJ Wildberger

There are three planar metrical geometries that fit together beautifully; the usual Euclidean geometry (which we call blue), and two relativistic geometries (red and green). Most of the fundamental theorems of the subject hold for all three. But even more remarkably, when we transcend Klei

From playlist Math Foundations