Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. Two of the most highly abstract areas of modern mathematics are category theory and model theory. (Wikipedia).

Teach Astronomy - The Role of Mathematics

http://www.teachastronomy.com/ The language of science is mathematics. Mathematics is a system for replacing the concrete with numbers and with abstract entities. It's a system that is unique and wonderful and we believe only developed by humans for making sense the natural world. Human

From playlist 01. Fundamentals of Science and Astronomy

Equivalence relations -- Proofs

This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.

From playlist Proofs

Functions, operators, and linearity: the language of abstract math (#SoME1)

Mathematicians and physicists often use abstract notation and terminology to reason about and describe problems at a level above the explicit details of the problem, but often take for granted that everyone already understands what they're doing and why. This video gives a short explanati

From playlist Summer of Math Exposition Youtube Videos

Field Definition (expanded) - Abstract Algebra

The field is one of the key objects you will learn about in abstract algebra. Fields generalize the real numbers and complex numbers. They are sets with two operations that come with all the features you could wish for: commutativity, inverses, identities, associativity, and more. They

From playlist Abstract Algebra

This video explains what a mathematical function is and how it defines a relationship between two sets, the domain and the range. It also introduces three important categories of function: injective, surjective and bijective.

From playlist Foundational Math

Definition of infinity In this video, I define the concept of infinity (as used in analysis), and explain what it means for sup(S) to be infinity. In particular, the least upper bound property becomes very elegant to write down. Check out my real numbers playlist: https://www.youtube.co

From playlist Real Numbers

Derivatives are the main object of study in differential calculus. They describe rates of change of functions. That makes them incredibly useful in all of science, as many models can be expressed by describing the changes over time (e.g. of physical quantities). However, the abstract defin

From playlist Summer of Math Exposition Youtube Videos

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

It's a concept which intrigues mathematicians, but scientists aren't so keen on it. More at http://www.sixtysymbols.com/

From playlist From Sixty Symbols

What Are Numbers? Philosophy of Mathematics (Elucidations)

What is mathematics about and how do we acquire mathematical knowledge? Mathematics seems to be about numbers, but what exactly are numbers? Are numbers and other mathematical objects something discovered or invented? Daniel Sutherland discusses some of these issues in the philosophy of ma

From playlist Logic & Philosophy of Mathematics

Mark Balaguer - How is Mathematics Truth and Beauty?

When mathematicians speak about their craft, why do they use terms of philosophy and art? What is it about mathematics that can penetrate trivial truth and reveal fundamental “Truth?” What are the characteristics of fundamental truth? What is it about mathematics that can elicit the descri

From playlist How is Mathematics Truth and Beauty? - CTT Interview Series

Edward Frenkel: Let's Stop Hating Math | Big Think

Edward Frenkel: Let's Stop Hating Math Watch the newest video from Big Think: https://bigth.ink/NewVideo Join Big Think Edge for exclusive videos: https://bigth.ink/Edge ---------------------------------------------------------------------------------- Mathematician Edward Frenkel knows w

From playlist Interviews

LambdaConf 2015 - The Abstract Method, In General Gershom Bazerman

“Programming is about abstraction.” But what is abstraction about? Surely not just programming. Why do we need it, why do we keep reinventing it? How do we even recognize it when we see it, across so many domains? When something is more abstract, it is not necessarily more useless, even th

From playlist LambdaConf 2015

4. Calculus: One of the Most Successful Technologies

(October 22, 2012) Professor Keith Devlin discusses how calculus is truly one of the most useful discoveries of all time. Originally presented in the Stanford Continuing Studies Program. Stanford University: http://www.stanford.edu/ Stanford Continuing Studies Program: https://continuin

From playlist Lecture Collection | Mathematics: Making the Invisible Visible

Is Mathematics Invented or Discovered? | Episode 409 | Closer To Truth

Mathematics describes the real world of atoms and acorns, stars and stairs, with remarkable precision. So is mathematics invented by humans-like chisels and hammers and pieces of music? Or is mathematics discovered-always out there, somewhere, like mysterious islands waiting to be found? F

From playlist Closer To Truth | Season 4

William Lane Craig Retrospective V: God and Abstract Objects | Closer To Truth Chats

Analytic philosopher and Christian apologist William Lane Craig talks about God’s absolute sovereignty, self-sufficiency, and how Abstract Objects – forms, numbers, logic – threaten an autonomous God. Craig has authored or edited over thirty books, including God Over All: Divine Aseity and

From playlist Big Questions About God - Closer To Truth - Core Topic

Category Theory 1.1: Motivation and Philosophy

Motivation and philosophy

From playlist Category Theory

Lecture 1 | The Theoretical Minimum

(January 9, 2012) Leonard Susskind provides an introduction to quantum mechanics. Stanford University: http://www.stanford.edu/ Stanford Continuing Studies: http://continuingstudies.stanford.edu/ Stanford University Channel on YouTube: http://www.youtube.com/stanford

From playlist Lecture Collection | The Theoretical Minimum: Quantum Mechanics

Lecture 2. Homomorphisms and ideals

From playlist Abstract Algebra 2