It all begins here. What is the object upon which all of mathematics rests? The empty set
How many numbers are there? Replying 'infinitely many!' would be like asking how many fingers you have, and hearing back 'some number!'
Finite numbers come in different flavors, and infinite numbers do as well. What is a number, anyway? The answer depends on what you choose to believe about the Higher Infinite
Picture a circle, and choose a single point inside. What's its area? Every point inside a circle has area exactly 0 —zero, nada—, and yet, somehow, the sum of all those 0's makes up for an area that is not 0?
How can this be? It took over two thousand years to give a proper —i.e. rigorous— answer. It turns out summing 0 + 0 + 0 ... is not enough
We need to understand the depths of the Continuum, which will lead us to the limits of reasoning itself
There are objects —like a square— that don't change appearance when you rotate them by 90 degrees. This invariance is called a symmetry
Symmetries, in turn, form abstract structures whose study goes to the deepest branches of reason, giving rise to groups, rings, fields, homomorphisms, domains, adeles, ideles...
Even the fundamental nature of reality exhibits such order
Take a cup of Play-Doh and form a square. Now deform it to make a circle. In Topology, these two are equal, i.e. they're homeomorphic to each other
While you're at it, make a sphere. Can you deform it and make it pass through itself without tearing a hole through it? You may not, but there's a way
Prepare to delve into higher dimensions
Geometry dominated the minds of Greeks in antiquity. The universe was harmonic, and numbers embodied such essence
The primitives of existence were special kinds of polyhedra, out of which all beauty took form
But their dreams of acme were not meant to be
Highlights
Two New Yorkers have the same number of hairs
What happens when an 'obvious' theorem has non-obvious consequences?
Elementary mathematics: summary
A place to glance over stuff you know but not really learn anything new. Work in progress!
Kids explain infinity (the Alephs)
We explained the main ideas of Set Theory to kids. Then we had them explain them back.
From eqyptian pyramids to hyperdimensional rings
Beneath the symbols describing polynomials, there's something deep lurking.
Barb the barber shaves people
In a town, which shall remain nameless, there’s a barber, which shall be named Barb. He shaves. Everyone. Except he doesn't. But he does. But he...
Once upon revolutionary times
These were dark, Enlightened days. Chants of freedom swept the kingdom. A fierce kid challenged reason. But he failed
Mathematics allows the mind to soar
Mathematics is the backbone of technological innovation. Behind every running computer, every burning engine, every shining city, and every flying airplane, there rest hundreds of theorems, painstakingly discovered over centuries
mathIsART is committed to spreading mathematical knowledge freely and intuitively across the globe and —soon, hopefully— beyond