Let’s admit it, most of us are intellectually lazy, and thinking is a strange and hard activity, because it requires effort. Even philosophy, the prime discipline devoted to thinking, mostly circulates the ideas of other people, and has some similarity to journalism. Real thinking...

Here are two computer-animated videos from Youtube that demonstrate some properties of knots, the relations between knots and space, and how we arrive at hyperbolic space. Easy to watch, and very informative. Part Two:

Source: Kurt Gödel, Collected Works, Volume III (1961) publ. Oxford University Press, 1981. The Complete lecture reproduced here. I would like to attempt here to describe, in terms of philosophical concepts, the development of foundational research in mathematics since around...

Below there is an interesting video inspired by numbers, geometry and nature, created by Cristóbal Vila. Nature looks complex, but the underlying principles are simple, for instance the Fibonacci Series of numbers. The natural beauty and complexity we see all...

How do you turn a sphere inside out, without punching a hole into it? It is possible,as you can see in the transformations in this fascinating video:

How does mathematics and reality relate to each other? Some mathematicians believe that mathematical entities are “real”, and this view can be characterized as a version of Platonic philosophy. Quine and Putnam developed an argument in support of mathematical realism,...

Quoted from: Pi in the Sky, by John Barrow. Oxford University Press, 1992. p. 37-38 “English: one/first ; two/second ; three/third ; four/fourth French: un/premier ; deux/second or deuxième ; trois/troisième ; quatre/quatrième German: ein/erste ; zwei/ander or zweite ;...

"There are two approaches to mathematical infinity. It can be seen as defining limiting cases that can never be realized or as existing in some philosophical sense. These mathematical approaches parallel approaches to meaning and value that I call absolutist...

This is a simple computer-simulated rotation from a point, which has 0 dimensions, to a line (1 dimension), a square (2), a cube (3), all the way up to 6 dimensions, and then down. Multi-dimensional objects are much more complex...

The short video clip below shows the 3D rotations of a 4D object. The deeper questions concern the nature of a "dimension". How do we know what a dimension is, and if we live in a 3D universe, could we...

The Clay Institute offers a prize to anyone who can solve one of these Millenium problems. Here is a description of the problems from the Institute’s website: Birch and Swinnerton-Dyer Conjecture Mathematicians have always been fascinated by the problem of...

Definition A Fibonacci sequence is easily constructed: Start with 0 and 1, and for each following number, add the previous two: 0, 1, 0+1=1, 1+1=2, 1+2=3, and so on. Here is the beginning of the sequence: 0, 1, 1, 2, 3, 5,...

This explanation is quoted from the The Clay Mathematics Institute, Poincaré Conjecture. (solved by: Grigoriy Perelman, 2002-3) “If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving...

Knot theory is a very fast growing field of mathematics. Knots are not natural phenomena, and there exists only a finite number of distinct knots in three-dimensional space. Knots define spaces, because we can think of a knot as a...

Prime numbers are the basic units of the system of numbers, since every natural number is either a prime, or can be expressed as a composite of prime numbers. Here are some excerpts from the Wikipedia article about Prime numbers:...

Perfect numbers are gateways to the wonders of the mathematical world. Contemplating them, one realizes how small our human minds really are compared to the reality that surrounds us, and creates us. Definition A perfect number is a positive integer...