John Von Neumann: The Mathematician. 1947
John von Neumann wrote The Mathematician which was published in Works of the Mind Vol. I no. 1 (University of Chicago Press, Chicago, 1947), 180-196. It has also been published...
Category: Mathematics
Cool Number Tricks
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...
Hyperbolic Space
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...
Gödel: The modern development of the foundations of mathematics in the light of philosophy.
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...
Nature by Numbers
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...
Turning a Sphere inside-out
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:
Mathematical Realism
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...
Ordinals, Cardinals, Representation of Numbers in Language.
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 ;...
Mathematical Infinity
“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 and evolutionary. The absolutist sees ultimate meaning as something that exists most commonly in the form of an all powerful infinite God. The evolutionary sees life and all of a creation as an ever expanding journey with no ultimate or final goal. There is only the journey. There is no destination. This video argues for an evolutionary view in our sense of meaning and values and in our mathematical understanding. There is a deep connection between the two with profound implications for the evolution of consciousness and human destiny.” (Paul Budnik)
0 to 6 dimensions and back – simple rotation.
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...
Tesseracts: from 3 to 4 Dimensions.
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 possible also exist in a higher-dimensional universe? it is best to start with simple examples in order to train the mind to think about these questions. The clip below shows a tesseract, which is the four-dimensional analog of a cube. (In geomery, it is called a regular octachoron or cubic prism.) The tesseract is to the cube as the cube is to the square.
Unsolved Millenium Problems
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...
Fibonacci Sequence
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...
Poincaré Conjecture
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...
Knot theory
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...
Prime Numbers
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...
