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
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...
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)
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 spaces...
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...
Perfect 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...
The Number Pi
From Wikipedia: “π (sometimes written pi) is a mathematical constant that is the ratio of any circle’s circumference to its diameter. π is approximately equal to 3.14 in the usual...
Kurt Gödel
Here is the biographical sketch of Kurt Gödel’s life from the Stanford Encyclopedia of Philosophy: Biographical Sketch Kurt Gödel was born on April 28, 1906 in what was then the Austro-Hungarian...
Incompleteness Theorem
The Incompleteness Theorem is Gödel’s main contribution to 20th century thought. Gödel showed that within a logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are...
Continuum Hypothesis
Georg Cantor originally proposed that there is no infinite set with a cardinal number between that of the “small” infinite set of natural numbers (N) and the “large” infinite set...
A History of Set Theory
A set is a collection of things. A set can consist of numbers or letters (such as 1, 2, 3, 4 or a, b, c, d) or of objects (such...
