Category: Mathematics

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 in von Neumann’s Collected Works. A discussion of the nature of intellectual work is a difficult task in any field, even in fields which are not so far removed from the central area of our common human intellectual effort as mathematics still is. A discussion of the nature of any intellectual effort is difficult per se – at any rate, more difficult than the mere exercise of that particular intellectual...

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 concepts, the development of foundational research in mathematics since around the turn of the century, and to fit it into a general schema of possible philosophical world-views [Weltanschauungen]. For this, it is necessary first of all to become clear about the schema itself. I believe that the most fruitful principle for gaining an overall view of the possible world-views will be to divide them up according to the degree and...

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 and Putnam developed an argument in support of mathematical realism, which starts with the observation that mathematics is indispensable for almost every other science. Here is a short description of these indispensability arguments in the philosophy of mathematics. It is quoted from the  Stanford Encyclopedia of Philosophy, the entry is written by Mark Colyvan: “One of the most intriguing features of mathematics is its applicability to empirical science. Every branch...

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 ; trois/troisième ; quatre/quatrième German: ein/erste ; zwei/ander or zweite ; drei/dritter ; vier/vierte Italian: uno/primo ; due/secondo ; tre/terzo ; quattro/quarto In each of these four languages the words for ‘one’ and ‘first’ are quite distinct in form and emphasize the distinction between solitariness (one) and priority (being first). In Italian and the more old-fashioned German and French usage of ander and second, there is also a clear difference between...

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 Conjecture Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic equations like x2 + y2 = z2 Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert’s tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole...

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 the beginning of the sequence:  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … The Fibonacci sequence is named after Leonardo of Pisa, who is also known as Fibonacci. In 1202 he wrote a book entitled “Liber Abaci“,  which introduces the sequence to Western European mathematics. It had been described earlier, however, in Indian mathematics.  Fibonacci number sequences are everywhere in nature, and the math can quickly get...

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 we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface...

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 because we can think of a knot as a way in which different dimensions can be connected. Mathematicians are working on notation systems for knots, which leads to a form of arithmetic for knots. This has  fascinating consequences for other disciplines, and for our understanding of reality in general.  What is a knot? Complex knots can oftentimes be simplified with a few moves, which the German mathematician Reidemeister organized into...

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 are some excerpts from the Wikipedia article about Prime numbers: Definition “A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime, as only 1 and 5 divide it, whereas 6 is composite since it has...

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 creates us. Definition A perfect number is a positive integer that is equal to the sum of its positive divisors excluding itself.  The first perfect number is six. Six is the number of sides to each cell in the bee’s honeycomb,  or the number of points of all snowflakes.  ALL SNOWFLEKES HAVE SIX CORNERS, OR POINTS. Amazing, isn’t it? There is an infinite number of snowflakes, each different from every...

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 decimal notation. Many formulae in mathematics, science, and engineering involve π, which makes it one of the most important mathematical constants.For instance, the area of a circle is equal to π times the square of the radius of the circle. π is an irrational number, which means that its value cannot be expressed exactly as a fraction having integers in both the numerator and denominator. Consequently, its decimal representation never...

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 city of Brünn, and what is now Brno in the Czech Republic. Gödel’s father Rudolf August was a businessman, and his mother Marianne was a well-educated and cultured woman to whom Gödel remained close throughout his life, as witnessed by the long and wide-ranging correspondence between them. The family was well off, and Gödel’s childhood was an uneventful one, with one important exception; namely, from about the age of four...

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 undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions. It appears to destroy the hope of mathematical certitude through the use of the obvious methods. It also deconstructs an ideal of science, that we...

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 of real numbers C (the “continuum”).  “There is no set whose cardinality is strictly between that of the integers and that of the real numbers.”  This means that N=Aleph0, C=Aleph1. Gödel showed that no contradiction arises if the continuum hypothesis is added to conventional Zermelo-Fraenkel set theory. However, Paul Cohen proved in 1963 that no contradiction arises if the negation of the continuum hypothesis is added to set theory. Together,...

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 as chairs or books). Set theory is the field of mathematics that deals with the properties of sets that are independent of the things that make up the set. For example, a mathematician might be interested in knowing about sets S and T without being interested at all about the concrete elements of the sets. The following article addresses the history of set theory, but it also serves as an...

Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every positive integer can be written as a product of prime numbers in a unique way. The following description of the theorem and its proof is quoted from the Wikipedia encyclopedia: Definition “In mathematics, and in particular number theory, the fundamental theorem of arithmetic is the statement that every positive integer can be written as a product of prime numbers in a unique way. For instance, we can write 6936 = 23 · 3 · 172   or   1200 = 24 · 3 · 52 and there are no other such factorizations...

Georg Cantor

from the MacTutor History of Mathematics archive, article byJ J O’Connor and E F Robertson:  “Georg Cantor‘s father, Georg Waldemar Cantor, was a successful merchant, working as a wholesaling agent in St Petersburg, then later as a broker in the St Petersburg Stock Exchange. Georg Waldemar Cantor was born in Denmark and he was a man with a deep love of culture and the arts. Georg’s mother, Maria Anna Böhm, was Russian and very musical. Certainly Georg inherited considerable musical and artistic talents from his parents being an outstanding violinist. Georg was brought up a Protestant, this being the religion...

Georg Cantor Quotes

The final problem which Cantor grappled with was the realization that there could be no set containing everything, since, given any set, there is always a larger set — its set of subsets. Furthermore, he believes that infinity actually exists – it is not just a mathematical construct.  Cantor came to the conclusion that there is an Absolute Infinite that transcends transfinite numbers. It has mathematical properties, and he identified this concept with God. Subsequently, he believed that his new mathematics is actually a form of theology.  Here are some quotes by Georg Cantor: I have never proceeded from any...

Paul Erdös

from the MacTutor History of Mathematics archive, article byJ J O’Connor and E F Robertson:  Born: 26 March 1913 in Budapest, Hungary Died: 20 Sept 1996 in Warsaw, Poland Paul Erdös came from a Jewish family (the original family name being Engländer) although neither of his parents observed the Jewish religion. Paul’s father Lajos and his mother Anna had two daughters, aged three and five, who died of scarlet fever just days before Paul was born. This naturally had the effect making Lajos and Anna extremely protective of Paul. He would be introduced to mathematics by his parents, themselves both...