Philosophy Psychoanalysis Religion Theologie Theology Lacan Physics Mathematics Psychotherapy Thinking | | Miscellaneous Quotes about Mathematics Paul Erdös, the most prolific mathematician and problem-solver of the 20th century. He proved, for instance, that there is always a prime between n and 2n. The Prime Number Theorem describes the distribution of prime numbers. Euclid could prove that there is an infinite number of primes, but their location can only be predicted by statistical means, as an approximation. The Fundamental Theorem of Arithmetic and its proof. It states that every positive integer can be written as a product of prime numbers in a unique way. Georg Cantor, who discovered the transfinite numbers. The Beginnings of Set Theory. This text describes in (almost) plain English the history of the problems that led to Cantor's formulation of set theory. Transfinite numbers: aleph0 < c = 2aleph0 . Cantors argument: The cardinality of real numbers (c, for continuum, or aleph1) is infinitely larger than the countable infinity of natural numbers (aleph0). You can find a good exposition of this argument on infinite ink. Kurt Gödel: Incompleteness Theorem. The basic argument. 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 foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced. What happens if there are not only limits to that which is knowable, but if contradictions are unavoidable? The result is a new approach to mathematics, called inconsistent mathematics. Knots: Inspired by Lacan's attempt to develop a topological model of the psyche, I have developed an interest in topology and knot theory, which is a rapidly expanding field of mathematics. Knots are not natural phenomena, they define spaces, and there exist only a small finite number of distinct knots in three-dimensional space. This site offers a knot table with up to 9 crossings, and a java script will generate a movable knot in three-dimensional space. As an example for topology and the challenges it poses for our understanding, I include some information about Klein Bottles. Most containers have an inside and an outside, a Klein bottle is a closed surface with no interior and only one surface. It is unrealizable in 3 dimensions without intersecting surfaces. It can be realized in 4 dimensions, and it results from the joining of two Moebius strips. What is the relation between perfect numbers and primes? A number is perfect when it equals the sum of its divisors. 6 is the first perfect number, since the divisors, 1, 2, and 3, also equals 6. They have a relationship to primes, although they are always even, but there is no proof for it. It is not known to this day whether there are any odd perfect numbers. Euclid showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form. Very large numbers. They may have properties which are still entirely unknown to us, because our ways of representing and comprehending numbers (even with computers) is still marginal compared to the magnitude of mathematical objects. I think it is awe-inspiring to look at them, and therefore I made the effort to reproduce some of these very large numbers: The first 10.000 digits of Pi. , the first 10000 Primes. , and the first 23 perfect numbers. Some unsolved mathematical problems. The Clay Mathematics Institute posted seven unresolved problems in 2000; they also offer a prize of $1.000.000 for each solution.
External Links:Mathworld is back online! This site represents an enormous effort to create an online encyclopedia, was developed by Eric Weisstein, and is now hosted by Wolfram Research, the people who made Mathematica. It is definitely worth a visit! MacTutor. This is an archive for the History of Mathematics - very comprehensive and well-done. Infinite Ink: A site about math, science, computing, and philosophy. A little hard to maneuver, but excellent expositions of various problems. You will find for instance a very good explanation of the Continuum Hypothesis. Hilbert’s mathematical problems These are the 23 fundamental mathematical problems that Hilbert described in his famous speech from Aug 1900. Many of them are today solved, like Fermat's problem, but some are still open, like Goldachs conjecture that every even number greater than 2 can be written as the sum of two primes. Here is an overview of the current status. Participate in the Internet-collaborative search for prime numbers. Knots. At the KnotPlot site, you can see various knots and links. They pose great difficulty for our understanding, and knot-theory, as one of the disciplines within topology, may well be a science of the future! Cantor's famous diagonal argument, where he proves that the open interval (0, 1) is uncountable. Fibonacci Numbers: The Fibonacci Series (ThinkQuest 1999), Fibonacci Numbers and Nature , Golden Ratio (or Golden Mean), the Golden Rectangle, and the relation between the Fibonacci Sequence and the Golden Ratio. The Mandelbrot Set Explorer. A site about the mathematics of Mandelbrot sets (Fractals), with some experiments about fractal equations.
Last updated: 05/29/2004) |
