## Mathematics

### Mathematical Problems

A list of the currently unsolved Millenium Problems can be found at the Clay Institute Website. They also offer a prize of \$1.000.000 for each solution. One of the problems has been solved (Poincaré Conjecture - solved by Grigoriy Perelman, 2002-3).

Here is a list of some unsolved mathematical problems from the Wikipedia:

• Goldbach's conjecture: Can every even integer greater than 2 be written as a sum of two primes?

• Twin Prime Conjecture: A twin prime is a pair of primes with difference 2, such as 11 and 13. Are there infinitely many twin primes?

• Does the Fibonacci sequence contain an infinite number of primes?

• Are there infinitely many perfect numbers?

• Are there any odd perfect numbers?

### Mathematics and Reality?

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 of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics.

From the rather remarkable but seemingly uncontroversial fact that mathematics is indispensable to science, some philosophers have drawn serious metaphysical conclusions. In particular, Quine and Putnam have argued that the indispensability of mathematics to empirical science gives us good reason to believe in the existence of mathematical entities. According to this line of argument, reference to (or quantification over) mathematical entities such as sets, numbers, functions and such is indispensable to our best scientific theories, and so we ought to be committed to the existence of these mathematical entities. To do otherwise is to be guilty of what Putnam has called "intellectual dishonesty". Moreover, mathematical entities are seen to be on an epistemic par with the other theoretical entities of science, since belief in the existence of the former is justified by the same evidence that confirms the theory as a whole (and hence belief in the latter). This argument is known as the Quine-Putnam indispensability argument for mathematical realism."

### Quotes about Mathematics

• The most savage controversies are those about matters as to which there is no good evidence either way. Persecution is used in theology, not in arithmetic. ~Bertrand Russell

• Music is the pleasure the human mind experiences from counting without being aware that it is counting. ~Gottfried Leibniz

• Black holes result from God dividing the universe by zero. ~Author Unknown

• If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. ~John Louis von Neumann

• Even stranger things have happened; and perhaps the strangest of all is the marvel that mathematics should be possible to a race akin to the apes. ~Eric T. Bell, The Development of Mathematics

• Pure mathematics is, in its way, the poetry of logical ideas. ~Albert Einstein

• The essence of mathematics is not to make simple things complicated, but to make complicated things simple. ~S. Gudder

• A mathematician is a device for turning coffee into theorems. ~Paul Erdos

• Still more astonishing is that world of rigorous fantasy we call mathematics. ~Gregory Bateson

• To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts. Few realize that the world of modern mathematics is rich with vivid images and provocative ideas. ~Ivars Peterson

• With my full philosophical rucksack I can only climb slowly up the mountain of mathematics. ~Ludwig Wittgenstein, Culture and Value

• The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions. ~Ronald L. Graham

• As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. ~Albert Einstein, Sidelights on Relativity

• God does not care about our mathematical difficulties; He integrates empirically. ~Albert Einstein

• If there is a God, he's a great mathematician. ~Paul Dirac

• Infinity is a floorless room without walls or ceiling. ~Author Unknown

• Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover. ~Bertrand Russell

• The laws of nature are but the mathematical thoughts of God. ~Euclid

• In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generations adds a new story to the old structure. ~Hermann Hankel

• I like mathematics because it is not human and has nothing particular to do with this planet or with the whole accidental universe - because like Spinoza's God, it won't love us in return. ~Bertrand Russell, 1912

• The man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization. ~John Kemeny

• As for everything else, so for a mathematical theory: beauty can be perceived but not explained. Arthur Cayley

• How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality? Max Wilhelm Dehn

• Number is the within of all things. Pythagoras

• Numbers are the highest degree of knowledge. It is knowledge itself. Plato

• Poets do not go mad, but chess players do. Mathematicians go mad, and cashiers, but creative artists very seldom. I am not, as will be seen, in any sense attacking logic: I only say that this danger does lie in logic, not in imagination. -- G. K. Chesterton

• In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics. (Some have engaged in it for this reason alone.) ~ Stansilaw Ulam, Adventures of a Mathematician

Stephen Wolfram: A New Kind of Science. The book is online now!

Clay Mathematics Institute: Dedicated to increasing and disseminating mathematical knowledge

Infinite Ink: Writing and publishing about computing, mathematics, science, and philosophy.

Knotplot: Knot theory, pictures, links. (Edit) This page is part of the KnotPlot Site, where you'll find many more pictures of knots and links as well as MPEG animations and lots of things to download.

Math Answers: It's a very user-friendly site that answers your math questions, and has nice introductions.

Mathematical Sciences Research Institute: Established in 1982 in Berkeley, California Located above the campus of the University of Berkeley, it is an independent non-profit research institute Sponsored by approximately 100 universities and institutions around the world.

Mathologger: Educational Videos about Mathematics

Mountain Math Software: Information and speculation on physics, mathematics and philosophy. Has very good introductions.

Numberphile: Interesting video clips about Math. (Edit) Numberphile is produced by video journalist Brady Haran. The stars of the show include mathematicians and other guests from around the world.

The MacTutor History of Mathematics archive The history of mathematics, biographies of many mathematicians. Very comprehensive.

The Prime Pages Prime number research, records, and resources

Wolfram Alpha Computational Search Engine Wolfram|Alpha introduces a fundamentally new way to get knowledge and answers— not by searching the web, but by doing dynamic computations based on a vast collection of built-in data, algorithms, and methods.

Wolfram MathWorld: MathWorld is the web's most extensive mathematical resource, provided as a free service to the world's mathematics and internet communities as part of a commitment to education and educational outreach by Wolfram Research, makers of Mathematica.