Coastline Paradox

"From Wikipedia, the free encyclopedia

"The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines.

"The measured length of a coastline depends on the scale of measurement: the smaller the increment of measurement, the longer the measured length becomes. Since a landmass has features at all scales, from hundreds of kilometers in size to tiny fractions of a millimeter and smaller, there is no obvious limit to the size of the smallest feature that should not be measured around, and hence no single well-defined perimeter to the landmass.

"Over a wide range of measurement scales, down to the atomic, coastlines show a degree of self-similarity, and as the measurement scale is made smaller and smaller, the measured length continues to increase, tending towards infinity.

"An example of the coastline paradox. If the coastline of Great Britain is measured using fractal units 100 km long, then the length of the coastline is approximately 2800 km. With 50 km units, the total length is approximately 3400 km (600 km longer)."

Gödel and the Grundlagenkrise der Mathematik (and Knowledgelessness)

“From Wikipedia, the free encyclopedia

“Since the time of Pythagoras, mathematicians have wondered about the nature of mathematical truth, the ontology of mathematical entities and the reasons for the validity of proof and, more generally, mathematical knowledge. From the Enlightenment until the middle of the 19th century, the prevailing scientific ideology saw mathematics as the only way of reaching a truth that is final, absolute and totally independent of the human mind's capacity to understand it. The basic notions of mathematics were thought to reflect essential properties of the cosmos and the theorems to be the truths of a higher reality.... Yet, in the 19th century this traditional belief was undermined in the minds of some people and eventually led to a serious foundational crisis in mathematics. The first of the discoveries which caused the loss of faith, dating from the time of the Renaissance, was that of the imaginary numbers (i.e. those involving the square root of minus one)....

“The foundational crisis of mathematics (in German: Grundlagenkrise der Mathematik) was the early 20th century's term for the search for proper foundations of mathematics.

“After several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within mathematics itself began to be heavily challenged.

“One attempt after another to provide unassailable foundations for mathematics was found to suffer from various paradoxes (such as Russell's paradox) and to be inconsistent: an undesirable situation in which every mathematical statement that can be formulated in a proposed system (such as 2 + 2 = 5) can also be proved in the system.

“Various schools of thought on the right approach to the foundations of mathematics were fiercely opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which thought to ground mathematics on a small basis of a logical system proved sound by metamathematical finitistic means. The main opponent was the intuitionist school, led by L. E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols. The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time.

“Gödel's incompleteness theorems, proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent recursively axiomatizable system -- such as necessary to axiomatize the elementary theory of arithmetic on the (infinite) set of natural numbers -- a statement that can be shown to be true, but that does not follow from the rules of the system. It thus became clear that the notion of mathematical truth can not be reduced to a purely formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system was not powerful enough for proving its own consistency, let alone that a simpler system could do the job. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means... Meanwhile, the intuitionistic school had not attracted many adherents among working mathematicians, due to difficulties of constructive mathematics.

“In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC (the Zermelo–Fraenkel set theory with the axiom of choice), generally their preferred axiomatic system....

“It may or may not be the case that there is a fundamental limit to what humans can understand about numbers (i.e., there may be true number-theoretical principles which cannot be perceived as being true by any human), but Gödel's theorem does not tell us which of these is the case, and we have no way of knowing.”

www.TheSteppe.com

Did you vote?

"In every election, many people grapple with the nagging suspicion their vote doesn't count.... They are wrong. In fact, our democracy depends on every citizen recognizing the value of his or her vote.

"And here is the value of that vote. In the most recent presidential election 105,360,260 people cast ballots. That means each person's vote counted .000000949%.... So we can agree, your vote counts. It counts .000000949%."

--Stephen Colbert, "Of Course Your Vote Counts!," America (The Book): A Citizen's Guide to Democracy Inaction, 2004

Grelling–Nelson paradox

An example of Russell's paradox.

“From Wikipedia, the free encyclopedia

“Suppose one interprets the adjectives 'autological' and 'heterological' as follows:

“An adjective is autological if and only if it describes itself. For example 'short' is autological, since the word 'short' is short. 'English,' 'unhyphenated' and 'pentasyllabic' are also autological.

“An adjective is heterological if and only if it does not describe itself. Hence 'long' is a heterological word, as are 'abbreviated' and 'monosyllabic.'

“All adjectives, it would seem, must be either autological or heterological, for each adjective either describes itself, or it doesn't. The Grelling–Nelson paradox arises when we consider the adjective 'heterological'....

“Is 'heterological' a heterological word? If the answer is 'yes', 'heterological' is autological (leading to a contradiction). If the answer is 'no', 'heterological' is heterological (again leading to a contradiction).”

2008 International Snow Sculpture Championships in Colorado, Breckenridge

"We were pleased to see Cool Jazz, our prize-winning work from 2007, featured on the town's poster for the 2008 event."

--Stan Wagon, Macalester College mathematics professor, Team Minnesota

http://stanwagon.com/wagon/SnowSculptureRedirect/snowsculptureindex.html

"History of philosophy in Poland": Copernicus and others

"From Wikipedia, the free encyclopedia

"The history of philosophy in Poland parallels the evolution of philosophy in Europe generally. Polish philosophy drew upon the broader currents of European philosophy, and in turn contributed to their growth. Among the most momentous Polish contributions were made in the 13th century by the Scholastic philosopher and scientist Witelo; and in the 16th century, by the Renaissance polymath Nicolaus Copernicus.

"Subsequently the Polish-Lithuanian Commonwealth partook in the intellectual ferment of the Enlightenment, which for the multi-ethnic Commonwealth ended not long after the partitions and political annihilation that would last for the next 123 years, until the collapse of the three partitioning empires in World War I.

"The period of Messianism, between the November 1830 and January 1863 Uprisings, reflected European Romantic and Idealist trends, as well as a Polish yearning for political resurrection. It was a period of maximalist metaphysical systems.

"The collapse of the January 1863 Uprising prompted an agonizing reappraisal of Poland's situation. Poles gave up their earlier practice of 'measuring their resources by their aspirations,' and buckled down to hard work and study. '[A] Positivist,' wrote the novelist Bolesław Prus' friend, Julian Ochorowicz, was 'anyone who bases assertions on verifiable evidence; who does not express himself categorically about doubtful things, and does not speak at all about those that are inaccessible.'

"The 20th century brought a new quickening to Polish philosophy. There was growing interest in western philosophical currents. Rigorously trained Polish philosophers made substantial contributions to specialized fields—to psychology, the history of philosophy, the theory of knowledge, and especially mathematical logic. Jan Łukasiewicz gained world fame with his concept of many-valued logic and his 'Polish notation.' Alfred Tarski's work in truth theory won him world renown.

"After World War II, for over four decades, world-class Polish philosophers and historians of philosophy such as Władysław Tatarkiewicz continued their work, often in the face of adversities occasioned by the dominance of a politically enforced official philosophy. The phenomenologist Roman Ingarden did influential work in esthetics and in a Husserl-style metaphysics; his student Karol Wojtyła acquired a unique influence on the world stage as Pope John Paul II."

Macalester College Math Problem of the Week

http://mathforum.org/wagon/