Origins "Nihilism" comes from the Latin nihil, or nothing, which means not anything, that which does not exist. It appears in the verb "annihilate," meaning to bring to nothing, to destroy completely. Early in the nineteenth century, Friedrich Jacobi used the word to negatively characterize transcendental idealism.

However, the first to exhibit an interest in the foundations of mathematics were the ancient Greeks. Arithmetic or geometry Early Greek philosophy was dominated by a dispute as to which is more basic, arithmetic or geometry, and thus whether mathematics should be concerned primarily with the positive integers or the positive realsthe latter then being conceived as ratios of geometric quantities.

The Greeks confined themselves to positive numbersas negative numbers were introduced only much later in India by Brahmagupta. Underlying this dispute was a perceived basic dichotomynot confined to mathematics but pervading all nature: The Pythagorean school of mathematics, founded on the doctrines of the Greek philosopher Pythagoras, originally insisted that only natural and rational numbers exist.

The remarkable proof of this fact has been preserved by Aristotle. The contradiction between rationals and reals was finally resolved Foundations of classical civilizations Eudoxus of Cnidusa disciple of Plato, who pointed out that two ratios of geometric quantities are equal if and only if they partition the set of positive rationals in the same way, thus anticipating the German mathematician Richard Dedekind —who defined real numbers as such partitions.

Being versus becoming Another dispute among pre-Socratic philosophers was more concerned with the physical world. Parmenides claimed that in the real world there is no such thing as change and that the flow of time is an illusiona view with parallels in the Einstein-Minkowski four-dimensional space-time model of the universe.

Heracleituson the other hand, asserted that change is all-pervasive and is reputed to have said that one cannot step into the same river twice. Zeno of Eleaa follower of Parmenides, claimed that change is actually impossible and produced four paradoxes to show this. The most famous of these describes a race between Achilles and a tortoise.

Since Achilles can run much faster than the tortoise, let us say twice as fast, the latter is allowed a head start of one mile.

When Achilles has run one mile, the tortoise will have run half as far again—that is, half a mile. When Achilles has covered that additional half-mile, the tortoise will have run a further quarter-mile.

So how can Achilles ever catch up with the tortoise see figure? Mathematically speaking, his argument involves the sum of the infinite geometric progression no finite partial sum of which adds up to 2.

As Aristotle would later say, this progression is only potentially infinite. Universals The Athenian philosopher Plato believed that mathematical entities are not just human inventions but have a real existence.

For instance, according to Plato, the number 2 is an ideal object. The number 2 is to be distinguished from a collection of two stones or two apples or, for that matter, two platinum balls in Paris.

What, then, are these Platonic ideas? Already in ancient Alexandria some people speculated that they are words. Three dominant views prevailed: It would seem that Plato believed in a notion of truth independent of the human mind. The axiomatic method Perhaps the most important contribution to the foundations of mathematics made by the ancient Greeks was the axiomatic method and the notion of proof.

This notion survives today, except for some cosmetic changes. The idea is this: It may take considerable ingenuity to discover a proof; but it is now held that it must be possible to check mechanically, step by step, whether a purported proof is indeed correct, and nowadays a computer should be able to do this.

The mathematical statements that can be proved are called theoremsand it follows that, in principle, a mechanical device, such as a modern computer, can generate all theorems. Two questions about the axiomatic method were left unanswered by the ancients: Since the middle of the 20th century a gradually changing group of mostly French mathematicians under the pseudonym Nicolas Bourbaki has tried to emulate Euclid in writing a new Elements of Mathematics based on their theory of structures.

Unfortunately, they just missed out on the new ideas from category theory. Number systems While the ancient Greeks were familiar with the positive integers, rationals, and reals, zero used as an actual number instead of denoting a missing number and the negative numbers were first used in India, as far as is known, by Brahmagupta in the 7th century ce.

Much later, the German mathematician Carl Friedrich Gauss — proved the fundamental theorem of algebrathat all equations with complex coefficients have complex solutions, thus removing the principal motivation for introducing new numbers.

Still, the Irish mathematician Sir William Rowan Hamilton —65 and the French mathematician Olinde Rodrigues — invented quaternions in the midth century, but these proved to be less popular in the scientific community until quite recently. Currently, a logical presentation of the number system, as taught at the university level, would be as follows: Here the letters, introduced by Nicolas Bourbaki, refer to the natural numbers, integers, rationals, reals, complex numbers, and quaternions, respectively, and the arrows indicate inclusion of each number system into the next.

However, as has been shown, the historical development proceeds differently:Western dance: Western dance, history of Western dance from ancient times to the present and including the development of ballet, the waltz, and various types of modern dance.

The peoples of the West—of Europe and of the countries founded through permanent European settlement elsewhere—have a history of dance. Foundations Curriculum Guide Second Printing [Leigh A. Bortins] on pfmlures.com *FREE* shipping on qualifying offers. This is the 3rd edition, copyright The Greeks: Lost Civilizations [Philip Matyszak] on pfmlures.com *FREE* shipping on qualifying offers.

This book is a portrait of Ancient Greece—but not as we know it. Few people today appreciate that Greek civilization was spread across the Middle East. Foundations Period: 10, BCE- CE. Foundations: 3 Major Themes – Change from survival (physical needs) to internal peace (spiritual needs) • Civilizations Rise of Classical Civilizations.

Greek Achievements Age of Pericles; Direct Democracy, Golden age Art, architecture, sculpture. Learn what "big history" is and how scholars apply this approach to the story of humanity.

Gain new understanding of the complete sweep of human history, across all civilizations and around the world. Nihilism. Nihilism is the belief that all values are baseless and that nothing can be known or communicated.

It is often associated with extreme pessimism and a radical skepticism that condemns existence. A true nihilist would believe in nothing, have no loyalties, and no purpose other than, perhaps, an impulse to destroy.

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Foundations of mathematics - Wikipedia