Search results
Results from the WOW.Com Content Network
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements.Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions from these.
The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l.
In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption. First principles in philosophy are from first cause [1] attitudes and taught by Aristotelians, and nuanced versions of first principles are referred to as postulates by Kantians.
A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
El Bachraoui's proof is an extension of Erdős's arguments for the primes between n and 2n. Shevelev, Greathouse, and Moses (2013) discuss related results for similar intervals. [7] Bertrand’s postulate over the Gaussian integers is an extension of the idea of the distribution of primes, but in this case on the complex plane.
The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures.
Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry, Euclidean and hyperbolic.