Postulate in the context of Statement (logic)

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 attitudes and taught by Aristotelians, and nuanced versions of first principles are referred to as postulates by Kantians.

In mathematics and formal logic, first principles are referred to as axioms or postulates. In physics and other sciences, theoretical work is said to be from first principles, or ab initio, if it starts directly at the level of established science and does not make assumptions such as empirical model and parameter fitting. "First principles thinking" consists of decomposing things down to the fundamental axioms in the given arena, before reasoning up by asking which ones are relevant to the question at hand, then cross referencing conclusions based on chosen axioms and making sure conclusions do not violate any fundamental laws. Physicists include counterintuitive concepts with reiteration.

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his Elements, with the great innovation of proving all properties of the space as theorems, by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate).

In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature. It is a special case of a curve and an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points (its endpoints).

Euclid's Elements defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry.

Uniformitarianism, also known as the Doctrine of Uniformity or the Uniformitarian Principle, is the assumption that the same natural laws and processes that operate in our present-day scientific observations have always operated in the universe in the past and apply everywhere in the universe. It refers to invariance in the metaphysical principles underpinning science, such as the constancy of cause and effect throughout space-time, but has also been used to describe spatiotemporal invariance of physical laws. Though an unprovable postulate that cannot be verified using the scientific method, some consider that uniformitarianism should be a required first principle in scientific research.

In geology, uniformitarianism has included the gradualistic concept that "the present is the key to the past" and that geological events occur at the same rate now as they have always done, though many modern geologists no longer hold to a strict gradualism. Coined by William Whewell, uniformitarianism was originally proposed in contrast to catastrophism by British naturalists in the late 18th century, starting with the work of the geologist James Hutton in his many books including Theory of the Earth. Hutton's work was later refined by scientist John Playfair and popularised by geologist Charles Lyell's Principles of Geology in 1830. Today, Earth's history is considered to have been a slow, gradual process, punctuated by occasional natural catastrophic events.

Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical study of the relation of this framework with reality.

Luminiferous aether or ether (luminiferous meaning 'light-bearing') is the formerly postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), something that waves should not be able to do. The assumption of a spatial plenum (space completely filled with matter) of luminiferous aether, rather than a spatial vacuum, provided the theoretical medium that was required by wave theories of light.

The aether hypothesis was the topic of considerable debate throughout its history, as it required the existence of an invisible and infinite material with no interaction with physical objects. As the nature of light was explored, especially in the 19th century, the physical qualities required of an aether became increasingly contradictory. By the late 19th century, the existence of the aether was being questioned, although there was no physical theory to replace it.

Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties, initially called postulates and at present called axioms.

After the 17th-century introduction by René Descartes of the coordinate method, which was called analytic geometry, the term "synthetic geometry" was coined to refer to the older methods that were, before Descartes, the only known ones.

Albert Einstein derived the theory of special relativity in 1905, from principles now called the postulates of special relativity. Einstein's formulation is said to only require two postulates, though his derivation implies a few more assumptions.

The idea that special relativity depended only on two postulates, both of which seemed to follow from the theory and experiment of the day, was one of the most compelling arguments for the correctness of the theory (Einstein 1912: "This theory is correct to the extent to which the two principles upon which it is based are correct. Since these seem to be correct to a great extent, ...")