The spacetime interval is invariant over the Lorentz transformation (LT). The following is a proof of this for the inverse LT with length space axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, along with β = v/c and γ = 1/√(1 − β²): The invariant interval is […]
What follows are Lorentz and Ignatowski transformations and their duals with symmetric and vector forms for reference. For the (3+1) Lorentz transformation there are length spaceaxes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, with β = v/c and γ = 1/√(1 − β²): The symmetric form is
Vladimir Ignatowski (1875-1942) was a Russian physicist. “In 1910 he was to first who tried to derive the Lorentz transformation by group theory only using the relativity principle (postulate), and without the postulate of the constancy of the speed of light.” K M Browne gave a simplified derivation in the European Journal of Physics, 39
Taxonomy is the science of classification. Taxonomy applied to biology is a systematic approach to classifying organisms. It can be applied to all organisms at a particular time, throughout time, or within any context. Once a classification is determined, other questions arise such as whether there is an independent reason that organisms are in the
Begin with elements. Elements are a very general concept: they may be either members of sets or distinctions of classes. As a set is defined by its members, so a class is defined by its distinctions. So, the elements of sets are members and the elements of classes are distinctions. Sets may be divided into
At the highest level of classification, history may be divided into human history (better known simply as ‘history’) and physical history. The former is a large subject with many subdivisions, while the latter is usually turned over to the physical sciences. This is a pity since science and history are different disciplines (see posts here).
There is a well-known alternation of induction and deduction in science (click to enlarge): The induction phase consists of data collection, data analysis, and model development. The deduction phase consists of taking the model, making hypothetical inferences, and following up with experiments that lead to new data collection. Then the cycle repeats.
The following (long) excerpts are from Isaiah Berlin’s article “History and Theory: The Concept of Scientific History”, published in History and Theory 1 (1):1 (1960). Republished in Concepts and Categories: Philosophical Essays. NY: Viking Press, 1979. (online here). HISTORY, according to Aristotle, is an account of what individual human beings have done and suffered. In
For previous posts on history and science, see here. History and science are different kinds of knowledge. History is based on the particulars that go into narratives. Science is based on the universals that go into theories. History is focused on the matter and science is focused on the form, in the Aristotelian sense. The
These derivations follow that in ‘Hyperphysics’ here. Length Contraction The length of any object in a moving frame will appear foreshortened in the direction of motion or contracted. The length is maximum in the frame in which the object is at rest. If the length L0 = x2´ − x1´ is measured in the moving
