The mean round-trip speed, as in simple harmonic motion, is Galilean invariant. Start with the standard configuration in which motion is along the X and X′ axes in S and S′ respectively with velocity v in S′ relative to S. Then the speed in S′ in the x′ direction is: The return speed is: The …
The purpose of science is to discover laws, which are then applied to predict and explain phenomena, develop technology, and make things. This occurs through a cycle of material induction and formal deduction. Induction consists of making observations, defining terms, and proposing postulates. Deduction consists of taking the terms and definitions from induction, possibly with …
Dual Euclidean transformations are required to transform six dimensions of length and duration: one Euclidean transformation for length space with time and one Euclidean transformation for duration space with distance. The two Euclidean transformations are: x′ = x − vt and z′ = z − ws where x and x′ are length space vectors, t …
The Euclidean geometry is a category with point positions as the objects and Euclidean transformations as the morphisms. In kinematics there are two Euclidean geometries: that of length and that of duration. They are in turn categories: Length space is a category with point locations as the objects and Euclidean transformations of points as the …
The traditional Aristotelian square of opposition is like that of first-order logic apart from existential import: Or in words: Outer negation is the contradictory, i.e., affirm/deny, and inner negation is the contrary, i.e., all/none. For quantifiers (or other operators) there is a duality square: Outer negation is negation of the whole quantifier; inner negation is …
Attempted derivations of the Lorentz transformation in the previous post here, which is similar to the light wavefronts approach here, do not work. The reason is that independent and dependent variables are treated alike, but they are not. I suspect this applies to all derivations of the Lorentz transformation. Let us look at the first …
The Merton Rule, which dates to the Middle Ages, relates a uniform change rate to its initial and final rates. Because of its main application, it is also called the Mean Speed Theorem, which in modern language states that a uniformly accelerating body over a period of time traverses the same distance as the product …
What is moral exists without a necessary opposite. Moral truth, goodness, and beauty are defined as those that exist on their own, without the necessity of a contrary (inner) or contradictory (outer) opposite. God is moral because God exists without a necessary opposite. Whatever is of God is also moral. Whatever contradicts God or something …
The units of quantities are conveniently ignored in the definition of a derivative, but they should not be. A derivative should be defined as a function of two quantities, both with their own units: where r’ is a vector function of two quantites and r is a vector functon of one quantity. The second derivative is …
Historically, there are two kinds of induction, called here the postulational and the hypothetical. Postulational induction (cf. material induction) is the induction practiced in ancient and early modern times in which empirical induction leads to essential definitions and universal postulates for subsequent deduction. This is the Socratic view of induction: “in modern philosopher’s technical terms—the …
