Context and question: Exchanging "+" and "" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.
Which group cannot be expressed using radicals?
Answer: unanswerable
Context and question: Einstein showed in his thought experiments that people travelling at different speeds, while agreeing on cause and effect, measure different time separations between events, and can even observe different chronological orderings between non-causally related events. Though these effects are typically minute in the human experience, the effect becomes much more pronounced for objects moving at speeds approaching the speed of light. Many subatomic particles exist for only a fixed fraction of a second in a lab relatively at rest, but some that travel close to the speed of light can be measured to travel farther and survive much longer than expected (a muon is one example). According to the special theory of relativity, in the high-speed particle's frame of reference, it exists, on the average, for a standard amount of time known as its mean lifetime, and the distance it travels in that time is zero, because its velocity is zero. Relative to a frame of reference at rest, time seems to "slow down" for the particle. Relative to the high-speed particle, distances seem to shorten. Einstein showed how both temporal and spatial dimensions can be altered (or "warped") by high-speed motion.
What exist for a second in a lab?
Answer: unanswerable
Context and question: This is not to say that Whitehead's thought was widely accepted or even well-understood. His philosophical work is generally considered to be among the most difficult to understand in all of the western canon. Even professional philosophers struggled to follow Whitehead's writings. One famous story illustrating the level of difficulty of Whitehead's philosophy centers around the delivery of Whitehead's Gifford lectures in 192728 following Arthur Eddington's lectures of the year previous which Whitehead would later publish as Process and Reality:
Who also struggled to follow Whitehead's writings?
Answer:
professional philosophers