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the development of Mathematics provides a tightly connected network of formal rules, concepts, and systems. Nodes of this network are closely bound to procedures useful in human activities and to questions arising in science. The transition from activities to the formal Mathematical systems is guided by a variety of general insights and ideas.
Similarly, every cut of reals is identical to the cut produced by a specific real number (which can be identified as the smallest element of the B set). In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps.
Using a technique called Free Timing, a conductor will use either (a) a stopwatch or studio size stop clock, or (b) watch the film on a screen or video monitor while conducting the musicians to predetermined timings. These are represented visually by vertical lines (streamers) and bursts of light called punches.