In language, a thing is continuous if it exists without gaps or pauses. In mathematics, a line is continuous if it has no breaks. Space then is continuous if there are no holes in it, however small, but this creates problems for physics, as Zeno’s paradoxes (Mazur, 2008) illustrated two thousand years ago:
1. If a tortoise running from a hare sequentially occupies infinite points of space, the hare can never catch it, because every time it gets to where the tortoise was, it has moved a little further on; OR
2. If space and time aren’t continuous, there must be an instant when the arrow from a bow is at a fixed unmoving position. If so, how can many such instants beget movement?
To deny the first paradox exposes one to the second, and vice-versa. Zeno’s paradoxes resurface today as infinities in field equations. For example, continuity requires an electron to exist at a point with no size, which makes its mass infinitely dense. Physics avoids these infinities by a mathematical trick called renormalization, which attributes all particle interactions to other particles. Dirac described this tactic as follows:
“Sensible mathematics involves neglecting a quantity when it turns out to be small – not neglecting it just because it is infinitely great and you do not want it!” (Kragh, 1990), p. 184.
Feynman was even blunter:
“No matter how clever the word, it is what I call a dippy process! … I suspect that renormalization is not mathematically legitimate.” (Richard Feynman, 1985), p128.
Physicists found a way to ignore the infinities of continuity because they wanted to, not because there was a good reason to do so:
“… although we habitually assume that there is a continuum of points of space and time this is just an assumption that is … convenient … There is no deep reason to believe that space and time are continuous, rather than discrete…” (Barrow, 2007) p57.
There no deep reason to think space is continuous, but there is a good reason to think it isn’t because quantum theory requires a minimum length. Shorter wavelengths of light have more energy but only up to the limit of Planck’s constant, so the quantum of quantum theory requires a shortest length, which is called the Planck length.
Planck length predicts that repeatedly dividing space gives a smallest point that can’t be divided further. Just as closely inspecting a TV screen gives irreducible dots, closely inspecting our space gives only irreducible Planck lengths, so it is digital not continuous. And not only does repeatedly dividing our space give a pixel that can’t be split, repeatedly dividing our time gives a cycle that can’t be paused. That space and time are digital then answers Zeno’s paradoxes as follows:
There is indeed an instant when the arrow is in a fixed, unmoving position but there is still movement, because the next cycle generates the next physical state. Equally the hare cannot get closer to the turtle forever, because there is a minimum pixel distance that can’t be divided, so the hare catches the turtle.
A digital world of irreducible pixels and indivisible ticks makes the infinities of continuity disappear, like ghosts in the day, because denying the infinitely small avoids the infinitely large. If the physical world is a virtual reality, its resolution is a Planck length of 10-33 meters, and its refresh rate is a Planck time of 10-43 seconds. The quantum network is very fine but not infinitely so, and quantum processing is very fast but not infinitely so either.