In language, a continuous thing exists without gaps or pauses, and in mathematics, a continuous line has no breaks. Space then is continuous if there are no gaps or holes in it, however small. Yet continuity makes physics illogical, 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 ignore the infinities of continuity because they want to, not because there is a 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 is no reason to believe that space is continuous, but there is a reason to think it isn’t. The quantum of quantum theory means that shorter wavelengths of light only have more energy up to the limit of Planck’s constant. This limit, called the Planck length, is according to quantum theory, the shortest length possible.
It follows that repeatedly dividing space gives a smallest length that can’t be divided further. Just as closely inspecting a TV screen gives irreducible dots, closely inspecting our space gives irreducible Planck lengths, so it is digital not continuous. Not only does repeatedly dividing space give a pixel that can’t be split, repeatedly dividing time also gives a processing cycle that can’t be paused, so both space and time are digital. This 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 position. 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, as denying the infinitely small avoids the infinitely large. If the physical world is a virtual reality, its resolution is 10-33 meters and its refresh rate is 10-43 seconds, or Planck length and time respectively. The quantum network is very fine, but not infinitely so, and quantum processing is very fast, but not infinitely so either.