What Did Pythagoras Mean By "All Things Are Number"?

Most of us first heard the name “Pythagoras” when we were in high school geometry class, but few recall that there was an actual person named Pythagoras who lived and died around 570–495 BCE. Very little is known about the real Pythagoras, or rather a lot is “known” about him, but most of it is surely wrong. The documentary trail is littered with contradictions. It combines the sublime, the ridiculous, the unbelievable, and the just plain weird.
Pythagoras was said to be the son of Apollo, to have a golden thigh, and to glow. He may or may not have advocated vegetarianism. Among his most notorious sayings is an injunction not to eat beans, because “beans have a soul.” Yet several early sources explicitly deny that Pythagoras said or believed anything of the sort. More reliably, Pythagoras believed in, and taught, reincarnation. Several stories—each dubious, to be sure—corroborate this.
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The Stanford Encyclopedia of Philosophy—a free and extremely valuable online resource, by the way—sums it up as follows:

“The popular modern image of Pythagoras is that of a master mathematician and scientist. The early evidence shows, however, that, while Pythagoras was famous in his own day and even 150 years later in the time of Plato and Aristotle, it was not mathematics or science upon which his fame rested. Pythagoras was famous as:
1. An expert on the fate of the soul after death, who thought that the soul was immortal and went through a series of reincarnations
2. An expert on religious ritual
3. A wonder-worker who had a thigh of gold and who could be two places at the same time
4. The founder of a strict way of life that emphasized dietary restrictions, religious ritual and rigorous self discipline

A few things seem clear. The historical Pythagoras was born on the Greek island of Samos, traveled widely, and became the founder of and inspiration for an unusual religious movement. His cult flourished briefly in Crotone, in southern Italy, and developed chapters in several other places before being everywhere suppressed. The Pythagoreans formed secret societies, on which the initiates’ lives centered. These communities, which included both men and women, promoted a kind of intellectual mysticism that seemed marvelous, yet strange and threatening, to most of their contemporaries. Their worldview centered on worshipful admiration of numbers and musical harmony, which they saw as reflecting the deep structure of reality. (As the next two thousand years have shown, they were on to something.)
Here again is the Stanford Encyclopedia:

http://io9.com/what-did-pythagoras-mean-by-all-things-are-number-1717748417