Numbers as language
Ordinarily, what a number refers to is more important than the number itself. 153 fish is a lot of fish. 2 small copper coins is a tiny amount of money. 14 years is a certain period of time. But to quote Bill James, sometimes numbers "have acquired the powers of language." When that happens, the numbers themselves begin to have meaning independent of what is being counted.1
God began using numbers literally from the beginning to communicate to humanity—our week is a continual reminder of Who ordered the universe and how.2 The number 12 carries particular significance since it was the number of Israelite tribes. Moses went so far as to separate out Joseph's inheritance into two half-tribes so that the number of territories was kept at twelve3. The early church made replacing Judas a top priority in order to bring the number of apostles back to twelve.
Sometime during and after the Exile, a new genre of Jewish writing appeared that is labeled "apocalyptic". As a genre, it has close links to dream interpretation, such as we see with Joseph in Genesis 37, 40, and 41. Notice that numbers are of critical significance in each of the dreams and have a significance beyond simply counting things up. The second half of Daniel, a textbook case of the apocalyptic genre, extends the dream interpretation structure to reveal God's purpose, not just for one person, but for all of humanity. Revelation, which clearly uses Daniel imagery, tells us to "calculate" 666 in the most famous use of numerology.
Numbers as superstition
Numbers surround us and some of them tell a story. Unfortunately, not all numbers are meaningful. We tend to give more credence to ideas that make good stories and so we sometimes assume more about numbers than is warranted.4 If 12 is "good" or "lucky" or "blessed", than it seems reasonable that 13 (or 11) is the opposite. God uses numbers as language, but not all numbers and not at all times. When we assume otherwise, we start down the path of superstition.
Numbers for discipline
The question, however, brings up a somewhat different use for numbers: as a discipline or reminder. To remember that, for instance, Jesus was tempted in the desert for 40 days by observing Lent, seems only fitting. As long as numbers themselves point to God and do not replace Him, I think they can be worth observing.
Having posted this on Friday the 13th, it's sure to be downvoted and probably will destroy the internet. I'm sorry you asked.
1. Baseball is a great example of this. To a fan, the title the movie 61* is completely descriptive. We know exactly what the story will be about without reading the summary. SciFi (and humor) nerds don't need to be told the significance of 42 (as pointed out in the comments). In the US, we all recognize 911, 1776, and 13 as being important symbols, though there might be several interpretations of their meanings.
2. Depending on how literal or not you take Genesis 1, this could be seen as a just-so story, but that does not alter the fact that the number 7 acquired meaning independent of what was counted.
3. Levi received no land for an inheritance.
4. A good place to explore the narrative fallacy is Nassim Nicholas Taleb's The Black Swan and Fooled by Randomness.