These people were all wrong of course. There are times when mathematical symmetries show up in very compelling ways in the world, but in each case they show up because they follow naturally from underlying phenomena. Crystals come about from the ways that molecules can be packed together. Circles and ellipses in planetary motions arise from the specific formula that describes the strength of gravity. There is no “top down” imposition of mystical mathematical properties.
Have we outgrown our superstitious phase about math? In one sense I think we have: prior to the scientific revolution, human thought was dominated by philosophy, religion and superstition: subjects which it is extremely easy for us to superimpose unsubstantiated patterns one. On the other hand, we still seek out patterns: we crave them actually, and if there isn’t a legitimate one we will hallucinate one. It is well established that humans will see trends and symmetries in white noise: it is a psychological bias we have.
Unchecked by scientific reality checks, this need to see patterns (even if they aren’t there) quickly morphs into superstition. The most compelling patterns can take on spiritual significance, and we will seek them out everywhere we look. We now have science, but those instincts are just as strong as they ever were, and they continue to hurt us in subtle ways.
Look at people who believe they are “on a winning streak” at a casino: if you play long enough you will certainly have a winning streak now and then just by the laws of chance. Many people understand that a casino is very carefully stacked in the house’s favor will fall into the same trap when it comes to their investment portfolio. Many people believe in "lucky numbers", such as multiples of three, as guiding principles for all aspects of their lives. It's true that mathematical patterns often describe the world, but