This note is a part of my Zettelkasten. What is below might not be complete or accurate. It is also likely to change often.
Math is a science of not being wrong about things, its techniques and habits hammered out by centuries of hard work and argument. With the tools of mathematics in hand, you can understand the world in a deeper, sounder, and more meaningful way.
An example of this is from the second world war, when officers in the US Air Force were trying to figure out which parts of the airplanes need to be armored the most against enemy fire. The data they had to take this decision was the distribution of bullet holes in aircrafts after their engagements. One solution would be to spread armor in the same manner as the bullet holes are distributed - basically, the areas where there are more bullet holes should have more armor.
This however is a fallacy - because the underlying assumption is that the planes are being shot specifically in those areas and the planes which are analysed are a random subset of all planes. However, it makes much more sense that the bullets are randomly distibuted and that the planes which are analysed are a subset of all planes which weren't shot in the sensitive areas. This is an example of Survivorship bias
Mathematics starts with common sense and is extended beyond by other means. Formal mathematics without common sense—without the constant interplay between abstract reasoning and our intuitions about quantity, time, space, motion, behavior, and uncertainty — would just be a sterile exercise in rule-following and bookkeeping.
As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from “reality” it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art.