In large part the study of mathematics relies on the definitions put forth and what may be sowed from their birth. A mathematician is an artist that carefully weaves mathematical theory from the underlying definitions. Hence, mathematics becomes similar to a language where we have certain rules that must be followed in order to construct comprehensive, precise, and unambiguous statements.
A mathematician can not simply state that a particular number is positive while treating it as a negative number. They must follow certain rules regardless of their likes or dislikes. For instance, in general, (x+y)^2 is not x^2 + y^2, much to the chagrin of many students!
The axioms of mathematics provide the backbone that all mathematics flows from. I marvel at the incredible utility and ubiquity of mathematics in all areas of science. I hope you share that sentiment. Nevertheless, why does mathematics work so well? Why is it so reliable in its descriptions? Why is the universal language mathematics? In other words, why is the universe so darn mathematical?
My children love playing with legos, just as I did as a child (still do!). They love to build and create. Ships, houses, vehicles, etc. Mathematics is analogous to legos, in that, we build and create. Indeed, both were created. However, were both created by man?
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