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**Folly Inc**:

But how do you think those Axioms were derived? like it or not, anything in this plane of existence is tied to reality in one way or another. Math was not something that popped into existence because someone liked the idea of numbers. It was practical to real life.

The point that .999... does not equal 1 simply because it -is not one- is true. You CAN say, that within the limitation of certain (and most) mathematics that .999... can equal 1 for the purpose of the equation.

**Something like .999... as you are thinking of it never occurs in the real world**. It's a purely mathematical construct called a "geometric series", one which happens to equal 1. .999... does with absolute certainty = 1.

The historical tale of axiomatic systems is much much more complicated than you are painting it to be. There are plenty of other systems that don't relate very well (or at all) to real life. Take non-euclidean geometries, modal logic, any of the Zn groups, for example. Also when we try to tie math to the real world things often go horribly wrong, go read up on "russell's paradox". It arose because we tried to model set theory axioms off of what "made the most sense" in the real world. Originally one of the axioms of set theory was the axiom of unrestricted comprehension which makes tons of intuitive sense, sadly it turned out it wasn't mathematically valid and had to be replaced with schema of separation which makes very little sense in the real world.

Historically we originally did math off of common sense and intuition. With the belief that the only mathematical systems that were valid were ones that could be represented in reality. Slowly over time formal systems were adopted, and the view switched to the total opposite. Now pure mathematicians develop systems with no care or concern for what exist in the real world, then it's the applied mathematicians job to find a use for them.