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## Main Question or Discussion Point

Hello, I have a few questions about mathematics as a deductive system in which mathematical statements follow as a logical consequence from axioms and other statements. In particular, consider the real number system and Euclidean geometry.

Is doing arithmetic with numbers and working on theorems that follow deductively from axioms of Euclidean geometry simply a "game of logic"? The real numbers and Euclidean geometry are useful and accurate in real life. Why is that so? One idea I could think of is that axioms of real numbers and Euclidean geometry are not chosen without any meaning. They are set up with the aim of mathematizing natural observations. So how do the axioms eventually lead to mathematizing natural observations?

Say the axioms are literally descriptions of our world. Then I guess the deductions of the axioms will also describe our world closely. What happens if we derive from the axioms a statement that is different from observations in our world? (Have this occurred in Euclidean geometry?)

Is doing arithmetic with numbers and working on theorems that follow deductively from axioms of Euclidean geometry simply a "game of logic"? The real numbers and Euclidean geometry are useful and accurate in real life. Why is that so? One idea I could think of is that axioms of real numbers and Euclidean geometry are not chosen without any meaning. They are set up with the aim of mathematizing natural observations. So how do the axioms eventually lead to mathematizing natural observations?

Say the axioms are literally descriptions of our world. Then I guess the deductions of the axioms will also describe our world closely. What happens if we derive from the axioms a statement that is different from observations in our world? (Have this occurred in Euclidean geometry?)

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