As long as you can realize the difference between analytic statements and synthetic statements, you can know what pure mathematics is.

Pure Mathematics is a system of useful analytic statements.[10]

Pure Mathematics is a system of useful tautologies, whether obvious or unobvious.

In Physics, every generation of physicists have to update the previous theories. For example, Einstein’s theory of gravity has updated Newton’s, explaining what Newton’s gravitation could not explain. But for Pure Maths, although every generation of mathematicians also create new mathematics, the new theories do not and cannot contradict the old ones. For example, “1+1=2” is always true, even in thousands of years later.[11]

1.1.7 Why maths is always true but physics is not?[12]

Pure Maths is a system of analytics statements. Analytic statements say nothing about the world. When you say nothing, you cannot be wrong.

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[10] In Philosophy of Mathematics, this is called the Formalist’s theory of Mathematics. There is a bug in the formalist’s system. It is about the status of the axiom of infinity. For reference, see Bertrand Russell’s Introduction to Mathematical Philosophy.

[11] Mathematics is eternal, as it is timeless, or outside time.

[12] Mr. Lee

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2008.05.20 Tuesday